Find Nash Equilibrium for Mixed Strategy 3 x 3 Game: A Step-by-Step Guide

Game theory is a fascinating area of economics and strategic decision-making, and one of its pivotal concepts is the Nash Equilibrium. In this guide, we will focus on finding the Nash Equilibrium for mixed strategies in a 3 x 3 game. Whether you’re a student, a professional, or just someone interested in understanding game theory better, this step-by-step guide will walk you through the process.

Understanding Nash Equilibrium

Before we dive into the step-by-step approach, let’s define what we mean by Nash Equilibrium. A Nash Equilibrium occurs when players in a game choose strategies such that no player can benefit by changing their strategy while the others keep theirs unchanged. In the case of mixed strategies, players randomize their choices among available strategies.

Step 1: Set Up the Game

The first step is to clearly set up your 3 x 3 game matrix. For example, let’s consider two players, Player A and Player B, each with three strategies:

• Player A: A1, A2, A3
• Player B: B1, B2, B3

Create a payoff matrix where the rows represent Player A’s strategies and the columns represent Player B’s strategies. The entries will contain the respective payoffs for each strategy combination. Here’s a simplified version of what the matrix might look like:

B1 B2 B3
-----------------------
A1 | (2, 2) (0, 1) (1, 0)
A2 | (1, 0) (3, 3) (0, 2)
A3 | (0, 1) (1, 0) (2, 2)

The first number in each pair represents Player A’s payoff, while the second represents Player B’s payoff.

Step 2: Identify Pure Strategy Nash Equilibriums (if any)

Before addressing mixed strategies, identify any pure strategy Nash Equilibriums (PSNE) by examining best responses. A pure strategy Nash Equilibrium occurs when both players are playing their optimal strategies.

1. For each of Player A’s strategies, find Player B’s best response.
2. For each of Player B’s strategies, find Player A’s best response.

If any entries in the matrix satisfy both conditions, they represent pure strategy Nash Equilibriums.

Step 3: Analyze Mixed Strategies

If no pure strategies exist, or if you want to find mixed Nash Equilibriums, we move to the next step – calculating mixed strategies. In mixed strategies, players randomize their moves.

Consider the probabilities assigned to each strategy:

• Let ( p_1, p_2, p_3 ) be Player A’s probabilities of playing A1, A2, A3 respectively.
• Let ( q_1, q_2, q_3 ) be Player B’s probabilities of playing B1, B2, B3 respectively.

Step 4: Set Up the Probability Equations

To find the mixed Nash Equilibrium, you need to ensure that each player is indifferent among their strategies given the mixed strategies of the other player. This requires setting up equations based on expected payoffs.

1. Expected Payoff for Player A: Calculate the expected payoff for each of A’s strategies based on B’s probabilities. Set them equal to each other.

2. Expected Payoff for Player B: Do the same for Player B’s strategies based on A’s probabilities.

For example, if we want to calculate Player A’s expected payoff for A1 against B’s mixed strategy:
[ E(A1) = 2q_1 + 0q_2 + 1q_3 ]

Repeat for A2 and A3 and set them equal to each other to find relations among the probabilities.

Step 5: Solve the Equations

Once you have the system of equations, solve for the probabilities ( p ) and ( q ) that ensure both players are indifferent among their strategies.

A simple example might yield relations like ( q_1 + 2q_2 + 0.5q_3 = 0.5 ) for Player A, leading to equations that can be solved simultaneously.

Step 6: Check the Validity of Results

After calculating the probabilities, check that they sum to 1, as they need to represent a valid probability distribution.

Conclusion

Finding the Nash Equilibrium in mixed strategies for a 3 x 3 game can be complex, but by systematically setting up your game, identifying payoffs, formulating probability equations, and solving them, you can uncover the equilibrium point. Practice with different matrices to build your understanding and intuition. Game theory has far-reaching implications, making these skills valuable not just academically but in real-world strategic decision-making as well.

Happy strategizing!