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Multiplayer games where there is an adversary trying to make you lose.1Utility function assigns scores to leaves. 2For zero sum games, you have an utility function that assigns values according to point of view of player 1 (X).Player 1 is trying to maximize this utility fn, Player 2 tries to minimize.Important NOTE: Here we assume players can see to the bottom of the search tree and play optimallyOne ply is a move from one of the players. This is a 2 ply tree: Max makes one move and Min makes one move.Max is 1stplayer who tries to maximize utility. Represented by triangle upMin is 2ndplayer who tries to minimize utility. Represented by triangle down(Notation: up triangle is MAX, inverted triangle is MIN)345Never worse. Basic argument: if we are facing a suboptimal opponent and they could force us to take a worse outcome by being stupid, then the adversary would be better than the optima, contradiction. More formal argument: use induction on the definition of minimax values (base case: leaves).6If it is player X’s turn, then player X will try to maximize his or her score.7Answer to rhetorical question: “No”. We do not need to look at the other nodes from the middle branch after seeing “2”, because we know our MIN opponent will select something that has at most a payout of 2 for us, and we can already get 3 by going left.8Alpha and beta are initalized to negative and positive in910We set the beta of this min node to the max value of its successor.1112We have determined the value of the left move node is 3, so we set alpha = beta = 3 and propogate upwards.1314We pass down the 3 to our middle node search, because we know we can achieve it.15If alpha >= beta, then prune!1617In this branch, we might be able to get 141819Now we see 5.2021Now we see 2. If the right branch had more nodes, or if we looked at that node firstwe could prune as our alpha >= beta22Imagine the m > n. We can terminate our search in the squiggly branch.23Here we do not need to look at 4 or 6, because we already have found 2 < 3. If the order were 6,4,2 we would need to expand all the nodes in the middle branch.24Can we prune at every level? No. Only every other level. Need to do half the branching as before.Can establish average-case bounds by considering that we’ll find the max or min value after b/2 expansions (halfway between best and worst).2535^10 is 10-ply lookahead.26Characteristics we want in our heuristic: fast, good (not necessarily an underestimate).In general, eval gets better closer to the leaves. If eval was consistently accurate, then there would be little incentive to look far down the tree. You would just look at successors27Thought experiment: where do the values of pawns, knights, bishops, etc. come from? Can you think of ways to generate these values?28According to Manuela, the first big-time computer chess competition took place in the Wean Hall lounge.293031Sometimes we call chance nodes moves by the “nature” player.32Calculate the expected values at chance


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CMU CS 15381 - lecture

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