Monotonicity of the core and value in dynamic cooperative games

Rosenthal, Edward C.

doi:10.1007/bf01753707

Cited by 24 publications

(7 citation statements)

References 25 publications

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“…Intuitively, cross-monotonicity requires that the price charged to any individual in a group does not increase as the group expands. There is a large body of literature [Dutta 1990;Dutta and Ray 1989;Hokari 2000;Jain and Vazirani 2002;Moulin 1999;Rosenthal 1990;Shapley 1953;Sprumont 1990] on cross-monotonic cost-sharing schemes for submodular cost functions, 1 a subclass of cost functions of particular interest. Many mechanisms exist, prominent among them the Shapley value [Shapley 1953], which minimizes the worst-case efficiency loss, and the Dutta-Ray solution [Dutta and Ray 1989].…”

Section: Introductionmentioning

confidence: 99%

Limitations of cross-monotonic cost-sharing schemes

Immorlica

Mahdian

Mirrokni³

2008

ACM Trans. Algorithms

136

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A cost-sharing scheme is a set of rules defining how to share the cost of a service (often computed by solving a combinatorial optimization problem) amongs serviced customers. A cost-sharing scheme is cross-monotonic if it satisfies the property that everyone is better off when the set of people who receive the service expands. In this article, we develop a novel technique for proving upper bounds on the budget-balance factor of cross-monotonic cost-sharing schemes or the worst-case ratio of recovered cost to total cost. We apply this technique to games defined, based on several combinatorial optimization problems, including the problems of edge cover, vertex cover, set cover, and metric facility location and, in each case, derive tight or nearly-tight bounds. In particular, we show that for the facility location game, there is no cross-monotonic cost-sharing scheme that recovers more than a third of the total cost. This result, together with a recent 1/3-budget-balanced cross-monotonic cost-sharing scheme of Pál and Tardos [2003] closes the gap for the facility location game. For the vertex cover and set cover games, we show that no cross-monotonic cost-sharing scheme can recover more than a O(n −1/3 ) and O( 1 n ) fraction of the total cost, respectively. Finally, we study the implications of our results on the existence of group-strategyproof mechanisms. We show that every groupstrategyproof mechanism corresponds to a cost-sharing scheme that satisfies a condition weaker than cross-monotonicity. Using this, we prove that group-strategyproof mechanisms satisfying additional properties give rise to cross-monotonic cost-sharing schemes and therefore our upper bounds hold.

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Section: Introductionmentioning

confidence: 99%

Limitations of cross-monotonic cost-sharing schemes

Immorlica

Mahdian

Mirrokni³

2008

ACM Trans. Algorithms

136

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“…Proof: Rosenthal (1990) shows that the Shapley value on convex games satisfies populationmonotonicity. In other words, addition of new players expands opportunities of all players, and all players in the new game are better off.…”

Section: Shapley Value Allocationsmentioning

confidence: 99%

Stable Group Purchasing Organizations

2010

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“…It does not hold on the class of either convex or concave games since the Shapley value 4 (Shapley 1953) satisfies all the axioms on these domains (Sprumont 1990;Rosenthal 1990 (1, 3) = 12, and v (1, 2, 3) = 20. It is easy to check that v and v are super-additive games, ϕ Sh (v) = (6, 4), and ϕ Sh (v ) = (5.5, 6, 8.5).…”

Section: Proof Of Claim 3 Suppose By Way Of Contradiction That For Allmentioning

confidence: 99%

Population solidarity, population fair-ranking, and the egalitarian value

Chun

Park

2011

Int J Game Theory

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We investigate the implications of two axioms specifying how a value should respond to changes in the set of players for TU games. Population solidarity requires that the arrival of new players should affect all the original players in the same direction: all gain together or all lose together. On the other hand, population fair-ranking requires that the arrival of new players should not affect the relative positions of the original players. As a result, we obtain characterizations of the egalitarian value, which assigns to each player an equal share over an individual utility level. It is the only value satisfying either one of the two axioms together with efficiency, symmetry, and strategic equivalence.

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Monotonicity of the core and value in dynamic cooperative games

Cited by 24 publications

References 25 publications

Limitations of cross-monotonic cost-sharing schemes

Limitations of cross-monotonic cost-sharing schemes

Stable Group Purchasing Organizations

Population solidarity, population fair-ranking, and the egalitarian value

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