<p>Once again, thanks to denesp (and Herr K.) for the help. I thought that I'd try to answer the question just for reference purposes, and I guess that it's also a step toward improving the "answers per question" metric on Economics SE.</p>

<p><img src="https://i.sstatic.net/TEwap.jpg" alt="enter image description here"></p>

<p>As aforementioned, the PSNE is $(u,r)=(3,3)$.</p>

<p><strong>Player 2</strong></p>

<blockquote>
  <ul>
  <li>If $P1$ plays $u$, $P2$ can play either $l$ or $r$.</li>
  <li>$l$ would give $P2$ a payoff of $2$ whilst $r$ would give $P2$ a    payoff of $3$.</li>
  <li>Therefore, $P2$ will play $r$ with probability $\pi=1$</li>
  </ul>
</blockquote>

<p><strong>Player 1</strong></p>

<blockquote>
  <ul>
  <li>If $P2$ plays $r$, $P1$ can play either $u$ or $d$.</li>
  <li>$u$ would give $P1$ a payoff of $3$ whilst $d$ would give $P1$ a    payoff of $2$</li>
  <li>Therefore, $P1$ will play $u$ with probability $1-\rho=1$, $\rho=0$.</li>
  </ul>
</blockquote>

<p><strong>The result</strong></p>

<p>$(\pi^*,\rho^*)=(1,0)$</p>

CC BY-SA 3.0
2015-04-26T12:07:08.757Z
2015-04-26T12:07:08.757Z
Five σ
<p><img src="https://i.sstatic.net/xg6Fo.jpg" alt="enter image description here"></p>

<p>The question I'm dealing with:</p>

<blockquote>
  <p>Let $x=3$, find any Nash equilibrium in pure or mixed strategies.</p>
</blockquote>

<p>I have observed that the pure strategy Nash equilibrium is $(u,r)=(3,3)$.</p>

<p>The answer to this question is:</p>

<p>$(\pi^*,\rho^*)=(1,0)$</p>

<p>Am I correct in interpreting the above as:</p>

<p>Given the PSNE of $(u,r)$, the row player will play $u$ with probability $1$ and the column player will play $r$ with probability $1-0=1$?</p>

<p>Thanks</p>

CC BY-SA 3.0
2015-04-23T22:14:34.397Z
The other way around. The row player will play $u$, and column will play $r$. Other than this your interpretation is correct, probability of $u$ is $\pi = 1$, probability of $r$ is $1 - \rho = 1 - 0 = 1$.
Giskard
CC BY-SA 3.0
2015-04-23T22:23:05.890Z
@denesp, the row/column mistake was a typo. I appreciate the help you've given me with regards to the several game theory questions that I've asked.
Five σ
CC BY-SA 3.0
2015-04-24T17:23:10.313Z
You could also observe that $u$ and $r$ are strictly dominant strategies when $x=3$. Thus $(u,r)$ is the unique NE of this game.
Herr K.
CC BY-SA 3.0
2015-04-25T11:07:29.287Z
@denesp You should make your comment into an answer as it answers the question.
The Almighty Bob
CC BY-SA 3.0
2015-04-25T18:55:23.260Z
I felt that Five $\sigma$ did most of the work so he should get credit for the answer.
Giskard
<game-theory>
Finding a nash equilibrium in pure or mixed strategies
Five σ
# Finding a nash equilibrium in pure or mixed strategies

(Asked by: username_0 on 2015-04-23)

![enter image description here](https://i.sstatic.net/xg6Fo.jpg)

The question I'm dealing with:

> Let $x=3$, find any Nash equilibrium in pure or mixed strategies.

I have observed that the pure strategy Nash equilibrium is $(u,r)=(3,3)$.

The answer to this question is:

$(\pi^\*,\rho^\*)=(1,0)$

Am I correct in interpreting the above as:

Given the PSNE of $(u,r)$, the row player will play $u$ with probability $1$ and the column player will play $r$ with probability $1-0=1$?

Thanks

**Tags:** game-theory

> **Comments:**
> **username_1**: The other way around. The row player will play $u$, and column will play $r$. Other than this your interpretation is correct, probability of $u$ is $\pi = 1$, probability of $r$ is $1 - \rho = 1 - 0 = 1$.  
> **username_0**: @denesp, the row/column mistake was a typo. I appreciate the help you've given me with regards to the several game theory questions that I've asked.  
> **username_2**: You could also observe that $u$ and $r$ are strictly dominant strategies when $x=3$. Thus $(u,r)$ is the unique NE of this game.  
> **username_3**: @denesp You should make your comment into an answer as it answers the question.  
> **username_1**: I felt that Five $\sigma$ did most of the work so he should get credit for the answer.  

---

## 1 answers

### Answer by username_0 on 2015-04-26. Score: 2

Once again, thanks to denesp (and username_2) for the help. I thought that I'd try to answer the question just for reference purposes, and I guess that it's also a step toward improving the "answers per question" metric on Economics SE.

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

As aforementioned, the PSNE is $(u,r)=(3,3)$.

**Player 2**

> * If $P1$ plays $u$, $P2$ can play either $l$ or $r$.
> * $l$ would give $P2$ a payoff of $2$ whilst $r$ would give $P2$ a payoff of $3$.
> * Therefore, $P2$ will play $r$ with probability $\pi=1$

**Player 1**

> * If $P2$ plays $r$, $P1$ can play either $u$ or $d$.
> * $u$ would give $P1$ a payoff of $3$ whilst $d$ would give $P1$ a payoff of $2$
> * Therefore, $P1$ will play $u$ with probability $1-\rho=1$, $\rho=0$.

**The result**

$(\pi^\*,\rho^\*)=(1,0)$


---

