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Int J Game Theory (2010) 39:585602
DOI 10.1007/s00182-010-0239-5
ORIGINAL PAPER
Accepted: 4 April 2010 / Published online: 30 May 2010 The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract The ShapleyIchiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system denes the class. Application of this general theorem to the class of convex games yields an alternative proof of the ShapleyIchiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal dening system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.
Keywords TU games Core Linearity regions Computation of Q-sets
J. Kuipers (B)
Department of Knowledge Engineering, Maastricht University, P.O.Box 616, 6200 MD Maastricht, The Netherlandse-mail: [email protected]
D. Vermeulen
Department of Quantitative Economics, Maastricht University, P.O.Box 616, 6200 MD Maastricht, The Netherlandse-mail: [email protected]
M. Voorneveld
Department of Economics, Stockholm School of Economics, P.O.Box 6501, 113 83 Stockholm, Sweden e-mail: [email protected]
A generalization of the ShapleyIchiishi result
Jeroen Kuipers Dries Vermeulen
Mark Voorneveld
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586 J. Kuipers et al.
1 Introduction
In his study of convex games, Shapley (1971) proved that the core of a convex game is the convex hull of the marginal vectors of that game. A decade later, Ichiishi (1981) showed that the converse of this statement is also true: if the core of a game is equal to the convex hull of marginal vectors, then the game is convex. The combined Shapley Ichiishi result thus relates the dening system of linear inequalities of convex games to the combinatorial structure of the core common to all convex games. The interesting combinatorial properties of convex games has led to a vast literature in which convex games play a role. Examples are unanimity games, bankruptcy games (ONeill 1982;...