My question concerns continuous games $((S_i)_{1 \leq i \leq n}, (u_i)_{1 \leq i \leq n})$, where $S_i$ are (continuous) compact strategy sets and $u_i$ denotes the utility function of agent $i$.
From Glicksberg's theorem, we know that if the utility functions $u_i$ are continuous in the agents' strategies, then there exists a (mixed) Nash equilibrium. However, my question is rather about the general existence of $\varepsilon$-Nash equlibria.
Is it true that any continuous game with compact strategy sets posses an $\varepsilon$-Nash equilibrium for all $\varepsilon > 0$?
Note that no continuity of the utilities $u_i$ has been assumed here.
Uh, unless I am misreading something, the game referenced in the Glicksberg's theorem wikipedia page does not have an $\epsilon$-equilibrium.
Quoting the game here for completeness:
"Here is the game: Players I and II choose numbers x and y respectively, between 0 and 1. The payoff to player I is $$K(x,y)= \begin{cases} -1 & \text{if } x<y<x+1/2, \\ 0 & \text{if } x=y \text{ or } y=x+1/2,\\ 1 & \text{otherwise.} \end{cases}$$"
and also
"There is no epsilon equilibrium for sufficiently small $\varepsilon$." (as the minmax is 3/7 and maxmin is 1/3.)