Elsevier

Operations Research Letters

Volume 45, Issue 6, November 2017, Pages 616-619
Operations Research Letters

Weakly balanced contributions and solutions for cooperative games

https://doi.org/10.1016/j.orl.2017.09.008Get rights and content

Abstract

We explore a relaxation of the balanced contributions property for solutions for TU games that requires the direction (sign) of one player’s change of payoffs when another player leaves the game to equal the direction (sign) of the latter player’s change of payoffs when the former leaves the game. There exists a large class of solutions that satisfy both efficiency and this weak balanced contributions property. The Shapley value is the unique solution that also obeys weak differential marginality.

Introduction

The Shapley value [13] probably is the most eminent one-point solution concept for cooperative games with transferable utility (TU games). Besides its original axiomatic foundation by Shapley himself, alternative foundations of different types have been suggested later on. Important direct axiomatic characterizations are due to [10] and [16]. [8] suggests an indirect characterization as the marginal contributions of a potential (function). [12] shows that the Shapley value can be understood as a von Neumann–Morgenstern utility. As a contribution to the Nash program, which aims at building bridges between cooperative and non-cooperative game theory, [11] implements the Shapley value as the outcome of the subgame perfect equilibria of a combined bidding and proposing mechanism, which is modeled by a non-cooperative extensive form game.

Among the direct characterizations, the one by  [10] stands out by invoking only two properties, efficiency and the balanced contributions property. Efficiency says that the worth generated by the grand coalition is distributed without gains or losses among the players. The balanced contributions property requires that the amount one player gains or loses when another player leaves the game equals the amount the latter player gains or loses when the former player leaves the game.

We suggest a relaxation of the balanced contributions property called the weak balanced contributions property that relaxes the former in the same vein as weak differential marginality [6] relaxes differential marginality [4]. This property requires that the direction (sign) of the change of one player’s payoff when another player leaves the game equals the direction (sign) of the change of the latter player’s payoff when the former player leaves the game. It turns out that there exists a huge class of solutions that satisfy efficiency and the weak balanced contributions property, among them the subclass of (simply) weighted Shapley values [13].

As our main result, we show that the Shapley value is the unique solution that satisfies efficiency, the weak balanced contributions property, and weak differential marginality (Theorem 2). This result partly rests on the fact that efficiency and the weak balanced contributions property together imply the standard dummy player property and the dummy player out property [14]: whenever a dummy player leaves a game, then the other players’ payoffs remain unaffected (Lemma 1).

The remainder of this paper is organized as follows. In Section 2, we give basic definitions and notation. In Section 3, we present our discussion and results. Some remarks conclude this paper.

Section snippets

Basic definitions and notation

Let U be a countably infinite set, the universe of players, and let N denote the set of all finite subsets of U. A (finite TU) game on the player set NN is given by a coalition function v:2NR, v=0, where 2Ndenotes the power set of N. Subsets of N are called coalitions; vS is called the worth of coalition S. The set of all games on N is denoted by VN.

For NN, TN, and vVN, the coalition function v|TVT is given by v|TS=vS for all ST. For NN, v,wVN and αR, the coalition

Weakly balanced contributions

[10] suggests the balanced contributions property for solutions and shows that the Shapley value is the unique solution that satisfies efficiency and the balanced contributions property.

Balanced contributions, BC. For all NN, vVN, and i,jN, we have φivφiv|Nj=φjvφjv|Ni.This property requires player j’s contribution to the payoff of player i, φivφiv|Nj, to equal player i’s contribution to the payoff of player j, φjvφjv|Ni. In this sense, the players’

Concluding remarks

We suggest the weak balanced contributions property as a considerable relaxation of the balanced contributions property. When combined with efficiency, then its implications are much weaker than for the original balanced contribution property. This distinguishes it from weak differential marginality, which essentially keeps the strength of differential marginality when combined with efficiency and the null player property. The reason for this seems to be that the (weak) balanced contributions

Acknowledgments

We are grateful to participants of the 4th Workshop on Cooperative Game Theory in Business Practice at HHL Leipzig Graduate School of Management for comments on this paper. Financial support by the Deutsche Forschungsgemeinschaft (grant CA 266/4-1) is gratefully acknowledged.

References (16)

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