Subjective expected utility
Encyclopedia
Subjective expected utility is a method in decision theory
in the presence of risk
, promoted by L. J. Savage in 1954 following previous work by Ramsey
and von Neumann
. The theory of subjective expected utility combines two subjective concepts: first, a personal utility
function, and second a personal probability
distribution (based on Bayesian probability
theory).
Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes
each with a utility to you of 
then your choices can be explained as arising from a function in which you believe that there is a subjective probability of each outcome is 
, and your subjective expected utility is the expected value
of the utility,
You may be able to make a decision which changes the possible outcomes to
in which case your subjective expected utility will become
Which decision you prefer depends on which subjective expected utility is higher. Different people may make different decisions because they may have different utility functions or different beliefs about the probabilities of different outcomes.
Savage assumed that it is possible to take convex combinations of decisions and that preferences would be preserved. So if you prefer
to 
and 
to 
then you will prefer 
to 
, for 
.
Experiments have shown that many individuals do not behave in a manner consistent with subjective expected utility, e.g. most prominently Allais
(1953)
and Ellsberg
(1961)
.
Decision theory
Decision theory in economics, psychology, philosophy, mathematics, and statistics is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision...
in the presence of risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...
, promoted by L. J. Savage in 1954 following previous work by Ramsey
Frank P. Ramsey
Frank Plumpton Ramsey was a British mathematician who, in addition to mathematics, made significant and precocious contributions in philosophy and economics before his death at the age of 26...
and von Neumann
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
. The theory of subjective expected utility combines two subjective concepts: first, a personal utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
function, and second a personal probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
distribution (based on Bayesian probability
Bayesian probability
Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities. The Bayesian interpretation of probability can be seen as an extension of logic that enables reasoning with propositions, whose truth or falsity is...
theory).
Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes
each with a utility to you of then your choices can be explained as arising from a function in which you believe that there is a subjective probability of each outcome is , and your subjective expected utility is the expected valueExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
of the utility,
You may be able to make a decision which changes the possible outcomes to
in which case your subjective expected utility will becomeWhich decision you prefer depends on which subjective expected utility is higher. Different people may make different decisions because they may have different utility functions or different beliefs about the probabilities of different outcomes.
Savage assumed that it is possible to take convex combinations of decisions and that preferences would be preserved. So if you prefer
to and to then you will prefer to , for .Experiments have shown that many individuals do not behave in a manner consistent with subjective expected utility, e.g. most prominently Allais
Allais paradox
The Allais paradox is a choice problem designed by Maurice Allais to show an inconsistency of actual observed choices with the predictions of expected utility theory.-Statement of the Problem:...
(1953)
and Ellsberg
Ellsberg paradox
The Ellsberg paradox is a paradox in decision theory and experimental economics in which people's choices violate the expected utility hypothesis.An alternate viewpoint is that expected utility theory does not properly describe actual human choices...
(1961)
.