The Shapley value of coalitions to other coalitions

Kjell Hausken

doi:10.1057/s41599-020-00586-9

Abstract

The Shapley value for an n-person game is decomposed into a 2n × 2n value matrix giving the value of every coalition to every other coalition. The cell ϕIJ(v, N) in the symmetric matrix is positive, zero, or negative, dependent on whether row coalition I is beneficial, neutral, or unbeneficial to column coalition J. This enables viewing the values of coalitions from multiple perspectives. The n × 1 Shapley vector, replicated in the bottom row and right column of the 2n × 2n matrix, follows from summing the elements in all columns or all rows in the n × n player value matrix replicated in the upper left part of the 2n × 2n matrix. A proposition is developed, illustrated with an example, revealing desirable matrix properties, and applicable for weighted Shapley values. For example, the Shapley value of a coalition to another coalition equals the sum of the Shapley values of each player in the first coalition to each player in the second coalition.

Highlights

Lloyd Shapley (1923–2016) is perhaps best known for his socalled Shapley value (Shapley, 1953b), interpreted by Roth (1988b, p. 6) as “player i’s ‘fair share’ in the game.” Three other interpretations are a player’s expected marginal contribution, the weighted average of his marginal contributions to the coalition of all n players involved, and what player i can “reasonably” command to himself
Whereas Hausken and Mohr (2001) present the value of a player to another player, this article generalizes to determine the value of row coalition I to column coalition J in a 2n × 2n value matrix
Since an n-person game has 2n possible coalitions, including the null coalition {0} and the set N 1⁄4 f1; 2; 1⁄4 ; ng of all players, the Shapley value of row coalition I to column coalition J is exhaustively expressed by a 2n 2n matrix

Summary

Introduction

Lloyd Shapley (1923–2016) is perhaps best known for his socalled Shapley value (Shapley, 1953b), interpreted by Roth (1988b, p. 6) as “player i’s ‘fair share’ in the game.” Three other interpretations are a player’s expected marginal contribution, the weighted average of his marginal contributions to the coalition of all n players involved, and what player i can “reasonably” command to himself. The sum of the elements of any row or column in the n × n matrix equals the Shapley value of the respective player in an n-person game. Aside from Hausken and Mohr’s (2001) and Owen’s (1972) contributions, the authors are unaware of other work considering the value of a player or coalition to another player or coalition. Whereas Hausken and Mohr (2001) present the value of a player to another player, this article generalizes to determine the value of row coalition I to column coalition J in a 2n × 2n value matrix. Two non-overlapping coalitions in a game may find it useful to know their values to each other. If the value is negative, both coalitions may have an interest in excluding the other from the game, or ensuring that alternative coalitions are formed.

Literature review

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Conclusion