Pure Pairwise Nash Equilibriums in multiplayer games

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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.
 
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I'm afraid it is not as easy. Search for "multiplayer Nash equilibrium", there are plenty of seminar or lecture papers.
 

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