We propose a new geometric approach for the analysis of cooperative games. A cooperative game is viewed as a real valued function $u$ defined on a finite set of points in the unit simplex. We define the \emph{concavification} of $u$ on the simplex as the minimal concave function on the simplex which is greater than or equal to $u$. The concavification of $u$ induces a game which is the \emph{totally balanced cover} of the game. The concavification of $u$ is used to characterize well-known classes of games, such as balanced, totally balanced, exact and convex games. As a consequence of the analysis it turns out that a game is convex if and only if each one of its sub-games is exact.
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Find related papers by JEL classification: C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory D8 - Microeconomics - - Information, Knowledge, and Uncertainty
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Branzei, Rodica & Dimitrov, Dinko & Tijs, Stef, 2006.
"Convex games versus clan games,"
Discussion Paper
58, Tilburg University, Center for Economic Research.
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Other versions:
Rodica Branzei & Dinko Dimitrov & Stef Tijs, 2006.
"Convex games versus clan games,"
Working Papers
381, Bielefeld University, Institute of Mathematical Economics.
[Downloadable!]
Csóka, Péter & Herings, P. Jean-Jacques & Kóczy, László Á., 2007.
"Balancedness Conditions for Exact Games,"
Research Memoranda
039, Maastricht : METEOR, Maastricht Research School of Economics of Technology and Organization.
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