How to Calculate Mixed Strategy Nash Equilibrium

This article provides a straightforward, step-by-step guide on how to calculate a mixed strategy Nash Equilibrium in game theory. You will learn the fundamental theory behind mixed strategies, the core principle of indifference, and a concrete mathematical example using a standard 2x2 payoff matrix to find the equilibrium probabilities for both players.

Understanding Mixed Strategy Nash Equilibrium

In game theory, a mixed strategy Nash Equilibrium occurs when at least one player randomizes their choices by assigning a probability distribution to each available pure strategy. Unlike a pure strategy equilibrium, where players choose one specific action with certainty, a mixed strategy equilibrium is used when no player can benefit by unilaterally changing their probability distribution.

The key to finding a mixed strategy Nash Equilibrium is the Indifference Principle. This principle states that for a player to be willing to play a mixed strategy, they must be indifferent to the outcomes of the pure strategies they are mixing. In other words, the expected payoff for each of their chosen pure strategies must be equal.

Step-by-Step Calculation Example

To calculate a mixed strategy Nash Equilibrium, we will use a classic 2x2 coordination game.

1. Define the Payoff Matrix

Consider two players, Player 1 and Player 2, with the following options and payoffs (Player 1 payoff, Player 2 payoff):

• Player 1 chooses between Up (with probability \(p\)) and Down (with probability \(1-p\)).
• Player 2 chooses between Left (with probability \(q\)) and Right (with probability \(1-q\)).

2. Calculate Player 2’s Probabilities (Using Player 1’s Payoffs)

To find the probability \(q\) that Player 2 chooses “Left,” we must make Player 1 indifferent between choosing “Up” and “Down.” This means the expected payoff of Player 1 playing “Up” must equal the expected payoff of Player 1 playing “Down.”

• Expected Payoff for Player 1 playing Up:
  \[E(\text{Up}) = 2(q) + 0(1-q) = 2q\]

• Expected Payoff for Player 1 playing Down:
  \[E(\text{Down}) = 0(q) + 1(1-q) = 1 - q\]

Set these two expected payoffs equal to each other: \[2q = 1 - q\] \[3q = 1\] \[q = \frac{1}{3}\]

Therefore, Player 2 must play Left with a probability of \(1/3\) and Right with a probability of \(2/3\) (\(1 - 1/3\)).

3. Calculate Player 1’s Probabilities (Using Player 2’s Payoffs)

To find the probability \(p\) that Player 1 chooses “Up,” we must make Player 2 indifferent between choosing “Left” and “Right.” The expected payoff of Player 2 playing “Left” must equal the expected payoff of Player 2 playing “Right.”

• Expected Payoff for Player 2 playing Left:
  \[E(\text{Left}) = 1(p) + 0(1-p) = p\]

• Expected Payoff for Player 2 playing Right:
  \[E(\text{Right}) = 0(p) + 2(1-p) = 2 - 2p\]

Set these two expected payoffs equal to each other: \[p = 2 - 2p\] \[3p = 2\] \[p = \frac{2}{3}\]

Therefore, Player 1 must play Up with a probability of \(2/3\) and Down with a probability of \(1/3\) (\(1 - 2/3\)).

4. State the Final Mixed Strategy Nash Equilibrium

The mixed strategy Nash Equilibrium is the pair of probability distributions where neither player has an incentive to deviate.

The equilibrium is: * Player 1 plays: Up with probability \(\frac{2}{3}\), Down with probability \(\frac{1}{3}\). * Player 2 plays: Left with probability \(\frac{1}{3}\), Right with probability \(\frac{2}{3}\).