13nashb合作博弈.doc

Bargaining Introduction A Nash bargaining is a form of cooperative game, which analyses the choice of a particular point on the Pareto efficiency frontier (Harsanyi). Consider the case of two individuals: i = 1, 2. Let ui denote the utility level of the i-th individual ci denote the utility obtained by the i-th individual in case of conflict (ci is often called the threat point). Derivations Make the following assumptions: 1/ individual rationality: ui ? ci 2/ joint efficiency: Pareto optimality: h(u1, u2) = 0 as a representation of the Pareto efficiency frontier. 3/ symmetry: If the bargaining game is symmetric (i.e. if h(u1, u2) = 0 and (c1, c2) are symmetric with respect to the line u1 = u2), then the solution of the bargaining process is on the line u1 = u2. 4/ ui represents a cardinal utility function, i.e. a function defined up to a positive linear transformation. An example is the utility function when decision makers maximize expected utility under risk. 5/ independence of irrelevant alternatives: The bargaining solution is invariant to changing irrelevant restrictions on the feasible space. Proposition: Under assumptions 1-5, the bargaining game always has a unique solution (u1*, u2*) given by: Max(u1,u2) {(u1-c1)(u2-c2), subject to h(u1, u2) = 0, ui ? ci, i = 1, 2}. Implications: ?ui*/?ci ? 0, ?ui*/?cj ? 0 for i ? j. Note: These results generalize to n agents, where the solution of the Nash bargaining game is given by: Maxu {?i (ui-ci), subject to h(u1, ..., un) = 0, ui ? ci, i = 1, ..., n}. Zeuthens bargaining scheme Consider a bargaining process where, at each stage k of the process, player 1 has proposed an agreement A1k to player 2, while player 2 has proposed another agreement A2k to player 1. If they fail to agree, then a conflict situation C will develop and the players will receive the conflict payoffs ui(C) = ci, i = 1, 2. Assume that ui(C) ui(Ajk) ui(Aik) , i = 1, 2, i ? j. At stage k, the i-th player can: either accept his