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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.

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1 Answer 1

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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 ;-)

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Thank you so much, you saved our bacon :) –  Paul Slevin Mar 10 '12 at 17:04
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