# convex for nash equilibrium

I have trouble understanding this question,

the first question to my understanding is asking me that for a fixed p , (p,q) is nash equilibrium, prove that all (p,q) are convex.

and for the second, what I understand is that for (p,q) as nash equilibrium. prove that (p,q) is not convex.

If someone could reword the question for me , that would be much appreciated.

• Is $(p,q)$ the probability distribution players 1 and 2 put on their $m$ and $n$ actions in equilibrium? – Pburg Nov 13 '14 at 1:44

I am assuming we are supposed to think of $(p,q)$ as the probability distribution on actions from two players?
Take the 2x2 pure coordination game as an example. Let pure strategies available to player 1 be $\{A,B\}$ and let the pure strategies available to player 2 be $\{a,b\}$. Then we attach payoffs $u_i(A,a)=u_i(B,b)=1$ for $i=1,2$ and utility is zero from any other outcome.
It should be obvious that $\{A,a\}$ is a Nash equilibrium and the same goes for $\{B,b\}$. Then degenerate mixed strategies $(0,0)$ and $(1,1)$ are in the set of Nash equilibria. Then, we can find a convex combination $(.3,.3)$ that is not a Nash equilibria. If player $1$ chooses $A$ strictly more than $50\%$ of the time, then player $2$ will choose $a$ $100\%$ of the time. So, $(.3,.3)$ can't be a Nash equilibrium. Therefore, the set described in b.) is not convex.