Weakly balanced contributions and solutions for cooperative games
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 be a countably infinite set, the universe of players, and let denote the set of all finite subsets of . A (finite TU) game on the player set is given by a coalition function , where denotes the power set of . Subsets of are called coalitions; is called the worth of coalition . The set of all games on is denoted by
For , and , the coalition function is given by for all For , and , 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 , and , we have This property requires player ’s contribution to the payoff of player to equal player ’s contribution to the payoff of player 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.
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2022, Journal of Mathematical EconomicsCitation Excerpt :By Corollaries 9.10 and 9.12, and Theorem 9.2, we present our last corollary. André Casajus has made significant contributions to cooperative game theory in recent years by introducing qualitative versions (sign symmetry, weak balanced contributions (Casajus, 2017), and weak differential marginality (Casajus and Yokote, 2017)) of standard axioms such as symmetry, balanced contributions, and differential marginality (Casajus, 2011). This allowed Casajus (2017) to ‘qualitatively’ improve the axiomatization of the Shapley value in Myerson (1980), Casajus (2018) to improve the axiomatization in Young (1985) (see Theorem 8.1), and Casajus (2019) to improve the axiomatization in Shapley (1953b) (see Theorem 6.1).
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2021, Journal of Mathematical EconomicsCitation Excerpt :Instead, they satisfy the weak balanced contributions property: 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 (Casajus, 2017a). Casajus (2017a, Lemma 1) already establishes some joint implications of the weak balanced contributions property and efficiency. In particular, he shows that these properties imply the dummy player property and the dummy player out property.
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2018, Journal of Economic TheoryCitation Excerpt :An appendix contains the proof of our main result. In this section, we introduce superweak differential marginality and explore its relation to kindred properties, marginality (Young, 1985) and the weak balanced contributions property (Casajus, 2017). van den Brink (2001) shows that the Shapley value is characterized by efficiency, the null player property, and a fairness property.
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