Mixed-Strategy Nash Equilibria in 2x2 Games

Introduction

Welcome back! In our last lesson, we established why players might choose to randomize their actions. We learned that in games with no pure-strategy Nash equilibrium, being predictable is a strategic disadvantage. The solution is to play a mixed strategy, where you choose your actions based on a set of probabilities.

The core logic we uncovered was the indifference principle: you choose your probability mix to make your opponent indifferent between the pure strategies they are actively using. This prevents them from finding a single best response to exploit.

Today, we will turn this principle into a concrete, step-by-step mathematical method. Your goal for this lesson is to master the algorithm for calculating the mixed-strategy Nash equilibrium (MSNE) in any 2x2 game. Given your interest in mathematics, you'll see how a conceptual puzzle in strategic thinking transforms into a straightforward problem of solving a system of linear equations.

The Mixed Strategy Algorithm

The most direct way to learn the calculation is to see it in action. The following video provides a concise and clear explanation of what it calls the "mixed strategy algorithm." It will walk you through the entire process for a sample 2x2 game.

Game Theory 101: The Mixed Strategy Algorithm

This video from William Spaniel's 'Game Theory 101' series breaks down the calculation of a mixed strategy Nash equilibrium into a clear, repeatable algorithm. Pay close attention to how finding one player's strategy involves using the other player's payoffs.

Please watch the entire video (from 01:12 to 06:19). It's short and covers the complete procedure for both players.

Formalizing the Algorithm

The video provided an excellent demonstration. Now, let's formalize the steps with algebraic notation to create a general-purpose tool.

Consider a generic 2x2 game:

	Player 2: Left	Player 2: Right
Player 1: Up	( $u_{11}, u_{21}$ )	( $u_{12}, u_{22}$ )
Player 1: Down	( $u_{13}, u_{23}$ )	( $u_{14}, u_{24}$ )

Let's assume this game has no pure-strategy Nash equilibrium, so an MSNE must exist.

Player 1 plays Up with probability $p$ and Down with probability $1 - p$ .
Player 2 plays Left with probability $q$ and Right with probability $1 - q$ .

The algorithm to find the equilibrium probabilities $(p^{*}, q^{*})$ is as follows:

Step 1: Find Player 2's equilibrium strategy $(q^{*})$ by making Player 1 indifferent.

Player 1's expected utility from playing Up must equal their expected utility from playing Down.

$E_{1} [Up] = q \cdot u_{11} + (1 - q) \cdot u_{12}$
$E_{1} [Down] = q \cdot u_{13} + (1 - q) \cdot u_{14}$

Set them equal:

q \cdot u_{11} + (1 - q) \cdot u_{12} = q \cdot u_{13} + (1 - q) \cdot u_{14}

Now, solve this equation for $q$ . The value you find is $q^{*}$ .

Step 2: Find Player 1's equilibrium strategy $(p^{*})$ by making Player 2 indifferent.

Player 2's expected utility from playing Left must equal their expected utility from playing Right.

$E_{2} [Left] = p \cdot u_{21} + (1 - p) \cdot u_{23}$
$E_{2} [Right] = p \cdot u_{22} + (1 - p) \cdot u_{24}$

Set them equal:

p \cdot u_{21} + (1 - p) \cdot u_{23} = p \cdot u_{22} + (1 - p) \cdot u_{24}

Solve this equation for $p$ . The value you find is $p^{*}$ .

Step 3: State the Mixed Strategy Nash Equilibrium.

The MSNE is the profile of mixed strategies:
( $σ_{1}^{*} = (p^{*} on Up, 1 - p^{*} on Down)$ , $σ_{2}^{*} = (q^{*} on Left, 1 - q^{*} on Right)$ ).

This "crossing" of payoffs and strategies is the mathematical embodiment of the indifference principle. To find your mix, you look at your opponent's payoffs.

Practice: The Tennis Match

In the last lesson, you saw a segment of a Yale lecture that introduced the indifference principle. We're now going to return to the central example from that lecture: a tennis match between Venus and Serena Williams.

Serena is at the net, and Venus must decide whether to hit the ball to Serena's left (L) or right (R). Simultaneously, Serena must decide whether to lean left (L) or right (R). The payoffs represent the percentage chance of winning the point for each player.

	Serena: L	Serena: R
Venus: L	(50, 50)	(80, 20)
Venus: R	(90, 10)	(20, 80)

First, confirm for yourself that this game has no pure-strategy Nash equilibrium. Then, apply the algorithm we just defined.

Your Task:
Calculate the mixed-strategy Nash equilibrium for this game.

Let $p$ be the probability Venus chooses L.
Let $q$ be the probability Serena chooses L.
Find $p^{*}$ and $q^{*}$ .

Take a few minutes to work through the algebra on your own.

Solution

Let's walk through the calculation together.

1. Find Serena's mix ( $q^{*}$ ) by making Venus indifferent.
Venus's expected payoff for playing L must equal her payoff for playing R.

E_{V} [L] = E_{V} [R]

q \cdot 50 + (1 - q) \cdot 80 = q \cdot 90 + (1 - q) \cdot 20

50 q + 80 - 80 q = 90 q + 20 - 20 q

80 - 30 q = 70 q + 20

60 = 100 q

q^{*} = 0.6

So, Serena must lean Left 60% of the time and Right 40% of the time.

2. Find Venus's mix ( $p^{*}$ ) by making Serena indifferent.
Serena's expected payoff for leaning L must equal her payoff for leaning R.

E_{S} [L] = E_{S} [R]

p \cdot 50 + (1 - p) \cdot 10 = p \cdot 20 + (1 - p) \cdot 80

50 p + 10 - 10 p = 20 p + 80 - 80 p

40 p + 10 = 80 - 60 p

100 p = 70

p^{*} = 0.7

So, Venus must hit to the Left 70% of the time and to the Right 30% of the time.

3. State the MSNE.
The unique MSNE is:
(Venus plays L with 70% prob, Serena leans L with 60% prob).

Now, let's watch the section of the lecture where Professor Polak performs this exact calculation. See how your steps match his.

9. Mixed strategies in theory and tennis

Here is the segment of the Yale lecture where the professor calculates the MSNE for the tennis game. His explanation of the 'trick'—using one player's payoffs to find the other's strategy—is a powerful reinforcement of the algorithm.

Watch from 40:27 to 52:58. This covers the setup and calculation for both Serena's mix and Venus's mix.

For Further Study

To solidify your understanding, it is highly valuable to see the algorithm applied to different 2x2 games. The following reading from an MIT course provides rigorous, worked-out examples for two classic games.

[PDF] Chapter 6 Nash Equilibrium

This chapter from an MIT OpenCourseWare text on game theory provides more formal examples of calculating mixed-strategy equilibria. It's an excellent way to test your understanding on new problems.

I recommend reading through two sections: first, the example calculation for the Battle of the Sexes game (starts on page 88, labeled 'Example – Battle of the Sexes'); second, the calculation for the Hawk-Dove game (starts on page 90, in section 6.4). You don't need to read the evolutionary dynamics part that follows, just focus on the equilibrium calculation.

Conclusion

You have now learned the complete mechanical process for finding a mixed-strategy Nash equilibrium in 2x2 games. The procedure may seem counter-intuitive at first—using player 1's payoffs to find player 2's strategy—but it follows directly from the logic of the indifference principle.

Key Takeaways:

To find a player's mixed strategy, you set up an equation that makes the other player indifferent between their pure strategies.
This process involves calculating each of the opponent's expected utilities as a function of the player's unknown probability mix.
Setting the opponent's expected utilities equal to each other creates a linear equation that can be solved to find the player's equilibrium probability mix.
Repeating this for all players yields the full Mixed Strategy Nash Equilibrium.

We have now answered what a mixed strategy is, why it's needed, and how to calculate it. In our next lesson, we will delve deeper into the final question: what do these probabilities really mean? We will explore different interpretations of mixed strategies, bridging the gap between mathematical calculation and real-world strategic behavior.

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