Pure Pairwise Nash Equilibriums in multiplayer games

Xian

Hey everyone,

Not sure if this is the right section to post this in, but makes the most sense to me so here goes.

Consider a game with k players. Now suppose that if only two of the players are allowed to make changes to their strategies (i.e. k-2 players are frozen) then those two players have a pure strategy nash equilibrium. Phrased another way; every pair of players has a pure nash equilibrium.

My question is does this imply anything about a global nash equilibrium, specifically; if pairwise pure nash equilibriums exist, must there also exist a nash equilibrium for all k players? Do we need strengthen our assumptions to determine anything? What about in the case of 3 players?

Thanks.