On $α$-roughly weighted games

Gvozdeva, Hemaspaandra, and Slinko (2011) have introduced three hierarchies for simple games in order to measure the distance of a given simple game to the class of (roughly) weighted voting games. Their third class Cα consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least 1 and each losing coalition a weight of at most α. For a given game the minimal possible value of α is called its critical threshold value. We continue the work on the critical threshold value, initiated by Gvozdeva et al., and contribute some new results on the possible values for a given number of voters as well as some general bounds for restricted subclasses of games. A strong relation beween this concept and the cost of stability, i.e. the minimum amount of external payment to ensure stability in a coalitional game, is uncovered.

[1]  Dan S. Felsenthal,et al.  Ternary voting games , 1997, Int. J. Game Theory.

[2]  Sascha Kurz,et al.  On the inverse power index problem , 2012, ArXiv.

[3]  Sascha Kurz,et al.  On minimum sum representations for weighted voting games , 2011, Ann. Oper. Res..

[4]  Josep Freixas,et al.  Complete simple games , 1996 .

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

[6]  Lane A. Hemaspaandra,et al.  Three hierarchies of simple games parameterized by “resource” parameters , 2013, Int. J. Game Theory.

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

[8]  J. Brenner,et al.  The Hadamard Maximum Determinant Problem , 1972 .

[9]  Josep Freixas,et al.  A power analysis of linear games with consensus , 2004, Math. Soc. Sci..

[10]  Timo Berthold,et al.  On the computational impact of MIQCP solver components , 2011 .

[11]  Josep Freixas,et al.  Dimension of complete simple games with minimum , 2008, Eur. J. Oper. Res..

[12]  Jörg Rothe,et al.  The Cost of Stability in Coalitional Games , 2009, SAGT.

[13]  Ton Storcken.  Effectivity functions and simple games , 1997, Int. J. Game Theory.

[14]  Timo Berthold,et al.  Extending a CIP framework to solve MIQCPs , 2012 .

[15]  Josep Freixas Bosch,et al.  Weighted games without a unique minimal representation in integers , 2009 .

[16]  Josep Freixas,et al.  Weighted voting, abstention, and multiple levels of approval , 2003, Soc. Choice Welf..

[17]  M. Osborne.  Introduction to Game Theory: International Edition , 2009 .

[18]  Daniel Granot,et al.  On Some Network Flow Games , 1992, Math. Oper. Res..

[19]  Rocco A. Servedio,et al.  Improved Approximation of Linear Threshold Functions , 2009, Computational Complexity Conference.

[20]  Adam N. Letchford,et al.  Reformulating mixed-integer quadratically constrained quadratic programs , 2011 .

[21]  Tatiana Gvozdeva,et al.  Weighted and Roughly Weighted Simple Games , 2009 .

[22]  Jesús Mario Bilbao,et al.  The distribution of power in the European Constitution , 2007, Eur. J. Oper. Res..

[23]  Jeffrey S. Rosenschein,et al.  The Cost of Stability in Network Flow Games , 2009, MFCS.

[24]  J. R. Isbell,et al.  A class of simple games , 1958 .

[25]  Bezalel Peleg,et al.  Voting by Count and Account , 1992 .

[26]  Yoram Bachrach,et al.  The Least-Core of Threshold Network Flow Games , 2011, MFCS.

[27]  Gerhard J. Woeginger,et al.  On the dimension of simple monotonic games , 2006, Eur. J. Oper. Res..

[28]  Taylor Alan,et al.  Weighted Voting, Multicameral Representation, and Power , 1993 .