When solving for Player 2's q, whose payoffs are made equal?

Q: When solving for Player 2's q, whose payoffs are made equal?

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. This is because Player 2's mixture must make Player 1 indifferent between their own actions in order for Player 1 to be willing to randomize.

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. This is because Player 2's mixture must make Player 1 indifferent between their own actions in order for Player 1 to be willing to randomize.

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.

Reasoning, step by step

Identify the player whose mixing probability is being solved for (Player 2).
Identify the opponent (Player 1).
Calculate the opponent's (Player 1's) expected payoff for each of their pure strategies.
Set the opponent's expected payoffs equal to each other.
Solve the equation for the mixing probability $q$ .

Example

To find $q$ , the board computes $\Pi_1(R) = 3 - 3q$ and $\Pi_1(L) = 4q$ , and then sets $\Pi_1(R) = \Pi_1(L)$ to get $3 - 3q = 4q$ . Player 1's payoffs are equalized, not Player 2's.

Common misconceptions

A common error is to use Player 2's own payoffs when solving for Player 2's $q$ . The indifference principle requires making the *other* player indifferent.
Confusing which player's payoffs are used to solve for which player's mixing probability.

Watch the explanation

YouTubeMixed Strategies Nash Equilibrium: The Math

2:38 – 2:45Watch this moment ↗

Explore next

Answers are generated from source material and independently checked. Consult the original video or creator if something is unclear.