Potential, Value, and Consistency

Hart, Sergiu; Mas-Colell, Andreu

doi:10.2307/1911054

Cited by 612 publications

(446 citation statements)

References 17 publications

(23 reference statements)

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“…While this method has been extended to more general spaces than pN A (see [21] and [26]), the extension of games to the ideal coalitions needs some kind of non-atomiqueness for the games. Here we obviate this inconvenient by adopting the "potential" point of view undertaken for finite games by Hart and Mas-Colell [12].…”

Section: Outlinementioning

confidence: 99%

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Refinement Derivatives and Values of Games

Montrucchio

Semeraro

2008

Mathematics of OR

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A definition of set-wise differentiability for set functions is given through refining the partitions of sets. Such a construction is closely related to the one proposed by Rosenmuller (1977) as well as that studied by Epstein (1999) and Epstein and Marinacci (2001). We present several classes of TU games which are differentiable and study differentiation rules. The last part of the paper applies refinement derivatives to the calculation of value of games. Following Hart and Mas-Colell (1989), we define a value operator through the derivative of the potential of the game. We show that this operator is a truly value when restricted to some appropriate spaces of games. We present two alternative spaces where this occurs: the spaces pM ∞ and P OT 2 . The latter space is closely related to Myerson's balanced contribution axiom.

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Section: Outlinementioning

confidence: 99%

“…In [12] the value of finite games is axiomatized by a potential function which assigns a real number to each game. Fitting their approach into our setting, this amounts to constructing a new game u, called the potential of ν, associated with the original game ν.…”

Section: Outlinementioning

confidence: 99%

Refinement Derivatives and Values of Games

Montrucchio

Semeraro

2008

Mathematics of OR

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“…Concerning the solution theory for cooperative TU-games, the paper is devoted to singlevalued solution concepts. [2], [6], [9], [11]) Let ψ be a solution on G.…”

Section: Consistency Property For Solutions That Admit a Potentialmentioning

confidence: 99%

“…[6], Theorem A, page 591) The Shapley value is the unique solution ψ on G that admits a potential and is efficient as well. Here the Shapley value Sh i (N, v) of player i ∈ N in the n-person game N, v is defined as follows (cf.…”

Section: Consistency Property For Solutions That Admit a Potentialmentioning

confidence: 99%

“…In their innovative paper, Hart and Mas-Colell (cf. [6]) showed that the well-known game-theoretic solution called Shapley value is the unique solution that has a potential representation and meets the efficiency principle as well. In the second stage (the nineties) of the potential research into the solution part of cooperative game theory, various researchers contributed different, but equivalent characterizations of (not necessarily efficient) solutions that admit a potential (cf.…”

Section: Introductionmentioning

confidence: 99%

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Consistency and Potentials in Cooperative TU-Games: Sobolev’s Reduced Game Revived

Driessen¹

Chapters in Game Theory

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It was a quarter of a century ago that Sobolev proved the reduced game (otherwise called consistency) property for the much-discussed Shapley value of cooperative TUgames. The purpose of this paper is to extend Sobolev's result in two ways. On the one hand the unified approach applies to the enlarged class consisting of game-theoretic solutions that possess a so-called potential representation; on the other Sobolev's reduced game is strongly adapted in order to establish the consistency property for solutions that admit a potential. Actually, Sobolev's explicit description of the reduced game is now replaced by a similar, but implicit definition of the modified reduced game; the characteristic function of which is implicitly determined by a bijective mapping on the universal game space (induced by the solution in question). The resulting consistency property solves an outstanding open problem for a wide class of game-theoretic solutions. As usual, the consistency together with some kind of standardness for two-person games fully characterize the solution. A detailed exposition of the developed theory is given in the event of dealing with so-called semivalues of cooperative TU-games and the Shapley and Banzhaf values in particular.

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The Shapley Value and Related Solution Concepts

Vlach¹

2011

Wiley Encyclopedia of Operations Research and Management Science

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The Shapley value not only belongs to the most important solution concepts of cooperative game theory but it is also an easily tractable mathematical object with a remarkably wide range of applications. After necessary preliminaries, we introduce basic properties and characterizations of the Shapley value for cooperative games with transferable utility. Then we deal with some of its extensions appearing frequently in the literature. In particular, we present the Aumann–Drèze and Owen generalizations to games with a coalition structure, and the Myerson extension to games with a graph structure. We conclude with the probabilistic values and a brief discussion of application to cost allocation problems.

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Potential, Value, and Consistency

Cited by 612 publications

References 17 publications

Refinement Derivatives and Values of Games

Refinement Derivatives and Values of Games

Consistency and Potentials in Cooperative TU-Games: Sobolev’s Reduced Game Revived

The Shapley Value and Related Solution Concepts

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