Characterization of Nash equilibria of large games

For a large game with traits, this paper characterizes its trait–action distributions in equilibrium by using an inequality under the following two sets of conditions: (i) both trait space and action space of the game contain at most countably many elements; (ii) the agent space is nowhere equivalent to the characteristic type space. Two examples are presented to show the failure of the characterization result when the conditions are not satisfied. Then, we also show the existence of the characterization inequality and the necessity of the two sets of conditions to the characterization results. Finally, the existence of Nash equilibria under the two sets of conditions comes naturally as a corollary.