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)

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

• maybe im being obtuse but this answer doesnt seem to address payout or uniqueness at all. Jan 14 at 21:19
• Please refer to minimax theorem or Nash theorem for non copperative games: the 2 theorems do not garantee uniqueness of solution. In case of simmetric Games, the payout for zero sum Games is equal to zero. Jan 14 at 23:17
• What do you mean uniqueness of solution? I'm well aware NE is not unique. But I'm asking about payout. I recall reading that is unique in the circumstances in the OP. You haven't really confirmed or denied this, and also the references in the comment and your answer are different, which is confusing to say the least. I'm really only asking about this one fact, whether payout is unique Jan 15 at 1:18
• At page n.47-48 in E. Burger, Introduction to the Theory of Games, Prentice-Hall, Inc., 1959 it is written: "Besides the question of existence, the question of the uniqueness of equilibrium points is also important. In connection with this, it is desiderable to consider two equilibrium point as equivalent if all of their respective payoffs agree. Theorem: In a costant-sum two-person game all equilibrium points are equivalent. Jan 15 at 8:29
• Ah. thank you so much! Jan 15 at 8:34