Uniqueness of payout of Nash equilibrium in a 2-player zero-sum game

Question

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)

Answer 1

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:

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

2. $g\left(s_1^e, s_2^e \right) \le \ g\left(s_1^e, s_{2,j} \right) \forall s_{2,j} \in S_2$

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