Weakly monotonic solutions for cooperative games

The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some player’s marginal contribution to all coalitions increase or stay the same, then this player’s payoff should not decrease. We investigate the class of values that satisfy effi ciency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity refl ects the nature of the egalitarian Shapley values in the same vein as strong monotonicity refl ects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payoffs as follows: First, the Shapley payoffs are taxed proportionally at a fi xed rate. Second, the total tax revenue is distributed equally among all players. Acknowledgement: We are grateful to Sylvain Béal, René van den Brink, Sylvain Ferrières, and Miklos Pintér for helpful comments on our paper. Financial support for Frank Huettner from the Deutsche Forschungsgemeinschaft (DFG) grant HU 2205/1-1 is gratefully acknowledged. Weakly monotonic solutions for cooperative games André Casajusa,b,c, , Frank Huettnera,b aChair of Economics and Information Systems, HHL Leipzig Graduate School of Management, Jahnallee 59, 04109 Leipzig, Germany bLSI Leipziger Spieltheoretisches Institut, Leipzig, Germany cInstitut für Theoretische Volkswirtschaftslehre, Universität Leipzig, Grimmaische Str. 12, 04109 Leipzig, Germany Abstract The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some player’s marginal contribution to all coalitions increase or stay the same, then this player’s payo¤ should not decrease. We investigate the class of values that satisfy e¢ ciency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity re‡ects the nature of the egalitarian Shapley values in the same vein as strong monotonicity re‡ects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payo¤s as follows: First, the Shapley payo¤s are taxed proportionally at a …xed rate. Second, the total tax revenue is distributed equally among all players. Keywords: Egalitarian Shapley values, redistribuition, solidarity, TU games, weak monotonicity 2010 MSC: 91A12 JEL: C71, D60The principle of weak monotonicity for cooperative games states that if a game changes so that the worth of the grand coalition and some player’s marginal contribution to all coalitions increase or stay the same, then this player’s payo¤ should not decrease. We investigate the class of values that satisfy e¢ ciency, symmetry, and weak monotonicity. It turns out that this class coincides with the class of egalitarian Shapley values. Thus, weak monotonicity re‡ects the nature of the egalitarian Shapley values in the same vein as strong monotonicity re‡ects the nature of the Shapley value. An egalitarian Shapley value redistributes the Shapley payo¤s as follows: First, the Shapley payo¤s are taxed proportionally at a …xed rate. Second, the total tax revenue is distributed equally among all players.