Non-Expected Utility Theory

BIBLIOGRAPHY

The expected utility/subjective probability model of risk preferences and beliefs has long been the preeminent model of individual choice under conditions of uncertainty. It exhibits a tremendous flexibility in representing aspects of attitudes toward risk, has a well-developed analytical structure, and has been applied to the analysis of gambling, games of strategy, incomplete information, insurance, portfolio and investment decisions, capital markets, and many other areas. This model posits a cardinal utility function over outcomes (usually alternative wealth levels) and assumes that an individual evaluates risky prospects on the basis of the expected value of his or her utility function. In situations of objective uncertainty (e.g., roulette wheels), this expectation is based on the objective probabilities involved. In situations of subjective uncertainty (e.g., horse races) likelihood beliefs are represented by the individual’s personal or subjective probabilities of the various alternative occurrences. First proposed by the Dutch mathematician Daniel Bernoulli in 1738 as a solution to the well-known Saint Petersburg paradox, the expected utility model has since been axiomatized under conditions of both objective and subjective uncertainty. Many consider these axioms and the resulting model to be the essence of rational risk preferences and beliefs.

In spite of its flexibility, the expected utility/subjective probability model has refutable implications, and beginning in the 1950s, psychologists and economists have uncovered a growing body of experimental evidence that individuals do not necessarily conform to many of the key axioms or predictions of the model. One well-known example, first demonstrated by the French economist Maurice Allais in 1953, consists of asking subjects to express their preferred option from each of two pairs of objective gambles. The majority of subjects express preferences that are inconsistent with expected utility, and they directly violate its primary empirical axiom, the so-called independence axiom. Although initially dismissed as an isolated example, the Allais paradox has been replicated by numerous researchers and found to be a special case of at least two forms of systematic violations of the independence axiom. Such departures have also been replicated using real-money gambles.

Starting in the early 1960s, researchers have also uncovered a class of systematic violations of the subjective probability hypothesis. The most well-known example, offered by Daniel Ellsberg in 1961, consists of an urn with ninety balls, thirty of which are red, with the remaining sixty being black or yellow in an unknown proportion. Subjects are asked to select from each of two pairs of bets on this urn, and they typically select in a manner inconsistent with well-defined likelihood beliefs in regard to obtaining a black versus a yellow ball. This finding was also originally dismissed, but the phenomenon has since been replicated by many researchers in a number of different examples. Choices in such experiments reveal a general preference for betting on objective rather than subjective events, a phenomenon that has been termed “ambiguity aversion.”

In response to these empirical violations, researchers have developed, axiomatized, and analyzed a number of alternative models of risk preferences and beliefs, most of which replace the expected utility formula with alternative formulas that individuals are assumed to maximize. The earliest of these models, proposed by Ward Edwards in the 1950s and adopted by Daniel Kahneman and Amos Tversky in the 1970s as part of their well-known “prospect theory,” was found to generate implausible predictions (namely that individuals would select some gambles with lower payoffs than other gambles). Economists have since developed and axiomatized non-expected utility models of risk preferences that avoid these difficulties, are consistent with the broad class of Allais-type violations of the independence axiom, and are capable of formal analysis and application to economic and other decisions. The most notable of these is the “rank-dependent expected utility model” of the Australian economist John Quiggin.

Researchers have also developed models of preferences over subjective prospects that are consistent with both Allais-type departures from expected utility risk preferences and Ellsberg-type departures from probabilistic beliefs. One such model, long informally discussed in the literature, axiomatized by Itzhak Gilboa and David Schmeidler, and known as “maximin expected utility,” posits a utility function and a set of subjective probability distributions over events. It assumes that individuals evaluate each bet on the basis of its minimum expected utility over this class of distributions. Another important model, again axiomatized by Gilboa and Schmeidler and known as “Choquet expected utility,” posits a utility function but replaces the classical (i.e., additive) probability measure of subjective expected utility with a nonadditive measure over events. It also replaces the standard expected utility formula with an alternative notion of expectation in respect to this nonadditive measure. Both models have been successfully applied to economic decision-making.

SEE ALSO Expected Utility Theory; Probability; Probability, Subjective; Prospect Theory; Rationality; Risk; Uncertainty

BIBLIOGRAPHY

Kahneman, Daniel, and Amos Tversky. 1979. Prospect Theory: An Analysis of Decision under Risk. Econometrica 47 (2): 263–291.

Machina, Mark. 1987. Choice under Uncertainty: Problems Solved and Unsolved. Journal of Economic Perspectives 1 (1): 121–154.

Quiggin, John. 1982. A Theory of Anticipated Utility. Journal of Economic Behavior and Organization 3 (4): 323–343.

Schmeidler, David. 1989. Subjective Probability and Expected Utility without Additivity. Econometrica 57 (3): 571–587.

Mark J. Machina