On the characterization of Nash equilibrium action distributions of large distributional games

In a large distributional game, we show that an action distribution is a Nash equilibrium action distribution if and only if for any subset of actions, the proportion of players playing actions within the subset is no more than the proportion of players having their best response in it. We also show that given an action set, a symmetric Nash equilibrium action distribution in every atomless large distributional game with the given action set can be characterized if and only if the action set is at most countable.