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 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 (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 formula_1 each with a utility of formula_2 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, formula_3 The subjective expected utility is the resulting expected value of the utility, If instead of choosing formula_1 the person were to choose formula_6 the person's subjective expected utility would be 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 formula_8 to formula_9and formula_10 to formula_11 then that person will prefer formula_12 to formula_13, for any formula_14. 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).