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On The Computation Of The Nucleolus Of A Cooperative Game
, 1999
"... We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly ..."
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We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel.
Complexity in cooperative game theory
"... Abstract. We introduce cooperative games (N,v) with a description polynomial in n, where n is the number of players. In order to study the complexity of cooperative game problems, we assume that a cooperative game v:2N → Q isgivenbyanoraclereturning v (S) for each query S ⊆ N. Finally, we consider s ..."
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Cited by 11 (0 self)
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Abstract. We introduce cooperative games (N,v) with a description polynomial in n, where n is the number of players. In order to study the complexity of cooperative game problems, we assume that a cooperative game v:2N → Q isgivenbyanoraclereturning v (S) for each query S ⊆ N. Finally, we consider several cooperative game problems and we give a list of complexity results. Key words. Computational complexity, cooperative game AMS subject classifications. 91A12 1
An Efficient Algorithm for Nucleolus and Prekernel Computation in some Classes of TUGames
, 1998
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Submodularity of some classes of the combinatorial optimization games
 Mathematical Methods of Operations Research
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Computing Shapley Value in Supermodular Coalitional Games
"... Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair ..."
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Coalitional games allow subsets (coalitions) of players to cooperate to receive a collective payoff. This payoff is then distributed “fairly” among the members of that coalition according to some division scheme. Various solution concepts have been proposed as reasonable schemes for generating fair allocations. The Shapley value is one classic solution concept: player i’s share is precisely equal to i’s expected marginal contribution if the players join the coalition one at a time, in a uniformly random order. In this paper, we consider the class of supermodular games (sometimes called convex games), define and survey computational results on other standard solution concepts, and contrast these results with new results regarding the Shapley value. In particular, we give a fully polynomialtime randomized approximation scheme (FPRAS) to compute the Shapley value to within a (1 ± ε) factor in monotone supermodular games. We show that this result is tight in several senses: no deterministic algorithm can approximate Shapley value as well, no randomized algorithm can do better, and both monotonicity and supermodularity are required for the existence of an efficient (1 ± ε)approximation algorithm. We also argue that, relative to supermodularity, monotonicity is a mild assumption, and we discuss how to transform supermodular games to be monotonic.
Multilevel revenue sharing for viral marketing
 In Proceedings of ACM NetEcon 2011
, 2011
"... In this paper we present the design and analysis of revenue sharing schemes for viral marketing over social networks. The increasing need for monetizing social networks more effectively is causing social network platforms to look for alternatives to online behavioral targeting. Specifically, we turn ..."
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Cited by 2 (0 self)
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In this paper we present the design and analysis of revenue sharing schemes for viral marketing over social networks. The increasing need for monetizing social networks more effectively is causing social network platforms to look for alternatives to online behavioral targeting. Specifically, we turn to cooperative game theory and the Shapley value to design revenue sharing schemes to incentivize users to help the social network platform for more effective viral marketing. Our goal is to identify mechanisms that achieve desirable objectives in terms of computability, individual rationality, and potential reach. In particular, we propose multilevel revenue sharing for referralbased and viral marketing over online social networks. We show via simulations that users have more incentive to collaborate with the social network platform in implementing the campaign when the revenue or discount is shared across multiple levels rather than the commonly used singlelevel model. For this purpose, we design the graphbased model, for which we show that computing the Shapley value is #Phard. However, we show that in a variation of that model, which we call the treebased model, computing the Shapley value becomes polynomial time. We also show that the revenue function is supermodular only in the treebased model. Supermodularity of the revenue function entails desirable corollaries. 1.
Traveling salesman games with the Monge property
, 2001
"... Several works indicate the relationship between wellsolved combinatorial optimization problems and the core nonemptiness of cooperative games associated with them. In this note, we consider the core of traveling salesman games. We show that the core of traveling salesman games in which the dist ..."
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Several works indicate the relationship between wellsolved combinatorial optimization problems and the core nonemptiness of cooperative games associated with them. In this note, we consider the core of traveling salesman games. We show that the core of traveling salesman games in which the distance matrix is a Monge matrix is nonempty. This is the first result for traveling salesman games related with a wellsolved structure. Moreover, we show that the problem of testing the core nonemptiness of a given traveling salesman game is NPhard.
Fair Cost Allocations under Conflicts  A Gametheoretic Point of View
, 2003
"... Optimization theory resolves problems to minimize the total costs when the agents are involved in some conflicts. In this paper, we consider how to allocate the minimized total cost among the agents. To do that, the allocation is required to be fair in a certain sense. We use a gametheoretic point ..."
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Optimization theory resolves problems to minimize the total costs when the agents are involved in some conflicts. In this paper, we consider how to allocate the minimized total cost among the agents. To do that, the allocation is required to be fair in a certain sense. We use a gametheoretic point of view, and provide algorithms to compute fair allocations in polynomial time for a certain conflict situation. More specifically, we study a minimum coloring game, introduced by Deng, Ibaraki & Nagamochi [X. Deng, T. Ibaraki and H. Nagamochi: Algorithmic aspects of the core of combinatorial optimization games. Math. Oper. Res. 24 (1999) 751766], and investigate the core, the nucleolus, the #  value, and the Shapley value. In particular, we provide the following four results. (1) The characterization of the core for a perfect graph in terms of its extreme points. This leads to polynomialtime algorithms to compute a vector in the core, and to determine whether a given vector belongs to the core. (2) A polynomialtime algorithm to compute the #value for a perfect graph. (3) A characterization of the nucleolus for some classes of the graphs, including the complete multipartite graphs and the chordal graphs. This leads to a polynomialtime algorithm to compute the nucleolus for these classes of graphs. (4) A polynomialtime algorithm to compute the Shapley value for a forest. The investigation of this paper gives several insights to the relationship of algorithm theory with cooperative games.
Submodularity of MinimumCost Spanning Tree
"... We give a necessary condition and a sufficient condition for a minimumcost spanning tree game introduced by Bird to be submodular (or convex). When the cost is restricted to two values, we give a characterization of submodular minimumcost spanning tree games. We also discuss algorithmic issues. ..."
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We give a necessary condition and a sufficient condition for a minimumcost spanning tree game introduced by Bird to be submodular (or convex). When the cost is restricted to two values, we give a characterization of submodular minimumcost spanning tree games. We also discuss algorithmic issues.