This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low prob-abilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling. 1.

Observed preference reversal (PR) cannot be adequately explained by violations of independence, the reduction axiom, or transitivity. The primary cause of PR is the failure of procedure invariance, especially the overpricing of low-probability high-payoff bets. This result violates regret theory and generalized (nonindependent) utility models. PR and a new reversal involving time preferences are explained by scale compatibility, which implies that payoffs are weighted more heavily in pricing than in choice.

To accommodate the observed pattern of risk-aversion and risk-seeking, as well as common violations of expected utility (e.g., the certainty effect), we introduce and characterize a weighting function according to which an event has greater impact when it turns impossibility into possibility, or possibility into certainty, than when it merely makes a possibility more or less likely. We show how to compare such weighting functions (of different individuals) with respect to the degree of departure from expected utility, and we present a method for comparing an individual's weighting functions for risk and for uncertainty.

Two probabilistic theories of choice behavior (the model of independent random utility and the model of elimination by aspects) imply a testable property, the multiplicative inequality, according to which the probability of selecting an alternative x from an offered set A u B is at least as large as the product of the probabilities of selecting x from A and from B. WHEN FACED WITH a choice among complex alternatives (e.g., commodity bundles, investment plans, job offers) people often exhibit inconsistency. That is, they do not always select the same alternative under seemingly identical conditions. In order to accommodate this fact and obtain an adequate conception of choice behavior, psychologists and economists (e.g., Thurstone [18 and 19], GeorgescuRoegen [8], Luce [12], and Marschak [14]) developed models of choice in which the traditional concept of preference is replaced by the notion of choice probability. These models were investigated by many authors (e.g., Davidson and Marschak [5], Debreu [6], Chipman [4], Luce and Suppes [13], and Tversky [20]) from both mathematical and experimental standpoints, and they have also been applied to various aspects of economic theory such as equilibrium analysis (Hildenbrand [10] and Bhattacharya and Majumdar [2]) and consumer behavior (GeorgescuRoegen [9], Quandt [17], Mossin [16], and McFadden and Richter [15]). Two general forms of probabilistic choice models, called random utility and constant utility, were investigated (see Luce and Suppes [13]). In the random utility form, the subjective values undergo random fluctuations, and the alternative with the highest momentary value is selected. In the constant utility form, choice probability is expressed as a function of some (constant) scale values. Thus, the two forms differ regarding the locus of the random element in the choice process. The random utility form attributes uncertainty to the determination of subjective value, whereas the constant utility form attributes uncertainty to the application of the decision rule. (Some choice models, however, can be expressed in either form.) This paper investigates one prominent example of each form: the independent random utility model and the model of elimination by aspects. It shows that both models satisfy two testable properties which provide (fairly tight) upper and lower bounds for all choice probabilities. To formulate the results, we introduce the following definitions: