Subjective expected utility

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Subjective expected utility is a method in decision theory in the presence of risk originally put forward by L. J. Savage in 1954. It combines two distinct subjective concepts: a personal utility function and a personal probability analysis based on Bayesian probability theory.

If you believe an uncertain event has possible outcomes {xi} each with a utility to you of u(xi) and where you believe that the probability of each outcome is P(xi), then your subjective expected utility will be

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

You may be able to make a decision which changes the possible outcomes to {yj} in which case your subjective expected utility will become

\sum_j \; u(y_j) \; P(y_j).

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 was possible to take convex combinations of decisions and that preferences would be preserved. So if you prefer x( = {xi}) to y and s to t then you will prefer λx + (1 − λ)s to λy + (1 − λ)t, for 0 < λ < 1.

Experiments involving offering people lottery tickets have suggested that many individuals do not seem to have personally consistent utility functions in the face of risk; Savage's response was not that this showed a flaw in his method, but that applying his method allowed individuals to improve their decision taking.

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