Let \(X, Y \subseteq \mathbb {R}\) and let \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\). Suppose that \(X\) and \(Y\) are nonempty, that \(X\) is compact and convex, that \(Y\) is convex, that the map \(x \mapsto f(x,y)\) is continuous and convex on \(X\) for every \(y \in Y\), that the map \(y \mapsto f(x,y)\) is concave on \(Y\) for every \(x \in X\), and that \(f\) is bounded above and below on \(X \times Y\). Suppose moreover that for every nonempty finite subset \(u \subseteq X\),

and that for every \(\varepsilon \gt 0\) there exists a nonempty finite subset \(u \subseteq X\) with

Then

Let \(X, Y \subseteq \mathbb {R}\) be nonempty, with \(X\) compact and convex and \(Y\) convex, and let \(f : \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) be bounded above and below on \(X \times Y\), with \(f(\cdot , y)\) continuous and convex on \(X\) for every \(y \in Y\) and \(f(x, \cdot )\) concave on \(Y\) for every \(x \in X\). Suppose in addition that \(f\) is uniformly equicontinuous in its first argument, uniformly over the columns: for every \(\varepsilon \gt 0\) there is \(\rho \gt 0\) with \(|f(x,y) - f(x',y)| \lt \varepsilon \) whenever \(x, x' \in X\), \(y \in Y\), and \(\operatorname{dist}(x, x') \lt \rho \). Then for every \(\varepsilon \gt 0\) there exists a nonempty finite sample \(u \subseteq X\) such that

that is, restricting the row player to a suitable finite subset of \(X\) raises the lower value of the game by at most \(\varepsilon \).

Let \(X, Y \subseteq \mathbb {R}\) and \(f : \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) satisfy the convex-compact minimax hypotheses, and suppose in addition that \(Y\) is compact and that \((x,y) \mapsto f(x,y)\) is continuous on \(X \times Y\). Then \(f\) is uniformly equicontinuous in its first argument, uniformly over all columns: for every \(\varepsilon \gt 0\) there exists \(\rho \gt 0\) such that for all \(x, x' \in X\) and all \(y \in Y\),

Let \(X, Y \subseteq \mathbb {R}\) and \(f : \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) satisfy the convex-compact minimax hypotheses, and suppose in addition that \(Y\) is compact and that \(f\) is jointly continuous on \(X \times Y\). Then for every \(\varepsilon \gt 0\) there is a nonempty finite subset \(u \subseteq X\) such that

that is, restricting the row player to a suitable finite sample raises the lower value by at most \(\varepsilon \).

Let \(X, Y \subseteq \mathbb {R}\) satisfy the convex-compact minimax hypotheses for a payoff \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\), and suppose in addition that \(f\) is normalized, \(0 \le f(x,y) \le 1\) for all \(x \in X\) and \(y \in Y\). Fix an integer \(M \ge 2\) and a finite row sample \(x : \mathrm{Fin}\, M \to X\), writing \(x_i\) for its \(i\)-th point. Then

Let \(X, Y \subseteq \mathbb {R}\) and \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\) satisfy the convex-compact minimax hypotheses, and suppose the payoff is normalized so that \(0 \le f(x,y) \le 1\) for all \(x \in X\) and \(y \in Y\). Then for every nonempty finite subset \(u \subseteq X\),

that is, the game’s upper value is bounded above by the lower value of the game in which the row player is restricted to the finite sample \(u\) while the column player still ranges over all of \(Y\).

Let \(X, Y \subseteq \mathbb {R}\) be nonempty, compact, convex sets, and let \(f : \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) be jointly continuous on \(X \times Y\) and normalized so that \(0 \le f(x,y) \le 1\) for all \(x \in X\) and \(y \in Y\). Suppose that \(f\) is bounded above and below on \(X \times Y\), that for every \(y \in Y\) the map \(x \mapsto f(x,y)\) is continuous and convex on \(X\), and that for every \(x \in X\) the map \(y \mapsto f(x,y)\) is concave on \(Y\). Then

Let \(X, Y \subseteq \mathbb {R}\) and \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\) satisfy the convex-compact minimax hypotheses, and suppose \(f\) is normalized so that \(0 \le f(x,y) \le 1\) for all \(x \in X\) and \(y \in Y\). Fix an integer \(M \ge 2\), a row sample \(x : \mathrm{Fin}\, M \to X\), a real \(\varepsilon \gt 0\), and a finite column sample \(v \subseteq Y\). Write

for the indexed sampled lower value. Then the set

is nonempty; that is, there is a row \(\bar{x} \in X\) with \(f(\bar{x}, y) \le K + \varepsilon \) for every \(y \in v\). This is the finite \(\varepsilon \)-net step in the proof of Cesa-Bianchi–Lugosi Theorem 7.1.