Zermelo's Theorem states that "Every finite zero-sum 2-player game is determined (one of the two players has a winning strategy)." I was wondering if anyone has investigated the generalization of this theorem to more than 2-player games. A proof, reference, or counterexample to this theorem for more than 2-players would be an answer that I am looking for.
Thanks!
On
That is actually not what Zermelo has shown, ase explained here.
A winning strategy is not even defined for other games.
What is closely related to how many people (wrongly) believed Zermelo has shown such a thing is Kuhn's "algorithm": In games of perfect information, one gets a Nash equilibrium by backward induction (that also satisfies subgame perfection). One gets a unique Nash equilibrium this way, when no player is indifferent between any two plays.
I am no expert, but it seems to me that with at least three players things get messy.
Consider the following trivial game with players $A,B,C$: the first two moves of $A,B$ are irrelevant, and then $C$ chooses between two options where payoffs are $(1,-1,0)$ (i.e. $A$ wins) and $(-1,1,0)$ (i.e. $B$ wins), and the game ends there. Then, the fate of $A$ and $B$ depend on the decision by $C$, who has no particular incentive to choose either, so surely there is no hope for a winning strategy for anyone. Would that be a counterexample you are looking for?