A Simple Expression for the Shapley Value in a Special Case

Littlechild, Stephen; Owen, Guillermo

doi:10.1287/mnsc.20.3.370

Cited by 405 publications

(207 citation statements)

References 2 publications

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“…In the literature, various classes of games that are fully determined by the worths of these special coalitions can be found. For example, in auction games (see Graham et al, 1990) and bankruptcy games (see O'Neill, 1982;Aumann and Maschler, 1985), the game is fully described by the worth of the 'grand coalition' N and the worths of all coalitions of size n − 1, while an airport game (see Littlechild and Owen, 1973) is fully described by the n worths of the singletons. Although the solutions studied in this article are especially useful for these classes of games, we support these solutions also in other applications of games, even if the game depends on the worths of more coalitions.…”

Section: Discussionmentioning

confidence: 99%

Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games

Brink

Funaki

2008

Theory Decis

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ABSTRACT. A situation, in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (point-valued) solution for TU-games assigns a payoff distribution to every TU-game. In this article we discuss a class of equal surplus sharing solutions consisting of all convex combinations of the CIS-value, the ENSC-value and the equal division solution. We provide several characterizations of this class of solutions on variable and fixed player set. Specifications of several properties characterize specific solutions in this class.

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Section: Discussionmentioning

confidence: 99%

Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games

Brink

Funaki

2008

Theory Decis

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“…In this paper we consider this issue in the context of cooperative games with transferable utility. Cooperative games are widely applied in economic allocation problems such as auction games (see Graham et al 1990), airport landing fee problems (see Littlechild and Owen 1973), water distribution problems (see Ambec and Sprumont 2002), polluted river problems (see Ni and Wang 2007), sequencing problems (see Curiel et al 1989) and queueing problems (see Maniquet 2003). For these games the trade-off between marginalism and egalitarianism can be seen as the trade-off between allocating according to the Shapley value or the equal division solution.…”

Section: Introductionmentioning

confidence: 99%

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

2011

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One of the main issues in economic allocation problems is the trade-off between marginalism and egalitarianism. In the context of cooperative games this trade-off can be framed as one of choosing to allocate according to the Shapley value or the equal division solution. In this paper we provide three different characterizations of egalitarian Shapley values being convex combinations of the Shapley value and the equal division solution. First, from the perspective of a variable player set, we show that all these solutions satisfy the same reduced game consistency. Second, on a fixed player set, we characterize this class of solutions using monotonicity properties. Finally, towards a strategic foundation, we provide a non-cooperative implementation for these solutions which only differ in the probability of breakdown at a certain stage of the game. These characterizations discover fundamental differences as well as intriguing connections between marginalism and egalitarianism. R. van den Brink (B)

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“…The Harsanyi constrained core is always nonempty and generalizes 1 Besides the examples discussed in Section 3, other examples are, e.g. auction games (see Graham et al (1990)), dual airport games (see Littlechild and Owen (1973)), telecommunication games (see van den Nouweland et al (1996)) and queueing games (see Maniquet (2003)). …”

Section: Introductionmentioning

confidence: 89%

“…In the DR polluted river game v the worth of a coalition S is equal to the sum of all costs from the most upstream player in S to the most downstream player n. In fact, v is the airport game corresponding to the airport situation with costs ∑ n j=i c j for player (airplane) i ∈ N , see Littlechild and Owen (1973). 13 In general an airport game, and so v, is not totally positive.…”

Section: Polluted River Gamesmentioning

confidence: 99%

Constrained core solutions for totally positive games with ordered players

2013

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In many applications of cooperative game theory to economic allocation problems, such as river games, polluted river games and sequencing games, the game is totally positive (i.e., all dividends are nonnegative), and there is some ordering on the set of the players. A totally positive game has a nonempty core. In this paper we introduce constrained core solutions for totally positive games with ordered players which assign to every such a game a subset of the core. These solutions are based on the distribution of dividends taking into account the hierarchical ordering of the players. The Harsanyi constrained core of a totally positive game with ordered players is a subset of the core of the game and contains the Shapley value. For special orderings it coincides with the core or the Shapley value. The selectope constrained core is defined for acyclic orderings and yields a subset of the Harsanyi constrained core. We provide a characterization for both solutions.

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A Simple Expression for the Shapley Value in a Special Case

Cited by 405 publications

References 2 publications

Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games

Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

Constrained core solutions for totally positive games with ordered players

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