Theorem Ville Theorem
theorem admitted

Ville Theorem

Let $I=J=[0,1]$ and let $f:I\times J\to\mathbb{R}$ be continuous. The mixed extension over Borel probability measures on $[0,1]$ has a value, and each player has an optimal strategy. Moreover, for every $\varepsilon>0$, each player has an $\varepsilon$-optimal strategy with finite support.

The mixed payoff is $$ f(\sigma,\tau)=\int_{[0,1]\times[0,1]} f(x,y)\,d\sigma(x)\,d\tau(y), $$ where $\sigma$ and $\tau$ are Borel probability measures on $[0,1]$.

Proof

One discretizes the square $[0,1]\times[0,1]$ by finer and finer grids, applies finite minimax to each matrix game, and then extracts weakly convergent subsequences of optimal probability measures. Uniform continuity transfers the finite-grid guarantees to the continuous game. The finite-support $\varepsilon$-optimal strategies come from the same grid approximations.

References

  • [MFoGT, Thm. 2.5.3] Laraki, Renault, and Sorin, Mathematical Foundations of Game Theory. Continuous zero-sum game on [0,1] has a value in mixed strategies.

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