I know that in a 2 player zero sum game all equilibria give each player the same expected value, but is it the case that they also induce the exact same distribution over payoffs?
Or might there be higher and lower variance equilibria with the same expected value?
On
I guess this is actually somewhat trivial.
Just a game like "Player 1 choses to play rock paper scissors or not, if he chooses to play they player a round of RPS and the winner gets a dollar from the loser, if he chooses not the play they both get 0" has two equilibrium.
In one they play RPS and both play nash and each player wins 1 dollar and loses 1 dollar with probability 1/2. In the other player 1 decides not to play and they both get $0 with certainty.
You've got the right idea. As a further example, consider what you get when you cram two matching pennies games together as follows. There's obviously one mixed equilibrium across A/a and B/b and another across C/c and D/d, both of which have the same expected payoff but different variances. \begin{array}{ccccc} & a & b & c & d \\ A & 1,-1 & -1,1 & 0,0 & 0,0 \\ B & -1,1 & 1,-1 & 0,0 & 0,0 \\ C & 0,0 & 0,0 & 2,-2 & -2,2 \\ D & 0,0 & 0,0 & -2,2 & 2,-2 \end{array}