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.