Yes – if the players are allowed to mix with finitely additive probability measures ( “charges’’ on the power set of F₂ ) the game always has a saddle point, hence a Nash equilibrium, no matter what subset
W ⊂ F₂ is chosen. In particular the phenomenon has nothing to do with amenability; it holds for every group and for every infinite two–person zero–sum matrix game once finitely additive mixed strategies are admitted.

Why it is always solvable

1. Strategy spaces.
   Let M := ba₁(F₂) b...

Throughout let
 S­­₍F₎ = { x ∈ ℝ : 0 = a₁x^{e₁}+⋯+aₙx^{eₙ} for some n≥1 with pair-wise different exponents eᵢ∈F and non–zero coefficients aᵢ∈F }.
Put A₀ = ℚ and A_{n+1}=S(A_{n}).
The question is whether the inclusions A₀⊊A₁⊊A₂⊊… stay proper for every finite n, whether every A_{n} is a field, and whether numbers such as e ever lie in one of the A_{n}.

1. The first two steps
    A₁ is exactly the field of algebraic real numbers ℚ̄.

 Proof. If the exponents that occur are rational, wri...

Write “Flex-term’’ for a (closed) expression that you can build out of

• the variable n, • decimal numerals 0,1,2,…, • the operations + , − , × , ÷ (ordinary rational division), • the unary symbol ⌊·⌋, and • the binary symbol ^ (exponentiation),

using the usual brackets to specify the order of evaluation.
The value of a Flex-term τ at n∈ℕ is the natural number that you obtain by carrying out the ordinary real‐number operations and taking the floor when a ⌊·⌋ occurs. The class 𝔽lex is t...

A straightforward way to obtain the value of

    1

I(m)= ───── ∬{-π}^{π} ∬{-π}^{π} 4π m cosx cosy – cosx – cosy ------------------------------------------------ dx dy (1) [ sin²x+sin²y+( m – cosx – cosy )² ]^{3/2}

is to look at it as an ordinary flux–integral and to keep track of the places where the denominator can vanish.
Everything can be done with nothing more than multivariable calculus, the divergence theorem and the elementary...