The Shapley Value for Directed Graph Games

Khmelnitskaya, Anna Borisovna; Selçuk, Özer; Talman, Dolf

doi:10.2139/ssrn.2513257

Cited by 6 publications

(10 citation statements)

References 8 publications

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“…Remark 4.9. The values obtained in Example 4.8 are different from the Shapley values with precedence constraints in Faigle and Kern [10], Khmelnitskaya et al [18], which are ( 1 2 , 1 4 , 1 4 ). These take the approach of averaging over feasible permutations, a generalization of (2).…”

Section: Weighted Decompositions and Restricted Cooperationmentioning

confidence: 61%

“…The precedence constraints of Faigle and Kern [10] impose a partial ordering on N , so that some players are constrained to join the coalition prior to others. Khmelnitskaya et al [18] have recently generalized this to so-called digraph games, where precedence is determined by a digraph on N that (unlike the Faigle and Kern [10]) may contain cycles; a player i may be required to precede another player j in some coalitions but not others. (For another recent model of restricted cooperation, see Koshevoy et al [20].…”

Section: Weighted Decompositions and Restricted Cooperationmentioning

confidence: 99%

“…This may be seen as modeling variable willingness or unwillingness of players to join certain coalitions, as in some models of restricted cooperation. In the latter case, we compare the resulting solution concepts with other "Shapley values" for games with cooperation restrictions, such as the precedence constraints of Faigle and Kern [10] and the even more general digraph games of Khmelnitskaya et al [18].…”

Section: Introductionmentioning

confidence: 99%

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Hodge decomposition and the Shapley value of a cooperative game

Stern

Tettenhorst

2019

Games and Economic Behavior

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We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.

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Section: Weighted Decompositions and Restricted Cooperationmentioning

confidence: 61%

Section: Weighted Decompositions and Restricted Cooperationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hodge decomposition and the Shapley value of a cooperative game

Stern

Tettenhorst

2019

Games and Economic Behavior

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“…The Shapley value is also used to compute the bargaining power of players and has been intensively studied as part of cooperative game theory by authors such as McCain [2] and Vidal-Puga [3]. Other papers focused on different types of games where the Shapley value can be applied as is the case with graph games (for more details, see Khmelnitskaya et al [4]), bicooperative games (Bilbao et al [5]), or in diverse fields such as airport and irrigation games (Márkus et al [6]).…”

Section: Introductionmentioning

confidence: 99%

An Analysis of Countries’ Bargaining Power Derived from the Natural Gas Transportation System Using a Cooperative Game Theory Model

Roman

Stanculescu

2021

Energies

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A large consumption of natural gas accompanied by reduced production capabilities makes Europe heavily dependent on imports from Russia. More than half of Russian gas is exported by transiting Ukraine, so in the context of the underlying conflict between the two, this is considered uncertain. Therefore, in this article, we modeled the natural gas transportation system using cooperative game theory in order to determine the bargaining power of the major players (Russia, Ukraine, Germany, and Norway) by using a form of the Shapley value. We described the interaction between countries as network games where utilities from transport routes are considered and proposed three scenarios where the gas flow from Russia to Ukraine is either diminished or completely interrupted, with the purpose of finding out how the bargaining power on this market is shifted in case of network redesign. In this context, we included in the analysis the scenario where the Nord Stream 2 pipeline will be finished. Results showed that Russia dominates the market in any scenario, and by avoiding Ukraine, its position is even further strengthened. Moreover, Germany’s position remains stable considering its diverse imports and large storage capabilities, and its bargaining power increases in the case of diminishing or avoiding the Ukrainian gas pipelines.

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“…In particular, the combinatorial Hodge decomposition has recently been applied to game theory in various contexts, including noncooperative games (Candogan et al [1]), cooperative games (Stern and Tettenhorst [20]), and also other interesting problems in economics, e.g., ranking of social preferences (Jiang et al [7]). In addition, there have been efforts to model various cooperation restrictions, e.g., Faigle and Kern [5], Khmelnitskaya et al [9], and Koshevoy et al [11]. On the other hand, computational aspects of Shapley theory have been studied by Castro et al [2,3] and Deng and Papadimitriou [4].…”

Section: Introductionmentioning

confidence: 99%

A Hodge theoretic extension of Shapley axioms

Lim¹

2021

Preprint

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Lloyd S. Shapley [16,17] introduced a set of axioms in 1953, now called the Shapley axioms, and showed that the axioms characterize a natural allocation among the players who are in grand coalition of a cooperative game. Recently, Stern and Tettenhorst [20] showed that a cooperative game can be decomposed into a sum of component games, one for each player, whose value at the grand coalition coincides with the Shapley value. The component games are defined by the solutions to the naturally defined system of least squares linear equations via the framework of the Hodge decomposition on the hypercube graph.In this paper we propose a new set of axioms which characterizes the component games given by Stern and Tettenhorst, thereby suggesting that the component values for every coalition state may also serve for a valid measure of fair allocation among the players in each coalition. Our axioms may be seen as a completion of Shapley's in view of this characterization of the Hodge-theoretic component games. In addition, we provide a path integral representation of the component games which may be seen as an extension of the Shapley formula.

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The Shapley Value for Directed Graph Games

Cited by 6 publications

References 8 publications

Hodge decomposition and the Shapley value of a cooperative game

Hodge decomposition and the Shapley value of a cooperative game

An Analysis of Countries’ Bargaining Power Derived from the Natural Gas Transportation System Using a Cooperative Game Theory Model

A Hodge theoretic extension of Shapley axioms

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