Zero-Sum Games: A Comprehensive Guide

Zero-Sum Games are a cornerstone of Game Theory, modeling scenarios where one player’s gain is exactly balanced by another’s loss, resulting in a net payoff of zero. Pioneered by John von Neumann in 1928 and formalized with Oskar Morgenstern in their 1944 book "Theory of Games and Economic Behavior," these games capture the essence of pure competition.

From chess and poker to economic rivalries and military strategies, zero-sum games provide a framework for analyzing adversarial interactions. This MathMultiverse guide explores their definitions, mathematical frameworks, examples, strategies, and applications, enriched with equations and visualizations.

Mathematical Framework

Zero-Sum Games are defined by their payoff structure, where the sum of players’ payoffs is zero, and analyzed using tools like the minimax theorem.

Definition

For two players \( A \) and \( B \), with strategies \( s_A \in S_A \), \( s_B \in S_B \), and payoffs:

\[ u_A(s_A, s_B) + u_B(s_A, s_B) = 0 \]

Thus, \( u_B = -u_A \). The game is represented as:

\[ G = (N, S_A, S_B, u_A), \quad N = \{A, B\} \]

Payoff Matrix

For \( A \), with strategies \( s_{A_i} \), \( s_{B_j} \):

\[ M = [u_A(s_{A_i}, s_{B_j})] \]

Minimax Theorem

Von Neumann’s theorem guarantees a game value \( v \):

\[ v = \max_{s_A} \min_{s_B} u_A(s_A, s_B) = \min_{s_B} \max_{s_A} u_A(s_A, s_B) \]

Saddle Point

A strategy pair \( (s_A^*, s_B^*) \) is a saddle point if:

\[ u_A(s_A, s_B^*) \leq u_A(s_A^*, s_B^*) \leq u_A(s_A^*, s_B) \]

This is a Nash equilibrium with \( u_A(s_A^*, s_B^*) = v \).

Mixed Strategies

Players use probability distributions \( p_A \), \( p_B \). Expected payoff:

\[ E[u_A] = \sum_{i,j} p_A(i) p_B(j) u_A(s_{A_i}, s_{B_j}) \]

Equilibrium value:

\[ v = \max_{p_A} \min_{p_B} E[u_A] = \min_{p_B} \max_{p_A} E[u_A] \]

Rock-Paper-Scissors Payoffs

Bar chart showing expected payoffs for Player A in Rock-Paper-Scissors under mixed strategy equilibrium (\( p_A = p_B = (1/3, 1/3, 1/3) \)).

Examples

Classic zero-sum games illustrate strategic dynamics.

Rock-Paper-Scissors

Payoff for Player \( A \):

\[ \begin{array}{c|ccc} & R & P & S \\ \hline R & 0 & -1 & 1 \\ P & 1 & 0 & -1 \\ S & -1 & 1 & 0 \end{array} \]

Mixed equilibrium: \( p_A = p_B = (1/3, 1/3, 1/3) \), \( v = 0 \).

Matching Pennies

Payoff for \( A \):

\[ \begin{array}{c|cc} & H & T \\ \hline H & 1 & -1 \\ T & -1 & 1 \end{array} \]

Equilibrium: \( p_A = p_B = (1/2, 1/2) \), \( v = 0 \).

Chess (Simplified)

Win (+1), Loss (-1), Draw (0):

\[ v = 0 \text{ (perfect play)} \]

Poker (Simplified)

Bluff (B) vs. Call (C) or Fold (F):

\[ \begin{array}{c|cc} & C & F \\ \hline B & -2 & 3 \\ F & 0 & 0 \end{array} \]

Mixed strategy equilibrium computed via linear programming.

Resource Allocation

Two firms divide a market share \( x \):

\[ u_A = x, \quad u_B = 1 - x \]

Strategies

Players optimize outcomes using strategic tools.

Minimax Strategy

Player \( A \) maximizes their minimum payoff:

\[ s_A^* = \arg\max_{s_A} \min_{s_B} u_A(s_A, s_B) \]

Maximin Strategy

Player \( B \) minimizes \( A \)'s maximum payoff:

\[ s_B^* = \arg\min_{s_B} \max_{s_A} u_A(s_A, s_B) \]

Linear Programming

Solve for mixed strategies:

\[ \max v \] \[ \text{s.t. } \sum_i p_A(i) u_A(i, j) \geq v, \quad \sum_i p_A(i) = 1 \]

Graphical Solution

For 2x2 games, plot payoff lines to find the equilibrium intersection.

Applications

Zero-sum games model economic competition (market share battles), sports (tennis strategies), and military tactics (resource allocation).