lecture4c - CMSC 498T, Game Theory 4c. Lookahead Pathology...

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Nau: Game Theory 1 Updated 10/5/11 CMSC 498T, Game Theory 4c. Lookahead Pathology Dana Nau University of Maryland
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Nau: Game Theory 2 Updated 10/5/11 Motivation When discussing game-tree search, I said that Ø looking farther ahead (larger d ) usually gives better decisions For many years, people assumed that it would always give better decisions For my Ph.D. work in 1979, I tried to figure out why that would be true Ø What I found is that it isn’t true Ø There are pathological games in which deeper search gives worse decisions During the past 32 years, people have continued to do research on this topic Ø Much more is known about the conditions under which deeper search helps or hurts I’ll summarize the results. More details here if you want them: Ø D. S. Nau, M. Lu š trek, A. Parker, I. Bratko, and M. Gams. When is it better not to look ahead? Artificial Intelligence, 2010 http://www.cs.umd.edu/~nau/papers/nau10when.pdf
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Nau: Game Theory 3 Updated 10/5/11 Example h = 4 P-games Ø A class of games invented by Judea Pearl in 1980 Ø Different (and easier to understand) class of games than the ones in my PhD work Playing board Ø A rectangular grid of squares like a chess board Ø But instead of 8 × 8, it’s 2 Σ h /2 Φ × 2 Ρ h /2 Τ To set up the playing board, randomly label each square as Ø “win” (green) with some probability p 0 Ø or “loss” (white) otherwise Two players take turns splitting the board into smaller and smaller pieces until only one square is left
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Nau: Game Theory 4 Updated 10/5/11 1 st player’s move: Ø remove either the left half or right half of the board 2 nd player’s move: Ø remove the top half or bottom half of the board Continue until one square is left Ø “win” square: player who moved there wins Ø “loss” square: player who moved there loses P-Games win loss loss win win loss win loss loss loss win win win win loss win h = 4
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Nau: Game Theory 5 Updated 10/5/11 Game ends after h moves Ø So the same player always makes the last move Ø Call that player Max and the other Min “win” square à win for Max à u = +1 “loss” square à loss for Max à u = –1 P-Games +1 +1 +1 +1 +1 –1 +1 +1 –1 –1 +1 +1 –1 +1 –1 –1 –1 +1 +1 +1 +1 –1 +1 Min Max Min Max +1 +1 +1 +1 –1 +1 +1 +1 h = 4
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Nau: Game Theory 6 Updated 10/5/11 Decision Quality Suppose we do minimax search from a node x Under what conditions does deeper search make it more likely that we’ll choose the best move? Under what conditions is Pr [best choice | deeper search] > Pr [best choice | shallower search] ? If both of x ’s children have the same utility value, then it makes no difference which of them we choose, so it makes no difference how deep we search Ø In this case, we say that x is non-critical It’s only when a node is critical (its children have different utility values) that the search can make a difference
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Nau: Game Theory 7 Updated 10/5/11 Probability that a Node is Critical Some games have more critical nodes than others
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lecture4c - CMSC 498T, Game Theory 4c. Lookahead Pathology...

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