The Shapley-Shubik power index for games with several levels of approval in the input and output

Voting systems with several levels of approval in the input and output are considered in this paper. That means games with n ≥ 2 players, j ≥ 2 ordered qualitative alternatives in the input level and k ≥ 2 possible ordered quantitative alternatives in the output. We introduce the Shapley-Shubik power index notion when passing from ordinary simple games or ternary voting games with abstention to this wider class of voting systems. The pivotal role of players is analysed by means of several examples and an axiomatization in the spirit of Shapley and Dubey is given for the proposed power index.

[1]  P. Dubey.  On the uniqueness of the Shapley value , 1975 .

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

[3]  Frank Steffen.  Voting Power in Games with Abstentions , 2002 .

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

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

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

[7]  Rafael Amer,et al.  Extension of values to games withmultiple alternatives , 1998, Ann. Oper. Res..

[8]  D. Felsenthal,et al.  The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes , 1998 .

[9]  Dieter Schmidtchen,et al.  Jahrbuch für Neue Politische Ökonomie , 1982 .

[10]  Edward M. Bolger,et al.  A consistent value for games with n players and r alternatives , 2000, Int. J. Game Theory.

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

[12]  P. Fishburn.  The Theory Of Social Choice , 1973 .

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

[14]  Pradeep Dubey,et al.  Mathematical Properties of the Banzhaf Power Index , 1979, Math. Oper. Res..

[15]  T. E. S. Raghavan,et al.  Shapley Value for Multichoice Cooperative Games, I , 1993 .

[16]  Edward M. Bolger,et al.  A value for games withn players andr alternatives , 1993 .

[17]  Guillermo Owen,et al.  Power Indices and Coalition Formation , 2001 .

[18]  Edward M. Bolger.  Power indices for multicandidate voting games , 1986 .