utility hypothesis means

Search Search Please fill out this field.

Expected Utility: Definition, Calculation, and Examples

James Chen, CMT is an expert trader, investment adviser, and global market strategist.

What Is Expected Utility?

"Expected utility" is an economic term summarizing the utility that an entity or aggregate economy is expected to reach under any number of circumstances. The expected utility is calculated by taking the weighted average of all possible outcomes under certain circumstances. With the weights being assigned by the likelihood or probability, any particular event will occur.

Key Takeaways

Expected utility refers to the utility of an entity or aggregate economy over a future period of time, given unknowable circumstances.
Expected utility theory is used as a tool for analyzing situations in which individuals must make a decision without knowing the outcomes that may result from that decision
The expected utility theory was first posited by Daniel Bernoulli who used it to solve the St. Petersburg Paradox.
Expected utility is also used to evaluate situations without immediate payback, such as purchasing insurance.

Understanding Expected Utility

The expected utility of an entity is derived from the expected utility hypothesis. This hypothesis states that under uncertainty, the weighted average of all possible levels of utility will best represent the utility at any given point in time.

Expected utility theory is used as a tool for analyzing situations in which individuals must make a decision without knowing the outcomes that may result from that decision, i.e., decision making under uncertainty. These individuals will choose the action that will result in the highest expected utility, which is the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

This theory also notes that the utility of money does not necessarily equate to the total value of money. This theory helps explain why people may take out insurance policies to cover themselves for various risks. The expected value from paying for insurance would be to lose out monetarily. The possibility of large-scale losses could lead to a serious decline in utility because of the diminishing marginal utility of wealth .

History of the Expected Utility Concept

The concept of expected utility was first posited by Daniel Bernoulli, who used it to solve the St. Petersburg Paradox .

The St. Petersburg Paradox can be illustrated as a game of chance in which a coin is tossed at each game's play. For instance, if the stakes start at $2 and double every time heads appear, once the first time tails appear the game ends, and the player wins whatever is in the pot.

Under such game rules, the player wins $2 if tails appear on the first toss, $4 if heads appear on the first toss and tails on the second, $8 if heads appear on the first two tosses and tails on the third, and so on.

Mathematically, the player wins 2 k dollars, where k equals the number of tosses ( k must be a whole number and greater than zero). Assuming the game can continue as long as the coin toss results in heads and, in particular, that the casino has unlimited resources, in theory, the sum is limitless. Thus the expected win for repeated play is an infinite amount of money.

Bernoulli solved the St. Petersburg Paradox by distinguishing between the expected value and expected utility, as the latter uses weighted utility multiplied by probabilities instead of using weighted outcomes.

Expected Utility vs. Marginal Utility

Expected utility is also related to the concept of marginal utility . The expected utility of a reward or wealth decreases when a person is rich or has sufficient wealth. In such cases, a person may choose the safer option as opposed to a riskier one.

For example, consider the case of a lottery ticket with expected winnings of $1 million. Suppose a person with comparatively fewer resources buys the ticket for $1. A wealthy person offers to buy the ticket off them for $500,000. Logically, the lottery holder has a 50-50 chance of profiting from the transaction. It is likely that they will opt for the safer option of selling the ticket and pocketing the $500,000. This is due to the diminishing marginal utility of amounts over $500,000 for the ticket holder. In other words, it is much more profitable for them to get from $0 - $500,000 than from $500,000 - $1 million.

Now consider the same offer made to a very wealthy person, possibly a millionaire. Likely, the millionaire will not sell the ticket because they hope to make another million from it.

A 1999 paper by economist Matthew Rabin argued that the expected utility theory is implausible over modest stakes. This means that the expected utility theory fails when the incremental marginal utility amounts are insignificant.

Example of Expected Utility

Decisions involving expected utility are decisions involving uncertain outcomes. An individual calculates the probability of expected outcomes in such events and weighs them against the expected utility before making a decision.

For example, purchasing a lottery ticket represents two possible outcomes for the buyer. They could end up losing the amount they invested in buying the ticket, or they could end up making a smart profit by winning either a portion of the entire lottery. Assigning probability values to the costs involved (in this case, the nominal purchase price of a lottery ticket), it is not difficult to see that the expected utility to be gained from purchasing a lottery ticket is greater than not buying it.

Expected utility is also used to evaluate situations without immediate payback, such as purchasing insurance. When one weighs the expected utility to be gained from making payments in an insurance product (possible tax breaks and guaranteed income at the end of a predetermined period) versus the expected utility of retaining the investment amount and spending it on other opportunities and products, insurance seems like a better option.

UC Berkeley. " Risk Aversion and Expected-Utility Theory: A Calibration Theorem ." Accessed May 7, 2021.

Terms of Service
Editorial Policy
Privacy Policy
Your Privacy Choices

Media Center

Decision Utility

The basic idea, theory, meet practice.

TDL is an applied research consultancy. In our work, we leverage the insights of diverse fields—from psychology and economics to machine learning and behavioral data science—to sculpt targeted solutions to nuanced problems.

When you’re making a decision – whether it’s a life-changing choice or what you want to eat for lunch – you’re usually guided by the question: What would be the best for me?

In other words, you’re considering what the most satisfying or useful decision would be. Every time we make a decision, we evaluate the potential outcomes and how useful they might be. This concept of “utility”—how beneficial an outcome will be for us—lies at the heart of the behavioural economic and psychological study.

Utility is a key term in economics that describes the benefit an agent receives from the consumption of goods or services. In traditional economics, people are generally expected to act rationally and make decisions based on maximizing an outcome’s utility. In theory, this process makes sense. In practice, utility is often difficult to quantify in real life.

In order to further refine the concept of utility, psychologists and economists have differentiated between two types of utility: our perceptions of utility before we experience it, or decision utility , and the actual experienced utility of a choice, called experienced utility .¹ Decision utility describes the usefulness that we perceive and use to make a decision, while experienced utility describes the lived consequences of the decision in reality. These different types of utility have driven new understandings of utility and its role in decision-making.

Utility is an important concept in economics, psychology, business, and our personal lives – it guides our every choice. If we can understand utility, we can understand why and how people come to their decisions, and even make predictions about how people will behave.

Maintaining one’s vigilance against biases is a chore—but the chance to avoid a costly mistake is sometimes worth the effort. – Daniel Kahneman

The beginnings

Utility as an economic principle goes back multiple centuries and was first described by 18th-century Swiss mathematician Daniel Bernoulli. Over time, however, economists and eventually psychologists developed more nuanced theories of utility, leading to the multiple conceptions of utility used today. Moreover, over the past century, a new branch in economics has emerged that developed a different understanding of our relationship to utility.

George Stigler, an American economist, and future Nobel laureate wrote a 1950 paper giving a historical survey of utility in economics.² His review of utility theory from 1776 to 1915 in this article served as a basis for many other researchers to build on. He begins with a theory developed by English philosopher Jeremy Bentham. In an influential 1789 paper, Bentham proposed measuring the amount of pleasure and pain in the context of developing a rationalist legal system. He gave four dimensions of these two feelings: intensity, duration, certainty, and propinquity. Bentham also realized that individual differences would change how a given person feels pleasure or pain in a specific situation. In this way, he described our process of evaluating utility as one of optimizing pleasure and minimizing pain. Although Bentham’s theory was justified by its convenience and approximating ability, it was not necessarily effective, since the philosopher did not provide a way to measure the pleasure and pain of a situation.

Utility theory did not become heavily discussed or studied in economics until the 1870s, when economists attempted to advance the idea of utility in different ways, such as studying the relationships between price and utility, and demand and utility. While various mathematical formulations and models were tested to estimate the utility of outcomes using variables like price, the quantity of product, and supply and demand, measurement remained a difficult goal of utility theory.

Expected utility theory

Then, in 1944, John Von Nuemann and Oskar Morgenstern developed the expected utility hypothesis , based on Daniel Bernoulli’s first description of how we make decisions by estimating the probability and utility of an outcome. By multiplying the probability of an outcome by the expected benefit of that outcome, we get the expected utility of that choice. We can then use this to form our decision—we choose what will give us the best-expected utility. By using Bayesian statistics and probability, the theory suggested that we make precise calculations about the optimal outcome and decision even when the outcome is uncertain. While this theory became massively influential, it worked best in scenarios where the expected gains and probabilities are easily calculated. For instance, this framework can be applied to games like poker, but not easily to most life decisions, where we have trouble estimating the outcomes and the likelihood that we’ll get a particular outcome.³

Behavioral economics

In 1969, by the time the expected utility hypothesis was well known among economists, two economists undertook further research into applying the theory to real life situations. Intrigued by a psychologist’s observation that people followed this logical principle in their decisions by estimating basic probability, Daniel Kahneman and Amos Tversky performed studies to test how people actually behaved in comparison to the predictions made by decision analysts based on the expected utility hypothesis. They found that people often did not follow the statistical predictions decision analysts used, instead opting for a more intuitive approach.⁴ The concept of utility in the expected utility hypothesis was then either flawed or failed to take into account certain kinds of utility.

Kahneman and Tversky continued studying utility together, and into the later 20th century, economists distinguished between two different kinds of utility: decision utility and experienced utility. Experienced utility was connected back to the utility consisting of pleasure and pain that Bentham described, and characterized as a hedonic quality , or relating to the pursuit of pleasure. Decision utility , on the other hand, was conceived as the “weight of an outcome in a decision”⁵, or the value we optimize in a decision.

In modern economics, experienced utility was largely ignored because of arguments that it could not be observed or measured, and that choices reveal the utility of the outcomes because rational agents optimize their utility. In their paper, Kahneman, Peter Wakker, and Rakesh Sarin argued that experienced utility could, in fact, be measured and was distinct from decision utility. They suggested that normal human cognition could result in our perceived utility being different from our experienced satisfaction from an outcome. They proposed a utility framework consisting of 4 different types of utility: predicted utility, decision utility, experienced utility and remembered utility. Decision utility is the utility present at the time of the decision, meaning it drives our decision-making.⁵ As a result of these different utilities, we may not always act in a way that actually maximizes the expected utility of our decision —even if we think we are behaving logically at the time—although this is what traditional economics alleges. As a result, behavioural economics developed as a separate branch of economics that accounts for psychological aspects of decision-making that may cause us to act irrationally, or away from the maximum utility.

Biological decision utility

Recently, the biological basis of decision utility has also been studied in neuroscience and connected to dopamine mechanisms in the brain. The importance of dopamine in motivation provides a biological basis for Bentham’s theory of “hedonic qualities” driving our decisions. Particular cues based on memory can also alter the utility of a particular action immediately after we encounter them, thanks to the release of dopamine.⁶ For example, when you get stressed, you might feel an overwhelming desire to smoke a cigarette. In other calm situations, however, we would feel no urge to do so. Our different reactions to these situations demonstrate how utility can change due to different levels of brain chemicals. The way our brains remember pleasurable experiences may drive us towards a desire, even if the experience fails to meet the remembered feeling.⁷

From a rather simplistic view in the late 18th century to a nuanced and humanistic perspective at the end of the 20th century, our understanding of utility has evolved dramatically. Today, our knowledge of utility as a complex, emotional, and changing quality can help us recognize short-sighted decisions and improve our choices.

Consequences

At the center of all decisions, utility is a core concept that we use every day, whether or not we’re conscious of it. Although it seems logical to assume we automatically resort to maximizing the utility of our actions, sometimes this does not appear to occur. As we experience the consequences of our decisions, big or small, it’s common to look back on ourselves and wonder “What were we thinking?”. It can seem like a different person was responsible for a poor decision, not us.

The distinction between decision utility and experienced utility can often be significant, so understanding the difference is critical to improving our decision-making. For instance, we tend to make faulty judgments of life decisions and their effect on overall satisfaction. One study about the perceived versus lived satisfaction of living in California demonstrated that we often fall prey to a “the grass is greener” mentality.⁸ The authors concluded that when thinking about differences in climate and culture, we overestimate the effect they will have on our satisfaction. In reality, these factors do not significantly impact our enjoyment of where we live—the study showed that we often believe that living in the sunny California weather will make us happier than it actually does.

The difference in decision utility and experienced utility can also be explained by something called a projection bias.⁹ We overestimate how much our future preferences will look like our current preferences. Understanding this tendency, we can recognize it when we make a poor prediction of what we need and account for the bias.

Additionally, the way decisions are framed can affect our perception of their utility. Kahneman and Tversky first demonstrated the influence of framing in a 1986 paper.¹⁰ We can know that the outcome will be the same—like if different discounts result in the same price reduction—yet, we will still be more attracted to the higher discount percentage . The psychological aspect of buying something on sale, even if it is the same price as another product of equal quality that is not on sale, is another utility that traditional economical utility does not consider. As core parts of human decision-making, we have to take our cognitive biases into account if we are going to understand how we form ideas of utility and apply it to evaluate outcomes.

Controversies

A problem often brought up with traditional economics and expected utility theory is their assumption of rationality. The term “ Homo Economicus ” describes the agent implied by traditional economics. While real humans – Homo sapiens – are significantly affected by cognitive biases and emotions, Homo economicus is rational and economically-driven. Homo economicus may evaluate utility in a narrow way which disregards the social or emotional utility involved in a decision, for instance.

At the turn of this century, Richard Thaler , an economist inspired by Kahneman and Tversky’s work, wrote a perspective piece on this issue, predicting that economics would pivot to incorporating human behaviour.¹¹ In the latter half of the 20th century, there had been a shift towards accounting for irrational human behaviour , and Thaler was correct in predicting this future in the development of behavioural economics. That said, economists have been careful to specify that the movement away from traditional economics does not mean that we are not rational beings; rather, the existing conceptions of rational behaviour fail to describe the logic humans operate by.

We might expect in making decisions, we automatically maximize utility, or go with the choice that leads to the most useful outcome, considering that earlier economists working on utility theory in the late 19th century consistently arrived at this conclusion.³ In a 2006 paper, however, Kahneman and Thaler refuted this hypothesis.¹² They found that because we do not always know what we like, as demonstrated in the California example, we make errors in predicting the future utility of outcomes. As a result, we do not maximize the utility of our decisions because we make erroneous judgments about what will be useful to us. We make intuitive decisions without really thinking things through. When we go to the grocery store on an empty stomach, we often purchase much more food than we actually need – and more than what was on our grocery list – because of how we feel at that moment.

Intuitive decision-making

This error could also be explained by a process of substitution in intuitive thinking, where we wind up answering a different question than the one we intend to address. For instance, when we’re shopping and hungry, we may be making optimized utility decisions for ourselves in that moment, because we would like to eat the food we’re buying. Although we think we’re dealing with food decisions for the week ahead, we are really just addressing our immediate food desires.

In their paper, Kahneman and Thaler addressed four situations where “hedonic forecasting”, or our ability to know what we want in the future, resulted in errors in decision-making:

Where the emotional or motivational state of the agent is very different at the time of the decision versus time of consumption
Where the nature of the decision focuses attention on aspects of the outcome that will not be relevant when it is actually experienced
When choices are made on the basis of flawed evaluations of past experiences
When people forecast their future adjustment to new life circumstances.

The first instance, as in the example of shopping while hungry, has been proven to result in different outcomes. A similar case has been observed with the current weather influencing the clothes people buy—on an abnormally cold day, it is less likely that people will buy clothes for warm weather, even for future use by ordering on the phone. “Anchoring” in the present moment can result in us making a different decision that we’re not actually conscious of, but our attention becomes focused on our present needs, so we unknowingly make a decision fit for that problem.

In the second case, the way decisions are presented to us can influence how we evaluate the utility of those choices. This is where biases like naive allocation can result in us making a non-optimized decision in terms of utility. Given different assortments of options, we choose differently.

The third case, where we carry imperfect judgments of past experiences, has to do with the way we remember past pain and pleasure. The peak/end rule suggests that our retrospective evaluation of an incident will be composed of the average of our feelings at the most extreme point and at the end of the experience. In other words, we do not remember the beginning or less extreme aspects of an experience as well as the peak/end when thinking about past incidents. Based on studies on different hedonic experiences, such as measuring pain during medical procedures, people’s evaluations of painful experiences can indeed be altered by manipulating the extreme point of the experience or changing the ending. When a procedure ended more gradually and with a period of lesser pain, patients rated it as less painful than procedures that ended abruptly with pain, even though the only difference was the length of the procedure.

The final case describes what happens when people try to envision themselves living in California—we judge our future lives by metrics that won’t actually end up mattering to us. Kahneman also found that we adapt better to situations than we expect. In fact, we often think something will be worse than it actually is. Kahneman compared how paraplegics felt about their lives after they became paralyzed against asking non-paraplegics to estimate their feelings if they became paraplegic. Interestingly, he found that non-paraplegics greatly overestimated the negative effect of the disability and that paraplegics reported doing much better than people imagined.

Thus, utility can still describe the motivation for our decisions, but past models have failed to account for all that we consider useful.

Tree Testing

Progressive Disclosure

Mobile First Design

Nielsen's Heuristics

Eager to learn about how behavioral science can help your organization?

Get new behavioral science insights in your inbox every month..

Quickonomics

Von Neumann–Morgenstern Utility Theorem

Definition of the von neumann–morgenstern utility theorem.

The Von Neumann–Morgenstern utility theorem is a foundational concept in the field of expected utility theory, which underpins much of modern economics and decision theory. It establishes that under certain axioms of rational behavior, an individual’s preferences over uncertain prospects (or “lotteries”) can be represented by a utility function. This utility function then enables the individual to rank these prospects based on their expected utilities, essentially quantifying the desirability of various outcomes.

Conceptual Example

To illustrate the Von Neumann–Morgenstern utility theorem, imagine an individual faced with choosing between two games: 1. Game A guarantees them $100. 2. Game B offers a 50% chance of winning $200 and a 50% chance of winning nothing.

According to the theorem, the individual can assign utility values to the outcomes of these games based on their preferences. If the individual is risk-neutral, they may assign a utility value of 1 to $100, 1.5 to $200, and 0 to $0, making the expected utility of Game B (0.5 * 1.5 + 0.5 * 0) higher than that of Game A. Thus, the theorem allows for a rational choice under uncertainty by comparing the expected utilities of different options.

Why the Von Neumann–Morgenstern Utility Theorem Matters

The Von Neumann–Morgenstern utility theorem is paramount in economics and decision theory for several reasons. Firstly, it provides a rigorous mathematical framework for understanding and predicting human behavior under risk and uncertainty. Through this framework, economists can analyze choices in various contexts, from investment decisions to policy-making.

Moreover, the theorem introduces the concept of expected utility, which has become a cornerstone in the theories of rational choice and welfare economics. It allows for the comparison of different policies or economic conditions based on their impact on the welfare or utility of individuals and societies as a whole. The theorem’s influence extends beyond economics into psychology, political science, and even bioethics, wherever decision-making under uncertainty is studied.

Frequently Asked Questions (FAQ)

What are the axioms of rational behavior underlying the theorem.

The Von Neumann–Morgenstern utility theorem rests on four axioms of rational preference ordering: completeness, transitivity, continuity, and independence. These axioms assume that individuals can consistently rank all possible choices, that their preferences are internally consistent, they prefer certain averages of lotteries to extreme ones, and their preferences between lotteries are invariant to irrelevant alternatives.

Can the utility function vary between individuals?

Yes, the utility function can vary significantly between individuals based on their risk tolerance. Some people are risk-averse, preferring certain outcomes over gambles with potentially higher payoffs but greater uncertainty. Others may be risk-seeking, favoring the excitement or potential for higher returns that uncertainty brings. The utility function captures these individual preferences and risk attitudes.

How does the theorem apply to real-world economic behavior?

In real-world economics, the theorem helps explain and predict individuals’ choices in situations involving risk, such as in financial investments, insurance markets, and gambling. It provides a theoretical foundation for the expected utility hypothesis, which asserts that individuals choose among risky projects to maximize their expected utility. This concept is applied in portfolio theory, the pricing of insurance, and in evaluating the economic effectiveness of public policies under uncertainty.

What are the limitations of the Von Neumann–Morgenstern utility theorem?

While influential, the theorem is not without critics. One limitation is its assumption of rational behavior, which may not always hold in real-world decision-making due to cognitive biases, lack of information, or other psychological factors. Additionally, the axioms of the theorem, especially the independence axiom, have been subjected to empirical scrutiny and found wanting in certain situations, as demonstrated in the Allais paradox and Ellsberg paradox. These limitations have prompted the development of alternative models that attempt to accommodate observed deviations from the theorem’s predictions about rational behavior.

The Von Neumann–Morgenstern utility theorem remains a fundamental pillar of economic theory, providing a mathematical basis for understanding choices under uncertainty. Despite its limitations, the theorem’s influence pervades economic analysis, decision theory, and beyond, highlighting its enduring relevance and utility in navigating the complexities of human decision-making.

To provide the best experiences, we and our partners use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us and our partners to process personal data such as browsing behavior or unique IDs on this site and show (non-) personalized ads. Not consenting or withdrawing consent, may adversely affect certain features and functions.

Click below to consent to the above or make granular choices. Your choices will be applied to this site only. You can change your settings at any time, including withdrawing your consent, by using the toggles on the Cookie Policy, or by clicking on the manage consent button at the bottom of the screen.

Search Menu
Browse content in Arts and Humanities
Browse content in Archaeology
Anglo-Saxon and Medieval Archaeology
Archaeological Methodology and Techniques
Archaeology by Region
Archaeology of Religion
Archaeology of Trade and Exchange
Biblical Archaeology
Contemporary and Public Archaeology
Environmental Archaeology
Historical Archaeology
History and Theory of Archaeology
Industrial Archaeology
Landscape Archaeology
Mortuary Archaeology
Prehistoric Archaeology
Underwater Archaeology
Urban Archaeology
Zooarchaeology
Browse content in Architecture
Architectural Structure and Design
History of Architecture
Residential and Domestic Buildings
Theory of Architecture
Browse content in Art
Art Subjects and Themes
History of Art
Industrial and Commercial Art
Theory of Art
Biographical Studies
Byzantine Studies
Browse content in Classical Studies
Classical Literature
Classical Reception
Classical History
Classical Philosophy
Classical Mythology
Classical Art and Architecture
Classical Oratory and Rhetoric
Greek and Roman Papyrology
Greek and Roman Archaeology
Greek and Roman Epigraphy
Greek and Roman Law
Late Antiquity
Religion in the Ancient World
Digital Humanities
Browse content in History
Colonialism and Imperialism
Diplomatic History
Environmental History
Genealogy, Heraldry, Names, and Honours
Genocide and Ethnic Cleansing
Historical Geography
History by Period
History of Emotions
History of Agriculture
History of Education
History of Gender and Sexuality
Industrial History
Intellectual History
International History
Labour History
Legal and Constitutional History
Local and Family History
Maritime History
Military History
National Liberation and Post-Colonialism
Oral History
Political History
Public History
Regional and National History
Revolutions and Rebellions
Slavery and Abolition of Slavery
Social and Cultural History
Theory, Methods, and Historiography
Urban History
World History
Browse content in Language Teaching and Learning
Language Learning (Specific Skills)
Language Teaching Theory and Methods
Browse content in Linguistics
Applied Linguistics
Cognitive Linguistics
Computational Linguistics
Forensic Linguistics
Grammar, Syntax and Morphology
Historical and Diachronic Linguistics
History of English
Language Evolution
Language Reference
Language Variation
Language Families
Language Acquisition
Lexicography
Linguistic Anthropology
Linguistic Theories
Linguistic Typology
Phonetics and Phonology
Psycholinguistics
Sociolinguistics
Translation and Interpretation
Writing Systems
Browse content in Literature
Bibliography
Children's Literature Studies
Literary Studies (Romanticism)
Literary Studies (American)
Literary Studies (Modernism)
Literary Studies (Asian)
Literary Studies (European)
Literary Studies (Eco-criticism)
Literary Studies - World
Literary Studies (1500 to 1800)
Literary Studies (19th Century)
Literary Studies (20th Century onwards)
Literary Studies (African American Literature)
Literary Studies (British and Irish)
Literary Studies (Early and Medieval)
Literary Studies (Fiction, Novelists, and Prose Writers)
Literary Studies (Gender Studies)
Literary Studies (Graphic Novels)
Literary Studies (History of the Book)
Literary Studies (Plays and Playwrights)
Literary Studies (Poetry and Poets)
Literary Studies (Postcolonial Literature)
Literary Studies (Queer Studies)
Literary Studies (Science Fiction)
Literary Studies (Travel Literature)
Literary Studies (War Literature)
Literary Studies (Women's Writing)
Literary Theory and Cultural Studies
Mythology and Folklore
Shakespeare Studies and Criticism
Browse content in Media Studies
Browse content in Music
Applied Music
Dance and Music
Ethics in Music
Ethnomusicology
Gender and Sexuality in Music
Medicine and Music
Music Cultures
Music and Media
Music and Culture
Music and Religion
Music Education and Pedagogy
Music Theory and Analysis
Musical Scores, Lyrics, and Libretti
Musical Structures, Styles, and Techniques
Musicology and Music History
Performance Practice and Studies
Race and Ethnicity in Music
Sound Studies
Browse content in Performing Arts
Browse content in Philosophy
Aesthetics and Philosophy of Art
Epistemology
Feminist Philosophy
History of Western Philosophy
Metaphysics
Moral Philosophy
Non-Western Philosophy
Philosophy of Language
Philosophy of Mind
Philosophy of Perception
Philosophy of Action
Philosophy of Law
Philosophy of Religion
Philosophy of Science
Philosophy of Mathematics and Logic
Practical Ethics
Social and Political Philosophy
Browse content in Religion
Biblical Studies
Christianity
East Asian Religions
History of Religion
Judaism and Jewish Studies
Qumran Studies
Religion and Education
Religion and Health
Religion and Politics
Religion and Science
Religion and Law
Religion and Art, Literature, and Music
Religious Studies
Browse content in Society and Culture
Cookery, Food, and Drink
Cultural Studies
Customs and Traditions
Ethical Issues and Debates
Hobbies, Games, Arts and Crafts
Lifestyle, Home, and Garden
Natural world, Country Life, and Pets
Popular Beliefs and Controversial Knowledge
Sports and Outdoor Recreation
Technology and Society
Travel and Holiday
Visual Culture
Browse content in Law
Arbitration
Browse content in Company and Commercial Law
Commercial Law
Company Law
Browse content in Comparative Law
Systems of Law
Competition Law
Browse content in Constitutional and Administrative Law
Government Powers
Judicial Review
Local Government Law
Military and Defence Law
Parliamentary and Legislative Practice
Construction Law
Contract Law
Browse content in Criminal Law
Criminal Procedure
Criminal Evidence Law
Sentencing and Punishment
Employment and Labour Law
Environment and Energy Law
Browse content in Financial Law
Banking Law
Insolvency Law
History of Law
Human Rights and Immigration
Intellectual Property Law
Browse content in International Law
Private International Law and Conflict of Laws
Public International Law
IT and Communications Law
Jurisprudence and Philosophy of Law
Law and Society
Law and Politics
Browse content in Legal System and Practice
Courts and Procedure
Legal Skills and Practice
Primary Sources of Law
Regulation of Legal Profession
Medical and Healthcare Law
Browse content in Policing
Criminal Investigation and Detection
Police and Security Services
Police Procedure and Law
Police Regional Planning
Browse content in Property Law
Personal Property Law
Study and Revision
Terrorism and National Security Law
Browse content in Trusts Law
Wills and Probate or Succession
Browse content in Medicine and Health
Browse content in Allied Health Professions
Arts Therapies
Clinical Science
Dietetics and Nutrition
Occupational Therapy
Operating Department Practice
Physiotherapy
Radiography
Speech and Language Therapy
Browse content in Anaesthetics
General Anaesthesia
Neuroanaesthesia
Clinical Neuroscience
Browse content in Clinical Medicine
Acute Medicine
Cardiovascular Medicine
Clinical Genetics
Clinical Pharmacology and Therapeutics
Dermatology
Endocrinology and Diabetes
Gastroenterology
Genito-urinary Medicine
Geriatric Medicine
Infectious Diseases
Medical Toxicology
Medical Oncology
Pain Medicine
Palliative Medicine
Rehabilitation Medicine
Respiratory Medicine and Pulmonology
Rheumatology
Sleep Medicine
Sports and Exercise Medicine
Community Medical Services
Critical Care
Emergency Medicine
Forensic Medicine
Haematology
History of Medicine
Browse content in Medical Skills
Clinical Skills
Communication Skills
Nursing Skills
Surgical Skills
Medical Ethics
Browse content in Medical Dentistry
Oral and Maxillofacial Surgery
Paediatric Dentistry
Restorative Dentistry and Orthodontics
Surgical Dentistry
Medical Statistics and Methodology
Browse content in Neurology
Clinical Neurophysiology
Neuropathology
Nursing Studies
Browse content in Obstetrics and Gynaecology
Gynaecology
Occupational Medicine
Ophthalmology
Otolaryngology (ENT)
Browse content in Paediatrics
Neonatology
Browse content in Pathology
Chemical Pathology
Clinical Cytogenetics and Molecular Genetics
Histopathology
Medical Microbiology and Virology
Patient Education and Information
Browse content in Pharmacology
Psychopharmacology
Browse content in Popular Health
Caring for Others
Complementary and Alternative Medicine
Self-help and Personal Development
Browse content in Preclinical Medicine
Cell Biology
Molecular Biology and Genetics
Reproduction, Growth and Development
Primary Care
Professional Development in Medicine
Browse content in Psychiatry
Addiction Medicine
Child and Adolescent Psychiatry
Forensic Psychiatry
Learning Disabilities
Old Age Psychiatry
Psychotherapy
Browse content in Public Health and Epidemiology
Epidemiology
Public Health
Browse content in Radiology
Clinical Radiology
Interventional Radiology
Nuclear Medicine
Radiation Oncology
Reproductive Medicine
Browse content in Surgery
Cardiothoracic Surgery
Gastro-intestinal and Colorectal Surgery
General Surgery
Neurosurgery
Paediatric Surgery
Peri-operative Care
Plastic and Reconstructive Surgery
Surgical Oncology
Transplant Surgery
Trauma and Orthopaedic Surgery
Vascular Surgery
Browse content in Science and Mathematics
Browse content in Biological Sciences
Aquatic Biology
Biochemistry
Bioinformatics and Computational Biology
Developmental Biology
Ecology and Conservation
Evolutionary Biology
Genetics and Genomics
Microbiology
Molecular and Cell Biology
Natural History
Plant Sciences and Forestry
Research Methods in Life Sciences
Structural Biology
Systems Biology
Zoology and Animal Sciences
Browse content in Chemistry
Analytical Chemistry
Computational Chemistry
Crystallography
Environmental Chemistry
Industrial Chemistry
Inorganic Chemistry
Materials Chemistry
Medicinal Chemistry
Mineralogy and Gems
Organic Chemistry
Physical Chemistry
Polymer Chemistry
Study and Communication Skills in Chemistry
Theoretical Chemistry
Browse content in Computer Science
Artificial Intelligence
Computer Architecture and Logic Design
Game Studies
Human-Computer Interaction
Mathematical Theory of Computation
Programming Languages
Software Engineering
Systems Analysis and Design
Virtual Reality
Browse content in Computing
Business Applications
Computer Games
Computer Security
Computer Networking and Communications
Digital Lifestyle
Graphical and Digital Media Applications
Operating Systems
Browse content in Earth Sciences and Geography
Atmospheric Sciences
Environmental Geography
Geology and the Lithosphere
Maps and Map-making
Meteorology and Climatology
Oceanography and Hydrology
Palaeontology
Physical Geography and Topography
Regional Geography
Soil Science
Urban Geography
Browse content in Engineering and Technology
Agriculture and Farming
Biological Engineering
Civil Engineering, Surveying, and Building
Electronics and Communications Engineering
Energy Technology
Engineering (General)
Environmental Science, Engineering, and Technology
History of Engineering and Technology
Mechanical Engineering and Materials
Technology of Industrial Chemistry
Transport Technology and Trades
Browse content in Environmental Science
Applied Ecology (Environmental Science)
Conservation of the Environment (Environmental Science)
Environmental Sustainability
Environmentalist Thought and Ideology (Environmental Science)
Management of Land and Natural Resources (Environmental Science)
Natural Disasters (Environmental Science)
Nuclear Issues (Environmental Science)
Pollution and Threats to the Environment (Environmental Science)
Social Impact of Environmental Issues (Environmental Science)
History of Science and Technology
Browse content in Materials Science
Ceramics and Glasses
Composite Materials
Metals, Alloying, and Corrosion
Nanotechnology
Browse content in Mathematics
Applied Mathematics
Biomathematics and Statistics
History of Mathematics
Mathematical Education
Mathematical Finance
Mathematical Analysis
Numerical and Computational Mathematics
Probability and Statistics
Pure Mathematics
Browse content in Neuroscience
Cognition and Behavioural Neuroscience
Development of the Nervous System
Disorders of the Nervous System
History of Neuroscience
Invertebrate Neurobiology
Molecular and Cellular Systems
Neuroendocrinology and Autonomic Nervous System
Neuroscientific Techniques
Sensory and Motor Systems
Browse content in Physics
Astronomy and Astrophysics
Atomic, Molecular, and Optical Physics
Biological and Medical Physics
Classical Mechanics
Computational Physics
Condensed Matter Physics
Electromagnetism, Optics, and Acoustics
History of Physics
Mathematical and Statistical Physics
Measurement Science
Nuclear Physics
Particles and Fields
Plasma Physics
Quantum Physics
Relativity and Gravitation
Semiconductor and Mesoscopic Physics
Browse content in Psychology
Affective Sciences
Clinical Psychology
Cognitive Psychology
Cognitive Neuroscience
Criminal and Forensic Psychology
Developmental Psychology
Educational Psychology
Evolutionary Psychology
Health Psychology
History and Systems in Psychology
Music Psychology
Neuropsychology
Organizational Psychology
Psychological Assessment and Testing
Psychology of Human-Technology Interaction
Psychology Professional Development and Training
Research Methods in Psychology
Social Psychology
Browse content in Social Sciences
Browse content in Anthropology
Anthropology of Religion
Human Evolution
Medical Anthropology
Physical Anthropology
Regional Anthropology
Social and Cultural Anthropology
Theory and Practice of Anthropology
Browse content in Business and Management
Business Ethics
Business History
Business Strategy
Business and Technology
Business and Government
Business and the Environment
Comparative Management
Corporate Governance
Corporate Social Responsibility
Entrepreneurship
Health Management
Human Resource Management
Industrial and Employment Relations
Industry Studies
Information and Communication Technologies
International Business
Knowledge Management
Management and Management Techniques
Operations Management
Organizational Theory and Behaviour
Pensions and Pension Management
Public and Nonprofit Management
Strategic Management
Supply Chain Management
Browse content in Criminology and Criminal Justice
Criminal Justice
Criminology
Forms of Crime
International and Comparative Criminology
Youth Violence and Juvenile Justice
Development Studies
Browse content in Economics
Agricultural, Environmental, and Natural Resource Economics
Asian Economics
Behavioural Finance
Behavioural Economics and Neuroeconomics
Econometrics and Mathematical Economics
Economic History
Economic Methodology
Economic Systems
Economic Development and Growth
Financial Markets
Financial Institutions and Services
General Economics and Teaching
Health, Education, and Welfare
History of Economic Thought
International Economics
Labour and Demographic Economics
Law and Economics
Macroeconomics and Monetary Economics
Microeconomics
Public Economics
Urban, Rural, and Regional Economics
Welfare Economics
Browse content in Education
Adult Education and Continuous Learning
Care and Counselling of Students
Early Childhood and Elementary Education
Educational Equipment and Technology
Educational Strategies and Policy
Higher and Further Education
Organization and Management of Education
Philosophy and Theory of Education
Schools Studies
Secondary Education
Teaching of a Specific Subject
Teaching of Specific Groups and Special Educational Needs
Teaching Skills and Techniques
Browse content in Environment
Applied Ecology (Social Science)
Climate Change
Conservation of the Environment (Social Science)
Environmentalist Thought and Ideology (Social Science)
Natural Disasters (Environment)
Social Impact of Environmental Issues (Social Science)
Browse content in Human Geography
Cultural Geography
Economic Geography
Political Geography
Browse content in Interdisciplinary Studies
Communication Studies
Museums, Libraries, and Information Sciences
Browse content in Politics
African Politics
Asian Politics
Chinese Politics
Comparative Politics
Conflict Politics
Elections and Electoral Studies
Environmental Politics
European Union
Foreign Policy
Gender and Politics
Human Rights and Politics
Indian Politics
International Relations
International Organization (Politics)
International Political Economy
Irish Politics
Latin American Politics
Middle Eastern Politics
Political Behaviour
Political Economy
Political Institutions
Political Theory
Political Methodology
Political Communication
Political Philosophy
Political Sociology
Politics and Law
Public Policy
Public Administration
Quantitative Political Methodology
Regional Political Studies
Russian Politics
Security Studies
State and Local Government
UK Politics
US Politics
Browse content in Regional and Area Studies
African Studies
Asian Studies
East Asian Studies
Japanese Studies
Latin American Studies
Middle Eastern Studies
Native American Studies
Scottish Studies
Browse content in Research and Information
Research Methods
Browse content in Social Work
Addictions and Substance Misuse
Adoption and Fostering
Care of the Elderly
Child and Adolescent Social Work
Couple and Family Social Work
Developmental and Physical Disabilities Social Work
Direct Practice and Clinical Social Work
Emergency Services
Human Behaviour and the Social Environment
International and Global Issues in Social Work
Mental and Behavioural Health
Social Justice and Human Rights
Social Policy and Advocacy
Social Work and Crime and Justice
Social Work Macro Practice
Social Work Practice Settings
Social Work Research and Evidence-based Practice
Welfare and Benefit Systems
Browse content in Sociology
Childhood Studies
Community Development
Comparative and Historical Sociology
Economic Sociology
Gender and Sexuality
Gerontology and Ageing
Health, Illness, and Medicine
Marriage and the Family
Migration Studies
Occupations, Professions, and Work
Organizations
Population and Demography
Race and Ethnicity
Social Theory
Social Movements and Social Change
Social Research and Statistics
Social Stratification, Inequality, and Mobility
Sociology of Religion
Sociology of Education
Sport and Leisure
Urban and Rural Studies
Browse content in Warfare and Defence
Defence Strategy, Planning, and Research
Land Forces and Warfare
Military Administration
Military Life and Institutions
Naval Forces and Warfare
Other Warfare and Defence Issues
Peace Studies and Conflict Resolution
Weapons and Equipment

The Handbook of Rational and Social Choice

< Previous chapter
Next chapter >

CHAPTER 1 CHAPTER 1 Expected Utility Theory

Published: January 2009
Cite Icon Cite
Permissions Icon Permissions

This chapter reviews classic normative expected utility theory. The goal is to frame the subsequent chapters (which consider more modern extensions to and deviations from this classic theory) in a way that is accessible to the nonspecialist but also useful to the specialist. The chapter starts from scratch with a revealed preference approach to the existence of a utility function. It then presents the mathematical structure of additive and linear utility representations and their axiomatizations, in the context of abstract choice theory and using intertemporal choice as a source of examples. The chapter is thus able to focus on this mathematical structure without interference the specific interpretation and notation for decision under uncertainty. Furthermore, this approach allows the chapter to focus on the interpretation of the axioms when it turns to decision under uncertainty.

Signed in as

Institutional accounts.

Google Scholar Indexing
GoogleCrawler [DO NOT DELETE]

Personal account

Sign in with email/username & password
Get email alerts
Save searches
Purchase content
Activate your purchase/trial code

Institutional access

Sign in with a library card Sign in with username/password Recommend to your librarian
Institutional account management
Get help with access

Access to content on Oxford Academic is often provided through institutional subscriptions and purchases. If you are a member of an institution with an active account, you may be able to access content in one of the following ways:

IP based access

Typically, access is provided across an institutional network to a range of IP addresses. This authentication occurs automatically, and it is not possible to sign out of an IP authenticated account.

Sign in through your institution

Choose this option to get remote access when outside your institution. Shibboleth/Open Athens technology is used to provide single sign-on between your institutionâ€™s website and Oxford Academic.

Click Sign in through your institution.
Select your institution from the list provided, which will take you to your institution's website to sign in.
When on the institution site, please use the credentials provided by your institution. Do not use an Oxford Academic personal account.
Following successful sign in, you will be returned to Oxford Academic.

If your institution is not listed or you cannot sign in to your institutionâ€™s website, please contact your librarian or administrator.

Sign in with a library card

Enter your library card number to sign in. If you cannot sign in, please contact your librarian.

Society Members

Society member access to a journal is achieved in one of the following ways:

Sign in through society site

Many societies offer single sign-on between the society website and Oxford Academic. If you see â€˜Sign in through society siteâ€™ in the sign in pane within a journal:

Click Sign in through society site.
When on the society site, please use the credentials provided by that society. Do not use an Oxford Academic personal account.

If you do not have a society account or have forgotten your username or password, please contact your society.

Sign in using a personal account

Some societies use Oxford Academic personal accounts to provide access to their members. See below.

A personal account can be used to get email alerts, save searches, purchase content, and activate subscriptions.

Some societies use Oxford Academic personal accounts to provide access to their members.

Viewing your signed in accounts

Click the account icon in the top right to:

View your signed in personal account and access account management features.
View the institutional accounts that are providing access.

Signed in but can't access content

Oxford Academic is home to a wide variety of products. The institutional subscription may not cover the content that you are trying to access. If you believe you should have access to that content, please contact your librarian.

For librarians and administrators, your personal account also provides access to institutional account management. Here you will find options to view and activate subscriptions, manage institutional settings and access options, access usage statistics, and more.

Our books are available by subscription or purchase to libraries and institutions.

About Oxford Academic
Publish journals with us
University press partners
What we publish
New features
Open access
Rights and permissions
Accessibility
Advertising
Media enquiries
Oxford University Press
Oxford Languages
University of Oxford

Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide

Copyright © 2024 Oxford University Press
Cookie settings
Cookie policy
Privacy policy
Legal notice

This Feature Is Available To Subscribers Only

This PDF is available to Subscribers Only

For full access to this pdf, sign in to an existing account, or purchase an annual subscription.

An official website of the United States government

The .gov means it's official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you're on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Publications
Account settings
Browse Titles

NCBI Bookshelf. A service of the National Library of Medicine, National Institutes of Health.

Kohlhammer VW, author; Schildmann J, Buch C, Zerth J, editors. Defining the Value of Medical Interventions: Normative and Empirical Challenges [Internet]. Stuttgart (DE): W. Kohlhammer GmbH; 2021.

Cover of Defining the Value of Medical Interventions

Defining the Value of Medical Interventions: Normative and Empirical Challenges [Internet].

The assessment of value in health economics: utility and capability.

Jasper Ubels .

There is a discussion within the field of health economics about the appropriate informational base on which to assess value. One method to assess value in medical interventions is with the quality-adjusted life year (QALY). In the calculation of QALYs, the value of a life year is adjusted with a utility value. Multiple conceptualizations of utility exist. In one of these conceptualizations, utility represents a positive mental state; in another, utility reflects the preferences of individuals for certain things.

However, according to Nobel Laureate Amartya Sen, these conceptualizations of utility have limitations. Essentially, he argues that things might have an economic value beyond utility for a multitude of reasons. The first reason being that people might place value on their ability to choose between different alternatives beyond the availability of an alternative that maximizes an individual's utility, for the sake of choice itself. Second, individuals might choose things that go against their personal preferences for a variety of different motivations. The third reason is related to the positive mental health state conceptualization of utility. One particular problem with this conceptualization is that people adapt to limitations. This leads to individuals with severe disabilities reporting higher levels of subjective wellbeing than expected. In short, utility itself is too limited of a concept to use as the informational base for the assessment of value.

Instead, Sen argues that the value assessment should be based on the capabilities of individuals. Capabilities are understood as the freedom of individuals to do or to be. That what people are or are doing with their freedom, is called functionings. Sen argues that the use of capabilities as an informational base is preferred over utility and functioning. Namely, through measurement of capability, it is possible to measure all the alternative options available to an individual. This includes the utility derived from those options as well as the value of being able to choose between options. Furthermore, adaptation by individuals to limitations does not influence the assessment of value, since the informational base of capability is concerned with the freedom of individuals to do or be. Thus, by using capabilities as the informational base, it is possible to assess the value of medical interventions without the problems posed by using utility.

However, is the measurement of capabilities alone sufficient to assess value? Based on theoretical considerations by Fleurbay and Clark, as well as observational studies in patients affected by the locked-in syndrome, a medical condition in which a patient is aware but cannot communicate due to muscle paralysis, this chapter concludes that this is not the case. Capability, functioning and utility are all, when used individually, insufficient to estimate the value of a medical intervention. Instead, information about capability, functioning and utility needs to be combined for an appropriate assessment of the value of medical interventions.

1. Introduction

This chapter will dive deeper into the theory underlying the assessment of value in the context of evaluating medical interventions. 1 Section one presents the informational base that is used to assess value in conventional health economics. Informational base refers to the kind of information used to assess value. In section two, the fundamental critique of conventional economics by Amartya Sen and his theory about the appropriate informational base on which to assess value are introduced. In section three, Sen's theory is scrutinized. The chapter concludes with the assertion that Sen's theory of the appropriate informational base of value and certain understandings of utility need to be combined to have a complete informational base on which assess value.

2. The informational base in conventional health economics

The QALY is a measure that combines information about both the length and the quality of life. QALYs are applied in the valuation of medical interventions by assessing the increase in life years as a result of a medical interventions along with the utility adjusted quality of each of those life years ( Torrance, 1986 ). 2

The concept utility itself can be understood in a variety of different ways – for a discussion, see Sen ( 1985b ) and Richardson ( 1994 ). In one conceptualization, utility is viewed as the happiness of an individual ( Sen, 1985b ). In this understanding, utility is a representation of a person's positive mental health state. The value of a medical intervention would be expressed by its effect on the happiness of an individual. Medical interventions that, for example, increase the happiness of an individual create utility, thus providing value.

An alternative way of looking at the utility value of something is by expressing utility in terms of the preferences of individuals. These preferences can be observed by the individuals’ choice behaviour ( Luce and Raiffa, 1958 ) 3 . From such observations, an ordinal scale can be generated. The following example illustrates how this works in practice. Take a set of snacks consisting of chocolate bars, lollipops, chewing gum and broccoli. From this set, you can observe an individual choosing a bar of cholate. In this case, it is possible to say that a chocolate bar has a higher utility compared to the lollipops, chewing gum, and broccoli.

However, based on this observation, it is not possible to attribute numerical values to this utility. By observing the choice of an individual, we can only know which snack is preferred over the other by that individual. We can thus create a ranking of the snacks in terms of utility. However, by doing so we only know the order of preferences for the snacks. We have no information about how much stronger the preference for one snack is over another.

In order to attribute numerical values to utility that represent the strength of a preference, a more sophisticated framework with further assumptions is necessary. In the context of the example presented above, it is possible to translate the preference of an individual for a certain type of snack into a numerical utility value assigned to that snack. By doing so, a cardinal scale is created. One way of eliciting these numerical values is by introducing a gamble, which might consists of two choices ( Drummond et a.l, 2015 ; Gafni, 1994 ): choice one is a certain probability of the individual eating a chocolate bar, let's say probability p is p = 0.4. This probability is coupled with the probability of not eating anything, which is p = 0.6. Choice two is eating broccoli. Then through an iterative process, it is possible to find that probability of eating chocolate compared to not eating something at all where someone is indifferent between choice one and choice two. This point could be at p = 0.2 of eating a chocolate bar (with an associated probability p = 0.8 of not eating anything). Then, it is possible to say that from a scale from 0, which means not eating anything at all, to 1, which is eating chocolate, broccoli has a utility value of 0.2. A similar exercise can be repeated separately for lollipops and chewing gum. This way, by using eating a chocolate bar and not eating anything as anchor points for a scale, it is possible to create a numerical scale for the utility of eating broccoli, lollipops and chewing gum. This numerical scale gives the numerical value of eating broccoli, lollipops and chewing gum on a scale from eating nothing to eating chocolate. The above presented exercise is called the “standard gamble”.

An example of how the standard gamble is used in health economics is illustrated by how utility values are generated for the EQ-5D ( Rabin and Charro, 2001 ). The EQ-5D is a standardized instrument to assess the value of a medical intervention. It measures health related quality of life and consists of five different health related domains: mobility, self-care, usual activities, pain/discomfort and anxiety/depression. Per domain there are response categories representing different levels of health. For example, the domain pain/discomfort contains response categories ranging from “I have extreme pain or discomfort” to “I have no pain or discomfort”. One standard approach of using the EQ-5D is by measuring the change in the score of the different domains of questionnaire in a group of people before and after a new medical intervention is applied. This change is compared to the change in a control group where a standard treatment is administered. These domain scores can be translated into so-called utility values. The difference in the magnitude of change of the domain scores between the group where the medical intervention is applied and the control group with a standard treatment is the utility value of that medical intervention. This utility value is usually associated with a timeframe, which is typically a year. In this way, the effect of a medical intervention on the quality of a life year is assessed and QALYs can be calculated. But how are these utility values for domain scores generated? One method for doing so is by applying the standard gamble exercise to translate domain scores of the EQ-5D into utility values.

A standard gamble exercise for the elicitation of utility values for the different combinations of domain scores consists of presenting a sample of the population with a health state that is a combination of answer categories of these domains. An exemplary health state can be found in table 1 .

Example of a health state based on the EQ5D.

Then, participants are asked to either live in that health state for the rest of their lives, or choose the alternative option. This alternative option is to be immediately healthy for the rest of their lives, with a probability of dying. That point where the probability of becoming healthy immediately or die is equal to living in a certain health state for a number of years represents the utility value for that health state 4 .

The theoretical framework on which the standard gamble is based is the von Neumann-Morgenstern theory of utility, otherwise known as expected utility theory ( Gafni, 1994 ; Torrance, 1986 ). This theory is developed to explain how people should make rational decisions under uncertainty, while taking into account the strength of the preferences of individuals. It is important to be aware of the theoretical assumptions underlying the elicitation of preferences, because failure to meet one of the assumptions might result in the elicited preferences being invalid. One important assumption in the context of preference measurement is that the preferences themselves are assumed to be “complete” ( Warren et al., 2011 ). This means, that an individual knows what kind of options are available, and is able to provide a subjective value to those options, which are expressed when an individual makes a choice.

As with any framework, there are proponents and opponents discussing the merits and flaws of the use of expected utility theory to assess the value of things 5 . However, in the conventional health economic practice, utility plays an important role as the informational base for the assessment of value.

3. Broadening the informational base with the capability approach

3.1 amartya sen and the capability approach.

In the previous section an introduction was given about the use of utility as the informational base on which value is assessed in the field of health economics. We also mentioned that there are critics of the use of utility. One of the most prominent criticists is Amartya Sen. Sen criticizes the methods used to elicit utility, but also argues that utility as an informational base to assess value is limited in the first place ( Sen, 1985a ).

To best appreciate Sen's critique, it is necessary to understand his alternative theory that, amongst other things, justifies a broader informational base for the assessment of value. Sen calls this theory the capability approach. In this approach, Sen argues that the informational base that is used to evaluate the life of an individual should not be limited to what an individual is or does. Instead, the assessment of wellbeing should focus on the freedom of what an individual can be or can do ( Sen, 1985a ). The value of medical intervention can then, amongst other things, be based on its effects on the freedom of an individual. The benefits of extending the informational base for the assessment of value to the freedom of individuals is illustrated in the following example:

Imagine two individuals: Karla and Pierre. Both Karla and Pierre are losing weight. However, there is a difference between Karla and Pierre. Karla lives in an affluent area with enough possibilities to eat, but chooses to fast. Pierre lives in an impoverished area that is affected by famine and is starving. As a consequence, both Karla and Pierre are equally hungry because of their food intake. By only considering what Karla and Pierre are actually eating, it is impossible to say who is better off. However, when taking into account what Karla and Pierre can eat, it is clear that Karla is better off than Pierre.

In the context of the capability approach, the actual level of food intake by Karla and Pierre would be their functioning; the freedom to eat, a capability. By extending the informational base for the assessment of value to the capabilities of individuals, proponents of the capability approach argue that a more complete picture of the wellbeing of an individual can be assessed, which also results in an improved assessment of value of interventions aimed at increasing how well-off individuals are.

3.2 The advantages of using capability over utility

But what are the advantages of using capability as an informational base for the assessment of value over utility? One of the critiques of Sen on the use of expected utility is that things might have value beyond the preferences of individuals ( Sen, 1985b ). Take for example again, the scenario with the chocolate, the lollipop, chewing gum, and broccoli. Even if a person is only interested in eating chocolate, it is still possible that the mere ability to choose from the different snacks (or not eat anything at all) may lead to higher levels of wellbeing. In other words, if all the other snack options are taken away except for chocolate (which is possibly the preferred choice over the other snacks), one might still experience being worse off, since choice itself can be seen as part of life ( Sen, 1993 ).

Additionally, by extending the evaluative space to capability, one takes into account the fact that people might choose things that might not be their immediate preference or create happiness ( Sen, 1985b ). For example, a father may choose to eat broccoli instead of chocolate, all the while preferring chocolate, in order to set a good example for his child. In this case, the choice is not only limited to the father's own interests, but also takes the interest of another into account. This contradicts the conclusions that are drawn from the utility elicitation methods presented in section one, since people value (and have reason to value) things beyond their own preferences and their personal happiness.

Furthermore, can we trust that people have complete preferences? It is very much possible that people are not aware of all the choice options available to them, and they might also be unable to have a value for each of these options. To illustrate these points in the context of the assessment of an individual's health, Sen ( 2002 ) compared the life expectancy of the United States and various states of India with the incidence of self-reported morbidity of the United States and those respective states of India. Self-reported morbidity is a way of assessing disease in a population, where you ask people what kind of diseases or symptoms of diseases they have. Sen observed that people who live in regions with higher levels of education, better medical care and higher life expectancy report more comorbidities. Vice versa, the inhabitants of regions with lower levels of education, worse medical care might and lower life expectancy report less comorbidities. Sen concluded, that relying on the self-reported information of people often results in misleading evaluations of the health of those people. Furthermore, it is also questionable if the self-reported preferences of individuals are a trustworthy source of information – see ( Warren et al., 2011 ) for a discussion regarding preference construction.

Moreover, when utility is understood as reflecting a positive mental health state, another problem appears: that of adaptation. According to Sen, people can be happy and consider themselves to be well off even under dire circumstances ( Sen, 1985b ). This adaptation phenomenon occurs because people can adapt to limitations in their capabilities. A practical example of this can be found in patients affected by the so called “locked-in syndrome” (LIS). Patients affected by LIS are severely impaired in their movement abilities, but are otherwise not severely cognitively impaired ( Smith and Delargy, 2005 ). Patients affected by LIS can have a life expectancy of up to several decades ( Laureys et al., 2005 ). Patients can typically be divided in certain subgroups. These subgroups range from “incomplete” LIS, where patients have some rudimentary movement ability left, such as the movement of a foot or a finger, to complete LIS, where patients are completely unable to move, including movement of their eyes ( Smith and Delargy, 2005 ).

Surprisingly, patients affected by LIS report reasonably high levels of subjective wellbeing ( Bruno et al., 2011 ; Rousseau et al., 2015 ). Bruno et al. ( 2011 ) measured subjective wellbeing on an 11-point scale, with answers ranging from “as bad as in the worst period in my life” to “as well as in the best period in my life 6 “. Despite their physical limitations, patients considered themselves to be well off. Furthermore, the patients’ reported wellbeing remained stable over a longer period of time ( Rousseau et al., 2015 ). This observation is a defence for the assessment of the wellbeing of individuals in terms of their capabilities, as one could argue that even though the subjective wellbeing of patients affected by LIS is high, people without LIS are, nevertheless, better off. Thus, by looking at the disadvantages of using utility as the informational base on which to assess value and the advantages of focusing on capability, it seems that capabilities are a more appropriate informational base on which to assess value.

Finally, according to Sen, by assessing value in terms of capability, it is also possible to capture the utility that people derive from having that capability. For example, by evaluating the capability to eat certain snacks, one also includes the evaluation of the utility, since the evaluative space covers all the possible preferences of the individual, as well as the happiness derived from eating certain snacks ( Sen, 1985a ). Thus, capabilities are argued to be a sufficient informational base to assess value. Unfortunately, Sen does not provide an in depth explanation of how capability is related to utility, particularly in the context of utility conceptualized as a positive mental state. This lack of explanation has been considered a limitation of the capability approach ( Clark, 2005 ).

4. The need to integrate utility and the capability approach

In the last section we discussed the limitations of this use of utility as an informational base for the assessment of value based on the work of Sen. Furthermore, we introduced the capability approach as an alternative theory which argues for the extending the informational base of the evaluation of value to the capabilities of individuals. This was followed up with a discussion of the advantages of extending the informational base of value to capability.

However, should it be concluded that capabilities are a sufficient informational base to assess value? The example of Karla and Pierre illustrates, of course, that the use of capability adds additional information beyond utility that we have reason to value. However, authors have pointed out the limitations of solely using capability as the informational base on which to assess value ( Clark, 2005 ; Fleurbaey, 2006 ). According to Fleurbaey ( 2006 ), the hunger and fasting example does not show that capabilities themselves are a sufficient base of information, but merely show the added benefit of extending the informational base to capability in the assessment of wellbeing. Fleurbaey ( 2006 ) argues that in order to show that the use of capabilities is sufficient, an alternative thought experiment is necessary. In this thought experiment, the capabilities of two individuals are the same, but due to the choices made by the individuals, the resulting functionings are different. If the result of this thought experiment shows that there is no difference in the level of wellbeing between these individuals, then we can conclude that capability is indeed a sufficient informational base.

For this thought experiment 7 , we take two hypothetical individuals: Ronald and Norris. Ronald and Norris have very similar backgrounds. Both are college educated, middle-class men living with a small family in the same suburb. Also in terms of their capabilities, Ronald and Norris are similar. Near the houses of Norris and Ronald you can find a fast-food outlet. Here, there is a difference between Norris and Ronald. Norris decides to eat every day in this fast-food outlet, where Ronald only visits the outlet infrequently. Over the years, Norris’ unhealthy choices have resulted in his having health issues, while Ronald remains fit, in part due to his choice to eat healthy food. In this case, Ronald and Norris achieve certain functionings, given the capabilities which they have and the choices they make within those capabilities. However, the functionings achieved are very different for Ronald and Norris. Given this, are Ronald and Norris equally well off? Fleurbaey ( 2006 ) argues that capabilities themselves are an insufficient informational base to answer this question. Instead, he proposes that the assessment of wellbeing should combine information about an individual's capabilities with information about the individual's achieved functionings, to get a full picture of the wellbeing of an individual. In the case of Ronald and Norris, you could then argue that in the end, Ronald is indeed better off, given that he does not have any health problems.

Is an informational base of value based on functioning and capability than sufficient? No, according to Clark ( 2005 ). Clark argues that the capability approach, as conceptualized by Sen, leaves too little space for one understanding of utility to be assessed: that of mental states that people have reason to value (such as happiness or life satisfaction). By limiting the capability approach to the measurement of capability and functioning, as argued by Fleurbaey ( 2006 ), it is possible that valuable aspects of utility are not measured.

To illustrate this, we can revisit the snack example. Recall earlier in this chapter the example of the capability of eating a snack as the informational base to assess the value of that snack. Based on this example it was argued, that by evaluating the capability of eating a snack, it is also possible to capture the utility derived from the functioning of eating a snack. Now imagine that due to a pandemic, Karla and Pierre are required to stay quarantined at home for two months and are to refrain from any kind of human contact. Now imagine that the only food is available in Karla's and Pierre's homes are the snacks presented earlier in the chapter: chocolate, a lollipop, chewing gum and broccoli. Depending on the preferences of Karla and Pierre, the first couple of hours in quarantine be problematic with these types of food. In fact, they might enjoy the excuse to eat one of the snacks presented above, given that they are not able to access other types of food. However, after a while, Karla feels frustrated about eating the same snacks repeatedly. This is problematic for Karla, since due to the pandemic people are required to stay at home for two months and are not allowed to go out or meet with anyone. Thus, Karla and Pierre must keep eating these snacks to stay alive. Pierre, on the other hand, is happy to continue eating such snacks, especially due to his passion for chocolate.

How is the value of the snacks assessed in terms of capability, functioning and utility? In terms of capabilities, Karla's and Pierre's situation has not changed from the first day of quarantine to the last, since they still have a variety of snacks to choose from. Furthermore, their capabilities are comparable, since they are both forced to stay at home. Also in terms of their functionings, there has not been a change because by eating the snacks, Karla and Pierre always achieve a similar level of functioning. However, one difference can be identified. The difference between Karla and Pierre is their utility, understood as a positive mental health state or the fulfilment of desire. Karla feels significantly worse than Pierre, which is not captured by assessing their capabilities or their functionings. Thus, only the informational base of utility manages to capture the difference in wellbeing between Karla and Pierre in their two months of quarantine.

For a real-world example, consider again patients affected by LIS. As was noted before, patients affected by LIS generally experience good levels of well-being, especially considering their limitations in capability (and as a consequence their functioning) ( Bruno et al., 2011 ; Rousseau et al., 2015 ). Still, it is interesting to note that within the patient group affected by LIS, there is a large variation in the levels of subjective wellbeing. Recall that in these studies, the subjective wellbeing of the patients was assessed by letting people assess their own lives on a scale ranging from the worst period in their lives to the best period in their lives ( Bruno et al., 2011 ). Even under similar circumstances, in terms of capability and given the limitations due to LIS, two subgroups can be identified. One of these groups report “good” levels of subjective wellbeing, while the other groups reports “bad” levels of subjective wellbeing. Bruno et al. ( 2011 ) reported a significant difference between the two groups in terms of time spent in LIS (with shorter time relating to lower levels of subjective well-being), lack of recovery in speech, depression, anxiety, perceived ability to participate in recreational activities, the perceived adequate level of mobility in the patients community, perceived ability to cope with live events, attitude towards resuscitation in case of cardiac arrest, suicidal thoughts and intended euthanasia by patients.

Particularly a lack of recovery in speech, an inability to participate in recreational activities, and the perception of having an inadequate level of mobility in their community can be seen as factors that limit the capability of patients affected by LIS. These factors severely limit the capability of patients to participate in a variety of capabilities that they have reason to value. Still, the authors noted that the explanatory power of the total combination of variables included in their analysis (including ones that showed no significant difference between the group with good levels of subjective wellbeing and bad levels of subjective wellbeing 8 ) only explain a limited amount of the variance in the subjective wellbeing of the patients (38%) 9 . This practical example shows that the subjective wellbeing of an individual is not assessed if you focus on evaluating only the capability and functioning of individuals.

5. Conclusion

In summary, capability, functioning, and utility, in terms of mental state, taken separately, may be insufficient informational bases for the assessment of value. Instead, each bring their own kind of information to the table, which combined creates a complete picture of an individual's wellbeing. This has the consequence that in order to get a complete picture of the value of a medical intervention, information is needed about: 1) how the medical intervention influences the capability of an individual; 2) the influence of the medical intervention on what an individual does with that capability; and 3) how the individual experiences his or her capability. How can an informational base which is a combination of constructs be measured in the context of assessing the value of medical interventions? One way of measuring this is through the use of questionnaires with broad domains that reflect what people value. These questionnaires should be developed together with the people for which medical interventions are developed to improve their wellbeing. 10

How then, can we know how much a combination of capabilities, functioning and utility is worth? Or should monetary values even be assigned? These remain open questions. 11

Bruno M.-A, Bernheim J. L, Ledoux D, Pellas F, Demertzi A, Laureys S. “A survey on self-assessed well-being in a cohort of chronic locked-in syndrome patients: happy majority, miserable minority” BMJ open. 2011; Vol. 1 (No. 1):p. e000039. [ PMC free article : PMC3191401 ] [ PubMed : 22021735 ]
Clark D. A. “Sen's capability approach and the many spaces of human well-being” The Journal of Development Studies. 2005; Vol. 41 (No. 8):pp. 1339–1368.
Drummond M. F, Sculpher M. J, Claxton K, Stoddart G. L, Torrance G. W. Methods for the economic evaluation of health care programmes. Oxford University Press; 2015.
Fleurbaey M. “Capabilities, functionings and refined functionings” Journal of Human Development. 2006; Vol. 7 (No. 3):pp. 299–310.
Gafni A. “The standard gamble method: what is being measured and how it is interpreted” Health Services Research. 1994; Vol. 29 (No. 2):p. 207. [ PMC free article : PMC1069999 ] [ PubMed : 8005790 ]
Laureys S, Pellas F, Van Eeckhout P, Ghorbel S, Schnakers C, Perrin F, Berre J, Faymonville M.-E, Pantke K.-H, Damas F. “The locked-in syndrome: what is it like to be conscious but paralyzed and voiceless?” Progress in Brain Research. 2005; Vol. 150 :pp. 495–611. [ PubMed : 16186044 ]
Luce R. D, Raiffa H. New York: Wiley; 1958. “Games and decisions: Introduction and critical survey” pp. pp. 12–37.
Rabin R, Charro F. d. “EQ-SD: a measure of health status from the EuroQol Group” Annals of Medicine. 2001; Vol. 33 (No. 5):pp. 337–343. [ PubMed : 11491192 ]
Richardson J. “Cost utility analysis: what should be measured?” Social Science & Medicine. 1994; Vol. 39 (No.1):pp. 7–21. [ PubMed : 8066489 ]
Rousseau M.-C, Baumstarck K, Alessandrini M, Blandin V, De Villemeur T. B, Auquier P. “Quality of life in patients with locked-in syndrome: Evolution over a 6-year period” Orphanet Journal of Rare Diseases. 2015; Vol. 10 (No.1):p. 88. [ PMC free article : PMC4506615 ] [ PubMed : 26187655 ]
Sen A. Commodities and Capabilities. Amsterdam: North Holland; 1985a.
Sen A. “Well-being, agency and freedom: The Dewey lectures 1984” The Journal of Philosophy. 1985b; Vol. 82 (No.4):pp. 187–190.
Sen A. “Capability and Well-Being” In: Nussbaum M, Sen A, editors. The Quality of Life. New York: Oxford University Press; 1993.
Sen A. “Health: perception versus observation: self reported morbidity has severe limitations and can be extremely misleading” British Medical Journal Publishing Group. 2002; Vol. 324 :pp. 860–861. [ PMC free article : PMC1122815 ] [ PubMed : 11950717 ]
Smith E, Delargy M. “Locked-in syndrome” British Medical Journal Publishing Group. 2005; Vol. 330 (No. 7488):pp. 406–409. [ PMC free article : PMC549115 ] [ PubMed : 15718541 ]
Torrance G. W. ‘Measurement of health state utilities for economic appraisal: a review’ Journal of Health Economics. 1986; Vol. 5 (No. 1):pp. 1–30. [ PubMed : 10311607 ]
Warren C, McGraw A. P, Van Boven L. “Values and preferences: defining preference construction” Wiley Interdisciplinary Reviews: Cognitive Science. 2011; Vol. 2 (No. 2):pp. 193–205. [ PubMed : 26302010 ]

In the chapter by Mitchell in this publication, a general overview was provided how health economists conduct economic evaluations. Furthermore, it contained a general introduction in the theory and applications of the capability approach.

In the previous chapter by Mitchell, the use of questionnaires to estimate Quality Adjusted Life Years (QALYs) was explained.

Known as the “expected utility hypothesis”. For an excellent explanation of the axioms and assumptions underpinning this theory, see chapter two in the book Games and Decisions by Luce and Raiffa (1958, pp. 12–37).

Of course, other frames and methods can and are being used to elicit utility values in different contexts. See for an overview of the methods used in conventional health economics the book “Methods for the economic evaluation of health care programmes” by Drummond (2015).

See Richardson (1994) for a discussion of various utility elicitation methods. Furthermore, see Richardson (1994) and Luce and Raiffa (1958, pp. 12–37) for discussions about the tautological nature of utility in expected utility theory.

Bruno et al. ( 2011 ) slightly adapted the so called Anamnestic Comparative Self-Assessment Scale, by changing the two optimal answer categories” from “the best period in my life” to “the best period prior to LIS”.

Example is based on example introduced by Fleurbay (2006).

These are variables related to perceived adequacy of mobility in a variety of different contexts, comfortability of fulfilling self-care needs, ability to participate in work and social activities, ability to fulfil role in family needs, conformability with personal relationships, conformability with being in company of others and pain.

The dependent variable in this regression is the self-assessment of good life, with the independent variables: lack of recovery in speech, depression, anxiety, perceived ability to participate in recreational activities, the perceived adequate level of mobility in the patients community, perceived ability to cope with live events, attitude towards resuscitation in case of cardiac arrest, suicidal thoughts, intended euthanasia by patients, variables related to perceived adequacy of mobility in a variety of different contexts, comfortability of fulfilling self-care needs, ability to participate in work and social activities, ability to fulfil role in family needs, conformability with personal relationships, conformability with being in company of others and pain.

Examples of such questionnaires were introduced and discussed in the chapter by Mitchell in this publication.

Possible answers to these questions are explored in the next chapter of this publication by Himmler, which focuses on methods for eliciting monetary values for capabilities.

This is an open access article licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed.

Monographs, or book chapters, which are outputs of Wellcome Trust funding have been made freely available as part of the Wellcome Trust's open access policy

Cite this Page Ubels J. The assessment of value in health economics: utility and capability. In: Kohlhammer VW, author; Schildmann J, Buch C, Zerth J, editors. Defining the Value of Medical Interventions: Normative and Empirical Challenges [Internet]. Stuttgart (DE): W. Kohlhammer GmbH; 2021.

In this Page

Introduction
The informational base in conventional health economics
Broadening the informational base with the capability approach
The need to integrate utility and the capability approach

Other titles in this collection

Wellcome Trust–Funded Monographs and Book Chapters

Related information

PMC PubMed Central citations
PubMed Links to PubMed

Recent Activity

The assessment of value in health economics: utility and capability - Defining t... The assessment of value in health economics: utility and capability - Defining the Value of Medical Interventions

Your browsing activity is empty.

Activity recording is turned off.

Turn recording back on

Connect with NLM

National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894

Web Policies FOIA HHS Vulnerability Disclosure

Help Accessibility Careers

Table of Contents
Random Entry
Chronological
Editorial Information
About the SEP
Editorial Board
How to Cite the SEP
Special Characters
Advanced Tools
Support the SEP
PDFs for SEP Friends
Make a Donation
SEPIA for Libraries
Entry Contents

Bibliography

Academic tools.

Friends PDF Preview
Author and Citation Info
Back to Top

Normative Theories of Rational Choice: Rivals to Expected Utility

Expected utility theory , which holds that a decision-maker ought to maximize expected utility, is the prevailing theory of instrumental rationality. Nonetheless, four major challenges have arisen to the claim that the theory characterizes all rational preferences. These challenges are the phenomena of infinite or unbounded value, incommensurable goods, imprecise probabilities, and risk-aversion. The challenges have been accompanied by alternative theories that attempt to do better.

Expected utility theory consists of three components. The first is a utility function that assigns real numbers to consequences. The second is a probability function that assigns real numbers between 0 and 1 to every possible event. The final component is an “aggregation norm” that holds that the value of an act is its expected utility value relative to these two functions, and that rational preferences track expected utility value (see the entry on expected utility theory ). Each challenge to EU theory can be thought of as a rejection of one or more of the three components as normative, and alternatives generally replace or extend the relevant component.

It has long been observed that typical individuals do not in fact conform to expected utility theory, and in response, a number of descriptive alternatives have arisen, particularly in the field of economics (see Starmer 2000, Sugden 2004, Schmidt 2004 for surveys; see also the entry on descriptive decision theory ). This article will primarily discuss views that have been put forth as normative.

1.1 The Theory

1.2.1 utility, 1.2.2 probability, 1.2.3 expectation, 2.1 the challenge from infinite and unbounded utility, 2.2 proposals, 3.1 the challenge from incommensurability, 3.2 proposals: definitions, 3.3 proposals: decision rules, 4.1 the challenge from imprecise probabilities or ambiguity, 4.2 proposals: probability representations, 4.3.1 aggregative decision rules using sets of probabilities, 4.3.2 aggregative decision rules: choquet expected utility, 4.3.3 decision rules that select from a menu, 4.4 normative questions, 5.1 the challenge from risk aversion, 5.2 proposals, 5.3 normative issues, other internet resources, related entries, 1. expected utility theory.

Decision theory concerns individuals’ preferences among both consequences (sometimes called “outcomes”) and gambles . The theory as originally developed focused on decisions with monetary consequences (e.g., receiving $10 in a game of chance), but subsequent developments broadened their focus to include decisions with non-monetary consequences as well (e.g., eating a large omelet, eating a smaller omelet, being stuck in the rain without an umbrella, lugging an umbrella around in the sunshine). Most contemporary authors define consequences to include any facts about a decision-maker’s situation that matter to her—so monetary consequences must technically describe a decision-maker’s total fortune, and non-monetary consequences must technically describe the entire world that the decision-maker finds herself in—though these full descriptions are often omitted when a decision will not alter the surrounding facts. Let the consequence set be $\cX.$ A utility function $\uf: \cX \rightarrow \cR$ assigns values to consequences, with the constraint that the individual prefers (or should prefer), of two consequences, the one with the higher utility value, and is indifferent between any two consequences with the same utility value. Thus the utility function in some sense represents how the individual values consequences.

Gambles come in one of two forms, depending on whether we are dealing with the “objective probability” or “subjective probability” version of the theory. In the objective probability version, a gamble is a lottery that assigns probabilities to consequences. Consider, for example, a lottery that yields $100 with probability 0.5, $300 with probability 0.3, and $200 with probability 0.2. We can represent this lottery as {$100, 0.5; $300, 0.3; $200, 0.2}. More generally, lotteries have the form $L = \{x_1, p_1;\ldots; x_n, p_n\},$ where $x_i \in \cX$ and $p_i$ is the probability that consequence $x_i$ obtains. Lotteries needn’t be restricted to a finite set of consequences; they could instead be continuous.

In the subjective probability version, a gamble is an act (sometimes called as “Savage act”; Savage 1954) that assigns consequences to possible states of the world. Consider, for example, the act of cracking an extra egg into one’s omelet, when the egg may be rotten: if the egg is fresh, the consequence will be a large omelet, but if the egg is rotten, the consequence will be a ruined omelet. We can represent this act a as { extra egg is fresh , large omelet; extra egg is rotten , ruined omelet}, and we can represent the act of not cracking the extra egg as { extra egg is rotten or fresh , small omelet}. More generally, acts have the form $g = \{E_1, x_1;\ldots; E_n, x_n\},$ where $x_i \in\cX,$ $E_i \subseteq \cS$ is an event (a subset of the state space), and $x_i$ obtains when the true state of the world is in $E_i.$ Again, acts needn’t be restricted to a finite set of consequences. In the subjective probability version, the individual has a probability function $p$ that assigns to each $E_i$ a number between 0 and 1 (inclusive), which represents her subjective probabilities, also called “degrees of belief” or “credences”. The probability function is additive in the sense that if $E$ and $F$ are mutually exclusive, then $p(E v F) = p(E) + p(F).$ (For some of the discussion, it does not matter if we are talking about lotteries or acts, so I will use the variables A , B , C … to range over lotteries or acts.)

The core principle of expected utility theory concerns how the utility values of gambles are related to the utility values of consequences. In particular, the slogan of expected utility theory is that rational agents maximize expected utility . The expected utility (EU) of a lottery, relative to an individual’s utility function $\uf,$ is:

The expected utility of an act, relative to an individual’s utility function $\uf$ and probability function $p,$ is:

Continuous versions of these are defined using an integral instead of a sum.

Expected utility theory holds that an individual’s preferences order gambles according to their expected utility, or ought to do so: $A \succcurlyeq B$ iff $\EU(A) \ge \EU(B).$ Generally, weak preference $(\succcurlyeq)$ is taken to be basic and strict preference $(\succ)$ and indifference $(\sim)$ defined in the usual way ($A \succ B$ iff $A \succcurlyeq B$ and not-$(B \succcurlyeq A)$; $A \sim B$ iff $A \succcurlyeq B$ and $B \succcurlyeq A$).

We can take utility and probability to be basic, and the norm to tell us what to prefer; alternatively, we can take preferences to be basic, and the utility and probability functions to be derived from them. Say that a utility (and probability) function represents a preference relation $\succcurlyeq$ under EU- maximization just in case:

$u$ represents $\succcurlyeq$ under EU-maximization (objective probabilities): for all lotteries $L_1$ and $L_2,$

where EU is calculated relative to $u.$

$u$ and $p$ represent $\succcurlyeq$ under EU-maximization (subjective probabilities): for all acts $f$ and $g,$

where EU is calculated relative to $u$ and $p.$

A representation theorem exhibits a set of axioms (call it [ axioms ]) such that the following relationship holds:

Representation Theorem (objective probabilities): If a preference relation over lotteries satisfies [ axioms ], then there is a utility function $u,$ unique up to positive affine transformation, that represents the preference relation under EU-maximization.
Representation Theorem (subjective probabilities): If a preference relation over lotteries satisfies [ axioms ], then there is a utility function $u,$ unique up to positive affine transformation, and a unique probability function $p$ that represent the preference relation under EU-maximization.

“Unique up to positive affine transformation” means that any utility function $u'$ that also represents the preference relation can be transformed into $u$ by multiplying by a constant and adding a constant (to borrow an example from a different domain: temperature scales are unique up to positive affine transformation, because although temperature can be represented by Celsius or Fahrenheit, Celsius can be transformed into Fahrenheit by multiplying by 9/5 and adding 32).

The first and most historically important axiomatization of the objective probabilities version of expected utility theory is that of von Neumann and Morgenstern (1944). The axioms are as follows (slightly different versions appear in the original):

Completeness: For all lotteries $L_1$ and $L_2 : L_1 \succcurlyeq L_2$ or $L_2 \succcurlyeq L_1.$
Transitivity: For all lotteries $L_1, L_2,$ and $L_3$: If $L_1 \succcurlyeq L_2$ and $L_2 \succcurlyeq L_3,$ then $L_1 \succcurlyeq L_3.$
Continuity: For all lotteries $L_1, L_2,$ and $L_3$: If $L_1 \succcurlyeq L_2$ and $L_2 \succcurlyeq L_3,$ then there is some real number $p\in [0, 1]$ such that $L_2 \sim\{L_1, p$; $L_3, 1 - p\}$
Independence: For all lotteries $L_1,$ $L_2,$ $L_3$ and all $0 \lt p \le 1$: \[ L_1 \succcurlyeq L_2 \Leftrightarrow \{L_1, p; L_3, 1 - p\} \succcurlyeq \{L_2, p; L_3, 1 - p\} \]

The most historically important axiomatization of the subjective probabilities version of expected utility theory is that of Savage (1954), though other prominent versions include Ramsey (1926), Jeffrey (1965), Armendt (1986, 1988), and Joyce (1999). These generally include axioms that are analogues of the von Neumann-Morgenstern axioms, plus some additional axioms that will not be our focus.

The first two of the axioms that will be our focus are (as above) completeness and transitivity :

Completeness: For all acts $f$ and $g$: $f \succcurlyeq g$ or $f \succcurlyeq g.$
Transitivity: For all acts $f,$ $g,$ and $h$: If $f \succcurlyeq g$ and $g \succcurlyeq h,$ then $f \succcurlyeq h.$

The third is some version of continuity, sometimes called an Archimedean Axiom .

The final axiom is a separability axiom. Savage’s version of this axiom is known as the sure-thing principle. Where $f_E h$ is an act that agrees with f on event E and agrees with h elsewhere:

Sure-thing Principle: For all acts $f_E h,$ $g_E h,$ $f_E j,$ and $g_E j$:

In other words, if two acts agree on what happens on not- E , then one’s preference between them should be determined only by what happens on E. Other separability axioms include Jeffrey’s Averaging (1965) and Köbberling and Wakker’s Tradeoff Consistency (2003).

Because representation theorems link, on the one hand, preferences that accord with the axioms and, on the other hand, a utility (and probability) function whose expectation the individual maximizes, a challenge to one of the three components of expected utility theory must also be a challenge to one (or more) of the axioms.

1.2 Components

Since representation theorems show that a utility (and probability) function can be derived from preferences—that having a particular expectational utility function is mathematically equivalent to having a particular preference ordering—they open up a number of possibilities for understanding the utility function. There are two questions here: what the utility function corresponds to (the metaphysical question), and how we determine an individual’s utility function (the epistemic question).

The first question is whether the utility function corresponds to a real-world quantity, such as strength of desire or perceived desirability or perceived goodness, or whether it is merely a convenient way to represent preferences. The former view is known as psychological realism (Buchak 2013) or weak realism (Zynda 2000), and is held by Allais (1953) and Weirich (2008, 2020), for example. The latter view is known as formalism (Hansson 1988), operationalism (Bérmudez 2009), or the representational viewpoint (Wakker 1994), and is particularly associated with decision theorists from the mid-twentieth century (Luce & Raiffa 1957, Arrow 1951, Harsanyi 1977), and with contemporary economists.

The second question is what facts are relevant to determining someone’s utility function. Everyone in the debate accepts that preferences provide evidence for the utility function, but there is disagreement about whether there may be other sources of evidence as well. Constructivists hold that an individual’s utility function is defined by her preferences—utility is “constructed” from preference—so there are can be no other relevant facts (discussed in Dreier 1996, Buchak 2013); this view is also called strong realism (Zynda 2000). Non-constructive realists , by contrast, hold that there are other sources of evidence about the utility function: for example, an individual might have introspective access to her utility function. This latter view only makes sense if one is a psychological realist, though one can pair constructivism with either psychological realism or formalism.

A key fact to note about the utility function is that it is real-valued : each consequence can be assigned a real number. This means that no consequence is of infinite value , and all consequences are comparable . As we will see, each of these two properties invites a challenge.

Given that a probability function can also be derived from preferences, a similar question arises about the nature and determination of the probability function. One could hold that the probability function represents some real-world quantity, such as partial belief; or one could hold that the probability function is merely a way of representing some feature of betting behavior. There is also disagreement about what facts are relevant to determining someone’s probability function: some hold that it is determined from betting behavior or from the deliverances of a representation theorem while others take it to be primitive (see Eriksson & Hájek 2007).

Probability is real-valued and pointwise (“sharp”, “precise”), meaning that there is a unique number representing an individual’s belief in or evidence for an event. Again, this property will invite a challenge.

We can see the norm of expected utility in one of two ways: maximize expected utility, or have preferences that obey the axioms. Because of this, normative arguments for expected utility can argue either for the functional form itself or for the normativity of the axioms. Examples of the former include the argument that expected utility maximizers do better in the long run, though these arguments fell out of favor somewhat as the popularity of the realist interpretations of utility waned. Examples of the latter include the idea that each axiom is itself an obvious constraint and the idea that the axioms follow from consequentialist (or means-ends rationality) principles. Of particular note is a proof that non-EU maximizers will either be inconsistent or non-consequentialist over time (Hammond 1988); how alternative theories have fared under dynamic choice has been a significant focus of arguments about their rationality.

The idea that EU maximization is the correct norm can be challenged on several different grounds, as we will see. Those who advocate for non-EU theories respond to the arguments listed above by either arguing that the new norm doesn’t actually fall prey to the argument (e.g., provide a representation theorem with supposedly more intuitive axioms) or that it is nonetheless okay if it does.

2. Infinite and Unbounded Utility

The first challenge to EU maximization stems from two ways that infinite utility can arise in decision situations.

First, some particular outcome might have infinite utility or infinite disutility. For example, Pascal’s Wager is motivated by the idea that eternal life with God has infinite value, so one should “wager for God” as long as one assigns some non-zero probability to God’s existence (Pascal 1670). If a particular outcome has infinite (dis)value, then the Continuity Axiom or the Archimedean Axiom will not hold. (See discussions in Hacking 1972 and Hájek 2003, and a related issue for utilitarianism in Vallentyne 1993, Vallentyne & Kagan 1997, and Bostrom 2011.)

Second, all outcomes might have finite utility value, but this value might be unbounded , which, in combination with allowing that there can be infinitely many states , gives rise to various paradoxes. The most famous of these is the St. Petersburg Paradox , first introduced by Nicolas Bernoulli in a 1713 letter (published in J. Bernoulli DW ). Imagine a gamble whose outcome is determined by flipping a fair coin until it comes up heads. If it lands heads for the first time on the n th flip, the recipient gets $$2^n$; thus the gamble has infinite expected monetary value for the person who takes it (it has a $\lfrac{1}{2}$ probability of yielding $2, a $\lfrac{1}{4}$ probability of yielding $4, a $\lfrac{1}{8}$ probability of yielding $8, and so forth, and

While this version can be resolved by allowing utility to diminish marginally in money—so that the gamble has finite expected utility—if the payoffs are in utility rather than money, then the gamble will have infinite expected utility.

Related paradoxes and problems abound. One centers around a pair of games, the Pasadena game and the Altadena game (Nover & Hájek 2004). The Pasadena game is also played by flipping a fair coin until it lands heads; here, the player receives $\$({-1})^{n-1}(2^n/n)$ if the first heads occurs on the n th flip. Thus, its payoffs alternate between positive and negative values of increasing size, so that its terms can be rearranged to yield any sum whatsoever, and its expectation does not exist. The Altadena game is identical to the Pasadena game, except that every payoff is raised by a dollar. Again, its terms can be rearranged to yield any value, and again its expectation does not exist. However, it seems (contra EU maximization) that the Altadena game should be preferred to the Pasadena game, since the former state-wise dominates the latter—it is better in every possible state of the world (see also Colyvan 2006, Fine 2008, Hájek & Nover 2008). Similarly, it seems that the Petrograd game , which increases each payoff of the St. Petersburg game by $1, should be preferred to the St. Petersburg game, even though EU maximization will say they have the same (infinite) expectation (Colyvan 2008).

(See also Broome’s (1995) discussion of the two-envelope paradox; Arntzenius, Elga, and Hawthorne’s (2004) discussion of diachronic puzzles involving infinite utility; and McGee’s (1999) argument that the utility function ought to be bounded, which will dissolve the above paradoxes.)

Several proposals retain the basic EU norm, but reject the idea that the utility function ranges only over the real numbers. Some hold that the utility function can take hyper-real values (Skala 1975, Sobel 1996). Others hold that the utility function can take surreal values (Hájek 2003, Chen & Rubio 2020). These proposals allow for versions of the Continuity/Archimedean Axiom. Another alternative is to use a vector-valued (i.e., lexicographic ) utility function, which rejects these axioms (see discussion in Hájek 2003).

A different kind of response is to subsume EU maximization under a more general norm that also applies when utility is infinite. Bartha (2007, 2016) defines relative utility , which is a three-place relation that compares two outcomes or lotteries relative to a third “base point” that is worse than both. The relative utility of $A$ to $B$ with base-point $Z$ (written $(A, B; Z)$) will be:

If $A,$ $B$ and $Z$ are finitely valued gambles:

as in standard EU maximization

If $A$ is infinitely valued and $B$ and $Z$ are not: $\infty$

Relative utility ranges over the extended real numbers $\{\cR \cup \infty\}.$ “Finite” and “infinite” values can be determined from preferences. Furthermore, relative utility is expectational

and has a representation theorem consisting of the standard EU axioms minus Continuity. (See Bartha 2007 for application to infinite-utility consequences and Bartha 2016 for application to unbounded-utility consequences.)

When considering only the paradoxes of unbounded utility (not those of infinite utility), there are other ways to subsume EU maximization under a more general norm. Colyvan (2008) defines relative expected utility (unrelated to Bartha’s relative utility) of act $f = \{E_1, x_1;\ldots; E_n, x_n\}$ over $g = \{E_1, y_1;\ldots; E_n, y_n\}$ as:

In other words, one takes the difference in utility between $f$ and $g$ in each state, and weights this value by the probability of each state. Colyvan similarly defines the infinite state-space case as

The new norm is that $f \succcurlyeq g$ iff $\REU(f, g) \ge 0.$ This rule agrees with EU maximization in cases of finite state spaces, but also agrees with state-wise dominance; so it can require that the Altadena game is preferred to the Pasadena game and the Petrograd game is preferred to the St. Petersburg game. (See also Colyvan & Hájek 2016.)

A more modest extension of standard EU maximization is suggested by Easwaran (2008). He points out that although the Pasadena and Altadena games lack a “strong” expectation, they do have a “weak” expectation. (The difference corresponds to the difference between the strong and weak law of large numbers.). Thus, we can hold that a decision-maker is required to value a gamble at its weak expectation, which is equivalent to its strong expectation if the latter exists. (See also Gwiazda 2014, Easwaran 2014b; relatedly, Fine (2008) shows that these two games and the St. Petersberg paradox can be assigned finite values that are consistent with EU theory.).

Lauwers and Vallentyne (2016) combine an extension of Easwaran’s proposal to infinite weak expectations with an extension of Colyvan’s proposal to cardinal relative expectation that can be interval-valued . Meacham (2019) extends Colyvan’s proposal to cover cases in which the acts to be compared have utilities that are to be compared in different states, and cases in which probabilities are act-dependent; his difference minimizing theory re-orders each gamble from worst consequence to best consequence, before taking their relative expectation. A key difference between these two extensions is that difference minimizing theory adheres to stochastic dominance and a related principle called stochastic equivalence . (See also discussion in Seidenfeld, Schervish, & Kadane 2009; Hájek 2014; Colyvan & Hájek 2016.)

In a more radical departure from standard EU maximization, Easwaran (2014a) develops an axiomatic decision theory based on state-wise dominance, that starts with utility and probability and derives a normative preference relation. In cases that fit the standard parameters of EU maximization, this theory can be made to agree with EU maximization; but it also allows us to compare some acts with infinite value, and some acts that don’t fit the standard parameters (e.g., incommensurable acts, acts with probabilities that are comparative but non-numerical).

Finally, one could truncate the norm of EU maximization. Some have argued that that for a gamble involving very small probabilities, we should discount those probabilities down to zero, regardless of the utilities involved. When combined with a way of aggregating the remaining possibilities, this strategy will yield a finite value for the unbounded-utility paradoxes, and also allow people who attribute a very small probability to God’s existence to avoid wagering for God. (This idea traces back to Nicolaus Bernoulli, Daniel Bernoulli, d’Alambert, Buffon, and Borel [see Monton 2019 for a historical survey]; contemporary proponents of this view include Jordan 1994, Smith 2014, Monton 2019.)

3. Incommensurability

Another challenge to expected utility maximization is to the idea that preferences are totally ordered—to the idea that consequences can be ranked according to a single, consistent utility function. In economics, this idea goes back at least to Aumann (1962); in philosophy, it has been taken up more recently by ethicists. Economists tend to frame the challenge as a challenge to the idea that the preference relation is complete, and ethicists to the idea that the betterness relation is complete. I use $\succcurlyeq$ to represent whichever relation is at issue, recognizing that some proposals may be more compelling in one case than the other.

The key claim is that there are some pairs of options for which it is false that one is preferred to (or better than) the other, but it is also false that they are equi-preferred (or equally good). Proposed examples include both the mundane and the serious: a Mexican restaurant and a Chinese restaurant; a career in the military and a career as a priest; and, in an example due to Sartre (1946), whether to stay with one’s ailing mother or join the Free French. Taking up the second of these examples: it is not the case that a career in the military is preferred to (or better than) a career as a priest, nor vice versa ; but it is also not the case that they are equi-preferred (or equally good). Call the relation that holds between options in these pairs incommensurability .

Incommensurability is most directly a challenge to Completeness, since on the most natural interpretation of $\succcurlyeq,$ the fact that $A$ and $B$ are incommensurable means that neither $A \succcurlyeq B$ nor $B \succcurlyeq A.$ But incommensurability can instead be framed as a challenge to Transitivity, if we assume that incommensurability is indifference, or define $A \succcurlyeq B$ as the negation of $B \succ A$ (thus assuming Completeness by definition). To see this, notice that if two options $A$ and $B$ are incommensurable, then “sweetening” $A$ to a slightly better $A^+$ will still leave $A^+$ and $B$ incommensurable. For example, if $A$ is a career in the military and $A^+$ is this career but with a slightly higher salary, the latter is still incommensurable with a career as a priest. This pattern suffices to show that the relation $\sim$ is intransitive, since $A \sim B$ and $B \sim {A^+},$ but ${A^+} \succ A$ (de Sousa 1974).

There are four options for understanding incommensurability. Epistemicists hold that there is always some fact of the matter about which of the three relations $(\succ,$ $\prec,$ $\sim)$ holds, but that it is sometimes difficult or impossible to determine which one—thus incommensurability is merely apparent. They can model the decision problem in the standard way, but as a problem of uncertainty about values: one does not know whether one is in a state in which $A \succcurlyeq B,$ but one assigns some probability to that state, and maximizes expected utility taking these kinds of uncertainties into account. Indeterminists hold that it is indeterminate which relation holds, because these relations are vague; thus incommensurability is a type of vagueness (Griffin 1986, Broome 1997, Sugden 2009, Constantinescu 2012). Incomparabilists hold that in cases of incommensurability, $A$ and $B$ simply cannot be compared (de Sousa 1974, Raz 1988, Sinnott-Armstrong 1988). Finally, those who hold that incommensurability is parity hold that there is a fourth relation than can obtain between $A$ and $B$: $A$ and $B$ are “on a par” if $A$ and $B$ can be compared but it is not the case that one of the three relations holds (Chang 2002a, 2002b, 2012, 2015, 2016). (Taxonomy from Chang 2002b; see also Chang 1997.)

Aumann (1962) shows that if we have a partial but not total preference ordering, then we can represent it by a utility function (not unique-up-to-positive-affine-transformation) such that $A \succ B$ implies $\EU(A) \gt \EU(B),$ but not vice versa . Aumann shows that there will be at least one utility function that represents the preference ordering according to (objective) EU maximization. Thus, we can represent a preference ordering as the set of all utility functions that “one-way” represent the decision-maker’s preferences. Letting $\EU_u(A)$ be the expected utility of $A$ given utility function $u$:

If there is no incommensurability, then there will be a single (expectational) utility function in $\cU,$ as in standard EU theory. But when neither $A \succ B$ nor $B \succ A$ nor $A \sim B,$ there will be some $u \in\cU$ such that $\EU_u (A) \gt \EU_u (B),$ and some $u'\in\cU$ such that $\EU_{u'}(B) \gt \EU_{u'}(A)$; and vice versa .

Chang (2002a,b) proposes a similar strategy, but she takes value facts to be basic, and defines the betterness relation—plus a new “parity” relation we will denote “$\parallel$”—from them, instead of the reverse. In addition, she defines these relations in terms of the evaluative differences between $A$ and $B,$ i.e., $(A - B)$ is the set of all licensed differences in value between $A$ and $B.$ If $(A - B) = \varnothing,$ then $A$ and $B$ are incomparable ; however, if $(A - B) \ne \varnothing,$ the relevant relations are:

$A \succ B$ iff $(A - B)$ contains only positive numbers
$B \succ A$ iff $(A - B)$ contains only negative numbers
$A \sim B$ iff $(A - B)$ contains only 0
$A \parallel B$ otherwise

$(A - B)$ might be generated by a set of utility functions, each of which represents a possible substantive completion of the underlying value that utility represents (discussed in Chang 2002b); alternatively, it might be that there is parity “all the way down” (discussed in Chang 2016, where she also replaces the definition in terms of explicit numerical differences with one in terms of bias).

Rabinowicz (2008) provides a model that allows for both parity and grades of incomparability. On his proposal, the betterness relation is represented by a class $K$ of “permissible preference orderings”, each of which may be complete or incomplete. He defines:

(He defines $\succcurlyeq$ as the union of several “atomic” possibilities for $K.$) Letting $x\relT y$ hold iff $x \succ y$ or $y \succ x$ or $x \sim y,$ he then defines:

$x$ and $y$ are fully comparable iff $(\forall R \in K)(xT_R y)$
$x$ and $y$ are fully on a par iff they are fully comparable and $x\parallel y$
$x$ and $y$ are incomparable iff $(\forall R\in K)(\text{not-}(xT _Ry))$
$x$ and $y$ are weakly incomparable iff $(\exists R\in K)(\text{not-}(xT_Ry)$

Class $K$ won’t necessarily give rise to a utility function.

If the decision-maker’s preferences are represented by a set of utility functions $\cU,$ then a number of possible decision rules suggest themselves. All the proposed rules focus on selection of permissible options from a set of alternatives $\cS,$ rather than an aggregation function or a complete ranking of options (we can always recover the former from the latter, but not vice versa ). To understand the first three of these rules, we can imaginatively think of each possible utility function as a “committee member”, and posit a rule for choice based on facts about the opinions of the committee.

First, we might choose any option which some committee member endorses; that is, we might choose any option which maximizes expected utility relative to some utility function:

Levi (1974) terms this rule E-admissibility , and Hare (2010) calls it prospectism . (See section 4.3.3 for Levi’s full proposal, and for extensions of this rule to the case in which the decision-maker does not have a single probability function.)

Aumann suggests that we can choose any maximal option: any option that isn’t worse, according to all committee members, than some other particular option; that is, an option to which no other option is (strictly) preferred (assigned a higher utility by all utility functions):

This is a more permissive rule than E-admissibility: every E-admissible option will be maximal, but not vice versa . To see the difference between the two rules, notice that if the decision-maker has two possible rankings, $A \gt B \gt C$ and $C \gt B \gt A,$ then all three options will be maximal but only $A$ and $C$ will be E-admissible (no particular option is definitively preferred to $B,$ so $B$ is maximal; but it is definitive that something is preferred to $B,$ so $B$ is not E-admissible).

A final possibility is that we can choose any option that is not interval dominated by another act (Schick 1979, Gert 2004), where an interval dominated option has a lower “best” value than some other option’s “worst” value:

This is a more permissive rule than both E-admissibility and maximality.

A different type of rule first looks at the options’ possible utility values in each state, before aggregating over states; this is what Hare’s (2010) deferentialism requires. To find out if an option O is permissible under deferentialism, we consider how it fares if we, in each state, make the assumptions most favorable to it. First, “regiment” the utility functions in $\cU$ so that there’s some pair of consequences $\{x, y\}$ such that $(\forall u \in\cU)(u(x) = 1 \amp u(y) = 0)$; this allows only one representative utility function for each possible completion of the decision-maker’s preference. Next, take all the possible “state-segments”—the possible utility assignments in each state—and cross them together in every possible arrangement to get an “expanded” set of utility functions (for example, this will contain every possible utility function in $E$ coupled with every possible utility function not-$E$). Then $A$ is permissible iff $A$ maximizes expected utility relative to some utility function in this expanded set.

4. Imprecise Probabilities or Ambiguity

A third challenge to expected utility maximization holds that subjective probabilities need not be “sharp” or “precise”, i.e., need not be a single, point-wise function. (In economics, this phenomenon is typically called ambiguity .) There are three historically significant motivations for imprecise probabilities.

The first is that decision makers treat subjective (or unknown) probabilities differently from objective probabilities in their decision-making behavior. The classic example of this is the Ellsberg Paradox (Ellsberg 1961, 2001). Imagine you face an urn filled with 90 balls that are red, black, and yellow, from which a single ball will be drawn. You know that 30 of the balls are red, but you know nothing about the proportion of black and yellow balls. Do you prefer $f_1$ or $f_2$; and do you prefer $f_3$ or $f_4$?

$f_1$: $100 if the ball is red; $0 if the ball is black or yellow.
$f_2$: $100 if the ball is black; $0 if the ball is red or yellow.
$f_3$: $100 if the ball is red or yellow; $0 if the ball is black.
$f_4$: $100 if the ball is black or yellow; $0 if the ball is red.

Most people appear to (strictly) prefer $f_1$ to $f_2$ and (strictly) prefer $f_4$ to $f_3.$ They would rather bet on the known or objective probability than the unknown or subjective one—in the first pair, red has an objective probability of $\lfrac{1}{3}$, whereas black has a possible objective probability ranging from 0 to $\lfrac{2}{3}$; in the second pair, black or yellow has an objective probability of $\lfrac{2}{3}$ whereas red or yellow has a possible objective probability ranging from $\lfrac{1}{3}$ to 1. These preferences violate the Sure-thing Principle. (To see this, notice that the only difference between the two pairs of acts is that the first pair yields $0 on yellow and the second pair yields $100 on yellow .)

The second motivation for imprecise probability is that even if all the relevant probabilities are subjective, a decision-maker’s betting behavior might depend on how reliable or well-supported by evidence those probabilities are. Consider a decision-maker who may bet on three different tennis matches: in the first, she knows a lot about the players and knows they are very evenly matched; in the second, she knows nothing whatsoever about either player; and in the third, she knows that one of the two players is much better than the other, but she does not know which one. In each of the matches, the decision-maker should presumably assign equal probability to each player winning, since her information in favor of each is symmetrical; nonetheless, it seems rational to bet only on the first match and not on the other two (Gärdenfors & Sahlin 1982; see also Ellsberg 1961).

A final motivation for imprecise probability is that evidence doesn’t always determine precise probabilities (Levi 1974, 1983; Walley 1991; Joyce 2005; Sturgeon 2008; White 2009; Elga 2010). Assume a stranger approaches you and pulls three items out of a bag: a regular-sized tube of toothpaste, a live jellyfish, and a travel-sized tube of toothpaste; you are asked to assign probability to the proposition that the next item he pulls out will be another tube of toothpaste—but it seems that you lack enough evidence to do so (Elga 2010).

To accommodate imprecise probabilities in decision-making, we need both an alternative way to represent probabilities and an alternative decision rule that operates on the newly-represented probabilities. There are two primary ways to represent imprecise probabilities.

The first is to assign an interval , instead of a single number, to each proposition. For example, in the Ellsberg case:

The second is to represent the individual’s beliefs as a set of probability functions. For example, in the Ellsberg case:

This means, for example, that the probability distribution $p(\rR, \rB, \rY) = \langle \lfrac{1}{3}, 0, \lfrac{2}{3}\rangle$ and the probability distribution $\langle \lfrac{1}{3}, \lfrac{1}{3}, \lfrac{1}{3}\rangle$ are both compatible with the available evidence or possible “completion” of the individual’s beliefs.

Each set-probability representation gives rise to an interval representation (assuming the set of probability functions is convex); but the set-probability representation provides more structure to the relationships between propositions. (A different proposal retains a precise probability function but refines the objects over which utility and probability range (Bradley 2015; Stefánsson & Bradley 2015, 2019); see discussion in section 5.2 .)

4.3 Proposals: Decision Rules

We will examine rules for decision-making with imprecise probabilities in terms of how they evaluate the Ellsberg choices; for ease of exposition we will assume $u(\$0) = 0$ and $u(\$100) = 1.$ All proposals here are equivalent to EU maximization when there is a single probability distribution in the set, so all will assign utility $\lfrac{1}{3}$ to $f_1$ and $\lfrac{2}{3}$ to $f_4$ in the Ellsberg gamble; they differ in how they value the other acts.

The first type of decision rule associates to each act a single value, and yields a complete ranking of acts; call these aggregative rules . The rules in this section use sets of probabilities.

Before we discuss these rules, it will be helpful to keep in mind three aggregative rules that operate under complete ignorance, i.e., when we have no information whatsoever about the state of the world. The first is maximin, which says to pick the option with the highest minimum utility. The second is maximax, which says to pick the option with the highest maximum utility. The third, known as the Hurwicz criterion, says to take a weighted average, for each option, of the minimum and maximum utility, where the weight $\alpha \in [0, 1]$ represents a decision-maker’s level of optimism/pessimism (Hurwicz 1951a):

Using the set-probability representation, we can associate to each probability distribution an expected utility value, to yield a set of expected utility values. Let $\EU_p(f)$ be the expected utility of $f$ given probability distribution $p.$

One proposal is that the value of an act is its minimum expected utility value; thus, a decision-maker should maximize her minimum expected utility (Wald 1950; Hurwicz 1951b; Good 1952; Gilboa & Schmeidler 1989):

This rule is also sometimes known as MMEU. For the Ellsberg choices, the minimum expected utilities are, for $f_1$, $f_2$, $f_3$, and $f_4$, respectively: $\lfrac{1}{3},$ $0,$ $\lfrac{1}{3},$ and $\lfrac{2}{3},$. These values rationalize the common preference for $f_1 \gt f_2$ and $f_4 \gt f_3.$ Conversely, an individual who maximizes her maximum expected utility —who uses Γ-maximax—would have the reverse preferences.

Γ-maximin appears too pessimistic. We might instead use an EU-analogue of Hurwicz criterion: take a weighted average of the minimum expected utility and the maximum expected utility , with weight $\alpha \in[0, 1]$ corresponding to a decision-maker’s level of pessimism (Hurwicz 1951b; Shackle 1952; Luce & Raiffa 1957; Ellsberg 2001; Ghirardato et al. 2003):

In the Ellsberg choice, this model will assign $\alpha(\lfrac{2}{3})$ to $f_2$ and $(1 - \alpha)(\lfrac{1}{3}) + \alpha(1)$ to $f_3,$ making these acts disprefered to $f_1$ and $f_4,$ respectively, if $\alpha \lt \lfrac{1}{2}$; preferred to $f_1$ and $f_4$ if $\alpha \gt \lfrac{1}{2}$; and indifferent to $f_1$ and $f_4$ if $\alpha = \lfrac{1}{2}.$

Instead, we can assume the decision-maker considers two quantities when evaluating an act: the EU of the act, according to her “best estimate” at the probabilities ($\text{est}_p),$ and the minimum EU of the act as the probability ranges over $\cQ$; she also assigns a degree of confidence $\varrho \in[0, 1]$ to her estimated probability distribution. The value of an act will then be a weighted average of her best estimate EU and the minimum EU, with her best estimate weighed by her degree of confidence (Hodges & Lehmann 1952; Ellsberg 1961):

In the Ellsberg pairs, assuming the “best estimate” is that yellow and black each have probability $\lfrac{1}{3}$, this will assign $\varrho(\lfrac{1}{3})$ to $f_2$ and $\varrho(\lfrac{2}{3}) + (1 - \varrho)(\lfrac{1}{3})$ to $f_3,$ making these acts disprefered to $f_1$ and $f_4,$ respectively, as long as $\varrho \lt 1.$

We can also combine these two proposals (Ellsberg 2001):

This model will rationalize the common preferences for many choices of $\varrho$ and $\alpha$ (setting $\varrho = 0$ or $\alpha = 0$ yields previously-mentioned models).

We might add additional structure to the representation: to each probability function, the decision-maker assigns a degree of “reliability”, which tracks how much relevant information the decision-maker has about the states of nature (Good 1952; Gärdenfors & Sahlin 1982). A decision-maker selects a desired threshold level of epistemic reliability. She then considers all probability functions above this threshold, and maximizes the minimum expected utility (Γ-maximin) with respect to these probability functions. (In principle, a different decision rule could be used in this step.) For the decision-maker deciding whether to bet on tennis matches, the above-threshold probability functions for the first match may include only $p(P1 \text{WINS}) \approx 0.5,$ but for the second and third match may also include $p(P1 \text{WINS}) \approx 0$; thus betting on P1 in the first match will have a higher value than betting on P1 in the other matches.

A different kind of rule is Choquet expected utility , also known as cumulative utility (Schmeidler 1989; Gilboa 1987). This rule starts with a function $v$ which, like a probability function, obeys $v(E)\in[0, 1],$ $v(0) = 0,$ $v(1) = 1,$ and $A \subseteq B$ implies $v(A) \le v(A).$ Unlike a probability function, however, $v$ is non-additive; and $v$ is not straightforwardly used to calculate an expectation. (Many economists refer to this function as a “non-additive subjective probability function”.). Choquet expected utility is a member of the rank-dependent family (Quiggin 1982, Yaari 1987, Kahneman & Tversky 1979, Tversky & Kahneman 1992, Wakker 2010). Functions in this family let the weight of an event in an act’s overall value depend on both the probability-like element and the event’s position in the ordering of an act, e.g., whether it is the worst or best event for that act. Formally, let $g' = \{E_1, x_1;\ldots; E_n, x_n\}$ be a re-ordering of act $g$ from worst event to best event, so that $u(x_1) \le \ldots \le u(x_n).$ The Choquet expected utility of $g'$ (and therefore of $g$) is:

If $v$ is additive, then $v$ is an (additive) probability function and CEU reduces to EU. If $v$ is convex $(v(E) + v(F) \le v(EF) + v(E v F)),$ then the individual is uncertainty-averse .

In the Ellsberg example, we are given $p(\RED) = \lfrac{1}{3}$ and $p(\BLACK \lor \YELLOW) = \lfrac{2}{3},$ and so we can assume $v(\RED) = \lfrac{1}{3}$ and $v(\BLACK \lor \YELLOW) = \lfrac{2}{3}.$ A person who is “ambiguity averse” will assign $v(\BLACK) + v(\YELLOW) \le v(\BLACK \lor \YELLOW)$; let us assume $v(\BLACK) = v(\YELLOW) = \lfrac{1}{9}.$ Similarly, she will assign $v(\RED \lor \YELLOW) + v(\BLACK) \le 1$; let us assume $v(\RED \lor \YELLOW) = \lfrac{4}{9}.$

Then the values of the acts will be:

This assignment recovers the Ellsberg preferences.

Axiomatizations of CEU use a restricted version of the separability condition (the “Comonotonic” Sure-thing Principle or “Comonotonic” Tradeoff Consistency): namely, the condition only holds when all of the acts in its domain are comonotonic , i.e., when the worst-to-best ranking of the events coincides for all the acts (Gilboa 1987, Schmeidler 1989, Wakker 1989, Chew & Wakker 1996, Köbberling & Wakker 2003; see also Wakker 2010, who also notes the relationship between CEU and $\alpha$-maximin.)

Another different type of proposal focuses on selection of permissible options from a set of alternatives $\cS,$ rather than a complete ranking of options. As in section 3.3 , we can imaginatively think of each possible probability distribution in a set $\cQ$ as a “committee member”, and posit a rule for choice based on facts about the opinions of the committee. (The first three rules are versions of the rules for sets of utility functions in 3.3, and can be combined to cover sets of probability/utility pairs.)

The first possibility is that a decision-maker is permitted to pick an act just in case some committee member is permitted to pick it over all the alternatives: just in case it maximizes expected utility relative to some probability function in the set. This is known as E-admissibility (Levi 1974, 1983, 1986; Seidenfeld, Schervish, & Kadane 2010):

Levi in fact tells a more complicated story about what a decision-maker is permitted to choose, in terms of a procedure that rules out successively more and more options. First, the procedure selects from all the options just the ones that are E-admissible. Next, the procedure selects from the E-admissible options just the ones that are P-admissible : options that “do best” at preserving the E-admissible options (the idea being that a rational agent should keep her options open). Finally, the procedure selects from the P-admissible options just the ones that are S-admissible : options that maximize the minimum utility. (Note that this last step involves maximin, not Γ-maximin.)

A more permissive rule than E-admissibility permits a choice as long there is no particular alternative that the committee members unanimously (strictly) prefer. As in the case of utility-sets, this rule is known as maximality (Walley 1991):

(See section 3.3 for an example of the difference between E-admissibility and maximality.)

More permissive still is the rule that a choice is permissible as long as it is not interval dominated (Schick 1979, Kyburg 1983): its maximum value isn’t lower than the minimum value of some other act.

For a proof showing that Γ-maximax implies E-admissibility; Γ-maximin implies maximality; E-admissibility implies maximality; and maximality implies interval dominance, see Troffaes (2007).

A final approach is to interpret ambiguity as indeterminacy: one committee member has the “true” probability function, but it is indeterminate which one. If all probability functions agree that an option is permissible to choose, then it is determinately permissible to choose; if all agree that it is impermissible to choose, it is determinately impermissible to choose; and if some hold that it is permissible and others hold that it is impermissible, it is indeterminate whether it is permissible (Rinard 2015).

These rules in this section allow but do not obviously explain the Ellsberg choices unless supplemented by an additional rule (e.g., Levi’s more complicated story or one of the rules from section 4.3.1 ), since any choice between $f_1$ and $f_2$ and between $f_3$ and $f_4$ appears to be E-admissible, maximal, and non-interval-dominated.

For those who favor non-sharp probabilities, two sets of normative questions arise. The first set is epistemic: whether it is epistemically rational not to have sharp probabilities (White 2009; Elga 2010; Joyce 2011; Hájek & Smithson 2012; Seidenfeld et al. 2012; Mayo-Wilson & Wheeler 2016; Schoenfield 2017; Vallinder 2018; Builes et al 2022; Konek ms. – see Other Internet Resources).

The second set of questions question is practical. Some hold that ambiguity aversion is not well-motivated by practical reasons and so we have no reason to account for it (Schoenfield 2020). Others hold that some particular decision rule associated with non-sharp probabilities leads to bad consequences, e.g., in virtue of running afoul of the principles mentioned in section 1.2.3 . Of particular interest is how these decision rules can be extended to sequential choice (Seidenfeld 1988a,b; Seidenfeld et al. 1990; Elga 2010; Bradley and Steele 2014; Chandler 2014; Moss 2015; Sud 2014; Rinard 2015).

5. Risk Aversion

A final challenge to expected utility maximization is to the norm itself—to the idea that a gamble should be valued at its expectation . In particular, some claim that it is rationally permissible for individuals to be risk-averse (or risk-seeking) in a sense that conflicts with EU maximization.

Say that an individual is risk-averse in money (or any numerical good) if she prefers the consequences of a gamble to be less “spread out”; this concept is made precise by Rothschild and Stiglitz’s idea of dispreferring mean-preserving spreads (1972). As a special case of this, a person who is risk-averse in money will prefer $\$x$ to any gamble whose mean monetary value is $\$x.$ If an EU maximizer is risk-averse in money, then her utility function will be concave (it diminishes marginally, i.e., each additional dollar adds less utility than the previous dollar); if an EU maximizer is risk-seeking in money, then her utility function will be convex (Rothschild & Stiglitz 1972). Therefore, EU theory equates risk-aversion with having a diminishing marginal utility function.

However, there are intuitively at least two different reasons that someone might have for being risk-averse. Consider a person who loves coffee but cannot tolerate more than one cup. Consider another person whose tolerance is very high, such that the first several cups are each as pleasurable as the last, but who has a particular attitude towards risk: it would take a very valuable upside in order for her to give up a guaranteed minimum number of cups. Both will prefer 1 cup of coffee to a coin-flip between 0 and 2, but intuitively they value cups of coffee very differently, and have very different reasons for their preference. This example generalizes: we might consider a person who is easily saturated with respect to money (once she has a bit of money, each additional bit matters less and less to her); and another person who is a miser—he likes each dollar just as much as the last—but nonetheless has the same attitude towards gambling as our coffee drinker. Both will disprefer mean-preserving spreads, but intuitively have different attitudes towards money and different reasons for this preference. Call the attitude of the second person in each pair global sensitivity (Allais 1953, Watkins 1977, Yaari 1987, Hansson 1988, Buchak 2013).

This kind of example gives rise to several problems. First, if EU maximization is supposed to explain why someone made a particular choice, it ought to be able to distinguish these two reasons for preference; but if global sensitivity can be captured at all, it will have to be captured by a diminishing marginal utility function, identical to that of the first person in each pair. Second, if one adopts a view of the utility function according to which the decision-maker has introspective access to it, a decision-maker might report that she has preferences like the tolerant coffee drinker or the miser—her utility function is linear—but nonetheless if she maximizes EU her utility function will have to be concave; so EU maximization will get her utility function wrong. Finally, even if one holds that a decision-maker does not have introspective access to her utility function, if a decision-maker displays global sensitivity, then she will have some preferences that cannot be captured by an expectational utility function (Allais 1953, Hansson 1988, Buchak 2013).

A related worry displays a troubling implication of EU maximization’s equating risk-aversion with diminishing marginal utility. Rabin’s (2000) Calibration Theorem shows that if an EU-maximizer is mildly risk-averse in modest-stakes gambles, she will have to be absurdly risk-averse in high-stakes gambles. For example, if an individual would reject the gamble {−$100, 0.5; $110, 0.5} at any wealth level, then she must also reject the gamble {−$1000, 0.5; $n, 0.5} for any $n$ whatsoever.

Finally, the Allais Paradox identifies a set of preferences that are intuitive but cannot be captured by any expectational utility function (Allais 1953). Consider the choice between the following two lotteries:

$L_1 : \{\$5,000,000, 0.1; $0, 0.9\}$
$L_2 : \{\$1,000,000, 0.11; \$0, 0.89\}$

Separately, consider the choice between the following two lotteries:

$L_3 : \{\$1,000,000, 0.89; \$5,000,000, 0.1; \$0, 0.01\}$
$L_4 : \{\$1,000,000, 1\}$

Most people (strictly) prefer $L_1$ to $L_2,$ and (strictly) prefer $L_4$ to $L_3,$ but there are no values $u(\$0),$ $u(\$1\rM),$ and $u(\$5\rM)$ such that $\EU(L_1) \gt \EU(L_2)$ and $\EU(L_4) \gt \EU(L_3).$ The Allais preferences violate the Independence Axiom; when the lotteries are suitably reframed as acts (e.g., defined over events such as the drawing of a 100-ticket lottery), they violate the Sure-thing Principle or related separability principles.

There have been a number of descriptive attempts to explain global sensitivity, by those who are either uninterested in normative questions or assume that EU is the correct normative theory. The most well-known of these are prospect theory (Kahneman & Tversky 1979; Tversky & Kahneman 1992) and generalized utility theory (Machina 1982, 1983, 1987); others are mentioned in the discussion below. See Starmer (2000) for an overview.

Some normative proposals seek to accommodate global sensitivity within expected utility theory, by making the inputs of the utility function more fine-grained. Proponents of this “refinement strategy” hold that decision-makers prefer $L_4$ to $L_3$ because the $0 outcome in $L_3$ would induce regret that one forewent a sure $1M (alternatively, because $L_4$ includes psychological certainty) and therefore that the consequence-descriptions should include these facts. Thus, the correct description of $L_3$ is:

Once these gambles are correctly described, there is no direct conflict with EU maximization (Raiffa 1986, Weirich 1986, Schick 1991, Broome 1991, Bermúdez 2009, Pettigrew 2015, Buchak 2015). The problem of when two outcomes that appear the same should be distinguished is taken up by Broome (1991), Pettit (1991), and Dreier (1996).

The thought that the value of a consequence depends on what might have been is systematized by Bradley and Stefánsson (2017). Their proposal uses a version of expected utility theory developed by Jeffrey (1965) and axiomatized by Bolker (1966). Jeffrey replaces the utility function by a more general “desirability” function Des, which applies not just to consequences but also to prospects; indeed, it doesn’t distinguish between “ultimate” consequences and prospects, since its inputs are propositions. Bradley and Stefánsson propose to widen the domain of Des to include counterfactual propositions, thus allowing that preferences for propositions can depend on counterfacts. For example, a decision-maker can prefer “I choose the risky options and get nothing, and I wouldn”t have been guaranteed anything if I had chosen differently’ to “I choose the risky option and get nothing, and I would have been guaranteed something if I had chosen differently”, which will rationalize the Allais preferences. (Incidentally, their proposal can also rationalize preferences that seemingly violate EU because of fairness considerations, as in an example from Diamond (1967).)

In a different series of articles (Stefánsson & Bradley 2015, 2019), these authors again employ Jeffrey’s framework, but this time widen the domain of Des to include chance propositions (in addition to factual prospects), propositions like “the chance that I get $100 is 0.5”. They hold that a rational decision-maker can have a preference between various chances of $X,$ even on the supposition that $X$ obtains (she need not obey “Chance Neutrality”). They capture the idea of disliking risk as such by holding that even though a rational agent must maximize expected desirability, she need not have a $\Des$ function of $X$ that is expectational with respect to the Des function of chance propositions about $X$ (she need not obey “Linearity”). For example, $\Des(\text{“I get \$100”})$ need not be equal to $2(\Des(\text{“the chance that I get \$100 is 0.5”})).$ (This does not conflict with maximizing expected desirability, because it concerns only the relationship between particular inputs to the $\Des$ function, and does not concern the decision-maker’s subjective probabilities.). This proposal can also rationalize the Ellsberg preferences ( section 4.1 ), because it allows the decision-maker to assign different probabilities to the various chance propositions (see also Bradley 2015).

Other proposals hold that we should reject the aggregation norm of expected utility. The earliest of these came from Allais himself, who held that decision-makers care not just about the mean utility of a gamble, but also about the dispersion of values. He proposes that individuals maximize expected utility plus a measure of the riskiness of a gamble, which consists in a multiple of the standard deviation of the gamble and a multiple of its skewness. Formally, if $s$ stands for the standard deviation of $L$ and $m$ stands for the skewness of $L,$ then the utility value of $L$ will be (Allais 1953, Hagen 1979):

where $\varepsilon$ is an error term. He thus proposes that riskiness is an independently valuable property of a gamble, to be combined with (and traded off against) its expected utility. This proposal essentially treats the riskiness of a gamble as a property that is (dis)valuable in itself (see also Nozick 1993 on symbolic utility).

A final approach treats global sensitivity as a feature of the decision-maker’s way of aggregating utility values. It might be that a decision-maker’s utility and probability function are not yet enough to tell us what he should prefer; he must also decide how much weight to give to what happens in worse states versus what happens in better states. In risk-weighted expected utility (Buchak 2013), a generalization of Quiggin’s (1982) anticipated utility and a member of the rank-dependent family (see section 4.3.2 ), this decision is represented by his risk function .

Formally, let $g' = \{E_1, x_1;\ldots; E_n, x_n\}$ be a re-ordering of act $g$ from worst event to best event, so that $u(x_1) \le \ldots \le u(x_n).$ Then the risk-weighted expected utility of $g$ is:

with $0 \le r(p) \le 1,$ $r(0) = 0$ and $r(1) = 1,$ and $r(p)$ non-decreasing.

The risk function measures the weight of the top $p$-portion of consequences in the evaluation of an act—how much the decision-maker cares about benefits that obtain only in the top $p$-portion of states. (One could also think of the risk function as describing the solution to a distributive justice problem among one’s future possible selves—it says how much weight a decision-maker gives to the interests of the top $p$-portion of his future possible selves.) A risk-avoidant person is someone with a convex risk function: as benefits obtain in a smaller and smaller portion of states, he gives proportionally less and less weight to them. A risk-inclined person is someone with a concave risk function. And a globally neutral person is someone with a linear risk function, i.e., an EU maximizer.

Diminishing marginal value and global sensitivity are captured, respectively, by the utility function and the risk function. Furthermore, the Allais preferences can be accommodated by a convex risk function (Segal 1987, Prelec 1998, Buchak 2013; but see Thoma & Weisberg 2017). Thus, REU maximization holds that decision-makers have the Allais preferences because they care more about what happens in worse scenarios than better scenarios, or are more concerned with the minimum value than potential gains above the minimum.

The representation theorem for REU combines conditions from two existing theorems (Machina & Schmeidler 1992, Köbberling and Wakker 2003), replacing the separability condition with two weaker conditions. One of these conditions fixes a unique probability function of events (Machina & Schmeidler’s “Strong Comparative Probability”, 1992) and the other fixes a unique risk function of probabilities; the latter is a restricted version of the separability condition (Köbberling & Wakker’s [2003] “Comonotonic” Tradeoff Consistency; see section 4.3.2 ). Since the representation theorem derives a unique probability function, a unique risk function, and a unique (up to positive affine transformation) utility function, it separates the contribution of diminishing marginal value and global sensitivity to a given preference ordering. One can disprefer mean-preserving spreads as a result of either type of risk-aversion, or a combination of both.

Proposals that seek to retain EU but refine the outcome space face two particular worries. One of these is that the constraints of decision theory end up trivial (Dreier 1996); the other is that they saddle the decision-maker with preferences over impossible objects (Broome 1991).

For theories that reject the Sure-thing Principle or the Independence Axiom, several potential worries arise, including the worry that these axioms are intuitively correct (Harsanyi 1977, Savage 1954, Samuelson 1952; see discussion in McClennen 1983, 1990); that decision-makers will evaluate consequences inconsistently (Samuelson 1952, Broome 1991); and that decision-makers will reject cost-free information (Good 1967, Wakker 1988, Buchak 2013, Ahmed & Salow 2019, Campbell-Moore & Salow 2020). The most widely discussed worry is that these theories will leave decision-makers open to diachronic inconsistency (Raiffa 1968; Machina 1989; Hammond 1988; McClennen 1988, 1990; Seidenfeld 1988a,b; Maher 1993; Rabinowicz 1995, 1997; Buchak 2013, 2015, 2017; Briggs 2015; Joyce 2017; Thoma 2019).

Ahmed, Arif and Bernhard Salow, 2019, “Don’t Look Now”, The British Journal for the Philosophy of Science , 70(2): 327–350. doi:10.1093/bjps/axx047
Allais, Maurice, 1953, “Le Comportement de l’Homme Rationnel devant le Risque: Critique des Postulats et Axiomes de l’École Americaine”, Econometrica , 21(4): 503–546. doi:10.2307/1907921
Armendt, Brad, 1986, “A Foundation for Causal Decision Theory”, Topoi , 5(1): 3–19. doi:10.1007/BF00137825
–––, 1988, “Conditional Preference and Causal Expected Utility”, in William Harper and Brian Skyrms (eds.), Causation in Decision, Belief Change, and Statistics , Dordrecht: Kluwer, Volume II, pp. 3–24.
Arntzenius, Frank, Adam Elga, and John Hawthorne, 2004, “Bayesianism, Infinite Decisions, and Binding”, Mind , 113(450): 251–283. doi:10.1093/mind/113.450.251
Arrow, Kenneth, 1951, “Alternative Approaches to the Theory of Choice in Risk-Taking Situations”, Econometrica , 19: 404–437. doi:10.2307/1907465
Aumann, Robert J., 1962, “Utility Theory without the Completeness Axiom”, Econometrica , 30(3): 445–462. doi:10.2307/1909888
Bartha, Paul F.A., 2007, “Taking Stock of Infinite Values: Pascal’s Wager and Relative Utilities”, Synthese , 154(1): 5–52. doi:10.1007/s11229-005-8006-z
–––, 2016, “Making Do Without Expectations”, Mind , 125(499): 799–827. doi:10.1093/mind/fzv152
Bermúdez, José Luis, 2009, Decision Theory and Rationality , Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199548026.001.0001
Bernoulli, Jakob, [ DW ] 1975, Die Werke von Jakob Bernoulli , Band III, Basel: Birkhäuser. A translation from this by Richard J. Pulskamp of Nicolas Bernoulli’s letters concerning the St. Petersburg Game is available online .
Bolker, Ethan D., 1966, “Functions Resembling Quotients of Measures”, Transactions of the American Mathematical Society , 124(2): 292–312. doi:10.2307/1994401
Bostrom, Nick, 2011, “Infinite Ethics”, Analysis and Metaphysics , 10: 9–59.
Bradley, Richard, 2015, “Ellsberg’s Paradox and the Value of Chances”, Economics and Philosophy , 32(2): 231–248. doi:10.1017/S0266267115000358
Bradley, Richard and H. Orri Stefánsson, 2017, “Counterfactual Desirability”, British Journal for the Philosophy of Science , 68(2): 482–533. doi:10.1093/bjps/axv023
Bradley, Seamus and Katie Siobhan Steele, 2014, “Should Subjective Probabilities be Sharp?”, Episteme , 11(3): 277–289. doi:10.1017/epi.2014.8
Briggs, Rachael, 2015, “Costs of Abandoning the Sure-Thing Principle”, Canadian Journal of Philosophy , 45(5–6): 827–840. doi:10.1080/00455091.2015.1122387
Broome, John, 1991, Weighing Goods: Equality, Uncertainty, and Time , Oxford: Blackwell Publishers Ltd.
–––, 1995, “The Two-Envelope Paradox”, Analysis , 55(1): 6–11. doi:10.2307/3328613
–––, 1997, “Is Incommensurability Vagueness?”, in Ruth Chang (ed), Incommensurability, Comparability, and Practical Reason , Cambridge, MA: Harvard University Press, pp. 67–89.
Buchak, Lara, 2013, Risk and Rationality , Oxford: Oxford University Press. doi:10.1093/acprof:oso/9780199672165.001.0001
–––, 2015, “Revisiting Risk and Rationality: A Reply to Pettigrew and Briggs”, Canadian Journal of Philosophy , 45(5–6): 841–862. doi:10.1080/00455091.2015.1125235
–––, 2017, “Replies to Commentators”, Philosophical Studies , 174(9): 2397–2414. doi:10.1007/s11098-017-0907-4
Builes, David, Sophie Horowitz, and Miriam Schoenfield, 2022, “Dilating and Contracting Arbitrarily”, Noûs , 56(1): 3–20. doi:10.1111/nous.12338
Campbell-Moore, Catrin and Bernhard Salow, 2020, “Avoiding Risk and Avoiding Evidence”, Australasian Journal of Philosophy , 98(3): 495–515. doi:10.1080/00048402.2019.1697305
Chandler, Jake, 2014, “Subjective Probabilities Need Not Be Sharp”, Erkenntnis , 79(6): 1273–1286. doi:10.1007/s10670-013-9597-2
Chang, Ruth, 1997, “Introduction”, in Ruth Chang (ed.), Incommensurability, Incomparability, and Practical Reason , Cambridge, MA: Harvard University Press, pp. 1–34.
–––, 2002a, Making Comparisons Count , London and New York: Routledge, Taylor & Francis Group.
–––, 2002b, “The Possibility of Parity”, Ethics , 112(4): 659–688. doi:10.1086/339673
–––, 2012, “Are Hard Choices Cases of Incomparability?”, Philosophical Issues , 22: 106–126. doi:10.1111/j.1533-6077.2012.00239.x
–––, 2015, “Value Incomparability and Incommensurability”, in Oxford Handbook of Value Theory , Iwao Hirose and Jonas Olson (eds.), Oxford: Oxford University Press.
–––, 2016, “Parity: An Intuitive Case”, Ratio (new series), 29(4): 395–411. doi:10.1111/rati.12148
Chen, Eddy Keming and Daniel Rubio, 2020, “Surreal Decisions”, Philosophy and Phenomenological Research , 100(1): 54–74. doi:10.1111/phpr.12510
Chew, Soo Hong and Peter Wakker, 1996, “The Comonotonic Sure-Thing Principle”, Journal of Risk and Uncertainty , 12(1): 5–27. doi:10.1007/BF00353328
Colyvan, Mark, 2006, “No Expectations”, Mind , 115(459): 695–702. doi:10.1093/mind/fzl695
–––, 2008, “Relative Expectation Theory”, Journal of Philosophy , 105(1): 37–44. doi:10.5840/jphil200810519
Colyvan, Mark and Alan Hájek, 2016, “Making Ado without Expectations”, Mind , 125(499): 829–857. doi:10.1093/mind/fzv160
Constantinescu, Cristian, 2012, “Value Incomparability and Indeterminacy”, Ethical Theory and Moral Practice , 15(1): 57–70. doi:10.1007/s10677-011-9269-8
De Sousa, Ronald B., 1974, “The Good and the True”, Mind , 83(332): 534–551. doi:10.1093/mind/LXXXIII.332.534
Diamond, Peter, 1967, “Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparison of Utility: A Comment”, Journal of Political Economy , 75(5): 765–766. doi:10.1086/259353
Dreier, James, 1996, “Rational Preference: Decision Theory as a Theory of Practical Rationality”, Theory and Decision , 40(3): 249–276. doi:10.1007/BF00134210
Easwaran, Kenny, 2008, “Strong and Weak Expectations”, Mind , 117(467): 633–641. doi:10.1093/mind/fzn053
–––, 2014a, “Decision Theory without Representation Theorems”, Philosophers’ Imprint , 14: art. 27 (30 pages). [ Easwaran 2014 available online ]
–––, 2014b, “Principal Values and Weak Expectations”, Mind , 123(490): 517–531. doi:10.1093/mind/fzu074
Elga, Adam, 2010, “Subjective Probabilities Should be Sharp”, Philosophers Imprint , 10: art. 5 (11 pages). [ Elga 2010 available online ]
Ellsberg, Daniel, 1961, “Risk, Ambiguity, and the Savage Axioms”, Quarterly Journal of Economics , 75(4): 643–669. doi:10.2307/1884324
–––, 2001, Risk, Ambiguity, and Decision , New York: Garland.
Eriksson, Lina and Alan Hájek, 2007, “What Are Degrees of Belief?”, Studia Logica: An International Journal for Symbolic Logic , 86(2): 183–213. doi:10.1007/s11225-007-9059-4
Fine, Terrence L., 2008, “Evaluating the Pasadena, Altadena, and St Petersburg Gambles”, Mind , 177(467): 613–632. doi:10.1093/mind/fzn037
Gärdenfors, Peter and Nils-Eric Sahlin, 1982, “Unreliable Probabilities, Risk Taking, and Decision Making”, Synthese , 53(3): 361–386. doi:10.1007/BF00486156
Gert, Joshua, 2004, “Value and Parity”, Ethics , 114(3): 492–510. doi:10.1086/381697
Ghirardato, Paolo, Fabio Maccheroni, Massimo Marinacci, and Marciano Siniscalchi, 2003, “A Subjective Spin on Roulette Wheels”, Econometrica , 71(6): 1897–1908. doi:10.1111/1468-0262.00472
Gilboa, Itzhak, 1987, “Expected Utility with Purely Subjective Non-Additive Probabilities”, Journal of Mathematical Economics , 16(1): 65–88. doi:10.1016/0304-4068(87)90022-X
Gilboa, Itzhak and David Schmeidler, 1989, “Maximin Expected Utility Theory with Non-Unique Prior”, Journal of Mathematical Economics , 18(2):141–153. doi:10.1016/0304-4068(89)90018-9
Griffin, James, 1986, Well-Being: Its Meaning, Measurement, and Moral Importance , Oxford: Clarendon Press. doi:10.1093/0198248431.001.0001
Good, I. J., 1952, “Rational Decisions”, Journal of the Royal Statistical Society: Series B (Methodological) , 14(1): 107–114. doi:10.1111/j.2517-6161.1952.tb00104.x
–––, 1967, “On the Principle of Total Evidence”, British Journal for the Philosophy of Science , 17(4): 319–321. doi:10.1093/bjps/17.4.319
Gwiazda, Jeremy, 2014, “Orderly Expectations”, Mind , 123(490): 503–516. doi:10.1093/mind/fzu059
Hacking, Ian, 1972, “The Logic of Pascal’s Wager”, American Philosophical Quarterly , 9(2): 186–192.
Hagen, Ole, 1979, “Towards a Positive Theory of Preference Under Risk”, in Maurice Allais and Ole Hagen (eds.), Expected Utility Hypothesis and the Allais Paradox , Dordrecht: D. Reidel, pp. 271–302.
Hájek, Alan, 2003, “Waging War on Pascal’s Wager”, Philosophical Review , 112(1): 27–56. doi:10.1215/00318108-112-1-27
–––, 2014, “Unexpected Expectations”, Mind , 123(490): 533–567. doi:10.1093/mind/fzu076
Hájek, Alan and Harris Nover, 2008, “Complex Expectations”, Mind , 117(467): 643–664. doi:10.1093/mind/fzn086
Hájek, Alan and Michael Smithson, 2012, “Rationality and Indeterminate Probabilities”, Synthese , 187(1): 33–48. doi:10.1007/s11229-011-0033-3
Hammond, Peter J., 1988, “Consequentialist Foundations for Expected Utility”, Theory and Decision , 25(1): 25–78. doi:10.1007/BF00129168
Hansson, Bengt, 1988, “Risk-aversion as a Problem of Conjoint Measurement”, in Peter Gärdenfors and Nils-Eric Sahlin (eds.), Decision, Probability, and Utility , Cambridge: Cambridge University Press, pp. 136–158. doi:10.1017/CBO9780511609220.010
Hare, Caspar, 2010, “Take the Sugar”, Analysis , 70(2): 237–247. doi:10.1093/analys/anp174
Harsanyi, John C., 1977, “On the Rationale of the Bayesian Approach: Comments on Professor Watkins’s Paperm”, in Butts and Hintikka (eds.), Foundational Problems in the Special Sciences , Dordrecht: D. Reidel.
Hodges, J.L., Jr. and E.L. Lehman, 1952, “The Uses of Previous Experience in Reaching Statistical Decisions”, Annals of Mathematical Statistics , 23(3): 396–407. doi:10.1214/aoms/1177729384
Hurwicz, Leonid, 1951a, “A Class of Criteria for Decision-Making under Ignorance”, Cowles Commission Discussion Paper: Statistics No. 356 [ Hurwicz 1951a available online ].
–––, 1951b, “The Generalized Bayes-Minimax Principle”, Cowles Commission Discussion Paper: Statistics No. 355 , [ Hurwicz 1951b available online ].
Jeffrey, Richard, 1965, The Logic of Decision , New York: McGraw-Hill Inc.
Jordan, Jeff, 1994, “The St. Petersburg Paradox and Pascal’s Wager”, Philosophia , 23(1–4): 207–222. doi:10.1007/BF02379856
Joyce, James M., 1999, The Foundations of Causal Decision Theory , Cambridge: Cambridge University Press. doi:10.1017/CBO9780511498497
–––, 2005, “How Probabilities Reflect Evidence”, Philosophical Perspectives , 19(1): 153–179. doi:10.1111/j.1520-8583.2005.00058.x
–––, 2011, “A Defense of Imprecise Credences in Inference and Decision Making”, Philosophical Perspectives , 24: 281–323. doi:10.1111/j.1520-8583.2010.00194.x
–––, 2017, “Commentary on Lara Buchak’s Risk and Rationality”, Philosophical Studies , 174(9): 2385–2396. doi:10.1007/s11098-017-0905-6
Kahneman, Daniel and Amos Tversky, 1979, “Prospect Theory: An Analysis of Decision under Risk”, Econometrica , 47(2): 263–291. doi:10.2307/1914185
Köbberling, Veronika and Peter P. Wakker, 2003, “Preference Foundations for Nonexpected Utility: A Generalized and Simplified Technique”, Mathematics of Operations Research , 28(3): 395–423. doi:10.1287/moor.28.3.395.16390
Kyburg, Henry E., 1968, “Bets and Beliefs”, American Philosophical Quarterly , 5(1): 63–78.
–––, 1983, “Rational Belief”, Behavioral and Brain Sciences , 6(2): 231–245. doi:10.1017/S0140525X00015661
Lauwers, Luc and Peter Vallentyne, 2016, “Decision Theory without Finite Standard Expected Value”, Economics and Philosophy , 32(3): 383–407. doi:10.1017/S0266267115000334
Levi, Isaac, 1974, “On Indeterminate Probabilities”, The Journal of Philosophy , 71(13): 391–418. doi:10.2307/2025161
–––, 1983, The Enterprise of Knowledge , Cambridge, MA: MIT Press.
–––, 1986, “The Paradoxes of Allais and Ellsberg”, Economics and Philosophy , 2(1): 23–53. doi:10.1017/S026626710000078X
Luce, R. Duncan and Howard Raiffa, 1957, Games and Decisions , New York: John Wiley & Sons, Inc..
Machina, Mark J., 1982, “‘Expected Utility’ Analysis without the Independence Axiom”, Econometrica , 50(2): 277–323. doi:10.2307/1912631
–––, 1983, “Generalized Expected Utility Analysis and the Nature of Observed Violations of the Independence Axiom”, in B.P. Stigum and F. Wenstop (eds.) Foundations of Utility and Risk Theory with Applications , Dordrecht: D. Reidel, pp. 263–293.
–––, 1987, “Choice Under Uncertainty: Problems Solved and Unsolved”, Journal of Economic Perspectives , 1(1): 121–154. doi:10.1257/jep.1.1.121
–––, 1989, “Dynamic Consistency and Non-expected Utility Models of Choice Under Uncertainty”, Journal of Economic Literature , 27(4): 1622–1668.
Machina, Mark J. and David Schmeidler, 1992, “A More Robust Definition of Subjective Probability”, Econometrica , 60(4): 745–780. doi:10.2307/2951565
Maher, Patrick, 1993, Betting on Theories , Cambridge: Cambridge University Press. doi:10.1017/CBO9780511527326
Mayo-Wilson, Conor and Gregory Wheeler, 2016, “Scoring Imprecise Credences: A Mildly Immodest Proposal”, Philosophy and Phenomenological Research , 93(1): 55–87. doi:10.1111/phpr.12256
McClennen, Edward F., 1983, “Sure-thing doubts”, in B.P. Stigum and F. Wenstop (eds.), Foundations of Utility and Risk Theory with Applications , Dordrecht: D. Reidel, pp. 117–136.
–––, 1988, “Ordering and Independence: A Comment on Professor Seidenfeld”, Economics and Philosophy , 4(2): 298–308. doi:10.1017/S0266267100001115
–––, 1990, Rationality and Dynamic Choice: Foundational Explorations , Cambridge: Cambridge University Press. doi:10.1017/CBO9780511983979
McGee, Vann, 1999, “An Airtight Dutch Book”, Analysis , 59(4): 257–265. doi:10.1093/analys/59.4.257
Meacham, Christopher J.G., 2019, “Difference Minimizing Theory”, Ergo , 6(35): 999–1034. doi:10.3998/ergo.12405314.0006.035
Monton, Bradley, 2019, “How to Avoid Maximizing Expected Utility”, Philosophers’ Imprint , 19: art. 18 (25 pages). [ Monton 2019 available online ]
Moss, Sarah, 2015, “Credal Dilemmas”, Noûs , 49(4): 665–683. doi:10.1111/nous.12073
Nover, Harris and Alan Hájek, 2004, “Vexing Expectations”, Mind , 113(450): 237–249. doi:10.1093/mind/113.450.237
Nozick, Robert, 1993, “Decision-Value”, Chapter 2 of The Nature of Rationality , Princeton, NJ: Princeton University Press.
Pascal, Blaise, 1670, Pensées , Paris.
Pettigrew, Richard, 2015, “Risk, Rationality and Expected Utility Theory”, Canadian Journal of Philosophy , 45(5–6): 798–826. doi:10.1080/00455091.2015.1119610
Pettit, Philip, 1991, “Decision Theory and Folk Psychology”, in Michael Bacharach and Susan Hurley (eds.), Foundations of Decision Theory , Oxford: Basil Blackwell Ltd., pp. 147–175.
Prelec, Drazen, 1998, “The Probability Weighting Function”, Econometrica , 66(3): 497–527. doi:10.2307/2998573
Quiggin, John, 1982, “A Theory of Anticipated Utility”, Journal of Economic Behavior & Organization , 3(4): 323–343. doi:10.1016/0167-2681(82)90008-7
Rabin, Matthew, 2000, “Risk Aversion and Expected-Utility Theory: A Calibration Theorem”, Econometrica , 68(5): 1281–1292. doi:10.1111/1468-0262.00158
Rabinowicz, Wlodek, 1995, “To Have One’s Cake and Eat It, Too: Sequential Choice and Expected-Utility Violations:”, Journal of Philosophy , 92(11): 586–620. doi:10.2307/2941089
–––, 1997, “On Seidenfeld’s Criticism of Sophisticated Violations of the Independence Axiom”, Theory and Decision , 43(3): 279–292. doi:10.1023/A:1004920611437
–––, 2008, “Value Relations”, Theoria , 74: 18–49. doi:10.1111/j.1755-2567.2008.00008.x
Raiffa, Howard, 1968, Decision Analysis: Introductory Lectures on Choices Under Uncertainty , Reading, MA: Addison-Wesley.
Ramsey, Frank, 1926 [1931], “Truth and Probability”, Ch. VII of F. Ramsey, The Foundations of Mathematics and other Logical Essays , edited by R.B. Braithwaite, London: Kegan Paul Ltd., 1931, 156–198.
Raz, Joseph, 1988, “Incommensurability”, chapter 13 of The Morality of Freedom , Oxford: Oxford University Press, pp. 321–368.
Rinard, Susanna, 2015, “A Decision Theory for Imprecise Probabilities”, Philosophers’ Imprint , 15: art. 7 (16 pages). [ Rinard 2015 available online ]
Rothschild, Michael and Joseph E Stiglitz, 1972, “Addendum to ‘Increasing Risk: I. A Definition’”, Journal of Economic Theory , 5(2): 306. doi:10.1016/0022-0531(72)90112-3
Samuelson, Paul A., 1952, “Probability, Utility, and the Independence Axiom”, Econometrica , 20(4): 670–678. doi:10.2307/1907649
Sartre, Jean-Paul, 1946, L'Existentialisme est un humanisme , Paris: Nagel. Translated as Existentialism is a Humanism , Carol Macomber (trans.), New Haven, CT: Yale University Press, 2007.
Savage, Leonard, 1954, The Foundations of Statistics , New York: John Wiley and Sons.
Schick, Frederic, 1979, “Self-Knowledge, Uncertainty, and Choice”, British Journal for the Philosophy of Science , 30(3): 235–252. doi:10.1093/bjps/30.3.235
–––, 1991, Understanding Action: An Essay on Reasons , Cambridge: Cambridge University Press. doi:10.1017/CBO9781139173858
Schmeidler, David, 1989, “Subjective Probability and Expected Utility without Additivity”, Econometrica , 57(3): 571–587. doi:10.2307/1911053
Schmidt, Ulrich, 2004, “Alternatives to Expected Utility: Formal Theories”, chapter 15 of Handbook of Utility Theory , Salvador Barberà, Peter J. Hammond, and Christian Seidl (eds.), Boston: Kluwer, pp. 757–837.
Schoenfield, Miriam, 2017, “The Accuracy and Rationality of Imprecise Credences”, Noûs , 51(4): 667–685. doi:10.1111/nous.12105
–––, 2020, “Can Imprecise Probabilities be Practically Motivated? A Challenge to the Desirability of Ambiguity Aversion”, Philosophers’ Imprint , 20: art. 30 (21 pages). [ Schoenfield 2020 available online ]
Segal, Uzi, 1985, “Some Remarks on Quiggin’s Anticipated Utility”, Journal of Economic Behavior and Organization , 8(1): 145–154. doi:10.1016/0167-2681(87)90027-8
Seidenfeld, Teddy, 1988a, “Decision Theory without ‘Independence’ or without ‘Ordering’”, Economics and Philosophy , 4(2): 267–290. doi:10.1017/S0266267100001085
–––, 1988b, “Rejoinder”, Economics and Philosophy , 4(2): 309–315. doi:10.1017/S0266267100001127
Seidenfeld, Teddy, Mark J. Schervish, and Joseph B. Kadane, 1990, “Decisions Without Ordering”, in E. Seig (ed.), Acting and Reflecting: The Interdisciplinary Turn in Philosophy , Dordrecht: Kluwer: 143–70.
–––, 2009, “Preference for Equivalent Random Variables: A Price for Unbounded Utilities”, Journal of Mathematical Economics , 45(5–6): 329–340. doi:10.1016/j.jmateco.2008.12.002
–––, 2010, “Coherent Choice Functions Under Uncertainty”, Synthese , 172(1): 157–176. doi:10.1007/s11229-009-9470-7
–––, 2012, “Forecasting with Imprecise Probabilities”, International Journal of Approximate Reasoning , 53(8): 1248–1261. doi:10.1016/j.ijar.2012.06.018
Sinnott-Armstrong, Walter, 1988, Moral Dilemmas , Oxford: Blackwell.
Shackle, G.L.S., 1952, Expectations in Economics , Cambridge: Cambridge University Press.
Skala, Heinz J., 1975, Non-Archimedean Utility Theory , Dordrecht: D. Reidel.
Smith, Nicholas J.J., 2014, “Is Evaluative Compositionality a Requirement of Rationality?”, Mind , 123(490): 457–502. doi:10.1093/mind/fzu072
Sobel, Jordan Howard, 1996, “Pascalian Wagers”, Synthese , 108(1): 11–61. doi:10.1007/BF00414004
Starmer, Chris, 2000, “Developments in Non-Expected Utility Theory: The Hunt for a Descriptive Theory of Choice under Risk”, Journal of Economic Literature , 38(2): 332–382. doi:10.1257/jel.38.2.332
Stefánsson, H. Orri and Richard Bradley, 2015, “How Valuable Are Chances?”, Philosophy of Science , 82(4): 602–625. doi:10.1086/682915
–––, 2019, “What Is Risk Aversion?”, The British Journal for the Philosophy of Science , 70(1): 77–102. doi:10.1093/bjps/axx035
Sturgeon, Scott, 2008, “Reason and the Grain of Belief”, Noûs , 42(1): 139–165. doi:10.1111/j.1468-0068.2007.00676.x
Sud, Rohan, 2014, “A Forward Looking Decision Rule for Imprecise Credences”, Philosophical Studies , 167(1): 119–139. doi:10.1007/s11098-013-0235-2
Sugden, Robert, 2004, “Alternatives to Expected Utility: Foundations”, Chapter 14 of Handbook of Utility Theory , Salvador Barberà, Peter J. Hammond, and Christian Seidl (eds.), Boston: Kluwer, pp. 685–755.
–––, 2009, “On Modelling Vagueness—and on Not Modelling Incommensurability”, Aristotelian Society Supplementary Volume , 83: 95–113. doi:10.1111/j.1467-8349.2009.00174.x
Thoma, Johanna, 2019, “Risk Aversion and the Long Run”, Ethics , 129(2): 230–253. doi:10.1086/699256
Thoma, Johanna and Jonathan Weisberg, 2017, “Risk Writ Large”, Philosophical Studies , 174(9): 2369–2384. doi:10.1007/s11098-017-0916-3
Troffaes, Matthias C.M., 2007, “Decision Making under Uncertainty Using Imprecise Probabilities”, International Journal of Approximate Reasoning , 45(1): 17–29. doi:10.1016/j.ijar.2006.06.001
Tversky, Amos and Daniel Kahneman, 1992, “Advances in Prospect Theory: Cumulative Representation of Uncertainty”, Journal of Risk and Uncertainty , 5(4): 297–323. doi:10.1007/BF00122574
Wakker, Peter P., 1988, “Nonexpected Utility as Aversion of Information”, Journal of Behavioral Decision Making , 1(3): 169–175. doi:10.1002/bdm.3960010305
–––, 1989, Additive Representations of Preferences, A New Foundation of Decision Analysis , Dordrecht: Kluwer.
–––, 1994, “Separating Marginal Utility and Probabilistic Risk-aversion”, Theory and Decision , 36: 1–44. doi:10.1007/BF01075296
–––, 2010, Prospect Theory: For Risk and Ambiguity , Cambridge: Cambridge University Press. doi:10.1017/CBO9780511779329
Wald, Abraham, 1950, Statistical Decision Functions, New York: John Wiley & Sons.
Walley, Peter, 1991, “The Importance of Imprecision”, Ch. 5 of Statistical Reasoning with Imprecise Probabilities (Monographs on Statistics and Applied Probability 42), London: Chapman and Hall, pp. 207–281.
Watkins, J.W.N., 1977, “Towards a Unified Decision Theory: A Non-Bayesian Approach”, in R. Butts and J. Hintikka (eds.), Foundational Problems in the Special Sciences , Dordrecht: D. Reidel.
Weirich, Paul, 1986, “Expected Utility and Risk”, British Journal for the Philosophy of Science , 37(4): 419–442. doi:10.1093/bjps/37.4.419
–––, 2008, “Utility Maximization Generalized”, Journal of Moral Philosophy , 5(2): 282–299. doi:10.1163/174552408X329019
–––, 2020, Rational Responses to Risks , New York: Oxford University Press. doi:10.1093/oso/9780190089412.001.0001
White, Roger, 2009, “Evidential Symmetry and Mushy Credence”, in T. Szabo Gendler & J. Hawthorne (eds.), Oxford Studies in Epistemology , Oxford: Oxford University Press, pp. 161–186.
Vallentyne, Peter, 1993, “Utilitarianism and Infinite Utility”, Australasian Journal of Philosophy , 71(2): 212–217. doi:10.1080/00048409312345222
Vallentyne, Peter and Shelly Kagan, 1997, “Infinite Value and Finitely Additive Value Theory”, The Journal of Philosophy , 94(1): 5–26. doi:10.2307/2941011
Vallinder, Aron, 2018, “Imprecise Bayesianism and Global Belief Inertia”, The British Journal for the Philosophy of Science , 69(4): 1205–1230. doi:10.1093/bjps/axx033
Von Neumann, John and Oskar Morgenstern, 1944, Theory of Games and Economic Behavior , Princeton, NJ: Princeton University Press.
Yaari, Menahem E., 1987, “The Dual Theory of Choice under Risk”, Econometrica , 55(1): 95–115. doi:10.2307/1911158
Zynda, Lyle, 2000, “Representation Theorems and Realism about Degrees of Belief”, Philosophy of Science , 67(1): 45–69. doi:10.1086/392761

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

Konek, Jason, manuscript, “ Epistemic Conservativity and Imprecise Credence ”

Accessibility

Support SEP

Mirror sites.

View this site from another server:

Info about mirror sites

Library of Congress Catalog Data: ISSN 1095-5054

Our expert, award-winning staff selects the products we cover and rigorously researches and tests our top picks. If you buy through our links, we may get a commission. How we test ISPs

Home Internet

Net Neutrality Vote Coming Soon: What It Means for Everyday Internet and Streaming

The FCC wants to make broadband an essential utility, like water and power. Here's what that regulation could actually mean for you.

I've been covering technology and mobile for 12 years, first as a telecommunications reporter and assistant editor at ZDNet in Australia, then as CNET's West Coast head of breaking news, and now in the Thought Leadership team.

The FCC is set to vote at the end of April on net neutrality.

One of the longest-running debates about internet access is nearing a new resolution, and the outcome could affect everything you do online.

You might remember the net neutrality debate from a decade ago. Originally voted in by the Federal Communications Commission during the Obama administration , net neutrality guidelines were subsequently ended by Donald Trump's FCC in 2017 . Now there's a new push for the guidelines, under President Joe Biden .

In September, FCC Chair Jessica Rosenworcel proposed restoring net neutrality rules , and the agency is primed to vote on April 25 .

Locating local internet providers

Net neutrality is the principle that all internet traffic should be treated equally -- meaning your broadband provider won't slow down or speed up sites you visit according to whether those sites pay extra money to have their traffic prioritized, or whether they have a special relationship with your provider. For instance, if you get your internet through Comcast, then it shouldn't speed up access to its own streaming service Peacock while slowing down rival services like Netflix and Disney Plus .

Reinstating net neutrality rules should be a top priority, said Free Press Co-CEO Jessica J. González. Free Press is a media and technology watchdog.

"People across the country are demanding these open-internet safeguards, which will allow the FCC to ensure that everyone in the United States -- no matter their location, political persuasion, race or income -- has affordable, reliable and safe internet connections free from discrimination, blocking or other ISP manipulation," González said in a statement in October.

Broadband providers deny they prioritize or slow down traffic according to financial interests, but there have been incidents in the past. A full discussion also has to take into consideration the need to simply manage high volumes of traffic. New rules, however, could offer a safeguard against egregious throttling or site prioritization.

Net neutrality regulations would " protect the internet ," the Electronic Frontier Foundation said last year.

"The idea that ISPs could prevent access to certain sites, slow down rates and speeds for certain users, isn't just horrendous -- it's vastly unpopular," the EFF said. "When ISPs charge tolls or put up road blocks, it comes at the expense of all segments of society, and undermines internet access as a right."

What will the new net neutrality rules mean, and when will they kick in?

In keeping with the Obama-era rules, the FCC wants to reaffirm that broadband is an essential service much like water, power and phone services, by designating it a "common carrier" under Title II of the Communications Act of 1934 .

Free Press says this will give the FCC the authority to hold phone and cable companies like Verizon, AT&T and Xfinity accountable for outages and potential monopoly abuses, and to ensure the affordability and availability of internet services.

After soliciting comments and feedback, the FCC will be voting April 25 on the final rule that would restore net neutrality.

What impact could net neutrality rules have on you?

The concept of net neutrality means preventing broadband and wireless providers from acting as gatekeepers to what we can access, and how fast.

An open and accessible internet has become an essential part of democracy and everyday life, enabling free speech, political organization, activism, education, health care, shopping, entertainment and business opportunities.

The American Civil Liberties Union is in favor of net neutrality rules being reinstated , calling the internet one of the most important communications services and saying that everyone, regardless of income, race and ethnicity, should have access to affordable, fast and reliable broadband. During the COVID-19 pandemic, as more of our lives moved online, the digital divide became more apparent .

"The internet is our nation's primary marketplace of ideas, and it's critical that access to that marketplace is not controlled by the profit-seeking whims of powerful telecommunications giants," said Jenna Leventoff, ACLU senior policy counsel, in a statement.

What do ISPs say about net neutrality?

Many providers say they agree with these principles. Comcast's web page on net neutrality says it does "not block, slow down or discriminate against lawful content."

"We are for sustainable and legally enforceable net neutrality protections for our customers," said Comcast, which runs the home broardband service Xfinity.

Verizon also says it supports net neutrality and a free and open internet.

"We will not throttle or slow down any internet traffic based on its source or content," Verizon's broadband commitment says. "We will not accept payments from any company to deliver its traffic faster or sooner than other traffic on our consumer broadband service, nor will we deliver our affiliates' internet traffic faster or sooner than third parties'. We will not prioritize traffic in a way that harms competition or consumers."

So with pledges like that, do we need net neutrality rules? The ACLU says internet service providers were slowing down traffic to streaming services like YouTube and Netflix as recently as a few years ago, citing research from Northeastern University .

At the same time, the ACLU says, AT&T was allowing its customers to stream its own product, DirecTV Now, without it counting towards their monthly data cap. AT&T stopped this practice after California passed its net neutrality law in 2021.

Action at the state level has been driving net neutrality efforts in recent years. Legislation has been passed in California, Oregon, Washington, Vermont, Maine and Colorado, while executive orders mandate net neutrality in at least four other states.

The ACLU says a "subtle" form of breaching net neutrality, which is more easily found today, is when wireless providers discount or include certain streaming services for customers , saying those companies are "promoting specific web services over competitors."

How much control would the FCC have?

ISPs and affiliated trade associations are adamant that net neutrality regulation is unnecessary -- and say it could actually harm us.

Joel Thayer, president of advocacy group Digital Progress Institute, told CNET there's no reason for regulation after the 2017 demise of net neutrality.

"There hasn't been one instance of an ISP blocking a website or slowing down the access of any content you want to see since the repeal of those rules," he said. "All of the net neutrality violations are happening on the tech side of the network -- outside the FCC's reach."

What's happening outside of the FCC's reach? While internet providers are continuing to expand and diversify among cable operators, wireless companies and thousands of smaller providers, Thayer points to "four companies" controlling access and competition online: Amazon, Apple, Google and Meta.

"Google owns more than 90% of search and 80% of the ad-tech market," he said. "Apple has an iron grip on iPhone users."

Thayer points out that the Federal Trade Commission already handles consumer protection from monopolies. Indeed, Google is already facing an antitrust lawsuit about its alleged monopoly over online ads , and another over its alleged online search monopoly ; Apple won an antitrust lawsuit over its in-app payments earlier this year, and will be facing one about Apple Pay next; Meta has faced antitrust accusations over its alleged social media monopoly , while the FTC is reportedly gearing up to file an antitrust lawsuit against Amazon's online retail empire .

"The ISP industry is, I think, the least of customers' concerns. No customers are complaining about what's going on that front," AT&T CEO John Stankey said during AT&T's third-quarter earnings call on Oct. 19.

But those four tech giants aren't the companies running internet lines up to people's houses or apartments and charging them to get online (the modest footprint of Google Fiber notwithstanding).

While Apple , Google , Meta and Amazon were in favor of net neutrality rules back in 2017, none of them responded to requests for comment on the FCC's latest efforts.

Will net neutrality harm broadband access?

Thayer argues that net neutrality, by imposing blanket rules on prioritizing traffic, would interfere with legitimate network management concerns -- it would affect applications like gaming, for instance, that require higher throughput.

The same goes for streaming TV and movies. "The FCC's actions here can really only adversely impact those services due to the incredible amount of bandwidth," Thayer says. "All of them require high throughput and network prioritization given that most people are migrating over to wireless technologies to access these services via 5G networks."

The CTIA, a trade association that represents the wireless communications industry, mentions supporting an open internet on its net neutrality page. But in response to the FCC's decision, CTIA CEO Meredith Attwell Baker said net neutrality " undermines our ability to achieve those goals while also putting at risk American competitiveness."

"The FCC should instead focus on closing the digital divide, facilitating competition, and advancing access to the spectrum we need to invest and innovate," she said on Oct. 19.

The CEO of the trade association USTelecom, Jonathan Spalter, called on Congress to pass its own legislation on net neutrality, much as individual states and voters have done across the nation.

"Retrofitting outdated rules onto today's competitive broadband networks is simply the wrong approach," Spalter said. "Congress must step in to end this ludicrous regulatory rinse and repeat cycle."

Home Internet Guides

Best Internet Providers in Los Angeles
Best Internet Providers in New York City
Best Internet Providers in Chicago
Best Internet Providers in San Francisco
Best Internet Providers in Seattle
Best Internet Providers in Houston
Best Internet Providers in San Diego
Best Internet Providers in Denver
Best Internet Providers in Charlotte NC
Google Fiber Internet Review
Xfinity vs Verizon Fios
Verizon 5G vs. T-Mobile Home Internet
Verizon Internet Review
Xfinity Internet Review
Best Rural Internet
Best Cheap Internet and TV Bundles
Best Speed Tests
AT&T Home Internet Review
Best Satellite Internet
Verizon 5G Home Internet Review
T-Mobile Home Internet Review
Best Internet Providers
Frontier Internet Review
Best Mesh Wi-Fi Routers
Eero 6 Plus Review
TP-Link Review
Nest Wi-Fi vs. Google Wi-Fi
Best Wi-Fi Extender
Best Wi-Fi 6 Routers
Best Wi-Fi Routers
What is 5G Home Internet?
Home Internet Cheat Sheet
Your ISP May Be Throttling Your Internet Speed
How to Switch ISPs
Internet Connection Types
Internet for Apartments
Top 10 Tips for Wi-Fi Security
How to Save Money on Your Monthly Internet Bill
How Much Internet Speed Do You Need?
AT&T Internet Promo Codes
Verizon Fios Discounts
Comcast XFINITY Codes

Utility: Theories and Models

First Online: 13 May 2021

Cite this chapter

Murat Akkaya 5

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 306))

1151 Accesses

The aim of this study is to look at utility theory from a broad perspective. The main hypothesis in the theory of decision is that the person who is in the position of deciding is entitled to the “economic man.” Also, the individual acts rationally. Thus, utility is the ability to satisfy (eliminate) human needs of goods and services. Utility is basically a psychological concept and also is the basis of economics and finance. Three types of utility take place in the economics and finance literature: marginal utility, total utility, and average utility. In addition, two main approaches fall within utility comparison: cardinal utility theory and ordinal utility theory. Furthermore, expected utility theory forms the basis of traditional finance. Expected benefit theory assumes that people choose risky or uncertain opportunities by comparing the expected benefits from them. Allais and Ellsberg paradoxes criticize expected utility theory. Tversky and Kahneman (Econometrica, 47: 263–291, 1979) present that the expected utility axioms are violated for more reasonable lottery alternatives than in the Allais paradox and put a link between finance and psychology. The prospect theory of Tversky and Kahneman forms the basis of behavioral finance.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Available as PDF
Read on any device
Instant download
Own it forever
Available as EPUB and PDF
Compact, lightweight edition
Dispatched in 3 to 5 business days
Free shipping worldwide - see info
Durable hardcover edition

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

https://www.economicsdiscussion.net/utility/utility-meaning-characteristics-and-types-economics/13594 .

https://abs.cu.edu.tr/Dokumanlar/2016/EAS223/833448774_mikro_iktisat_i.pdf .

Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’école américaine. Econometrica: Journal of the Econometric Society, 1953 , 503–546.

Article Google Scholar

Bailey, R. E. (2002). Economics of finansal markets (pp. 15–20). Essex, UK: Department of Economics Universty of Essex.

Google Scholar

Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk (Comentarii Academiae Scientiarum Imperialis Petropolitanae). Econometrica, 22 , 22–36.

Ellsberg, D. (1961). Risk, ambiguity, and the Savage axioms. The Quarterly Journal of Economics, 1961 , 643–669.

Georgescu-Roegen, N. (1968). Utility. International Encyclopedia of the Social Sciences, 16 (1), 236–267.

Hanson, J. D., & Kysar, D. A. (1999). Taking behavioralism seriously: The problem of market manipulation. NYUL Rev, 74 , 630.

Jevons, W. S. (1871). The theory of political economy . London: Macmillan.

Kıyılar, M., & Akkaya, M. (2016). Davranışsal finans . İstanbul: Literatür Yayıncılık.

Langer, E. J. (1975). The illusion of control. Journal of Personality and Social Psychology, 32 (2), 311.

Menger, C. (1871). 1981. Principles of economics . New York: New York University Press.

Schoemaker, P. J. (1982). The expected utility model: Its variants, purposes, evidence and limitations. Journal of Economic Literature , 529–563.

Simon, H. A. (1947). Administrative behavior, a study of decision-making processes in administrative organization . New York: The Macmillan Co.

Simon, H. A. (1957). Models of man: Social and rational . New York: John Wiley and Sons, Inc..

Smith, A. (1937). The wealth of nations [1776].

Tversky, A., & Kahneman, D. (1979). Prospect theory: An analysis of decision under risk. Econometrica, 47 (2), 263–291.

Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior (commemorative edition) . Princeton: Princeton University Press.

Walras, L. (1874). 1954. Elements of pure economics . London: Allen and Unwin.

Additional Reading

Barberis, N., & Xiong, W. (2012). Realization utility. Journal of Financial Economics, 104 (2), 251–271.

Broome, J. (1991). Utility. Economics and Philosophy, 7 (1), 1–12.

Ellsberg, D. (1954). Classic and current notions of “measurable utility”. The Economic Journal, 64 (255), 528–556.

Fishburn, P. C. (1968). Utility theory. Management Science, 14 (5), 335–378.

Fishburn, P. C. (1970). Utility theory for decision making (No. RAC-R-105) . McLean, VA: Research Analysis Corp.

Book Google Scholar

Kahneman, D., & Snell, J. (1990). Predicting utility .

Kahneman, D., & Thaler, R. H. (2006). Anomalies: Utility maximization and experienced utility. Journal of Economic Perspectives, 20 (1), 221–234.

Layard, R., Mayraz, G., & Nickell, S. (2008). The marginal utility of income. Journal of Public Economics, 92 (8–9), 1846–1857.

MacCrimmon, K. R., & Larsson, S. (1979). Utility theory: Axioms versus ‘paradoxes’. In Expected utility hypotheses and the Allais paradox (pp. 333–409). Dordrecht: Springer.

Chapter Google Scholar

Sen, A. (1991). Utility: Ideas and terminology. Economics and Philosophy, 7 (2), 277–283.

Download references

Author information

Authors and affiliations.

T.C. Istanbul Arel University, Istanbul, Turkey

Murat Akkaya

You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murat Akkaya .

Editor information

Editors and affiliations.

Faculty of Transportation and Logistics, Istanbul University, Avcılar/Istanbul, Turkey

Burcu Adıgüzel Mercangöz

Rights and permissions

Reprints and permissions

Copyright information

About this chapter

Akkaya, M. (2021). Utility: Theories and Models. In: Mercangöz, B.A. (eds) Applying Particle Swarm Optimization. International Series in Operations Research & Management Science, vol 306. Springer, Cham. https://doi.org/10.1007/978-3-030-70281-6_1

Download citation

DOI : https://doi.org/10.1007/978-3-030-70281-6_1

Published : 13 May 2021

Publisher Name : Springer, Cham

Print ISBN : 978-3-030-70280-9

Online ISBN : 978-3-030-70281-6

eBook Packages : Business and Management Business and Management (R0)

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Publish with us

Policies and ethics

Find a journal
Track your research

IMAGES

13 Different Types of Hypothesis (2024)
Mean-Variance Expected Utility Hypothesis
Expected Utility Hypothesis
NM utility hypothesis
Relative Income Hypothesis
What is a Hypothesis

VIDEO

hypothesis testing of comparing means of two independent samples
Utility (How the World REALLY Works: The Economy)
What Is A Hypothesis?
1. Expected Utility Theory
UGC NET/JRF Economic: Unit 1: Concept of Utility (Theory of Consumer Behaviour)
Hypothesis Test for Means with Independent Samples

COMMENTS

Expected utility hypothesis
The expected utility hypothesis is a foundational assumption in mathematical economics concerning decision making under uncertainty. It postulates that rational agents maximize utility, meaning the subjective desirability of their actions. Rational choice theory, a cornerstone of microeconomics, builds this postulate to model aggregate social ...
Expected Utility Hypothesis Definition & Examples
Definition of Expected Utility Hypothesis. The Expected Utility Hypothesis is a theory in economics that suggests individuals choose between alternatives to maximize their expected utility—a measure of satisfaction or happiness derived from the outcomes of their choices. This hypothesis operates under the assumption that people are rational ...
Expected Utility: Definition, Calculation, and Examples
Expected utility is an economic term summarizing the utility that an entity or aggregate economy is expected to reach under any number of circumstances. The expected utility is calculated by ...
Expected Utility Theory
Definition. Expected utility theory (EUT) is an axiomatic theory of choice under risk that has held a central role in economic theory since the 1940s. The hypothesis is that, under certain assumptions, an individual's preferences towards lotteries can be represented as a linear function of the utility of each option multiplied by the ...
Normative Theories of Rational Choice: Expected Utility
Two-boxing dominates one-boxing: in every state, two-boxing yields a better outcome. Yet on Jeffrey's definition of conditional probability, one-boxing has a higher expected utility than two-boxing. There is a high conditional probability of finding $1 million is in the closed box, given that you one-box, so one-boxing has a high expected utility.
Decision Utility
Expected utility theory. Then, in 1944, John Von Nuemann and Oskar Morgenstern developed the expected utility hypothesis, based on Daniel Bernoulli's first description of how we make decisions by estimating the probability and utility of an outcome. By multiplying the probability of an outcome by the expected benefit of that outcome, we get ...
Expected Utility Theory
Expected utility theory is the dominant model of decision-making under uncertainty in law and economics. It posits that people choose among risky prospects, or lotteries, modeled as probability distributions over a set of possible outcomes, as if they assign a utility value to each outcome x according to a function u(x) and select the lottery that maximizes the expected value of u(x).
PDF Expected Utility Theory
Definition. A utility function U : P → has an expected utility form if. R. there exists a function u : C → such that. (p) = p (c) u (c) for all p ∈ P. c ∑. ∈C. In this case, the function U is called an expected utility function, and the function u is call a von Neumann-Morgenstern utility function.
Expected Utility Hypothesis
The expected utility hypothesis of behaviour towards risk is the hypothesis that the individual possesses (or acts as if possessing) a 'von Neumann-Morgenstern utility function' U(·) or 'von Neumann-Morgenstern utility index' {U i} defined over some set of alternative possible outcomes, and when faced with alternative risky prospects or 'lotteries' over these outcomes, will ...
Expected utility hypothesis
The expected utility hypothesis is a foundational assumption in mathematical economics concerning decision making under uncertainty. It postulates that rational agents maximize utility, meaning the subjective desirability of their actions. Rational choice theory, a cornerstone of microeconomics, builds this postulate to model aggregate social behaviour.
Expected Utility Theory
It is derived from the expected utility theory, a hypothesis stating that the weighted average of every possible utility level will best depict utility at any point in time under uncertainty. ... Maximizing this utility means choosing the option yielding the maximum average utility. In this case, the average utility is all utilities ...
Utility Hypothesis
This is the first of six chapters in Part II about demand and utility cost, a typical area for what is understood as choice theory. It discusses utility hypothesis and the theory of value. Its five sections are: needs of measurement (of utility); common practice and (William) Fleetwood; parallels in theory (as applied to utility construction ...
Expected Utility Hypothesis
(Empirically, this means analyzing market-level data rather than data from individuals.) One of the key conceptual innovations early in the study of asset pricing was the efficient markets hypothesis. Essentially, this hypothesis states that if markets are efficient, then prices should reflect all available information [61]. For example, in an ...
Von Neumann-Morgenstern Utility Theorem Definition & Examples
Definition of the Von Neumann-Morgenstern Utility Theorem. ... It provides a theoretical foundation for the expected utility hypothesis, which asserts that individuals choose among risky projects to maximize their expected utility. This concept is applied in portfolio theory, the pricing of insurance, and in evaluating the economic ...
CHAPTER 1 CHAPTER 1 Expected Utility Theory
1.1 Introduction. This Handbook is a modern look at decision theory and social choice. It emphasizes recent developments that go beyond the classic theory of utility and expected utility. However, a sensible starting point is the classical theory, a benchmark that will frame departures considered in subsequent chapters.
Von Neumann-Morgenstern utility theorem
The theorem is the basis for expected utility theory . In 1947, John von Neumann and Oskar Morgenstern proved that any individual whose preferences satisfied four axioms has a utility function; [1] such an individual's preferences can be represented on an interval scale and the individual will always prefer actions that maximize expected utility.
(PDF) Utility: Theories and Models
Chapter 1. Utility: Theories and Models. Murat Akkaya. Abstract The aim of this study is to look at utility theory from a broad perspective. The main hypothesis in the theory of decision is that ...
How Economists Came to Accept Expected Utility Theory: The Case of
mental evidence against the expected utility hypothesis prompted decision theorists Ivan Moscati is Associate Professor of Economics, University of Insubria, Varese, Italy. He ... alternatives by looking at the mean, the variance, and possibly other elements of the distribution of uncertain payoffs, rather than using expected utility (Hicks 1931).
The assessment of value in health economics: utility and capability
This means, that an individual knows what kind of options are available, and is able to provide a subjective value to those options, which are expressed when an individual makes a choice. ... Known as the "expected utility hypothesis". For an excellent explanation of the axioms and assumptions underpinning this theory, see chapter two in ...
Normative Theories of Rational Choice: Rivals to Expected Utility
Expected utility theory, which holds that a decision-maker ought to maximize expected utility, is the prevailing theory of instrumental rationality.Nonetheless, four major challenges have arisen to the claim that the theory characterizes all rational preferences. These challenges are the phenomena of infinite or unbounded value, incommensurable goods, imprecise probabilities, and risk-aversion.
Anomalies: Utility Maximization and Experienced Utility
In this column, we discuss a version of the utility maximization hypothesis that can be tested—and we find that it is false. Our analysis begins with a distinction between two senses of the term utility. Decision utility has also been called "wantability"; it is inferred from choices and used to explain choices.
Governor Cooper to Attend White House State Dinner Honoring Prime
Governor Roy Cooper and First Lady Kristin Cooper will travel to Washington, D.C. to attend a White House State Dinner on Wednesday night honoring America's relationship with Japan. President Biden will host Japanese Prime Minister Kishida Fumio and his wife Kishida Yuko, along with other members of the Japanese delegation. Following the State Dinner, Prime Minister and Mrs.
FUJIFILM Diosynth Biotechnologies Expands Holly Springs Facility to
When companies select a site located in a Tier 3 county such as Wake, their JDIG agreements move some of the new tax revenue into the state's Industrial Development Fund - Utility Account. Local communities in more economically challenged areas of the state use grants from the Utility Account to build public infrastructure projects, which ...
Risk and Expected Utility
We can now state the famous expected utility hypothesis: investments (or, more generally, risky prospects with multiple possible outcomes) are valued according to the expectation (i.e., statistical average) of the utility of the possible outcomes. In short, the hypothesis says that decisions are made on the basis of expected utility.
Net Neutrality Vote Coming Soon: What It Means for Everyday Internet
The concept of net neutrality means preventing broadband and wireless providers from acting as gatekeepers to what we can access, and how fast. An open and accessible internet has become an ...
UPDATED FOOTAGE/PHOTOS AVAILABLE: Governor Cooper Greets Prime Minister
On Thursday evening, Governor Roy Cooper and First Lady Kristin Cooper greeted Prime Minister of Japan Kishida Fumio and Mrs. Kishida Yuko upon their arrival at RDU-International Airport.
North Carolina Office of the State Auditor Issues Statewide Single
The North Carolina Office of the State Auditor (OSA) released its annual Statewide Single Audit of federal funds received by the state. For fiscal year 2023, the State of North Carolina spent $35 billion in federal awards that were distributed to 618 programs managed by 103 different state entities, including the university and community college systems.
High Path Avian Influenza detected in North Carolina dairy herd
Apr 10, 2024. The National Veterinary Services Laboratory has detected Highly Pathogenic Avian Influenza in a dairy herd in North Carolina. HPAI has previously been detected in dairy herds in Texas, Kansas, Michigan, Idaho, New Mexico, and Ohio. Movement of cattle from affected herds in these states to North Carolina has been suspended.
DEQ details next steps for water systems as EPA announces PFAS
DEQ has been working with public water systems to prepare for the regulation and assess PFAS levels in drinking water systems across the state. In 2023, DEQ sampled more than 530 small public water systems, those serving at least 15 connections or 25 individuals, to assess PFAS levels across the state. That effort follows the sampling in late ...
Utility: Theories and Models
The marginal utility of good x is MU x or U x.. 2. Ordinal utility theory: Ordinal utility theory assumes that benefit is an immeasurable magnitude. The assumptions of the cardinal utility theory that are far from reality and based on coercion have been criticized by economists, and these economists have stated that the idea of numerical measurement of the concept of abstract benefit is ...