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In Peleg (1969) it is shown that a game with finite strategy spaces and continuous utilities has a mixed Nash equilibrium for any cardinality of players.
Is the same true if the strategy spaces are compact (maybe under some extra conditions, like only countably many players)?
Lones Smith offers the following proof: if you have countably many players, arbitrarily enumerate them and call the original game with this enumeration $G$. Let $G_n$ be a game where the actions of players $k>n$ are fixed arbitrarily. By Glicksburg, $G_n$ has an equilibrium -- call it $x(n)$. Let $x$ be a limit point of the sequence $(x(n))_{n=1}^\infty$. Fix any player $i$. Now $x_i(n)$ is a best response to $x_{-i}(n)$ in $G_n$ for each $n$; to pass to limits, we appeal to the Theorem of the Maximum and continuity of the utility functions [1]. That shows the action $x_i$ is a best response to $x_{-i}$ in $G$. So $x$ is an equilibrium of $G$.
[1] This assumes a topology on the action space at least as coarse as the product topology, so that whenever $x(n)$ converges to $x$ in the product topology, it converges in the topology in question. The utilities should be continuous under this topology for actions. This is a big assumption and easily violated!
