Finding optimal mixed strategy

Question

I have to find an optimal mixed strategy for the 'column' player, who mixes with the probabilites $q_1,q_2,q_3$. What is known is the optimal mixing of the 'row" player. The game is a zero-sum game, each entry represents Row's gain, and Column's loss.

$\begin{bmatrix} &q_1 &q_2 &q_3\\ p=\frac{4}{7}& 4&5&2\\ 1-p=\frac{3}{7}&3& 1 &5 \end{bmatrix}$

I thought I should calculate the value of the game, so I wrote:

$\frac{4}{7}(4q_1+5q_2+2q_3)+\frac{3}{7}(3q_1+1q_2+5q_3)=$

$=\frac{25}{7}q_1+\frac{23}{7}q_2+\frac{23}{7}q_3$

But I don't know how to proceed.

Answer 1

Given the question as you've clarified it, [EDITED FROM HERE ON TO CORRECT FOR MY FAILURE TO READ THE UPDATED QUESTION CAREFULLY ENOUGH], yes in response to your comment, you want to minimize Row's payoff by having Column zero out $q_1$ and pick any $q_2$ and $q_3$ that add up to 1. This is because those two have the same coefficient, 23/7, and it's less than 25/7.

Answer 2

If you know that a strategy $p$ of the row player is part of a Nash equilibrium , you can find a strategy $q$ of the column player so that $(p,q)$ is a Nash equilibrium (even if the game is not zero-sum) as follows:

1. Find the set $Q$ of pure best responses to $p$. In this case $Q = \{2,3\}$, that is columns 2 and 3.

2. Check which mixtures over $Q$ make the pure strategies in the support of $p$ pure best responses (where the support of $p$ is those pure strategies that are played with positive probability, i.e. $\{i \mid p_i > 0\}$). In the example, the unique way to mix over columns 2 and 3 to make the row player indifferent (which is necessary and sufficient for both rows to be best responses) is to set $q = (0, 3/7, 4/7)$. This is an easy calculation that equated the expected payoff of the rows against a mixed strategy $q = (0,q_2,1-q_2)$. Since this game is zero sum and this was the unique way to make $p$ a best response, we have that $(p,q)$ is the unique Nash equilibrium of this game.

More generally, for solving $2 \times N$ bimatrix games see this answer to another question, which describes the "upper envelope method". For zero-sum games in particular, to look for equilibrium strategies you can look for maxmin strategies, which is easy for the row player in a $2 \times N$ game using the upper envelope method.

To explore other games and check that you can solve them correctly, you might find the following web-based software useful:

http://banach.lse.ac.uk

http://www.gametheoryexplorer.org

We actively develop the second, so feedback is welcome.