How to find mixed strategies in a game matrix?

To find mixed strategies in a game matrix, we use the concept of expected payoffs.

In a game matrix, a mixed strategy is a probability distribution over the pure strategies. To find the mixed strategy for a player, we need to calculate the expected payoff for each pure strategy and then find the probability distribution that maximizes the expected payoff.

For example, consider the following game matrix:

| | L | R |
|---|---|---|
| T | 2, 1 | 0, 0 |
| B | 0, 0 | 1, 2 |

To find the mixed strategy for Player 1, we calculate the expected payoff for each pure strategy:

- If Player 1 plays T with probability p and Player 2 plays L with probability q, the expected payoff for Player 1 is 2p + 0(1-p) = 2p.
- If Player 1 plays B with probability p and Player 2 plays L with probability q, the expected payoff for Player 1 is 0p + 1(1-p) = 1-p.
- Therefore, the expected payoff for Player 1 is 2p + (1-p)q if they play T and B with probabilities p and 1-p respectively.

To find the optimal mixed strategy for Player 1, we need to maximize the expected payoff. This can be done using calculus or by inspection. In this case, we can see that the optimal mixed strategy is to play T with probability 2/3 and B with probability 1/3.

Similarly, we can find the optimal mixed strategy for Player 2 by calculating the expected payoff for each pure strategy and finding the probability distribution that maximizes the expected payoff.