Computational Complexity of Weighted Threshold Games

Weighted threshold games are coalitional games in which each player has a weight (intuitively corresponding to its voting power), and a coalition is successful if the sum of its weights exceeds a given threshold. Key questions in coalitional games include finding coalitions that are stable (in the sense that no member of the coalition has any rational incentive to leave it), and finding a division of payoffs to coalition members (an imputation) that is fair. We investigate the computational complexity of such questions for weighted threshold games. We study the core, the least core, and the nucleolus, distinguishing those problems that are polynomial-time computable from those that are NP-hard, and providing pseudopolynomial and approximation algorithms for the NP-hard problems.

[1]  Vincent Conitzer,et al.  Complexity of constructing solutions in the core based on synergies among coalitions , 2006, Artif. Intell..

[2]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[3]  Jesús Mario Bilbao,et al.  Voting power in the European Union enlargement , 2002, Eur. J. Oper. Res..

[4]  Tamás Rapcsák,et al.  New Trends in Mathematical Programming , 1998 .

[5]  Tomomi Matsui,et al.  NP-completeness for calculating power indices of weighted majority games , 2001, Theor. Comput. Sci..

[6]  Yoav Shoham,et al.  Marginal contribution nets: a compact representation scheme for coalitional games , 2005, EC '05.

[7]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

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

[9]  Giorgio Gambosi,et al.  Complexity and Approximation , 1999, Springer Berlin Heidelberg.

[10]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[11]  Jerry S. Kelly,et al.  NP-completeness of some problems concerning voting games , 1990 .

[12]  Laurence Wolsey,et al.  The nucleolus and kernel for simple games or special valid inequalities for 0–1 linear integer programs , 1976 .

[13]  Bezalel Peleg,et al.  ON WEIGHTS OF CONSTANT-SUM MAJORITY GAMES. , 1968 .

[14]  T. Matsui,et al.  A SURVEY OF ALGORITHMS FOR CALCULATING POWER INDICES OF WEIGHTED MAJORITY GAMES , 2000 .

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