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The question I am dealing with is:

Find all Nash equilibria in pure or mixed strategies.

This is what I have done so far..

Pure strategy nash equilibria

By observing best responses, the PSNE are $(u2,l2)$ and $(d2,r2)$

Mixed strategies

Solving for $\tau$:

$10\tau +2(1-\tau) = 5\tau +8(1-\tau)$

$\tau^{*}=\frac{6}{11}$

Solving for $\sigma$:

$10\sigma +2(1-\sigma)=5\sigma+8(1-\sigma)$

$\sigma^*=\frac{6}{11}$

Therefore: $(\tau^*,\sigma^*)=\left(\frac{6}{11},\frac{6}{11}\right)$

The correct answer is apparently:

$(\tau^*,\sigma^*)=\color{red}{(1,1)},\color{red}{(0,0)}$ and $\left(\frac{6}{11},\frac{6}{11}\right)$

It seems that the correct answer has omitted my pure strategy nash equilibria. I also don't know how to get $(1,1)$ and $(0,0)$. Any chance you could help me out?

Thanks.

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The probability vectors you gave in red describe pure strategy NE. (They assign probability one to an action.)

$\tau$ shows the probability that the column player plays action $l_2$. If $\tau = 1$, then he will play $l_2$ with probability one. This is essentially another way of saying that he plays the pure strategy $l_2$.

One way to get these probability vectors is to find the PSNE as you have done. The PSNE $(u_2,l_2)$ means that the Row player plays $u_2$ with probability one, and the Column player plays $l_2$ with probability one. According to your notation this means that $(\tau, \sigma) = (1,1)$.

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  • $\begingroup$ Any chance you could explain what they mean within this specific context? I'm having trouble with understanding how they fit into the question. $\endgroup$ – Five σ Apr 22 '15 at 20:34
  • $\begingroup$ Is the current version satisfactory? If not, please clarify what part bothers you. To me this seems to be a misunderstanding of notation. $\endgroup$ – Giskard Apr 22 '15 at 20:41

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