Adversarial Search CS 171/271 (Chapter 6) Some text and images in these slides were drawn from Russel & Norvig’s published material

Games Multi-agent environment Games: Adversarial Search Problems 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)

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?)

Game Tree

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

Minimax Values

Minimax Algorithm

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

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

About α-β pruning Effectiveness is highly dependent on order in which successors are examined Can reduce effective tree depth to half its value

Other Considerations in 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

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.

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.