A minimax problem has two opposed variables: one player minimizes while the other maximizes. A saddle point is stronger than equality of the two optimal values because it supplies a pair of strategies that certify each other.
Lower and upper values
Definition. For \(L:X\times Y\to\overline{\mathbb R}\), define \[ \underline v=\sup_{y\in Y}\inf_{x\in X}L(x,y), \qquad \overline v=\inf_{x\in X}\sup_{y\in Y}L(x,y). \] A pair \((x^\star,y^\star)\) is a saddle point if \[ L(x^\star,y)\le L(x^\star,y^\star)\le L(x,y^\star) \qquad(x\in X,y\in Y). \]
Weak minimax inequality. For every function \(L\), \[ \underline v\le\overline v. \] If a saddle point exists, equality holds and \(L(x^\star,y^\star)=\underline v=\overline v\).
Proof. For every \(x,y\), \[ \inf_{u\in X}L(u,y)\le L(x,y)\le\sup_{v\in Y}L(x,v). \] Take the supremum over \(y\) on the left and the infimum over \(x\) on the right. If the saddle inequalities hold, then \[ \sup_y\inf_xL(x,y)\ge\inf_xL(x,y^\star) =L(x^\star,y^\star) =\sup_yL(x^\star,y)\ge\inf_x\sup_yL(x,y), \] so weak minimax forces equality throughout.
The converse requires attainment. Equality of the values alone need not produce a saddle pair when one of the infimum or supremum is not achieved.
Convex-concave minimax
Sion's minimax theorem. Let \(X\) be a compact convex subset of a linear topological space and \(Y\) a convex subset of another. Suppose \(L(\cdot,y)\) is lower semicontinuous and convex on \(X\) for every \(y\), and \(L(x,\cdot)\) is upper semicontinuous and concave on \(Y\) for every \(x\). Then \[ \sup_{y\in Y}\inf_{x\in X}L(x,y) = \inf_{x\in X}\sup_{y\in Y}L(x,y). \]
Proof outline. Fix \(\alpha<\inf_x\sup_yL(x,y)\). For each \(y\), let \(U_y=\{x:L(x,y)>\alpha\}\). Concavity in \(y\) and convexity in \(x\) imply the finite-intersection property needed to show that finitely many such sets cover \(X\) only if a convex combination of their \(y\)-values produces a single \(y_\alpha\) with \(\inf_xL(x,y_\alpha)>\alpha\). Compactness supplies the finite subcover step, and semicontinuity supplies open level sets. Hence \(\sup_y\inf_xL(x,y)\ge\alpha\); let \(\alpha\) increase to the upper value and combine with weak minimax.
Lagrangian saddle points and strong duality
Saddle certificate for convex programs. For a convex program with Lagrangian \(\mathcal L(x,\lambda,\nu)\), a primal-dual feasible triple \((x^\star,\lambda^\star,\nu^\star)\) satisfies KKT if and only if \[ \mathcal L(x^\star,\lambda,\nu) \le\mathcal L(x^\star,\lambda^\star,\nu^\star) \le\mathcal L(x,\lambda^\star,\nu^\star) \] for every \(x\) and every dual-feasible \((\lambda,\nu)\), with the equality-variable convention handled by unrestricted \(\nu\). Consequently the primal and dual values agree.
Proof. Stationarity and convexity make \(x^\star\) a global minimizer of \(\mathcal L(\cdot,\lambda^\star,\nu^\star)\), proving the right saddle inequality. Primal feasibility and complementary slackness imply \[ \mathcal L(x^\star,\lambda,\nu) \le f(x^\star) =\mathcal L(x^\star,\lambda^\star,\nu^\star), \] proving the left inequality. Conversely, varying \(x\) gives stationarity or subgradient stationarity, while varying each nonnegative multiplier enforces feasibility and complementary slackness.
A finite zero-sum game
Example. For a payoff matrix \(A\in\mathbb R^{m\times n}\), mixed strategies \(p\in\Delta_m,q\in\Delta_n\) give payoff \(p^\top Aq\). The function is linear in each variable and both simplices are compact convex, so \[ \max_{q\in\Delta_n}\min_{p\in\Delta_m}p^\top Aq = \min_{p\in\Delta_m}\max_{q\in\Delta_n}p^\top Aq. \] Linear programming duality computes the same equality. Randomization convexifies the pure strategies and removes a possible minimax gap.
The same geometry appears in Bayes and minimax decision theory: a rule minimizes risk while a least favorable prior maximizes Bayes risk. Adversarial learning and distributionally robust optimization use analogous saddle formulations.