# FORMING COALITIONS AND SHARING PAYOFFS IN n-PERSON NORMAL FORM GAMES

Degree granted by The University of Texas at Arlington

#### Abstract

For a given n-person normal form game, we form all possible sets of mutually exclusive and collectively exhaustive coalitions of the n players. For each set of coalitions, we define a coalitional semi-cooperative game as one in which these coalitions are taken as the players of this new game, each coalition tries to maximize the sum of its individual players’ payoffs, and the players within a coalition cooperate to do so. For any coalitional semi-cooperative game, the goal of the original n players is to improve their individual payoffs obtained in a Greedy Scalar Equilibrium (GSE) of the original game, where a GSE is an analog of the Nash equilibrium but always exists in pure strategies. We define a “best” such coalitional game as one that gives the n players their “best” possible payoffs among all possible coalitional semi-cooperative games. We present an algorithm for selecting such a “best” set of coalitions and present examples. Also, for a given n-person normal form game, we consider the situation where, for each strategy profile in the game, every player gives a pre-determined fraction of his payoff selfishly to himself and altruistically to the remaining n - 1 players. We show that the Nash equilibrium and Berge equilibrium are extreme cases of this situation.

*This paper has been withdrawn.*