What about three player games like 3-player isolation? >> What’s 3-player isolation? >> It’s the same as normal isolation, but with three players trying to be the last to move. >> So do the players form alliances against each other? >> They can, but there can be only one winner in the end. >> That could make the evaluation function difficult. >> Why don’t we ignore that for now and just #my_moves to make things simple? How would minimax work? >> Well, for multiplayer games we don’t use minimax anymore. Instead, we evaluate the game board from the perspective of each player and propagate the values up the tree. >> How does that work? >> Let’s a imagine a three player isolation game tree, where we search down to the level three of the tree. On the leftmost branch, we evaluate the resulting game board from each of the players’ perspectives. For player 1, the evaluation function returns a 1. For player 2, it’s a 2. And for player 3, the evaluation is a 6. >> I guess we evaluate each of the board nodes at this level, and then return triplet’s for each of them. >> Yep, and then we propagate the values up the tree. We first choose the max value at each of the level 2 branches from the player 3’s perspective. In the leftmost node, player 3 has a choice between 6 and 3, so, of course, we choose 6. In the next branch from the right on that level, there’s a choice between a 2 and a 1, so we chose 2. In the third branch from left, we choose between a 2 and a 1, so we choose the 2. And, finally, in the rightmost branch, we choose the 5. Okay and for the next level up, I guess we choose the maximum value from player 2’s perspective? >> Yep. >> So on the left branch we choose the branch with the 2, and on the right branch we chose the 5. >> Precisely, and at the top level we choose the maximum value from player 1’s perspective. In this example, both options are equally good, so we choose the one on the left by default. This version of a game tree, which we call a Max N, can work for multiplayer games with any number of players. >> Let’s try a quiz on this concept.