decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...

, subjective expected utility is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of

risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...

. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized 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 As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...

function, and second a personal

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

distribution (usually based on

Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...

theory). Savage proved that, if the decision-maker adheres to axioms of rationality, believing an uncertain event has possible outcomes

\

each with a utility of

u(x_i),

then the person's choices can be explained as arising from this utility function combined with the subjective belief that there is a probability of each outcome,

P(x_i).

The subjective expected utility is the resulting

expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...

of the utility, :

= \sum_i \; u(x_i) \; P(x_i) .

If instead of choosing

\

the person were to choose

\,

the person's subjective expected utility would be :

\Epsilon

(Y) A thumb signal, usually described as a thumbs-up or thumbs-down, is a common hand gesture achieved by a closed fist held with the thumb extended upward or downward in approval or disapproval, respectively. These gestures have become metaphors ...

= \sum_j \; u(y_j) \; P(y_j). Which decision the person prefers 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 combination In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other w ...

s of decisions and that preferences would be preserved. So if a person prefers

x(=\)

y (=\)

and

s(=\)

t(=\)

then that person will prefer

\lambda x + (1-\lambda )s

\lambda y + (1-\lambda )t

, for any

0<\lambda<1

. Experiments have shown that many individuals do not behave in a manner consistent with Savage's axioms of subjective expected utility, e.g. most prominently Allais (1953) and Ellsberg (1961).

Notes

References

* http://psychclassics.yorku.ca/Peirce/small-diffs.htm * Ramsey, Frank Plumpton; “Truth and Probability”
PDF
, Chapter VII in ''The Foundations of Mathematics and other Logical Essays'' (1931). * de Finetti, Bruno. "Probabilism: A Critical Essay on the Theory of Probability and on the Value of Science," (translation of 1931 article) in ''Erkenntnis,'' volume 31, September 1989. * de Finetti, Bruno. 1937, “La Prévision: ses lois logiques, ses sources subjectives,” Annales de l'Institut Henri Poincaré, : de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of th
1937 article
in French) in H. E. Kyburg and H. E. Smokler (eds), ''Studies in Subjective Probability,'' New York: Wiley, 1964. * de Finetti, Bruno. ''Theory of Probability'', (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974–5. * * *

External links

* {{DEFAULTSORT:Subjective Expected Utility Social philosophy Expected utility