<p><strong>Corrections</strong></p>

<p><strong>s cannot be greater than 1/2.</strong></p>

<p>If A chooses 0 -- if s>0, B chooses either -1 or 1 -- <strong>but</strong> if s=0, B chooses either -1 or 0 or 1. <strong>0 is only weakly dominated.</strong></p>

<p>If B chooses -1 -- if s>0, A chooses -1 -- <strong>but</strong> if s=0, A chooses either -1 or 0 or 1. <strong>0 and 1 are only weakly dominated.</strong></p>

<p>If B chooses 0 -- A chooses 0</p>

<p>If B chooses 1 -- if s>0, A chooses 1 -- <strong>but</strong> if s=0, A chooses either -1 or 0 or 1. <strong>-1 and 0 are only weakly dominated.</strong></p>

<p><strong>PSNE</strong></p>

<p><strong>if s=0</strong>, (0, -1) and (0, 0) and (0, 1) are pure strategy Nash equilibria.
Otherwise, there are none.</p>

<p><strong>Mixed Strategy</strong></p>

<p>B is indifferent when:</p>

<p>A plays $-1$ or $1$ with probability $-\frac{s}{s-2}=\frac{s}{2-s}$ (from 0 to 1/3), and $0$ with probability $\frac{3s-2}{s-2}$ (from 1 to 1/3)</p>

<p>A is indifferent when:</p>

<p>B plays $-1$ or $1$ with probability $\frac{s-1}{s-2}$ (from 1/2 to 1/3) and $0$ with probability $-\frac{s}{s-2}$ (from 0 to 1/3)</p>

CC BY-SA 4.0
2018-08-14T07:12:47.560Z
2018-08-14T07:12:47.560Z
Josh P
<p>The calculations look correct. </p>

<p>For the PSNE, I would just say since there is no mutual BRs in pure strategy, there is no PSNE, and leave it at that.</p>

<p>In the MSNE, A would play $-1$ with probability $\frac{1-s}{2-s}$, $0$ with $\frac{s}{2-s}$ and $1$ with $\frac{1-s}{2-s}$; B would play $-1$ with $\frac{s}{2-s}$, $0$ with $\frac{2-3s}{2-s}$ and $1$ with $\frac{s}{2-s}$. </p>

CC BY-SA 3.0
2017-10-22T20:21:30.497Z
2017-10-22T20:21:30.497Z
Herr K.
<p>Consider a spatial model in which two candidates A and B compete for office. The policy space ranges from -1 to 1 and each candidate can take one of three positions, -1, 0, and 1 (so that they have only three pure strategies). Voters vote for the candidate closer to their ideal points, but the location of the median voter is not known to the candidates: the median voter is located at either -1 or 1 with probability s &lt; 1/2 and is located at 0 with probability 1 - 2s. Assume that candidate A is slightly more advantaged than B: A prevails if A and B are equidistant from the median voter but B wins if he is closer to the median voter. Winning yields a payoff of 1 and losing 0. The candidates simultaneously choose a position. Answer the following questions.</p>

<ol>
<li>Draw a 3-by-3 payoff matrix and show if there are any pure-strategy Nash equilibria.</li>
</ol>

<p>My attempt:
$\begin{array}{r|ccc}
A\backslash B &amp; -1 &amp; 0 &amp; 1\\
\hline
-1 &amp; 1, 0 &amp; s, 1-2s+s &amp; 1-2s+s, s\\
0 &amp; 1-2s+s, s &amp; 1, 0 &amp; 1-2s+s, s\\
1 &amp; 1-2s+s, s &amp; s, 1-2s+s &amp; 1, 0   
\end{array}
$</p>

<p>This simplifies to
$\begin{array}{r|ccc}
A\backslash B &amp; -1 &amp; 0 &amp; 1\\
\hline
-1 &amp; 1, 0 &amp; s, 1-s &amp; 1-s, s\\
0 &amp; 1-s, s &amp; 1, 0 &amp; 1-s, s\\
1 &amp; 1-s, s &amp; s, 1-s &amp; 1, 0   
\end{array}
$</p>

<ul>
<li>Computing PSNE: </li>
<li>If A chooses -1
-- if s>1/2, B chooses 1
-- if s=1/2, B chooses either 1 or 0
-- if s&lt;1/2, B chooses 0</li>
<li>If A chooses 0
-- B chooses either -1 or 1</li>
<li>If A chooses 1
-- if s>1/2, B chooses -1
-- if s=1/2, B chooses either -1 or 0
-- if s&lt;1/2, B chooses 0</li>
<li>If B chooses -1
-- A chooses -1</li>
<li>If B chooses 0
-- A chooses 0</li>
<li>If B chooses 1
-- A chooses 1</li>
</ul>

<p>There's no PSNE. </p>

<p>Or can I say the following? Since the BR for either player should result in a strategy that helps them win the median voter, it can't happen that both players win the median voter. Therefore, there exists no PSNE.</p>

<ol start="2">
<li>Characterize a MSNE if it exists. Hint: let A's mixed strategy to choose -1, 0, and 1 be p, q, 1-p-q, respectively, and B's mixed strategy be x, y, and 1-x-y, respectively.</li>
</ol>

<p>For B to be indifferent, EU(-1)=EU(0)=EU(1). Thus, sq+s(1-p-q)=(1-s)p+(1-s)(1-p-q)=sp+sq</p>

<p>Solution: p=s/(2-s), q=(3s-2)/(s-2)</p>

<p>For A to be indifferent, EU(-1)=EU(0)=EU(1). Thus, x+sy+(1-s)(1-x-y)=(1-s)x+y+(1-s)(1-x-y)=(1-s)x+sy+(1-x-y)</p>

<p>Solutions: s=o, y=0, or
x=(s-1)/(s-2), y =s/(2-s)</p>

<p>I don't know how to translate these into MSNE. Can you also let me know if I have got everything right so far?</p>

<p>Thanks in advance.</p>

CC BY-SA 4.0
2018-08-14T18:24:45.563Z
"Winning yields a payoff of 1 and losing 0." Note that this is actually meaningless: positive affine transformations result in equivalent utility functions, so the payoffs for only two outcomes does not affect what is a NE (as long as the ordering of those results is preserved).
Acccumulation
<game-theory><nash-equilibrium><bayesian-game>
Compute Nash Equilibrium in median voter game
Emmie Ruta
# Compute Nash Equilibrium in median voter game

(Asked by: username_0 on 2017-10-22)

Consider a spatial model in which two candidates A and B compete for office. The policy space ranges from -1 to 1 and each candidate can take one of three positions, -1, 0, and 1 (so that they have only three pure strategies). Voters vote for the candidate closer to their ideal points, but the location of the median voter is not known to the candidates: the median voter is located at either -1 or 1 with probability s < 1/2 and is located at 0 with probability 1 - 2s. Assume that candidate A is slightly more advantaged than B: A prevails if A and B are equidistant from the median voter but B wins if he is closer to the median voter. Winning yields a payoff of 1 and losing 0. The candidates simultaneously choose a position. Answer the following questions.

1. Draw a 3-by-3 payoff matrix and show if there are any pure-strategy Nash equilibria.

My attempt:
$\begin{array}{r|ccc}
A\backslash B & -1 & 0 & 1\\
\hline
-1 & 1, 0 & s, 1-2s+s & 1-2s+s, s\\
0 & 1-2s+s, s & 1, 0 & 1-2s+s, s\\
1 & 1-2s+s, s & s, 1-2s+s & 1, 0
\end{array}
$

This simplifies to
$\begin{array}{r|ccc}
A\backslash B & -1 & 0 & 1\\
\hline
-1 & 1, 0 & s, 1-s & 1-s, s\\
0 & 1-s, s & 1, 0 & 1-s, s\\
1 & 1-s, s & s, 1-s & 1, 0
\end{array}
$

* Computing PSNE:
* If A chooses -1
  -- if s>1/2, B chooses 1
  -- if s=1/2, B chooses either 1 or 0
  -- if s<1/2, B chooses 0
* If A chooses 0
  -- B chooses either -1 or 1
* If A chooses 1
  -- if s>1/2, B chooses -1
  -- if s=1/2, B chooses either -1 or 0
  -- if s<1/2, B chooses 0
* If B chooses -1
  -- A chooses -1
* If B chooses 0
  -- A chooses 0
* If B chooses 1
  -- A chooses 1

There's no PSNE.

Or can I say the following? Since the BR for either player should result in a strategy that helps them win the median voter, it can't happen that both players win the median voter. Therefore, there exists no PSNE.

2. Characterize a MSNE if it exists. Hint: let A's mixed strategy to choose -1, 0, and 1 be p, q, 1-p-q, respectively, and B's mixed strategy be x, y, and 1-x-y, respectively.

For B to be indifferent, EU(-1)=EU(0)=EU(1). Thus, sq+s(1-p-q)=(1-s)p+(1-s)(1-p-q)=sp+sq

Solution: p=s/(2-s), q=(3s-2)/(s-2)

For A to be indifferent, EU(-1)=EU(0)=EU(1). Thus, x+sy+(1-s)(1-x-y)=(1-s)x+y+(1-s)(1-x-y)=(1-s)x+sy+(1-x-y)

Solutions: s=o, y=0, or
x=(s-1)/(s-2), y =s/(2-s)

I don't know how to translate these into MSNE. Can you also let me know if I have got everything right so far?

Thanks in advance.

**Tags:** game-theory, nash-equilibrium, bayesian-game

> **Comments:**
> **username_1**: "Winning yields a payoff of 1 and losing 0." Note that this is actually meaningless: positive affine transformations result in equivalent utility functions, so the payoffs for only two outcomes does not affect what is a NE (as long as the ordering of those results is preserved).  

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## 2 answers

### Answer by username_2 on 2018-08-14. Score: 1

**Corrections**

**s cannot be greater than 1/2.**

If A chooses 0 -- if s>0, B chooses either -1 or 1 -- **but** if s=0, B chooses either -1 or 0 or 1. **0 is only weakly dominated.**

If B chooses -1 -- if s>0, A chooses -1 -- **but** if s=0, A chooses either -1 or 0 or 1. **0 and 1 are only weakly dominated.**

If B chooses 0 -- A chooses 0

If B chooses 1 -- if s>0, A chooses 1 -- **but** if s=0, A chooses either -1 or 0 or 1. **-1 and 0 are only weakly dominated.**

**PSNE**

**if s=0**, (0, -1) and (0, 0) and (0, 1) are pure strategy Nash equilibria.
Otherwise, there are none.

**Mixed Strategy**

B is indifferent when:

A plays $-1$ or $1$ with probability $-\frac{s}{s-2}=\frac{s}{2-s}$ (from 0 to 1/3), and $0$ with probability $\frac{3s-2}{s-2}$ (from 1 to 1/3)

A is indifferent when:

B plays $-1$ or $1$ with probability $\frac{s-1}{s-2}$ (from 1/2 to 1/3) and $0$ with probability $-\frac{s}{s-2}$ (from 0 to 1/3)


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### Answer by username_3 on 2017-10-22. Score: 1

The calculations look correct.

For the PSNE, I would just say since there is no mutual BRs in pure strategy, there is no PSNE, and leave it at that.

In the MSNE, A would play $-1$ with probability $\frac{1-s}{2-s}$, $0$ with $\frac{s}{2-s}$ and $1$ with $\frac{1-s}{2-s}$; B would play $-1$ with $\frac{s}{2-s}$, $0$ with $\frac{2-3s}{2-s}$ and $1$ with $\frac{s}{2-s}$.


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