# Is a zero-sum game always a two player game?

Can a zero-sum game involve more than 2 players?

It does not seem to make sense to me, given that most books define a zero sum game as a game whereby the payoff matrices satisfy $$A + B = 0$$

Or $$B = -A$$.

Now, it is possible...to extend this to, $$A + B + C= 0$$

But this definition is not intuitive, because it could be that $$B = -A/2, C = -A/2$$, but $$B$$ and $$C$$ players are not necessarily against each other.

There could be some other non-standard definition of a zero-sum game perhaps I am missing.

Can someone clarify this for me?

• A zero sum game between $n$ players is a game where the net advantage of all the players is $0$, it need not be only between 2 players. Some examples of zero-sum games between more than 2 players would be most card games (eg, poker, bridge, etc). See here for an analysis of a 3-player zero-sum game model. – Prasun Biswas Jun 22 at 6:32

A zero-sum game need not be a two-player game.

Consider, for example, a game involving three players $$P, Q, R$$.

A coin is tossed. $$P$$ predicts the outcome. If $$P$$'s prediction comes true, $$Q$$ and $$R$$ give him a dollar each, else he pays a dollar to each one of them.

The definition of zero-sum applies immediately to any number of players, as others have noted.

However, the nice properties of two-player zero-sum games, like the minmax theorem, unique value, interchangeability of equilibrium strategies, do not hold for 3 or more player zero-sum games unless extra conditions are imposed.

In particular note that one can take any two-player non-zero-sum (aka general-sum) game and add a third dummy player, whose actions have no effect on the original two players, and whose payoffs are set to make the game zero-sum. Now any equilibrium of this three-player game immediately gives an equilibrium of the original two-player game, so three-player zero-sum games are as hard to solve and at least as general as two-player general-sum games.

A class of many-player zero-sum games in which a number, but not all, of the nice properties persist is the case of zero-sum polymatrix games -- see:

Yang Cai, Ozan Candogan, Constantinos Daskalakis, Christos H. Papadimitriou: Zero-Sum Polymatrix Games: A Generalization of Minmax. Math. Oper. Res. 41(2): 648-655 (2016).

A zero-sum game only requires the sum of the pay-offs to be zero. The number of players is not an issue. In fact, a game in which every time a player loses (or gains) something, the same is won (or lost) by another, is a zero-sum game.