Non-Convexity, Aggregation, and Pareto Optimality

Rothenberg, Jerome

doi:10.1086/258363

Cited by 23 publications

(7 citation statements)

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“…If the game happens to have an odd number of players, say n = 2r + 1, then in any Pareto-optimal payoff vector the most-favored player receives at least r/n units. 15 least-favored set of n -1 players will always get (r -1) -(r -1)/n or less, compared with the amount r -1 they can obtain in coalition. This must be compared with the amount r that they can obtain in coalition.…”

Section: Another Extension Of Theorem 3 Results From Observing That Imentioning

confidence: 99%

“…(This is the only use of the function Ko postulated in (15).) The important fact about this bound is that it is independent of k.…”

Section: B=m I I Min Ko(y*) 1 Subject To Y*e Convex Hull Of {Yh}l Y*mentioning

confidence: 99%

“…

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. As yet we have not found sharp conditions for the convergence of the c-cores, and our mathematical results in this paper deal only with the existence question.Recently, new attention has been focused on the economic implications of nonconvex preferences [5,9,12,15]. For more information about JSTOR, please contact support@jstor.org.

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confidence: 99%

“…Recently, new attention has been focused on the economic implications of nonconvex preferences [5,9,12,15]. Unfortunately, convexity is too often assumed in economics merely because tools for handling the opposite assumption are lacking.…”

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Quasi-Cores in a Monetary Economy with Nonconvex Preferences

Shapley¹,

Shubik²

1966

Econometrica

261

137

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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...

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Section: Another Extension Of Theorem 3 Results From Observing That Imentioning

confidence: 99%

“…(This is the only use of the function Ko postulated in (15).) The important fact about this bound is that it is independent of k.…”

Section: B=m I I Min Ko(y*) 1 Subject To Y*e Convex Hull Of {Yh}l Y*mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Quasi-Cores in a Monetary Economy with Nonconvex Preferences

Shapley¹,

Shubik²

1966

Econometrica

261

137

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“…From Proposition 1, it is possible to have an equilibrium and therefore efficient allocation without convexity (when M holds). However, in view of the central role of C in these theorems, the implications of relaxing this hypothesis have been examined intensively in recent years by Farrell [1959], Rothenberg [1960], Aumann [1966], and Starr [1969]. Their conclusions may be summarized as follows: Let C' be the weakened con- Thus, the only nonconvexities that are important for the present purposes are increasing returns over a range large relative to the economy.…”

Section: Competitive Equilibrium and Pareto Efficiencymentioning

confidence: 99%