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# gameL10 - Cooperative Game Theory Cooperative games are...

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Cooperative Game Theory Cooperative games are often de f ned in terms of a charac- teristic function, which speci f es the outcomes that each coalition can achieve for itself. For some games, outcomes are speci f ed in terms of the total amount of dollars or utility that a coalition can di- vide. These are games with transferable utility. For other games, utility is non-transferable, so we cannot characterize what a coalition can achieve with a single number. For example, if a coalition of consumers can re- allocate their endowments between themselves, then the "utility possibilities frontier" in general is not linear.

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Aumann (and others) think of cooperative games as dif- fering from non-cooperative games only in that binding agreements are possible before the start of the game. The primative notions de f ning a cooperative game are the set of players, the action sets, and the payo f s. The value to a coalition is what it can achieve by coordinating their actions. For the market game where players decide how much of their endowments to trade, the players outside a coali- tion cannot a f ect the trading within a coalition, so the action sets and payo f s uniquely determine the character- istic function of the cooperative game. For a Cournot game, where the players are the f rms who choose quantities, what a coalition can achieve depends on what is assumed about the behavior of the players outside the coalition.
De f nition 257.1: A coalitional game (game in char- acteristic function form) with transferable payo f consists of —a f nite set of players, N —a function v that associates with every nonempty subset S of N a real number v ( S ) , the "worth" of the coalition S . The interpretation of v ( S ) is the amount of money or utility that the coalition can divide between its members.

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De f nition 287.1: A coalitional game <N ,v> with transferable payo f is cohesive if we have v ( N ) K X k =1 v ( S k ) for every partition { S 1 ,...,S K } of N . Note: cohesiveness is a special case of the stronger as- sumption, superadditivity, which requires v ( S T ) v ( S )+ v ( T ) for all disjoint coalitions S and T
Cohesiveness guarantees that the equilibrium coalition is the one that should form. The worth of other coalitions will in f uence how v ( N ) will be divided among the play- ers. De F nition (feasibility): Given a coalitional game <N,v> with transferable payo f , for any pro F le of real numbers ( x i ) i N and any coalition S ,let x ( S ) P i S x i .Then ( x i ) i S is an S -feasible payo f vector if x ( S )= v ( S ) . An N -feasible payo f vector is a feasible payo f pro F le. De f nition 258.2: The core of a coalitional game with transferable payo f <N,v> is the set of feasible payo f pro F les ( x i ) i N for which there is no coalition S and S - feasible payo f vector ( y i ) i S such that y i >x i holds for all i S .

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For games with transferable payo f , the core is the set of feasible payo
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gameL10 - Cooperative Game Theory Cooperative games are...

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