Core Equivalence Theorems for Infinite Convex Games

We show that the core of a continuous convex game on a measurable space of players is a von Neumann-Morgenstern stable set. We also extend the definition of the Mas-Colell bargaining set to games with a measurable space of players, and show that for continuous convex games the core may be strictly included in the bargaining set but it coincides with the set of all countably additive payoff measures in the bargaining set. We provide examples which show that the continuity assumption is essential to our results.

[1]  R. Aumann.  Markets with a continuum of traders , 1964 .

[2]  Joseph Greenberg,et al.  On the sensitivity of von Neumann and Morgenstern abstract stable sets: The stable and the individual stable bargaining set , 1992 .

[3]  David Schmeidler,et al.  Cores of Exact Games, I* , 1972 .

[4]  Sergiu Hart,et al.  Formation of cartels in large markets , 1974 .

[5]  Freddy Delbaen.  Convex games and extreme points , 1974 .

[6]  A. W. Tucker,et al.  Advances in game theory , 1964 .

[7]  Itzhak Gilboa,et al.  Canonical Representation of Set Functions , 1995, Math. Oper. Res..

[8]  Debraj Ray,et al.  A consistent bargaining set , 1989 .

[9]  R. Aumann,et al.  THE BARGAINING SET FOR COOPERATIVE GAMES , 1961 .

[10]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.

[11]  William F. Lucas,et al.  Von Neumann-Morgenstern stable sets , 1992 .

[12]  Andrew B. Whinston,et al.  A Theory of Pricing under Decreasing Costs , 1978 .

[13]  R. Bott.  18. Symmetric Solutions to Majority Games , 1953 .

[14]  David Wettstein,et al.  Equivalence between bargaining sets and the core in simple games , 1996 .

[15]  Bezalel Peleg,et al.  A proof that the core of an ordinal convex game is a von Neumann-Morgenstern solution , 1986 .

[16]  M. Maschler.  The bargaining set, kernel, and nucleolus , 1992 .

[17]  Yakar Kannai,et al.  The core and balancedness , 1992 .

[18]  A. Mas-Colell.  An equivalence theorem for a bargaining set , 1989 .

[19]  M. Sion.  On general minimax theorems , 1958 .

[20]  L. Shapley,et al.  The kernel and bargaining set for convex games , 1971 .

[21]  K. Yosida,et al.  Finitely additive measures , 1952 .

[22]  Ezra Einy,et al.  Convex Games and Stable Sets , 1996 .

[23]  L. Shapley.  Cores of convex games , 1971 .

[24]  R. Aumann,et al.  Game theoretic analysis of a bankruptcy problem from the Talmud , 1985 .

[25]  The bargaining set and the core in mixed markets with atoms and an atomless sector , 1989 .

[26]  J. Neumann,et al.  Theory of Games and Economic Behavior. , 1945 .

[27]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[28]  D. Schmeidler.  Integral representation without additivity , 1986 .

[29]  Joseph Greenberg,et al.  The Theory of Social Situations: An Alternative Game-Theoretic Approach , 1990 .

[30]  Ron Holzman,et al.  Core and Stable Sets of Large Games Arising in Economics , 1996 .