Payoff Matrices: A Comprehensive Guide

Payoff matrices are powerful tools in game theory, visualizing the outcomes of strategic interactions between players. Introduced by John von Neumann and Oskar Morgenstern in 1944, they represent payoffs for each combination of strategies, enabling analysis of competitive and cooperative scenarios. This MathMultiverse guide explores their structure, construction, examples, and analytical techniques, including Nash equilibria, with equations and visualizations to enhance understanding.

From economics to biology, payoff matrices model decision-making in diverse fields, revealing optimal strategies and potential conflicts.

Structure of Payoff Matrices

Payoff matrices organize strategic outcomes in a tabular format.

Two-Player Game

For players with strategies \( S_1 = \{s_{11}, \ldots, s_{1m}\} \), \( S_2 = \{s_{21}, \ldots, s_{2n}\} \), the matrix is:

\[ M = \begin{bmatrix} (u_1(s_{11}, s_{21}), u_2(s_{11}, s_{21})) & \cdots & (u_1(s_{11}, s_{2n}), u_2(s_{11}, s_{2n})) \\ \vdots & \ddots & \vdots \\ (u_1(s_{1m}, s_{21}), u_2(s_{1m}, s_{21})) & \cdots & (u_1(s_{1m}, s_{2n}), u_2(s_{1m}, s_{2n})) \end{bmatrix} \]

Zero-Sum Games

Where \( u_1 + u_2 = 0 \):

\[ M = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad u_2 = -u_1 \]

Expected Payoff

For mixed strategies with probabilities \( p_1, p_2 \):

\[ E[u_1] = \sum_{i=1}^m \sum_{j=1}^n p_{1i} p_{2j} u_1(s_{1i}, s_{2j}) \]

Examples

Prisoner’s Dilemma

Strategies: Confess (C), Silent (S):

\[ M = \begin{array}{c|cc} & C & S \\ \hline C & (-5, -5) & (0, -10) \\ S & (-10, 0) & (-1, -1) \end{array} \]

Nash equilibrium: (C, C).

Pricing Game

Strategies: High (H), Low (L):

\[ M = \begin{array}{c|cc} & H & L \\ \hline H & (5, 5) & (1, 6) \\ L & (6, 1) & (2, 2) \end{array} \]

Rock-Paper-Scissors

Zero-sum:

\[ M = \begin{array}{c|ccc} & R & P & S \\ \hline R & (0, 0) & (-1, 1) & (1, -1) \\ P & (1, -1) & (0, 0) & (-1, 1) \\ S & (-1, 1) & (1, -1) & (0, 0) \end{array} \]

Prisoner’s Dilemma Payoffs

Payoffs for Player 1 in Prisoner’s Dilemma.

Analysis

Nash Equilibria

No player benefits from unilateral deviation:

\[ u_1(s_{1i}^*, s_{2j}^*) \geq u_1(s_{1k}, s_{2j}^*) \] \[ u_2(s_{1i}^*, s_{2j}^*) \geq u_2(s_{1i}^*, s_{2l}) \]

Dominant Strategies

Strategy \( s_{1i} \) dominates if:

\[ u_1(s_{1i}, s_{2j}) \geq u_1(s_{1k}, s_{2j}) \ \forall j \]

Minimax

For zero-sum games:

\[ v_1 = \max_i \min_j u_1(s_{1i}, s_{2j}) \]

Mixed Strategy

For Matching Pennies, equalize payoffs:

\[ p \cdot 1 + (1-p) \cdot (-1) = p \cdot (-1) + (1-p) \cdot 1 \] \[ p = 0.5 \]

Applications

Used in economics, biology, and negotiations.