Characterizations of weighted and equal division values

New and recent axioms for cooperative games with transferable utilities are introduced. The non-negative player axiom requires to assign a non-negative payoff to a player that belongs to coalitions with non-negative worth only. The axiom of addition invariance on bi-partitions requires that the payoff vector recommended by a value should not be affected by an identical change in worth of both a coalition and the complementary coalition. The nullified solidarity axiom requires that if a player who becomes null weakly loses (gains) from such a change, then every other player should weakly lose (gain) too. We study the consequence of imposing some of these axioms in addition to some classical axioms. It turns out that the resulting values or set of values have all in common to split efficiently the worth achieved by the grand coalition according to an exogenously given weight vector. As a result, we also obtain new characterizations of the equal division value.

[1]  S. Hart,et al.  Potential, value, and consistency , 1989 .

[2]  S. Tijs,et al.  Game theory and cost allocation problems , 1986 .

[3]  Jean Lemaire,et al.  Cooperative Game Theory and its Insurance Applications , 1991 .

[4]  H. Young.  Monotonic solutions of cooperative games , 1985 .

[5]  André Casajus,et al.  Null players, solidarity, and the egalitarian Shapley values , 2013 .

[6]  André Casajus,et al.  Nullifying vs. dummifying players or nullified vs. dummified players: The difference between the equal division value and the equal surplus division value , 2012 .

[7]  B. Peleg,et al.  Introduction to the Theory of Cooperative Games , 1983 .

[8]  David Wettstein,et al.  Sharing the surplus: An extension of the Shapley value for environments with externalities , 2007, J. Econ. Theory.

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

[10]  René van den Brink,et al.  Null or nullifying players: The difference between the Shapley value and equal division solutions , 2007, J. Econ. Theory.

[11]  R L Eisenman,et al.  A profit-sharing interpretation of Shapley value for N-person games. , 1967, Behavioral science.

[12]  André Casajus,et al.  Null, nullifying, or dummifying players: The difference between the Shapley value, the equal division value, and the equal surplus division value , 2014 .

[13]  Solidarity within a fixed community , 2014 .

[14]  L. Shapley.  A Value for n-person Games , 1988 .

[15]  Toru Hokari,et al.  Consistency implies equal treatment in TU-games , 2005, Games Econ. Behav..

[16]  E. Kalai,et al.  On weighted Shapley values , 1983 .

[17]  William Thomson,et al.  Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: An update , 2015, Math. Soc. Sci..

[18]  H. Peyton Young,et al.  Fair Representation: Meeting the Ideal of One Man, One Vote , 1982 .

[19]  René van den Brink.  Efficiency and Collusion Neutrality of Solutions for Cooperative TU-Games , 2009 .

[20]  Sylvain Béal,et al.  Preserving or removing special players: What keeps your payoff unchanged in TU-games? , 2015, Math. Soc. Sci..

[21]  René van den Brink,et al.  An axiomatization of the Shapley value using a fairness property , 2002, Int. J. Game Theory.

[22]  Hans Haller,et al.  Null Players Out? Linear Values for Games with Variable Supports , 1999, IGTR.

[23]  Sylvain Béal,et al.  A decomposition of the space of TU-games using addition and transfer invariance , 2015, Discret. Appl. Math..

[24]  R. Evans.  Value, Consistency, and Random Coalition Formation , 1996 .

[25]  Yukihiko Funaki,et al.  Consistency, population solidarity, and egalitarian solutions for TU-games , 2012 .

[26]  Y. Chun.  The proportional solution for rights problems , 1988 .

[27]  William Thomson,et al.  A characterization of a family of rules for the adjudication of conflicting claims , 2013, Games Econ. Behav..

[28]  William Thomson,et al.  ON THE AXIOMATICS OF RESOURCE ALLOCATION: INTERPRETING THE CONSISTENCY PRINCIPLE , 2012, Economics and Philosophy.

[29]  Hans Haller,et al.  Collusion properties of values , 1994 .

[30]  Boram Park,et al.  Population solidarity, population fair-ranking, and the egalitarian value , 2012, Int. J. Game Theory.

[31]  P. Solal,et al.  Characterization of the Average Tree solution and its kernel , 2015 .

[32]  Sylvain Béal,et al.  Axioms of invariance for TU-games , 2015, Int. J. Game Theory.

[33]  H. Moulin.  Equal or proportional division of a surplus, and other methods , 1987 .