How do you find Player 2's mixing probability q in a mixed-strategy Nash equilibrium?

To calculate the probability $p$ for Player 1's mixed strategy, you must determine the value of $p$ that makes Player 2 completely indifferent between choosing their pure strategies, Right (R) and Left (L). This is done by computing Player 2's expected payoff for each of their pure strategies as a function of $p$, setting those two expected payoffs equal to each other, and solving the resulting linear equation for $p$.

Conditions: The game is a finite normal-form game with a 2x2 payoff matrix.; Player 1 chooses R with probability $p$ and L with probability $1-p$.; Player 2's expected payoffs from playing R and L must be calculated using Player 1's mixed strategy.; The solution for $p$ must lie in the interval $[0, 1]$ to represent a valid mixed strategy.

In this game, $q = 3/7$ means that Player 2 chooses their pure strategy Right (R) with a probability of $3/7$, and consequently chooses their pure strategy Left (L) with a probability of $4/7$. At this specific mixture, Player 1 is completely indifferent between choosing R and L, receiving an expected payoff of $12/7$ regardless of their choice.

Conditions: The game is the displayed 2x2 normal-form game.; Player 2 mixes between R and L with probabilities $q$ and $1-q$.; The equilibrium concept is mixed-strategy Nash equilibrium.

The expected payoff for a specific pure strategy is calculated by multiplying the payoffs of that strategy against each of the opponent's possible pure strategies by the probabilities of the opponent choosing those strategies, and then summing the results. It is the weighted average of the outcomes.

Conditions: The opponent has a set of pure strategies.; The opponent assigns a specific probability to each of their pure strategies, and these probabilities sum to 1.; The payoffs for the target player are known for each combination of pure strategies.

We set the two expected payoffs equal to enforce the indifference principle. In a mixed strategy Nash equilibrium, a player randomizes over their pure strategies only if they are completely indifferent between them.

Conditions: The game is in a mixed-strategy Nash equilibrium.; The player is randomizing over multiple pure strategies.; The expected payoffs for the player's pure strategies are calculated based on the opponent's mixed strategy.

When solving for Player 2's mixing probability $q$, Player 1's expected payoffs are made equal. Specifically, you equalize Player 1's expected payoff from choosing their pure strategy R with Player 1's expected payoff from choosing their pure strategy L.

Conditions: The context is finding a mixed-strategy Nash equilibrium.; Player 2 is the one whose mixing probability $q$ is being solved for.; Player 1 is the opponent who must be made indifferent.