Mixed Strategies: A Comprehensive Guide

Mixed Strategies represent a sophisticated approach in Game Theory where players randomize their actions according to specific probability distributions to optimize their outcomes. Unlike pure strategies, where a single action is chosen deterministically, mixed strategies introduce uncertainty, making them essential for games lacking stable pure strategy equilibria. This concept, formalized by John von Neumann and Oskar Morgenstern, underpins the existence of Nash Equilibria in finite games.

By assigning probabilities to each possible action, players can balance their payoffs against opponents’ strategies, often leading to equilibrium states where no player benefits from deviating. Mixed strategies are particularly crucial in zero-sum games and scenarios with cyclic preferences, like Rock-Paper-Scissors. This guide provides an exhaustive exploration of mixed strategies, including their definitions, examples, calculation methods, and applications, enriched with detailed equations and analyses.

From economic competition to evolutionary dynamics, mixed strategies illuminate how randomness can stabilize strategic interactions across diverse domains.

Definition and Concepts of Mixed Strategies

Mixed strategies extend the framework of Game Theory by incorporating probabilistic decision-making. Below, we define key aspects with mathematical rigor.

Basic Definition

A mixed strategy for player \( i \) is a probability distribution \( p_i \) over their strategy set \( S_i = \{s_{i1}, s_{i2}, \ldots, s_{im}\} \), where:

\[ p_i(s_{ij}) \geq 0 \text{ and } \sum_{j=1}^m p_i(s_{ij}) = 1 \]

Expected Payoff

For a two-player game, the expected payoff for player 1 given mixed strategies \( p_1 \) and \( p_2 \):

\[ E[u_1] = \sum_{s_1 \in S_1} \sum_{s_2 \in S_2} p_1(s_1) p_2(s_2) u_1(s_1, s_2) \]

Similarly for player 2:

\[ E[u_2] = \sum_{s_1 \in S_1} \sum_{s_2 \in S_2} p_1(s_1) p_2(s_2) u_2(s_1, s_2) \]

Mixed Strategy Nash Equilibrium

A profile \( (p_1^*, p_2^*) \) is a Nash Equilibrium if:

\[ E[u_1(p_1^*, p_2^*)] \geq E[u_1(p_1, p_2^*)] \]
\[ E[u_2(p_1^*, p_2^*)] \geq E[u_2(p_1^*, p_2)] \]

For all \( p_1, p_2 \).

Indifference Principle

In equilibrium, a player’s mixed strategy makes their opponent indifferent among their pure strategies with positive probability:

\[ E[u_2(s_{2a}, p_1^*)] = E[u_2(s_{2b}, p_1^*)] \text{ for all } s_{2a}, s_{2b} \text{ with } p_2^*(s_{2a}), p_2^*(s_{2b}) > 0 \]

Support of a Mixed Strategy

The support \( \text{supp}(p_i) \) is the set of strategies with \( p_i(s_{ij}) > 0 \).

Detailed Examples of Mixed Strategies

Let’s examine mixed strategies in classic and extended games.

Example 1: Rock-Paper-Scissors

Payoff for Player 1:

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

Equilibrium: \( p_1 = p_2 = (1/3, 1/3, 1/3) \).

Example 2: Matching Pennies

Payoff:

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

Equilibrium: \( p_H = p_T = 1/2 \).

Example 3: Battle of the Sexes

Payoff:

\[ \begin{array}{c|cc} & F & O \\ \hline F & (2, 1) & (0, 0) \\ O & (0, 0) & (1, 2) \end{array} \]

Mixed equilibrium: Player 1: \( p_F = 2/3 \), Player 2: \( p_F = 1/3 \).

Example 4: Inspection Game

Payoff (Inspect, Evade):

\[ \begin{array}{c|cc} & I & N \\ \hline E & (-1, 1) & (1, -1) \\ C & (0, 0) & (0, 0) \end{array} \]

Equilibrium depends on payoff adjustments.

Example 5: Hawk-Dove

Payoff (V=2, C=3):

\[ \begin{array}{c|cc} & H & D \\ \hline H & (-0.5, -0.5) & (2, 0) \\ D & (0, 2) & (1, 1) \end{array} \]

Equilibrium: \( p_H = 2/3 \).

Calculation Methods for Mixed Strategies

Computing mixed strategy equilibria involves solving systems of equations based on indifference.

Two-Strategy Games

For Player 1 with \( p_1 = (p, 1-p) \), Player 2 with \( q_1 = (q, 1-q) \):

\[ E[u_1(A)] = q u_1(A, C) + (1-q) u_1(A, D) \]
\[ E[u_1(B)] = q u_1(B, C) + (1-q) u_1(B, D) \]
\[ E[u_1(A)] = E[u_1(B)] \]

Matching Pennies Calculation

Player 2’s indifference:

\[ p \cdot (-1) + (1-p) \cdot 1 = p \cdot 1 + (1-p) \cdot (-1) \]
\[ p = 1/2 \]

Graphical Method

Plot best response functions:

\[ p = BR_1(q), \ q = BR_2(p) \]

Linear Programming (Zero-Sum)

Maximize \( v \) subject to:

\[ \sum_{s_1} p(s_1) u_1(s_1, s_2) \geq v \]
\[ \sum_{s_1} p(s_1) = 1 \]

Iterative Approximation

Adjust probabilities until convergence.