An Algorithm for Payoff Space in C1-Games

Often in Game Theory the study of a normal-form game consists principally in the determination of the Nash equilibria in mixed strategies and in the analysis of their various stability properties (see for instance [17], [18] and [15]). Others (see for instance the books of J. P. Aubin [2] and [3]) feel the need to know the entire set of possibilities (consequences) of the players’actions, what we call the payoff space of the game; and moreover they introduce other form of non-cooperative solutions such as the pairs of conservative strategies. Nevertheless, only recently D. Carfì proposed a method to determine analytically the topological boundary of the payoff space and consequently to handle more consciously and precisely the entire payoff space. This method gives a complete and global view of the game, since, for instance, it allows to know the positions of the payoff profiles corresponding to the Nash equilibria in the payoff space of the game or the position of the conservative n-value of the game. The knowledge of these positions requires, indeed, the knowledge of the entire payoff space. Moreover, the knowledge of the entire payoff space becomes indispensable when the problem to solve in the game is a bargaining one: in fact, the determination of a bargaining solution (or of compromise solutions) needs the analytical determination of the Pareto boundaries or at least of the topological one. In the cited paper D. Carfì presented a general method to find an explicit expression of the topological boundary of the payoff-space of a Game and this latter boundary contains the two Pareto boundaries of the game.