A non-parametric approach to testing the axioms of the Shapley value with limited data

The unique properties of the Shapley value–efficiency, equal treatment of identical input factors, and marginality–have made it an appealing solution concept in various classes of problems. It is however recognized that the pay schemes utilized in many real-life situations generally depart from this value. We propose a nonparametric approach to testing the empirical content of this concept with limited datasets. We introduce the Shapley distance, which, for a fixed monotone transferable-utility game, measures the distance of an arbitrary pay profile to the Shapley pay profile, and show that it is additively decomposable into the violations of the classical Shapley axioms. The analysis has several applications. In particular, it can be used to assess the extent to which an income distribution or a cost allocation can be considered fair or unfair, and whether any particular case of unfairness is due to the violation of one or a combination of the Shapley axioms.

[1]  Josep Freixas,et al.  On ordinal equivalence of the Shapley and Banzhaf values for cooperative games , 2010, Int. J. Game Theory.

[2]  H. Varian.  Non-parametric Tests of Consumer Behaviour , 1983 .

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

[4]  Gerard van der Laan,et al.  An Efficient and Fair Solution for Communication Graph Games , 2011 .

[5]  L. Shapley.  Simple games: an outline of the descriptive theory. , 2007, Behavioral science.

[6]  Zéphirin Nganmeni,et al.  The Shapley–Shubik power index for dichotomous multi-type games , 2016 .

[7]  M. Shubik.  Incentives, Decentralized Control, the Assignment of Joint Costs and Internal Pricing , 1962 .

[8]  Bertrand Tchantcho,et al.  Voters' power in voting games with abstention: Influence relation and ordinal equivalence of power theories , 2008, Games Econ. Behav..

[9]  Guillermo Owen,et al.  Centrality and power in social networks: a game theoretic approach , 2003, Math. Soc. Sci..

[10]  Josep Freixas,et al.  Anonymous yes-no voting with abstention and multiple levels of approval , 2009, Games Econ. Behav..

[11]  Marcin Malawski.  “Procedural” values for cooperative games , 2013, Int. J. Game Theory.

[12]  Victor H. Aguiar,et al.  Slutsky matrix norms: The size, classification, and comparative statics of bounded rationality , 2017, J. Econ. Theory.

[13]  R. Nagel,et al.  The agencies method for coalition formation in experimental games , 2012, Proceedings of the National Academy of Sciences.

[14]  Kareen Rozen,et al.  Fairness Through the Lens of Cooperative Game Theory: An Experimental Approach , 2013, SSRN Electronic Journal.

[15]  John F. Nash,et al.  Some Experimental n-Person Games , 1952 .

[16]  Sylvain Béal,et al.  Efficient extensions of the Myerson value , 2015, Soc. Choice Welf..

[17]  Federico Valenciano,et al.  The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector , 1996 .

[18]  Robert J. Weber,et al.  Probabilistic Values for Games , 1977 .

[19]  L. Shapley,et al.  The Shapley Value , 1994 .

[20]  Roland Pongou,et al.  Valuing Inputs Under Supply Uncertainty: The Bayesian Shapley Value , 2016, Games Econ. Behav..

[21]  Martin Shubik,et al.  Ownership and the Production Function , 1967 .

[22]  Bezalel Peleg,et al.  Linear measures of inequality for cooperative games , 1991 .

[23]  Gary E. Bolton,et al.  How Communication Links Influence Coalition Bargaining: A Laboratory Investigation , 2003, Manag. Sci..

[24]  S. N. Afriat,et al.  On a system of inequalities in demand analysis , 1973 .

[25]  Bertrand Tchantcho,et al.  Properties of ladder tournaments , 2017, Eur. J. Oper. Res..

[26]  Roland Pongou,et al.  A policy-based rationalization of collective rules: Dimensionality, specialized houses, and decentralized authority , 2014 .

[27]  B. Peleg.  Game Theoretic Analysis of Voting in Committees , 1984 .

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

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

[30]  Josep Freixas,et al.  The Shapley-Shubik power index for games with several levels of approval in the input and output , 2005, Decis. Support Syst..

[31]  A. Khmelnitskaya.  Marginalist and efficient values for TU games , 1999 .

[32]  M. Yaari.  Rawls, edgeworth, shapley, nash: Theories of distributive justice re-examined , 1981 .

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

[34]  C. Chameni Nembua,et al.  Ordinal equivalence of values, Pigou-Dalton transfers and inequality in TU-games , 2016, Games Econ. Behav..

[35]  Miklós Pintér,et al.  Young’s axiomatization of the Shapley value: a new proof , 2008, Ann. Oper. Res..

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

[37]  James Konow,et al.  Which Is the Fairest One of All? A Positive Analysis of Justice Theories , 2003 .

[38]  Annick Laruelle,et al.  Voting and Collective Decision-Making - Bargaining and Power , 2008 .