Quasi-Cores in a Monetary Economy with Nonconvex Preferences

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 128.235.251.160 on Tue, A model of a pure exchange economy is investigated without the usual assumption of convex preference sets for the participating traders. The concept of core, taken from the theory of games, is applied to show that if there are sufficiently many participants, the economy as a whole will possess a solution that is sociologically stable-i.e., that cannot profitably be upset by any coalition of traders. The core of a game is the set of all undominated outcomes [11, 1]. A solution is any set of outcomes, mutually undominating, that collectively dominate all others [21]. The term "core" was introduced by Gillies and Shapley [13, 10] in studying properties of the von Neumann-Morgenstern solutions; the core as an independent solution concept was developed by the latter in lectures at Princeton in the fall of 1953. 805 This content downloaded from 128.235.251.160 on Tue, 24 Mar 2015 11:26:15 UTC All use subject to JSTOR Terms and Conditions 806 L. S. SHAPLEY AND M. SHUBIK have been obtained by Shubik [17], Debreu and Scarf [6, 7], Aumann [2], and Vind [19, 20], who all exploit to some degree the game-theoretic point of view. The preferences of the individual traders are usually assumed to be convex, since, as is well known, without convexity a competitive equilibrium may not exist. The present paper, on the other hand, is concerned especially with markets that possess neither a competitive equilibrium nor a core; indeed, in the presence of either one our chief theorems would be almost trivial to prove.3 The general tenor of our results is that nonconvexity in the preference sets is economically unimportant when the number of participating individuals is large. This may not be a new observation, but here we give it a mathematically precise expression in a fresh context.More specifically, we show under quite general conditions that even if the true core cannot be assured, because of nonconvexity, nevertheless certain quasi-cores will always appear when the number of traders in the market is large enough. These quasi-cores are characterized by the requirement that recontracting traders must show a definite positive profit before being permitted to block a proposed final allocation. A distinction must be made ("strong" vs. "weak" v-cores) depending on whether the c-threshold is applied to the whole set of recontracting traders, or to each one individually. The conditions for existence are somewhat different in the two cases (Theorems 2 and 4).There is reason to believe that the ?-cores, in a double limit process involving both c and the number of traders, would shrink down 'a la Edgeworth upon the competitive allocations of a certain related...

Algorithmic Cost Allocation Games: Theory and Applications

“…Unfortunately, the core in general may be empty, e.g., the ticket-pricing problem that we consider in Chapter 8. Because of this reason, Shapley and Shubik generalized the core concept to the strong ε-core and weak ε-core [56], which are non-empty with suitable parameter ε. Following this line of reasoning, Maschler, Peleg, and Shapley [45] introduced in 1979 the least core, which is the intersection of all nonempty strong ε-cores.…”

Cores of many-player games; nonemptiness and equal treatment

“…Approximate cores of economies with quasi-linear utility functions were introduced in Shapley and Shubik (1966), which showed that when the player set is replicated, then, for all sufficiently large replications, approximate cores are nonempty. 12 A contribution by Owen (1975) is also relevant.…”

“…It has also been recognized that in private goods economies with many agents, problems of nonconvexities disappear or become insignificant; some seminal contributions are Aumann (1964), Shapley and Shubik (1966), Aumann and Shapley (1974), and, for economies with production, Hurwicz and Uzawa (1977). More specifically, in a number of situations it has been shown that, in economies with many agents, cores are nonempty, price-taking equilibria exist, and equilibrium outcomes are close to core outcomes, where approximations, if any, become arbitrarily good as the economies become large.…”

Laws of scarcity for a finite game - exact bounds on estimations

“…Since Shapley and Shubik [19] showing nonemptiness of approximate cores of exchange economies with many players and quasi-linear utilities and Wooders [23], [24], showing nonemptiness of approximate cores of game with many players with and without side payments, there has been a number of further results. For parameterized collections of games, such results are demonstrated in [6], [7], [8] and [28].…”