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Minimax theorem

The minimax theorem is a cornerstone of , first proved by mathematician in 1928, stating that in any finite two-player represented by an m \times n payoff matrix A, there exist mixed strategies p for the row player (maximizer) and q for the column player (minimizer) such that \max_p \min_q p^T A q = \min_q \max_p p^T A q, defining a unique value of the game that each player can guarantee regardless of the opponent's actions. This result ensures the existence of an in mixed strategies for such games, resolving the strategic tension between players seeking to maximize and minimize the same payoff. Von Neumann introduced the theorem in his seminal paper "Zur Theorie der Gesellschaftsspiele" (On the Theory of Games of Strategy), published in Mathematische Annalen, where he formalized the concept of mixed strategies—probabilistic combinations of pure strategies—to address scenarios where pure strategies lead to suboptimal outcomes. His proof used a topological argument, demonstrating that the maximin value (the highest payoff the maximizer can force) equals the minimax value (the lowest payoff the minimizer can impose). This work marked the birth of as a mathematical discipline, influencing subsequent developments including the 1944 collaboration with in Theory of Games and Economic Behavior, which expanded its applications to economics. Beyond , the minimax theorem has profound implications in optimization, where it underpins the duality theory of : for a maximization problem, the minimization problem achieves the same optimal value under feasible conditions, enabling efficient algorithmic solutions like the simplex method. Generalizations, such as Maurice Sion's 1958 theorem, extend the result to infinite-dimensional settings for quasi-concave-quasi-convex functions over compact convex sets, broadening its utility in and . The theorem's principles also inform decision-making under uncertainty in fields like , where algorithms like search guide adversarial planning in games such as chess.

Background in game theory

Zero-sum games

A is a two-player in where the total payoff remains constant, typically zero, such that one player's gains exactly equal the other's losses across all possible outcomes. In such games, the interests of the players are strictly opposed, with no possibility of mutual benefit from the interaction. The structure is commonly represented by a payoff A = (a_{ij}), where the rows correspond to the pure strategies (actions) available to player 1 (the row player), the columns to those of player 2 (the column player), and each entry a_{ij} denotes the payoff received by player 1 when player 1 chooses row i and player 2 chooses column j; player 2 receives the negative payoff -a_{ij}. A classic example is the game of rock-paper-scissors, a simple with three actions per player. The payoff for player 1 is given by: \begin{array}{c|ccc} & \text{Rock} & \text{Paper} & \text{Scissors} \\ \hline \text{Rock} & 0 & -1 & 1 \\ \text{Paper} & 1 & 0 & -1 \\ \text{Scissors} & -1 & 1 & 0 \\ \end{array} Here, a tie yields 0, a win yields +1, and a loss yields -1, ensuring that the payoffs for both players sum to zero in every case (e.g., if player 1 plays rock and player 2 plays paper, player 1 gets -1 while player 2 gets +1). This illustrates the zero-sum property: no value is created or destroyed, only redistributed between opponents. Zero-sum games differ fundamentally from non-zero-sum games, where the total payoffs can exceed or fall short of zero, allowing for outcomes where both players gain or lose collectively. For instance, the is a non-zero-sum game in which two suspects interrogated separately can each receive severe penalties if both defect (total payoff low), lighter penalties if both cooperate (total higher), or mixed results otherwise, highlighting opportunities for mutual improvement absent in zero-sum settings. The constant-sum nature of zero-sum games emphasizes pure antagonism, as any advantage for one player directly diminishes the other's position. The term "zero-sum" was popularized by in his seminal 1928 paper "Zur Theorie der Gesellschaftsspiele," which formalized such games within the emerging field of . However, the underlying concept of interactions where one entity's gain mirrors another's loss predates this work, appearing in economic discussions as early as the mercantilist period (16th–18th centuries), when was often viewed as a fixed-pie contest favoring exports over imports to accumulate wealth at competitors' expense.

Mixed strategies

In zero-sum games, where one player's gains result only from the other's losses, players often employ mixed strategies to introduce and avoid predictable exploitation. A mixed strategy for the row player (player 1) is a p \in \mathbb{R}^m, where m is the number of pure strategies, satisfying p_i \geq 0 for all i and \sum_{i=1}^m p_i = 1, assigning probabilities to each pure strategy. Similarly, for the column player (player 2) with n pure strategies, a mixed strategy q \in \mathbb{R}^n satisfies q_j \geq 0 for all j and \sum_{j=1}^n q_j = 1. The expected payoff under mixed strategies p and q, given the payoff matrix A \in \mathbb{R}^{m \times n} where a_{ij} is the payoff to player 1 when choosing pure strategy i against j, is computed as the p^T A q = \sum_{i=1}^m \sum_{j=1}^n p_i a_{ij} q_j. This formulation, central to analyzing strategic uncertainty, allows players to evaluate outcomes probabilistically rather than deterministically. A classic illustration is the game of -paper-scissors, represented by the symmetric zero-sum payoff matrix A = \begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{pmatrix}, where rows and columns correspond to , , and , respectively. If both players adopt the uniform mixed strategy p = q = (1/3, 1/3, 1/3)^T, the expected payoff is p^T A q = 0, ensuring neither can gain an advantage through deviation. Mixed strategies enable the concept of security levels, quantifying a player's guaranteed outcome against an adversarial opponent. For player 1, the maximin security level is \max_p \min_q p^T A q, the highest payoff achievable regardless of player 2's response. For player 2, aiming to minimize player 1's payoff, the minimax security level is \min_q \max_p p^T A q, the lowest payoff they can force. These strategies are essential in games lacking pure strategy equilibria, such as rock-paper-scissors or , where committing to a single action allows exploitation by the opponent. By randomizing over pure strategies, a player renders their behavior unpredictable, securing the maximin value and preventing systematic losses.

's minimax theorem

Statement for finite games

introduced the minimax theorem in his 1928 paper "Zur Theorie der Gesellschaftsspiele," published in Mathematische Annalen, where he established the existence of optimal strategies in finite two-player zero-sum games through a proof establishing the existence of saddle points for quasiconcave-quasiconvex payoff functions over strategy spaces. The theorem applies to finite zero-sum games, where two players, say Player I (the maximizer) and Player II (the minimizer), have finite action sets of sizes m and n, respectively, and the game is represented by an m \times n payoff A with real-valued entries a_{ij}, such that the payoff to Player I from actions i and j is a_{ij} and to Player II is -a_{ij}. Mixed strategies allow each player to randomize over their actions: Player I chooses a p \in \Delta^m (the m-dimensional ), and Player II chooses q \in \Delta^n. The expected payoff is then p^T A q. Von Neumann's minimax theorem states that there exist optimal mixed strategies p^* \in \Delta^m and q^* \in \Delta^n, and a value v \in \mathbb{R}, such that \max_{p \in \Delta^m} \min_{q \in \Delta^n} p^T A q = \min_{q \in \Delta^n} \max_{p \in \Delta^m} p^T A q = p^{*T} A q^* = v. This equality implies the existence of a in mixed strategies: for any p \in \Delta^m and q \in \Delta^n, p^T A q^* \leq v \leq p^{*T} A q, so p^* and q^* form a Nash equilibrium where neither player benefits from unilateral deviation. A classic example is the game, where Player I wins 1 if both choose heads (H) or both tails (T), and loses 1 otherwise; Player II wins the opposite. The payoff matrix for Player I is A = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}. The value is v = 0, achieved by the uniform mixed strategies p^* = q^* = \left( \frac{1}{2}, \frac{1}{2} \right), ensuring each player's expected payoff is zero regardless of the opponent's choice.

Proof overview

Von Neumann's original proof of the minimax theorem, published in , used an analytic argument establishing existence, from which a can be derived; a simpler version employing was provided by in 1937 by considering the compact convex sets of mixed strategies for both players, thereby establishing the existence of a strategy pair where neither player can unilaterally improve their payoff, confirming the equality of maximin and minimax values. In 1938, Jean Ville offered a more algebraic proof that reframed the theorem using the geometry of simplices, invoking separating hyperplanes to demonstrate that if no exists—meaning no yields a strictly positive payoff against all opposing —then the maximin equals the minimax. Following George Dantzig's 1947 development of , a modern proof leverages LP duality, originally highlighted by : the row player's problem is formulated as maximizing a value v subject to p^T A \geq v \mathbf{1} for mixed p in the probability (where A is the payoff matrix and \mathbf{1} the all-ones vector), while the column player's dual minimizes v subject to A q \leq v \mathbf{1} for q in its ; theorem ensures the optimal values match, proving the minimax equality. A key insight unifying these approaches is the compactness of the finite-dimensional probability simplices for mixed strategies combined with the bilinear of the expected payoff, ensuring the infima and suprema are achieved as minima and maxima. These proofs hinge on the finiteness of action sets in the game, requiring further topological conditions like or convexity for generalizations beyond finite cases.

Generalizations to functions

Concave-convex functions

The minimax theorem extends beyond finite spaces to continuous defined on compact sets, particularly those that are in one variable and in the other. Consider a f: X \times Y \to \mathbb{R}, where X and Y are compact subsets of topological spaces. The f is assumed to be in its first argument x \in X for each fixed y \in Y and in its second argument y \in Y for each fixed x \in X, with ensuring well-behaved extrema. Under these conditions, the minimax theorem states that \sup_{x \in X} \inf_{y \in Y} f(x, y) = \inf_{y \in Y} \sup_{x \in X} f(x, y). Moreover, due to the of X and Y and the of f, the supremum over x and the infimum over y are attained, yielding values, respectively. This equality, combined with the concavity-convexity, establishes the existence of a (x^*, y^*) where f(x, y^*) \leq f(x^*, y^*) \leq f(x^*, y) for all x \in X, y \in Y, such that neither can unilaterally improve their outcome. In the context of zero-sum games, this formulation bridges finite matrix games to infinite settings by interpreting f(x, y) as the expected payoff to the maximizing player, with x and y representing mixed strategies over continuous or action spaces. A key example is the bilinear payoff f(x, y) = x^\top A y, where A is a payoff extended to infinite dimensions, and x, y lie in compact sets such as probability simplices; here, ensures concavity in x and convexity in y, directly invoking the duality. For a concrete illustration of applications in continuous games, the theorem applies to the continuous Colonel Blotto game, where strategies are distributions over an interval [0, 1] rather than discrete points; the payoff is determined by integrals over these strategy spaces, and the theorem guarantees that the value of the game equals the interchange of suprema and infima, attained at optimal mixed strategies. This generalization traces its roots to extensions of John 's 1928 finite-game theorem, building on the bilinear case discussed in von Neumann and Morgenstern's 1944 Theory of Games and Economic Behavior, with early general formulations for concave-convex functions appearing in optimization and literature during the early 1950s, such as Glicksberg (1952) and (1953), formalizing saddle-point equilibria in broader models.

Sion's theorem

Sion's minimax theorem, formulated by Maurice Sion in 1958, generalizes von Neumann's minimax theorem by relaxing the requirements on the payoff from strict concavity-convexity to quasi-concavity-quasi-convexity, thereby extending applicability to a broader class of and spaces. The theorem states that for a real-valued f: X \times Y \to \mathbb{R}, where X is a compact of a and Y is a of another , if f is quasi-concave in x for each fixed y \in Y and quasi-convex in y for each fixed x \in X, with appropriate conditions (upper semi-continuous in x and lower semi-continuous in y), then \min_{y \in Y} \max_{x \in X} f(x, y) = \max_{x \in X} \min_{y \in Y} f(x, y). Quasi-concavity of f in x means that for each fixed y \in Y and c, the upper level set \{x \in X : f(x, y) \geq c\} is , while quasi-convexity in y implies that for each fixed x \in X and c, the lower level set \{y \in Y : f(x, y) \leq c\} is . The key conditions include the compactness of X, which ensures the maxima over X are attained, and the of both X and Y, which supports the quasi-properties; the ensures the equality of the minimax values holds under these topological assumptions, though the minimum over Y may not be attained if Y is not compact. Published in the Pacific Journal of Mathematics, Sion's work unifies and extends earlier results, such as Ky Fan's 1953 minimax theorem for quasi-concave-convex functions on compact sets, by employing a approach based on the Knaster-Kuratowski-Mazurkiewicz theorem rather than fixed-point or separating methods. Unlike the stricter concave-convex case, which requires the function to be in one variable and in the other—implying full in many game-theoretic contexts—Sion's weaker assumptions accommodate non-smooth functions where level sets remain , broadening applications to infinite-dimensional settings. A representative application arises in zero-sum games with continuous action spaces, where the payoff function may not be bilinear but satisfies quasi-concavity in one player's strategies and quasi-convexity in the other's, allowing the theorem to guarantee the existence of a value for the game under compactness of one strategy set.

Applications

In optimization

The minimax theorem establishes a profound connection to () duality, where every LP problem admits a primal maximization form \max \{ c^T x \mid Ax \leq b, x \geq 0 \} and a minimization form \min \{ b^T y \mid A^T y \geq c, y \geq 0 \}. Under appropriate conditions, the theorem interprets the as an adversarial game: the player maximizes payoff subject to constraints, while the player minimizes by choosing "worst-case" constraint multipliers, ensuring where the maximum equals the minimum. This game-theoretic lens provides a proof of via the equality for finite zero-sum games, framing constraints as strategic choices. Historically, George Dantzig's , introduced in 1947, implicitly leverages this minimax framework to solve for game values in by reformulating them as LPs. Dantzig demonstrated the equivalence between solving two-person and LPs, allowing the to compute optimal mixed strategies and values efficiently. For instance, in a 2x2 with payoff A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}, the row player's optimal value v solves the LP \max v subject to p_1 a_{11} + p_2 a_{21} \geq v, p_1 a_{12} + p_2 a_{22} \geq v, p_1 + p_2 = 1, p \geq 0, where p = (p_1, p_2) is the mixed strategy. In robust optimization, problems are often formulated as \min_x \max_{u \in \mathcal{U}} f(x, u), where \mathcal{U} is an uncertainty set and u represents adversarial perturbations. guarantees the equality \min_x \max_{u \in \mathcal{U}} f(x, u) = \max_{u \in \mathcal{U}} \min_x f(x, u) under quasi-convex-quasi-concave conditions on f, enabling tractable reformulations and duality-based solutions for robust counterparts of LPs or convex programs. This application ensures computational guarantees for worst-case performance in uncertain environments, such as design or . Extensions of the minimax theorem to arise through saddle-point formulations, where a function L(x, y) is minimized over x and maximized over y, yielding \min_x \max_y L(x, y) = \max_y \min_x L(x, y) at saddle points under convexity-concavity. Penalty methods further incorporate this by constructing exact minimax penalties, such as \min_x \max_y \{ f(x) + \rho \|c(x) - y\|^2 \}, where \rho > 0 enforces constraints c(x) = 0 without altering the original for sufficiently large \rho. These approaches generalize duality to nonconvex settings, facilitating numerical solutions via alternating optimization or extragradient methods.

In decision theory and AI

In , the criterion addresses under by selecting the action that maximizes the minimum possible payoff, providing a robust strategy for risk-averse agents facing adversarial or unknown outcomes. This approach, formalized by as the maximin rule in statistical decision functions, prioritizes worst-case scenarios to minimize potential losses. Von Neumann's theorem extends this criterion to mixed strategies in zero-sum games, guaranteeing the existence of values and optimal randomized policies that align with expected maximization under complete . In , the minimax theorem underpins analysis in markets modeled as zero-sum interactions, influencing the Arrow-Debreu general equilibrium model by supplying foundational tools for proving existence through duality and fixed-point theorems analogous to game-theoretic . Debreu explicitly referenced von Neumann's minimax framework alongside activity analysis in developing the model's formal structure, enabling rigorous treatment of competitive pricing and . In , the algorithm operationalizes the theorem for adversarial search in two-player zero-sum games, recursively computing optimal moves by maximizing one's own score while assuming the opponent minimizes it. Pioneered in by Claude Shannon's 1950 analysis, which modeled chess as a vast search tree with an estimated game-tree complexity of $10^{120} (the ), the algorithm gained efficiency through alpha-beta pruning, which eliminates irrelevant branches to reduce evaluation time from exponential to near-square-root levels. This enabled early systems to compete in tactical domains like chess from the 1950s onward. The minimax principle also informs , particularly in value iteration for zero-sum Markov games, where agents learn policies approximating Nash equilibria via minimax updates to Q-values, balancing and in multi-agent environments. In modern applications, systems like (2016) integrate (MCTS) as a scalable approximation to full minimax, using neural networks to guide simulations and evaluate positions in high-branching-factor games like Go, achieving superhuman performance without exhaustive tree expansion. Another prominent application is in generative adversarial networks (GANs), introduced in 2014, where a generator and discriminator engage in a minimax game to improve generative modeling, with the objective \min_G \max_D V(D, G). Despite these advances, minimax's computational complexity—scaling exponentially with depth due to high branching factors in real games—renders exact solutions infeasible for large domains, necessitating approximations such as MCTS to focus simulations on promising paths and mitigate the "curse of dimensionality."