Let us denote $G$ set of all games (of some particular type), and then consider $N = \{ g \in G | \mbox{game $g$ admits (mixed) Nash equilibrium} \}$. I want to ask you for some good source describing the properties of set $N$ (is it convex, closed, what geometry does it have etc). Or rather, under wich additional conditions is $N$ convex, etc.
Particularly, let us consider the case of $G$ being the set of (possibly infinite) two-player zero-sum games that are symmetric, and $N$ being subset of those games admitting mixed Nash equilibrium. What are the properties of $N$ then?
Reacting to the comment of Yanko, by a convex combination of two games $G_1 = (S_i, u_i)_{i \in I}$ and $G_2 = (S_i, v_i)_{i \in I}$ having the same set of pure strategies I mean simply a game $H = (S_i, \lambda u_i + (1-\lambda) v_i )_{i \in I}$, i.e. a game with convex combination of their pay-off functions.
Thank you, cheers, Mirek