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TWO-STAGE LO1TERIES WITHOUT THE REDUCTION AXIOM1 BY Uzi SEGAL This paper analyzes preference relations over two-stage lotteries, i.e., lotteries having as outcomes tickets for other, simple, lotteries. Empirical evidence indicates that decision makers do not always behave in accordance with the reduction of compound lotteries axiom, but it seems that they satisfy a compound independence axiom (also known as the certainty equivalent mechanism). It turns out that although the reduction and the compound independence axioms together with continuity imply expected utility theory, each of them by itself is compatible with all possible preference relations over simple lotteries. By using these axioms I analyze three different versions of expected utility for two-stage lotteries. The second part of the paper is devoted to possible replacements of the reduction axiom. For this I suggest several different compound dominance axioms. These axioms compare two-stage lotteries by the probability they assign to the upper and lower sets of all simple lotteries X. (For a simple lottery X, its upper (lower) set is the set of lotteries that dominate (are dominated by) X by first order stochastic dominance.) It turns out that these axioms are all strictly weaker than the reduction of compound lotteries axiom. The main theoretical results of this part are: (1) an axiomatic basis for expected utility theory that does not require the reduction axiom and (2) a new axiomatization of the anticipated utility model (also known as expected utility with rank-dependent probabilities). These representation theorems indicate that to a certain extent the rank dependent probabilities model is a natural extension of expected utility theory. Finally, I show that some paradoxes in expected utility theory can be explained, provided one is willing to use the compound independence rather than the reduction axiom.

This paper assumes that in addition to conventional preferences over outcomes, players in a strategic environment have preferences over strategies. It provides conditions under which a player's preferences over strategies can be represented as a weighted average of the utility from outcomes of the individual and his opponents. The weight one player places on an opponent's utility from outcomes depends on the players' joint behavior. In this way, the framework is rich enough to describe the behavior of individuals who repay kindness with kindness and meanness with meanness. The paper identifies restrictions that the theory places on rational behavior.

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has provided an intriguing argument that social welfare can be expressed as a weighted sum of individual utilities. His theorem has been criticized on the grounds that a central axiom, that social preference satisfies the independence axiom, has the morally unacceptable implication that the process of choice and considerations of ex ante fairness are of no importance. This paper presents a variation of Harsanyi's theorem in which the axioms are compatible with a concern for ex ante fairness. The implied mathematical form for social welfare is a strictly quasi-concave and quadratic function of individual utilities.

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