Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider the following bimatrix game

$(2,6)\ \ \ (4,2) \\ (6,0) \ \ \ (0,4) $

I have been asked to compute all equilibria of this game, as well as the maximin strategies for both players. Now I used the Lemke-Howson algorithm to calculate $x = y =(\frac12,\frac12)$ to be a Nash equilibrium for this game.

My question is, how do I know if there are any more equilibria? Also, how is it possible to calculate the maximin strategies?

Any help would be greatly appreciated.

share|cite|improve this question
up vote 1 down vote accepted

For $2\times 2$ game this is not that hard. First express the pay of players as the function of strategies, i.e. $p_A(\alpha, \beta) = [\alpha, 1-\alpha] A [\beta, 1-\beta]^T$ where $\alpha$ is the probability of first player playing first action and $\beta$ is the probability of second player playing first action, $A$ is the payoff matrix for first player. Then you could calculate the best $\alpha$ as a function $m_\alpha$ of $\beta$, i.e. $m_\alpha(\beta) = \mathrm{maxarg}_\alpha\ p_A(\alpha,\beta)$ ($p_A$ is linear so this should be no problem). Finally, solve the set of quadratic equations $\alpha = m_\alpha(\beta), \beta = m_\beta(\alpha)$. To calculate maximin strategy you need to calculate $\mathrm{maxarg}_\alpha \min_\beta p_A(\alpha,\beta)$ and $\mathrm{maxarg}_\beta \min_\alpha p_B(\alpha,\beta)$ which can be done similarly.

Hope that helps ;-)

share|cite|improve this answer
Thank you so much, you saved our bacon :) – Paul Slevin Mar 10 '12 at 17:04

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.