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

<p>$\tau$ 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 $l_2$.</p>

<p>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 $(\tau, \sigma) = (1,1)$.</p>

CC BY-SA 3.0
2015-04-22T20:34:48.600Z
Any chance you could explain what they mean within this specific context? I'm having trouble with understanding how they fit into the question.
Five σ
CC BY-SA 3.0
2015-04-22T20:41:02.350Z
Is the current version satisfactory? If not, please clarify what part bothers you. To me this seems to be a misunderstanding of notation.
Giskard
CC BY-SA 3.0
2015-04-22T20:30:32.317Z
2015-04-22T20:40:05.683Z
2015-04-22T20:40:05.683Z
Giskard
<p><img src="https://i.sstatic.net/MpvMS.jpg" alt="enter image description here"></p>

<p>The question I am dealing with is:</p>

<blockquote>
  <p>Find all Nash equilibria in pure or mixed strategies.</p>
</blockquote>

<p>This is what I have done so far..</p>

<p><strong>Pure strategy nash equilibria</strong></p>

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

<p><strong>Mixed strategies</strong> </p>

<p>Solving for $\tau$:</p>

<p>$10\tau +2(1-\tau) = 5\tau +8(1-\tau)$</p>

<p>$\tau^{*}=\frac{6}{11}$</p>

<p>Solving for $\sigma$:</p>

<p>$10\sigma +2(1-\sigma)=5\sigma+8(1-\sigma)$</p>

<p>$\sigma^*=\frac{6}{11}$</p>

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

<p>The correct answer is apparently:</p>

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

<p>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?</p>

<p>Thanks.</p>

<game-theory>
Strategic form: mixed strategy nash equilibria?
Five σ
# Strategic form: mixed strategy nash equilibria?

(Asked by: username_0 on 2015-04-22)

![enter image description here](https://i.sstatic.net/MpvMS.jpg)

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.

**Tags:** game-theory

---

## 1 answers

### Answer by username_1 on 2015-04-22. Score: 3 | Accepted answer

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 $(\tau, \sigma) = (1,1)$.

> **Comments:**
> **username_0**: Any chance you could explain what they mean within this specific context? I'm having trouble with understanding how they fit into the question.  
> **username_1**: Is the current version satisfactory? If not, please clarify what part bothers you. To me this seems to be a misunderstanding of notation.  

---

