Extensions of Nash equilibria to games with infinite strategies In the Jehle and Reny textbook (which I should add I have not read much of beyond a few sections of interest), a theorem stating that there is always a (mixed) Nash equilibrium in finite strategic form games is proven. The book assumes that all players have the same number of actions available, but it's not difficult to imagine how this might be extended to the case where this isn't true.

What I'm interested in, however, is whether there is some extension of this to games, particularly those where there may be infinite choices. For instance, there's clearly no equilibrium in a game where a player wins by picking the highest number, but if we have, for instance, the same game, but where the number must be within the interval $[0, 100]$ (or any interval that contains its upper bound), the best response functions "converge". Similarly, I would also suspect that there need to be "well-behaved" cost and demand functions in competition models to get "good" results.

As such, I have two questions:


Is there any sort of well-defined setting in which a game with infinite strategy choices will have a Nash equilibrium?
What would relevant reading for this be?
Yes, there is such a setting. The result is that


  If each player's strategy space is
  
  
  convex
  compact
  
  
  and if payoffs are continuous then there exists at least one Nash equilibrium (possibly in mixed strategies).


This holds even when the set of possible actions is uncountably infinite. If one additionally assumes that payoffs are quasiconcave then the best-response correspondence will be convex even when we restrict attention to pure strategies so that we are then guaranteed to have at least one equilibrium in pure strategies in such a game.

I believe the original reference here is 


I.L. Glicksberg. "A further generalization of the Kakutani fixed
point theorem, with application to Nash equilibrium points."
Proceedings of the American Mathematical Society, 3(1):170–174, 1952.


The treatment in Glicksberg's paper, though, does not  seem very accessible. A good starting reference is more likely to be section 1.3 of Fudenberg &amp; Tirole's book "Game Theory".