A minimax inequality and its application to the existence of Pareto equilibria for multi-objective games in FC-spaces

Minimax inequality plays a key role in the field of nonlinear analysis, for example, optimization problem, fixed point problem, non-cooperative game problem etc. In this paper, by using the continuous unity partition theorem and a famous fixed point theorem due to Górniewicz (1975), an important minimax inequality in FC-spaces is proved. On the basis of this inequality, following the method introduced by Nikaido and Isoda (1955), this paper defines an aggregate payoff function and furthermore determines some restrictions on the aggregate function that guarantee the existence of Pareto equilibrium for multi-objective games with infinite countable players. Our results generalize and improve the known Nash equilibrium existence results for multi-objective games with finite players in the literature.