Let \(G\) be a finite two-player zero-sum game with \(M\) row actions and \(N\) column actions, let \(p\) be a mixed row strategy over the \(M\) rows, and let \(j\) be a pure column action. Then the expected payoff of \(p\) against column \(j\) is non-negative:

Let \(G\) be a finite two-player zero-sum game whose payoff matrix \(A\) has every entry in \([0,1]\), let \(p\) be a mixed row strategy with weights \(p_i\), and let \(j\) be a fixed pure column action. Then the row player’s expected payoff against column \(j\) satisfies

Let \(G\) be a finite two-player zero-sum game with \(M\) row actions and \(N\) column actions, whose payoff matrix \(A\) has entries satisfying \(0 \le A_{ij} \le 1\). Fix a pure row action \(i\) and a mixed column strategy \(q\), so its weights \(q_j\) are non-negative and sum to \(1\). Then the expected payoff of row \(i\) against \(q\) is non-negative:

Let \(G\) be a finite two-player zero-sum game with \(M\) row actions and \(N\) column actions, whose payoff matrix \(A\) satisfies \(0 \le A_{ij} \le 1\) for all entries, and let \(q\) be a mixed column strategy, that is, a probability distribution \(q_1, \dots , q_N\) over the columns. Then for every pure row action \(i\), the expected payoff of row \(i\) against \(q\) satisfies

Let \(G\) be a finite two-player zero-sum game with \(M\) row actions and \(N \ge 1\) column actions, whose payoff matrix \((A_{ij})\) satisfies \(0 \le A_{ij} \le 1\) for all \(i,j\). For a mixed row strategy \(p = (p_i)\) over the \(M\) rows, let \(L(p) = \inf _{j} \sum _i p_i A_{ij}\) be the least expected payoff against a pure column, taken over all \(N\) columns. Then the set \(\{ \, L(p) : p \text{ is a mixed strategy over the rows} \, \} \) is bounded above.

Let \(G\) be a finite two-player zero-sum game with \(M \ge 1\) row actions and \(N\) column actions, whose payoff matrix \(A\) satisfies \(A_{ij} \in [0,1]\) for all \(i,j\). For a mixed column strategy \(q\) (a probability distribution on the \(N\) columns), let

be the best expected payoff that a single pure row can achieve against \(q\). Then the set \(\{ \, v(q) : q \text{ a mixed column strategy}\, \} \) is bounded below.

Let \(G\) be a finite two-player zero-sum game with \(M \geq 1\) row actions and \(N \geq 1\) column actions, given by a payoff matrix \(A_{ij} \in [0,1]\) that the row player seeks to maximise and the column player to minimise. Then the lower value never exceeds the upper value:

where \(\Delta _M\) and \(\Delta _N\) denote the sets of mixed strategies (probability distributions) over the row and column actions, respectively.

Let \(S \subseteq \mathbb {R}\) be a convex set, let \(p\) be a mixed strategy over \(n\) actions with weights \(p_1, \dots , p_n \ge 0\) satisfying \(\sum _{i=1}^{n} p_i = 1\), and let \(x_1, \dots , x_n \in S\). Then the weighted average \(\sum _{i=1}^{n} p_i\, x_i\) again lies in \(S\).

Let \(S \subseteq \mathbb {R}\) be a convex set and let \(g : \mathbb {R}\to \mathbb {R}\) be convex on \(S\). Let \(p\) be a mixed strategy over \(n\) actions, that is, a family of weights \(p_1, \dots , p_n \ge 0\) with \(\sum _{i=1}^{n} p_i = 1\), and let \(x_1, \dots , x_n\) be points of \(S\). Then

Let \(S \subseteq \mathbb {R}\) and let \(g : \mathbb {R}\to \mathbb {R}\) be concave on \(S\). Let \(p = (p_1, \dots , p_n)\) be a mixed strategy over \(n\) actions, that is, a tuple of weights \(p_j \ge 0\) with \(\sum _{j=1}^{n} p_j = 1\), and let \(y_1, \dots , y_n\) be points of \(S\). Then

Let \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\) take values in \([0,1]\), and let \(x : [M] \to \mathbb {R}\) and \(y : [N] \to \mathbb {R}\) be finite families of row and column points, giving rise to the sampled game whose \((i,j)\) entry is \(f(x_i, y_j)\). Fix a column index \(j\), and suppose the section \(x' \mapsto f(x', y_j)\) is convex on a set \(X \subseteq \mathbb {R}\) with \(x_i \in X\) for every \(i\). Then for any mixed row strategy \(p\) over the \(M\) rows,

where the right-hand side is the expected payoff of the sampled game against column \(j\) under \(p\).

Let \(f:\mathbb {R}\times \mathbb {R}\to \mathbb {R}\) be a payoff function taking values in \([0,1]\), and let \(x:[M]\to \mathbb {R}\) and \(y:[N]\to \mathbb {R}\) be finite families of row and column points. Fix a row index \(i\) and a mixed column strategy \(q\), whose weights \(q_j\ge 0\) sum to \(1\). Suppose \(Y\subseteq \mathbb {R}\) contains every \(y_j\) and the section \(y'\mapsto f(x_i,y')\) is concave on \(Y\). Then the expected payoff of pure row \(i\) against \(q\) in the sampled game is at most the value of \(f\) at the \(q\)-averaged column point:

Let \(X, Y \subseteq \mathbb {R}\) be nonempty convex sets with \(X\) also compact, and let \(f : X \times Y \to \mathbb {R}\) be a payoff that is bounded above and below on \(X \times Y\), such that for every \(y \in Y\) the map \(x \mapsto f(x,y)\) is continuous and convex on \(X\), and for every \(x \in X\) the map \(y \mapsto f(x,y)\) is concave on \(Y\). Then

Let \(X \subseteq \mathbb {R}\) be a set, let \(f \colon \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) be a payoff function, and let \(y, c \in \mathbb {R}\). Then the minimax sublevel set \(\mathrm{sublevel}(X,f,y,c) = X \cap \{ x \in \mathbb {R}: f(x,y) \leq c\} \) is contained in \(X\).

Let \(X \subseteq \mathbb {R}\) be a closed set, let \(f\colon \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) be a payoff function, and fix a column point \(y \in \mathbb {R}\) and a threshold \(c \in \mathbb {R}\). If the map \(x \mapsto f(x,y)\) is continuous on \(X\), then the minimax sublevel set \(\mathrm{sublevel}(X,f,y,c) = X \cap \{ x \in \mathbb {R}: f(x,y) \le c\} \) is closed.

Let \(X \subseteq \mathbb {R}\) be compact, fix a payoff function \(f : \mathbb {R}\times \mathbb {R}\to \mathbb {R}\), a column point \(y \in \mathbb {R}\), and a threshold \(c \in \mathbb {R}\), and suppose the map \(x \mapsto f(x,y)\) is continuous on \(X\). Then the minimax sublevel set \(\mathrm{sublevel}(X,f,y,c) = X \cap \{ x \in \mathbb {R}: f(x,y) \le c\} \) is compact.

Let \(X, Y \subseteq \mathbb {R}\) together with a payoff function \(f : \mathbb {R}\to \mathbb {R}\to \mathbb {R}\) satisfy the convex-compact minimax hypotheses, and let \(c \in \mathbb {R}\). Suppose that for every finite subset \(u \subseteq Y\) the set \(X \cap \bigcap _{y \in u} \mathrm{sublevel}(X,f,y,c)\) is nonempty, where \(\mathrm{sublevel}(X,f,y,c) = X \cap \{ x \in \mathbb {R}: f(x,y) \le c\} \); that is, there is a common row point \(x \in X\) with \(f(x,y) \le c\) for all \(y\) in that finite set. Then there exists a single \(x \in X\) with \(f(x,y) \le c\) for every \(y \in Y\).