Subjective expected utility

In decision theory, subjective expected utility is the attractiveness of an economic opportunity as perceived by a decision-maker in the presence of risk. Characterizing the behavior of decision-makers as using subjective expected utility was promoted and axiomatized by L. J. Savage in 1954^[1][2] following previous work by Ramsey and von Neumann.^[3] The theory of subjective expected utility combines two subjective concepts: first, a personal utility function, and second a personal probability distribution (usually based on Bayesian probability theory).

Savage proved that, if the decision-maker adheres to axioms of rationality, believing an uncertain event has possible outcomes ${\displaystyle \{x_{i}\}}$ each with a utility of ${\displaystyle u(x_{i}),}$ 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, ${\displaystyle P(x_{i}).}$ The subjective expected utility is the resulting expected value of the utility,

${\displaystyle \mathrm {E} [u(X)]=\sum _{i}\;u(x_{i})\;P(x_{i}).}$

If instead of choosing ${\displaystyle \{x_{i}\}}$ the person were to choose ${\displaystyle \{y_{j}\},}$ the person's subjective expected utility would be

${\displaystyle \mathrm {E} [u(Y)]=\sum _{j}\;u(y_{j})\;P(y_{j}).}$

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 combinations of decisions and that preferences would be preserved. So if a person prefers ${\displaystyle x(=\{x_{i}\})}$ to ${\displaystyle y(=\{y_{i}\})}$and ${\displaystyle s(=\{s_{i}\})}$ to ${\displaystyle t(=\{t_{i}\})}$ then that person will prefer ${\displaystyle \lambda x+(1-\lambda )s}$ to ${\displaystyle \lambda y+(1-\lambda )t}$, for any ${\displaystyle 0<\lambda <1}$.

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)^[4] and Ellsberg (1961).^[5]