Coalition formation and payoff distribution in majority games

In this paper we provide a cooperative solution as well as a non-cooperative analysis to study coalition formation and payoff distribution in weighted majority games. The cooperative solu- tion concept proposed here, the Anonymous Core, keeps the spirit of Core-like competition (reflecting individual rationality as well as group rationality), but it is non-empty and it is precisely characterized for every vector of weights. Agents of the same type must receive equal treatment within the winning coalitions, and some consistency is required on blocking coalitions, following a similar logic to that underlying the Von-Neumann and Morgenstern’s Stable Set. We show that in any weighted majority game every agent within the winning coalition is expected to obtain a payoff share proportional to her bargaining power. The latter is what defines the different types, and it is obtained endogenously. We introduce a non-cooperative coalitional bargaining game which resembles the rules of the game describing the formation of coalitional governments. We obtain an algorithm for the computation of all the Symmetric Markov Sub- game Perfect Equilibria of such game, and show that the set of such equilibria has a one-to-one correspondence with the Anonymous Core for homogeneous weighted majority games.