What's the point of Nash Equilibrium in 3+-player games if it's unstable? For example, in a 3-player game, if 2 players deviate from the equilibrium strategy, there is no guarantees for the 3rd player that he maintains the equilibrium value playing a Nash Equilibrium strategy.
What exactly I'm asking is why it was defined as a set of mutually best responses for all the players, when in fact, each player plays against all the others. If we defined it as (Player1BestResponse; OthersBestResponse), which means player 1 tries to maximize his value and the others try to minimize it, we could maintain the properties of 2-player Nash Equilibrium. Player 1 could guarantee he would get a certain value playing this kind of strategy.
Best Answer
The reason one defines Nash equilibrium as the non-existence of profitable unilateral deviations for each player is that anything more requires coalitional thinking on behalf of the other players, which, in a non-cooperative context, there is no reason to presuppose.
In your example, certainly two players could 'work together' to try to play against the third in some fashion, but one would have to ask also if there is any incentive for this teamwork to happen. If there is no incentive for either member of the two-player coalition to work together, then your example would seem normatively unreasonable as a solution concept for a game: it would require those two players to act against their self-interest. If, however, there is incentive for them to act as a coalition against the third (in the sense that both of these players lack a profitable deviation from doing so) then your solution concept reduces to Nash equilibrium anyway.