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Complexity in Cooperative Game Theory
"... 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 :2 N ! Q is given by an oracle returning v (S) for each query S N: Finally, we consider sever ..."
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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 :2 N ! Q is given by an oracle returning v (S) for each query S N: Finally, we consider several cooperative game problems and we give a list of complexity results.
OPERATIONS RESEARCH GAMES: A SURVEY
, 2001
"... This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved. Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also fac ..."
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This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved. Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players. This interplay between optimisation and allocation is the main subject of the area of operations research games. It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.
Monotonicity in Bargaining Networks (extended abstract)
"... We study bargaining networks, discussed in a recent paper of Kleinberg and Tardos [KT08], from the perspective of cooperative game theory. In particular we examine three solution concepts, the nucleolus, the core center and the core median. All solution concepts define unique solutions, so they prov ..."
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We study bargaining networks, discussed in a recent paper of Kleinberg and Tardos [KT08], from the perspective of cooperative game theory. In particular we examine three solution concepts, the nucleolus, the core center and the core median. All solution concepts define unique solutions, so they provide testable predictions. We define a new monotonicity property that is a natural axiom of any bargaining game solution, and we prove that all three of them satisfy this monotonicity property. This is actually in contrast to the conventional wisdom for general cooperative games that monotonicity and the core condition (which is a basic property that all three of them satisfy) are incompatible with each other. Our proofs are based on a primaldual argument (for the nucleolus) and on the FKG inequality (for the core center and the core median). We further observe some qualitative differences between the solution concepts. In particular, there are cases where a strict version of our monotonicity property is a natural axiom, but only the core center and the core median satisfy it. On the other hand, the nucleolus is easy to compute, whereas computing the core center or the core median is #Phard (yet it can be approximated in polynomial time).