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Utility, Von Neumann-Morgenstern

BIBLIOGRAPHY

Von NeumannMorgenstern utility considers decision making under risk. It concerns choice when the probabilities of the possible outcomes of that choice are objectively known. This decision framework differs from decision making under certainty and decision making when probability is subjective. John von Neumann and Oskar Morgenstern (1947) follow a behavioral framework analogous to the revealed preference framework developed by Paul A. Samuelson (1938 and 1947). Quasi-operational experiments are envisioned where peoples behavior is held to satisfy key behavioral axioms.

Despite the infinite expected payoff, few would pay large sums to play this game. Bernoulli provided an answer to this paradox in terms of diminishing marginal utility. He argued that it is the utility of the gains and not the monetary gains from the game that are relevant to the persons decision whether or not to play the game. With diminishing marginal utility, the gains from the game could be finite and small.

Von NeumannMorgenstern utility can be illustrated by considering lotteries with two possible prizes, A and B, and a cost for playing the lottery, Y, with A Y B. Let P be the probability of the most favorable prize, B, and (1P) be the probability of the less favorable prize, A. Six key assumptions (Luce and Raiffa 1957) will assure that a utility function can be defined from the revealed preferences of players. These assumptions assure that a probability can be found for every lottery where the decision maker is indifferent between the price of the lottery ticket and the lottery. In other words,

U (Y ) = (1P )U (A ) + PU (B ), or

U (Y ) = U (A ) + (U (B )U (A ))P.

Suppose we rank every possible outcome of any lottery from W, the worst possible, to B, the best possible and assign an ordinal number to U (W ) and U (B ) representing this ranking. Then U (Y ) for any Y can be found as a linear transformation of these two extreme outcomes. Hence, Von NeumannMorgenstern utility is cardinal but only because it represents a linear transformation of the probabilities in the lotteries. It is not cardinal utility in any other context.

SEE ALSO Expected Utility Theory; Gambling; Revealed Preference; Utility, Objective; Utility, Subjective; Von Neumann, John

BIBLIOGRAPHY

Bernoulli, Daniel. 1738. Specimen Theoriae Novae de Mensura Sortis. Trans. Louise Sommer in Exposition of a New Theory on the Measurement of Risk. Econometrica 22, no. 1 (January 1954): 2336.

Luce, R. Duncan, and Howard Raiffa. 1957 Games and Decisions. New York: Wiley.

Samuelson, Paul A. 1938. A Note on the Pure Theory of Consumers Behavior. Economica 5: 6171.

Samuelson, Paul A. 1947. Foundations of Economic Analysis. Cambridge, MA: Harvard University Press.

Von Neumann, John, and Oskar Morgenstern. 1944. The Theory of Games and Economic Behavior, 2nd ed. Princeton, NJ: Princeton University Press, 1947.

Michael P. Shields