Combinatorial optimization games

We introduce a general integer programming formulation for a class of combinatorial optimization games, which immediately allows us to improve the algorithmic result for finding imputations in the core (an important solution concept in cooperative game theory) of the network flow game on simple networks by Kalai and Zemel [16]. An interesting result is a general theorem that the core for this class of games is nonempty if and only if a related linear program has an integer optimal solution, We study the properties for this mathematical condition to hold for several interesting problems, and apply them to resolve algorithmic and complexity issues for their cores along the line as put forward in [2,19]: d ecr ‘d e whether the core is empty; if the core is empty, find an imputation in the core; given an imputation x, test whether x is in the core. We also explore the properties of totally balanced games in this succinct formulation of cooperative games.

[1]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[2]  E. Kalai,et al.  Finite Rationality and Interpersonal Complexity in Repeated Games , 1988 .

[3]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[4]  V. Knoblauch.  Computable Strategies for Repeated Prisoner's Dilemma , 1994 .

[5]  Eitan Zemel,et al.  Totally Balanced Games and Games of Flow , 1982, Math. Oper. Res..

[6]  Richard M. Karp,et al.  Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems , 1972, Combinatorial Optimization.

[7]  Carl A. Futia.  The complexity of economic decision rules , 1977 .

[8]  Sándor P. Fekete,et al.  On the complexity of testing membership in the core of min-cost spanning tree games , 1997, Int. J. Game Theory.

[9]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[10]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[11]  A. Neyman.  Bounded complexity justifies cooperation in the finitely repeated prisoners' dilemma , 1985 .

[12]  Bernhard von Stengel,et al.  Fast algorithms for finding randomized strategies in game trees , 1994, STOC '94.

[13]  Imma Curiel,et al.  Cooperative game theory and applications , 1997 .

[14]  M. Padberg.  Linear Optimization and Extensions , 1995 .

[15]  Toshihide Ibaraki,et al.  Complexity of the Minimum Base Game on Matroids , 1997, Math. Oper. Res..

[16]  Eitan Zemel,et al.  Generalized Network Problems Yielding Totally Balanced Games , 1982, Oper. Res..

[17]  Ehud Kalai,et al.  Games, computers, and O.R. , 1996, SODA '96.

[18]  U. Faigle,et al.  On the complexity of testing membership in the core of min-cost spanning tree games , 1997 .

[19]  N. Megiddo.  On computable beliefs of rational machines , 1989 .