An act maps states of nature to outcomes: deterministic outcomes, as well as random outcomes, are included. Two acts f and g are comonotonic, by definition, if it never happens that f(s) > f(t) and g(t) > g(s) for some states of nature s and t. An axiom of comonotonic independence is introduced here. It weakens the von Neumann-Morgenstern axiom of independence as follows: If f > g and if f, g and h are comonotonic then $f + (1 - $)h > $g + (1 - $)h. If a nondegenerate, continuous, and monotonic (state independent) weak order over acts satisfies comonotonic independence, then it induces a unique non-(necessarily-) additive probability and a von Neumann-Morgenstern utility. Furthermore, one can compute the expected utility of an act with respect to the nonadditive probability, using the Choquet integral. This extension of the expected utility theory covers situations, such as the Ellsberg paradox, which are inconsistent with additive expected utility. The concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included. Copyright 1989 by The Econometric Society.(This abstract was borrowed from another version of this item.)

[This note provides a rigorous description of a game in a strategic form whose Nash equilibria are all strong equilibriaccoinciding with the Walras equilibria of the underlying Arrow-Breau pure exchange economy.]

In an atomless economy any allocation that is not blocked by coalitions is in the core. Hence, in such an economy, a competitive equilibrium is characterized by the blocking power of part of the coalitions which excludes all big coalitions. THE CORE of an economy consists of all the allocations that are not blocked by any coalition. In this note we prove that for any positive number 8, the core of an atomless economy coincides with the set of allocations that are not blocked by any coalition of measure less than s. This result implies that even if the large coalitions cannot be formed, any unblocked allocation is still in equilibrium with respect to some price system. In particular, the formation of the coalition of all traders or any large coalition is not needed to insure the Pareto-optimality of final allocation. We prove here directly that the core is equal to the set of allocations that are not blocked by small coalitions. The same result can also be obtained by proving that Aumann's [1], Vind's [5], or Hildenbrand's [2] equivalence theorems hold with the additional restriction on the measure of the blocking coalitions. In any case the proof is a simple application of Liapunov's convexity theorem [3 and 4] (for the statement of the theorem see also [2, Appendix, p. 451]).

It is shown that a market with a continuum of traders possesses a competitive equilibrium even when the preferences are not complete. This generalizes further a result of Aumann, ('Econometrica'; 32: 39-50 (1964); 34: 1-17(1966)) who showed that the convexity assumption may be dispensed within the presence of a continuum of traders. The proof is inspired by the Arrow-Debreu ('Econometrica; 22: 265-290(1954)) proof for the finite case. (Author)

Recent formulations of social decisions problems focus on rules by which a group of individuals (the society) arrives at a choice among available alternatives. Such a rule, which we shall call the outcome function (Gibbard's "game form, " see [ 3 J), specifies the

Although their goal is to separate a decision maker's underlying beliefs (their subjective probabilities of events) from their preferences (their attitudes toward risk), classic choice-theoretic derivations of subjective probability all rely upon some form of the Marschak-Samuelson "Independence Axiom" or the Savage "Sure-Thing Principle, " which is equivalent to requiring that the decision maker's preferences over lotteries conform to the expected utility hypothesis. This paper presents a choice-theoretic derivation of subjective probability which satisfies the axioms of classical probability theory, but which neither assumes nor implies that the decision maker's preferences over lotteries necessarily conform to the expected utility hypothesis.

[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]).