Uniqueness of payout of Nash equilibrium in a 2-player zero-sum game I'm convinced that in a 2 player zero sum game, any two NE strategies must have the same expected payout for both players. I'm even more convinced that this is not true for 3 or more players.
Where could I find a reference for this?
Is this still true if the game is not zero sum?
(you could add a dummy player to make it zero sum, so I suppose not)
 A: The couple of strategies $s_1^e$, $s_2^e$  is an equilibrium point (saddle point ) according to Von Neumann concept if the two following conditions hold:

*

*$g\left(s_1^e, s_2^e \right) \ge \ g\left(s_{1,i}, s_2^e \right) \forall s_{1,i} \in S_1$


*$g\left(s_1^e, s_2^e \right) \le \ g\left(s_1^e, s_{2,j} \right) \forall s_{2,j} \in S_2$
The definition of equilibrium has been formulated by v. Neumann for zero sum games.
The definition of equilibrium according to Von Neumann was given for zero-sum games and it can be seen as the "translation" of the concept of equilibrium in the sense of Nash given later for non-cooperative games. The following shows how to pass from the concept of Von Neumann equilibrium to the concept of Nash equilibrium: just remember that for a zero-sum game we have:
$g_1 = g$ and $g_2 = -g$
so it can be written that
$g_1 \left(s_1^e, s_2^e \right) \ge \ g_1 \left(s_{1,i}, s_2^e \right) \forall s_{1,i} \in S_1$
The first condition of equilibrium in the sense of Von Neumann is therefore the first condition of equilibrium in the sense of Nash referred to player 1.
For player 2 it becomes:
$-g_2 \left(s_1^e, s_2^e \right) \le \ -g_2 \left(s_1^e, s_{2,j} \right) \forall s_{2,j} \in S_2$
that is
$g_2\left(s_1^e, s_2^e \right) \ge \ g_2\left(s_1^e, s_{2,j} \right) \forall s_{2,j} \in S_2$
E. Burger, Introduction to the Theory of Games, Prentice-Hall, Inc., 1959
Non-constant sum games are implicitly non-zero-sum games; while all constant-sum games can be considered as zero-sum games without altering the outcome of the game (dummy player).
References:
J. von Neumann, Contributions to the theory of games. Vol. IV, Annals. of Mathematics Studies, no.40, Princeton Univ. Press, 1959
John von Neumann, Oskar Morgenstern (1944): Theory of Games and Economic Behavior, Princeton University Press; ed. 1953
