Here is a fairly standard “minimax‐in‐infinite‐dimensions’’ argument which shows that nothing special about amenability is needed – as soon as you allow arbitrary finitely‐additive probabilities on your countable group, the multiplication‐game always has a value and a saddle‐point (hence a Nash equilibrium). In particular one does not get a new characterization of amenability by asking “solvable for every W ⊆ G.”

1. Let G=F₂ be the free group, and let
   M := {μ a finitely additive probability on G}⊂ ba(G)
   where ba(G)=ℓ^∞(G)* is the Banach‐space dual of ℓ^∞(G). By Banach–Alaoglu the unit‐ball of ba(G) is weak*–compact, and M is a weak*–closed face of it, hence itself weak*–compact and convex.

2. Define the payoff function
   F : M×M→[0,1], 
   F(μ,ν) := ∫_x ∫_y 1_W(xy),dν(y),dμ(x).
   Here for fixed ν the inner map x↦∫_y1_W(xy)dν(y) is a bounded function on G (hence in ℓ^∞(G)), so the outer integral against μ is perfectly well defined; likewise in the other order.

3. By construction F is bilinear in (μ,ν) and in particular
   • for each fixed ν, μ↦F(μ,ν) is affine (hence both quasiconcave and quasiconvex) and weak*–continuous,
   • for each fixed μ, ν↦F(μ,ν) is affine and weak*–continuous.

4. Now apply the Ky‐Fan–Sion minimax theorem. Since M is convex and compact (in the weak*–topology) and F is quasiconcave–quasiconvex and separately continuous, one concludes

   sup_{μ∈M} inf_{ν∈M} F(μ,ν) = inf_{ν∈M} sup_{μ∈M} F(μ,ν).

5. Moreover one checks easily that
   g(μ)=inf_νF(μ,ν)
   is upper‐semicontinuous on the compact M (being the infimum of a family of continuous functions), so sup_μg(μ) is actually attained. Likewise
   h(ν)=sup_μF(μ,ν)
   is lower‐semicontinuous and its infimum is attained. Denote by μ* a maximizer of g and by ν* a minimizer of h. One then shows in the usual way that

   ∀ μ   F(μ,ν*) ≤ F(μ*,ν*) ≤ F(μ*,ν).

In other words (μ*,ν*) is a saddle‐point, i.e. a Nash equilibrium of the zero–sum multiplication game.

In particular nothing in the above used any special “invariance’’ or “amenability’’ of G. As long as your strategy‐spaces are the full set of finitely‐additive probability measures on G, they form compact convex sets in the weak*–topology and the payoff is bilinear‐continuous, so one gets existence of equilibrium for every W⊆G.

— References for this style of argument:
• M. Sion, “On general minimax theorems,” Pacific J. Math. 8 (1958) 171–176.
• Ky Fan’s original fixed‐point/minimax papers.