Compute Nash Equilibrium in median voter game 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.


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 &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}
$

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}
$


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&lt;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&lt;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.


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.
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 $\frac{s}{2-s}$.