Allocation of Cooperative Payoff in a Cooperative Game

Abstract

In a cooperative game, there may be a series of coalitions which may lead to the coalition equilibrium of the game. There is a mixed coalition equilibrium under the principle of maximum allocation in a cooperative game. If the allocation process of the coalitions can satisfy the competitive allocation condition, that's to say, the allocation that a player gets from his coalition is more than his escape-payoff but less than his marginal contribution to his coalition, the coalition equilibrium under the principle of maximum allocation equals to the coalition equilibrium under the principle of minimum escape-payoff. The allocation process of the cooperative payoff of a coalition is actually the bargaining process of the coalition members, that is, the bargaining game for the allocation of cooperative payoff surplus. The bargaining game based on mutual threats among the allied members of a coalition for the allocation of cooperative payoff surplus is an uncooperative game, in which every coalition member sets up his threat-point, if his request on the cooperative payoff surplus is accepted by other bargaining rivals, he will join in the cooperative coalition, or he will implement his threat to withdraw from the coalition. The cooperative payoff surplus of a coalition can be divided into the common payoffs of all its subsets. The Nash equilibrium in the allied bargaining game about the allocation of the common payoff of a member set (a subset of member set of the coalition) is that each member in the member set will get an equal share of the common payoff. This allocation scheme decides the allocation rule of cooperative payoff surplus of a coalition. In an allied bargaining game about the allocation of cooperative payoff surplus of a coalition, members would be allied and set up a series of cooperative teams. In the equilibrium situation of the allied bargaining game, the allocation that a certain team gets from the common payoff of a team set (a subset of the team set that includes all the cooperative teams formed in the equilibrium situation in the allied bargaining game) is just the same as that of another team. The aggregate allocation of cooperative payoff of the coalition that a certain team gets is the sum of its allocations of common payoff that it gets from all the team sets which it belongs to.