Continuity of the payoff functions

Summary. In his Nash equilibrium paper, Glicksberg states that the payoff functions are continuous. Such a function is defined on the product of mixed strategies, which are the Borel probability measures on a compactum, endowed with the product of the weak topologies. The continuity property is used in proving the existence of Nash equilibria. This note proves that the payoff functions are continuous, which is not immediate to establish.

[1]  Gerard Debreu,et al.  A Social Equilibrium Existence Theorem* , 1952, Proceedings of the National Academy of Sciences.

[2]  M. Ali Khan,et al.  Pure strategies in games with private information , 1995 .

[3]  J. Nash.  Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.