Complementary Weighted Multiple Majority Games

In this paper, we introduce a new family of simple games, whic h is referred to as the complementary weighted multiple majorit y game. For the two dimensional case, we prove that there are at most n+1 minimal winning coalitions (MWC for short), wheren is the number of players. An algorithm for computing all the MWCs is presented, with a running time of O(n logn). Computing the main power indices, i.e. Shapley-Shubik index, Banzhaf index, H oller-Packel index, and Deegan-Packel index, can all be done in polynomial time. Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. We also define a new kind of stability: the C-stability. Assum ing that allocation of the payoff among the winning coalition is proportional to players’ pow ers, we show that C-stable coalition structures are those that cont ain an MWC with the smallest sum of powers. Hence, C-stable coalition structur es can be computed in polynomial times.

[1]  Martin Shubik,et al.  A Method for Evaluating the Distribution of Power in a Committee System , 1954, American Political Science Review.

[2]  Tayfun Sönmez,et al.  Core in a simple coalition formation game , 2001, Soc. Choice Welf..

[3]  Jana Hajduková.  On Coalition Formation Games , 2004 .

[4]  S. Hart,et al.  On the endogenous formation of coalitions , 1983 .

[5]  Katarína Cechlárová,et al.  Stability in coalition formation games , 2000, Int. J. Game Theory.

[6]  P. P. Shenoy,et al.  On coalition formation: a game-theoretical approach , 1979 .

[7]  William S. Zwicker,et al.  Simple games - desirability relations, trading, pseudoweightings , 1999 .

[8]  Antonio Romero-Medina,et al.  Coalition Formation and Stability , 2006, Soc. Choice Welf..

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

[10]  Forming Coalitions and Measuring Voting Power , 1982 .

[11]  E. Maskin.  Bargaining, Coalitions and Externalities , 2003 .

[12]  M. Holler,et al.  Monotonicity of power and power measures , 2004 .

[13]  Anke Gerber.  Coalition formation in general NTU games , 2000 .

[14]  M. Chwe.  Farsighted Coalitional Stability , 1994 .

[15]  Lin Zhou,et al.  A New Bargaining Set of an N-Person Game and Endogenous Coalition Formation , 1994 .

[16]  Matthew O. Jackson,et al.  The Stability of Hedonic Coalition Structures , 2002, Games Econ. Behav..

[17]  William R. Zame,et al.  Bargaining in cooperative games , 1988 .

[18]  R. Myerson.  Values of games in partition function form , 1977 .

[19]  Andrew Postlewaite,et al.  Weak Versus Strong Domination in a Market with Indivisible Goods , 1977 .

[20]  R. J. Aumann,et al.  Cooperative games with coalition structures , 1974 .

[21]  Joan Magretta,et al.  The Power of Virtual Integration: An Interview with Dell Computer''s Michael Dell , 1998 .

[22]  Licun Xue,et al.  Farsighted Stability in Hedonic Games , 2000 .

[23]  W. Lucas,et al.  N‐person games in partition function form , 1963 .

[24]  J. Deegan,et al.  A new index of power for simplen-person games , 1978 .

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

[26]  L. S. Shapley,et al.  17. A Value for n-Person Games , 1953 .

[27]  Salvador Barberà,et al.  On coalition formation: durable coalition structures , 2003, Math. Soc. Sci..

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

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

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