Subjective expected utility

Subjective expected utility is a method in decision theory in the presence of risk, originally put forward by L. J. Savage in 1954 [Savage, Leonard J. 1954. "The Foundations of Statistics". New York, Wiley.] . It combines two distinct subjective concepts: a personal utility function and a personal probability analysis based on Bayesian probability theory.

Savage proved that, if you adhere to axioms of rationality, if you believe an uncertain event has possible outcomes $\{x\_i\}$ each with a utility to you of $u(x\_i)$ 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 $P(x\_i)$, and your subjective expected utility is the expected value of the utility,:$sum\_i\; ;\; u(x\_i)\; ;\; P(x\_i).$You may be able to make a decision which changes the possible outcomes to $\{y\_j\}$ 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 $x(=\{x\_i\})$ to $y$ and $s$ to $t$ then you will prefer $lambda\; x\; +\; (1-lambda\; )s$ to $lambda\; y\; +\; (1-lambda\; )t$, for

Experiments have shown that many individuals do not behave in a manner consistent with subjective expected utility, most prominently Allais (1953) [Allais, M. 1953. Le Comportement de l'Homme Rationnel Devant Le Risque: Critique des Postulats et Axiomes de L'Ecole Americaine. "Econometrica" 21(4):503-546.] and Ellsberg (1961). [Ellsberg, Daniel. 1961. Risk, Ambiguity and Savage Axioms. "Quarterly Journal of Economics" 75(4):643-79.] Savage's response was not that this showed a flaw in his method, rather that applying his method allowed individuals to improve their decision making.

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