# Pure Pairwise Nash Equilibriums in multiplayer games

by Xian
Tags: combinatorics, game theory, linear algebra
 P: 25 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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