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Not let's see whether we could simply replace all opponents by MIN players (this is suggested sometimes, so I found it useful from a didactic point of view). Now let's assume we have more than 2 players, say 3 for simplicity. Thus, both players were maximizing their own reward and hence played rational. This is going to be very important later when we talk about more than two players! What is rationality? Every player aims at maximizing their own reward(!) again assuming that all other players play rational as well.īack to 2-player zero-sum: We did exploit/assume rationality because P1 (MAX) was already maximizing its reward anyway (by definition), and MIN was also maximizing its own reward, because it was minimizing that of MAX (which is, because of zero-sum the same as maximizing MINs reward). Furthermore, minimizing payoff-P2 is the the same maximizing payoff-P1.įor almost all the games (except in real-life when psychological factors come into play, say revenge without caring for your own loss) we always assume that all agents play rational. Thus, we can always infer the win of the other and hence only represent the outcome from P1's perspective. However, since we usually consider zero-sum games, we know that their sum always equals zero, i.e., payoff-P1 + payoff-P2 = 0. So, to be super-formally correct, one would have to give a 2-tuple as game outcome per search node, say (payoff-P1,payoff-P2), with payoff-P1 being the outcome for P1 (MAX) and payoff-P2 being the outcome for P2 (MIN). However, technically, there are two! That of the MAX player and that of the MIN player. As a consequence, MiniMax is usually always described with only a single game outcome. So let's first consider whether one can simply turn 2-player MiniMax into an n-player MiniMax by considering all opponents as Min-Players, i.e., they will all try to minimize the reward of the MAX player.įirst, recall that MiniMax is always exclusively considering 2-player zero-sum games. Then, finally, I give references to the correct alternative, i.e., the Max^n algorithm. Anyway, below I explain why the standard 2-player MiniMax can indeed not be used in a multiplayer setting with 3 players or more.

The generalization to n players is thus called Max^n, which can still be regarded MiniMax, somehow.

If "MiniMax" refers to the "standard variant" of MiniMax then he is perfectly right! (And this is what he meant.) However, you can also regard MiniMax as MaxiMax, where two players maximize their own reward each (this is exactly what AlphaWolf wrote in the question). Mihai Maruseac is only partially right with saying that MiniMax cannot be used anymore.