[This paper presents an expected utility theory for state-dependent preferences. It proposes axioms that permit the joint derivation of subjective probabilities and utilities when the decision maker's preferences are not independent of the prevailing state of nature. In addition to the usual von Neumann-Morgenstern axioms, these axioms also include the requirement that the decision-maker's actual preferences are consistent with his preferences contingent on an hypothetical probability distribution over the states of nature. Two versions of the consistency axiom are introduced and their significance in the context of Bayesian decision theory is discussed.]
IT IS WELL KNOWN that for a finite exchange economy, where preferences are not assumed to be convex, there may be no price or even the core may be empty. For this reason it was proposed to enlarge the set of price and the core by introducing the concepts of equilibrium and core. The existence of approximate for exchange economies, where preferences are not assumed to be convex, has been investigated by R. Starr [6]. He showed that there exists a quasi-equilibrium, provided the number of participants is large enough and there is a bound on the degree of non-convexity [6, p. 30, Assumption D]. In this note we shall show the existence of equilibria (a stronger concept than the one considered by Starr [6, p. 31]) for large economies where the preferences are neither assumed to be convex nor complete. To obtain our result we shall assume that the preferences and the endowments of all participating agents belong to a compact set. In [5] Shapley and Shubik proved that, for a large replica of a given economy with transferable utility, the e-core is nonempty. We shall generalize this result to large economies without transferable utility by using the concept of ?-core as introduced by Kannai [2]. The nonemptiness of the e-core follows easily from the existence of approximate and a relationship between the set of approximate and e-core. The existence of c-core for large economies (with a fixed number of types) can also be deduced from Kannai's Theorem C' [2] in its stronger form (Theorem C in [3]).
SAMUELSON MADE THE CONJECTURE stated above in his 1967 paper [7]. He also formalized there the axiom of independence of irrelevant alternatives for cardinal preferences, used here. Preference are cardinal if their representation by a numerical function is invariant under, and only under, positive linear transformations. One may think that the disregard for intensity of preferences, embedded in Arrow's treatment of profiles of ordinal rankings of alternatives, leads to the impossibility result. Samuelson's conjecture points out that this is not the way to refute the conclusions of Arrow's theorem. There is also interest per se in aggregation of cardinal preferences. Such preferences are usually considered as von Neumann-Morgenstern utility, i.e., numerical representation of preferences over lotteries [11]. Since uncertainty is the rule and not the exception whenever decisions are involved, it is of some importance to obtain a social N-M utility over risky outcomes. Given such a utility, the society will be able to choose a best alternative among the several feasible risky actions (i.e., lotteries). However it is not necessary to restrict the interpretation of cardinal preferences to those induced by ordinal ranking over lotteries. One can think of cardinal preferences derived from comparisons between pairs of alternatives (as in an axiomatization of a regret relation). See Alt [1] for an early work of this kind. When working with cardinal preferences a continuity assumption is needed, in addition to unanimity and independence (see the example at the end of the next section). A standard reference for Arrow's theorem is the last chapter of his book [2]. For a general discussion of aggregation of cardinal preferences, see ShapleyShubik [10]. Some other impossibility results involving different notions of cardinal preferences appear in the works of Sen [9], DeMeyer-Plott [4], Schwartz [8], and Fishburn [5]. A model dealing with aggregation of cardinal preferences into social cardinal prefereiLces, as here, is that of Harsanyi [6]. However he is interested in 1 The work of the first author was done at Northwestern University and the work of the second author began at the University of Illinois in Urbana-Champaign and it was completed at the University of Minnesota in Minneapolis. Both authors are on leave from Tel-Aviv University. The authors wish to express their thanks to E. A. Pazner, M. A. Satterthwaite, J. Kelly and the referees for helpful comments. This research was partly supported by NSF Grant # SOC-75-05317.
We argue that the notion of Pareto dominance is not as compelling in the presence of uncertainty as it is under certainty. In particular, voluntary trade based on differences in tastes is commonly accepted as desirable, because tastes cannot be wrong. By contrast, voluntary trade based on incompatible beliefs may indicate that at least one agent entertains mistaken beliefs. We propose and characterize a weaker, No-Betting, notion of Pareto domination which requires, on top of unanimity of preference, the existence of shared beliefs that can rationalize such preference for each agent.