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Adversarial Search

CS 171/271(Chapter 6)

Some text and images in these slides were drawn fromRussel & Norvig’s published material

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Games Multi-agent environment

Agent needs to consider actions of other agents

Games: Adversarial Search Problems Considerations

Many possible moves of other player Time (need to optimize, or approximate)

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Game as a Search Problem Initial State Successor Function

Note the turn-taking aspect (“ply”) Terminal test

“Goal”: game over (leaf nodes) Utility Function

Score or outcome (examples?)

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Game Tree

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Infallible Opponent Assumption

Strategy: select the best move that assumes the your opponent will make the best play Need to consider all possible opponent moves

Minimax value of a node in the game tree Leaf node: minimax value = utility value Agent (called MAX) picks a move that results in

a state with maximum utility; minimax value of the node is that maximum

Opponent picks the move that minimizes utility for the agent; minimax value of the node is that minimum

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Minimax Values

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Minimax Algorithm

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α-β (alpha-beta) Pruning May skip examination of some nodes If a node has no impact on the min/max

choice at upper levels, prune that node Need to maintain

α -> highest valued choice so far along path for MAX

β -> lowest valued choice so far along path for MIN

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α-β pruning: omit examination of these nodes;Minimum of 2 cannot yield a maximum higher than 3

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About α-β pruning Effectiveness is highly dependent

on order in which successors are examined

Can reduce effective tree depth to half its value

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Other Considerationsin Games Because of time constraints, may

have to settle with estimate of utility (evaluation function) Non-terminal nodes turned into

leaves Elements of chance

e.g., dice and cards Min, max, and chance nodes

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State of the Art Checkers: Chinook ended 40-year-reign of

human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.

Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.

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State of the Art Othello: human champions refuse to

compete against computers, who are too good.

Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.