Adversarial Search: Game Playing Reading: Chess paper
2 Agenda Review of game-playing approaches A how-to example: Checkers Your homework
3 Minimax Algorithm Alternate levels of tree represent MAX (computer) and MIN (opponent) Depth first search for optimal strategy for MAX Assume both players make optimal move at each point Back minimax values up the tree MAX MIN
4 Search Formulation States: board configurations Operators (Successor function): legal moves Goal test: (for max) a terminal state with high utility Utility function: numeric values for final states. E.g., win, loss, draw with values 1, - 1, 0
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6 MAX MIN
7 MAX MIN
8 MAX MIN
9 MAX MIN 1
10 MAX MIN
11 MAX MIN
12 Move: A turn by P1 and response by P2 Ply: A level in the tree corresponding to a single turn Ply is used to specify how deep a program searches
13 Which values are necessary? XX
14 value is a lower-bound on the actual value of a MAX node value is an upper-bound on actual value of a MIN node ≤ 3 ≥ 3
15 ≤ 3 ≥ 3 ≤ 2 > β so no need to look further
16 Pruning Methods Alpha-Beta pruning Heuristic evaluation functions Evaluate a board state to produce an estimate of the utility at end game Order nodes by evaluation function results
17 Heuristics: evaluation functions Bound the depth of search, and use an evaluation function to estimate value of current board configurations E.g., Othello: #white pieces - #black pieces E.g., Chess: Value of all white pieces – Value of all black pieces Typical values from –infinity (lost) to +infinity (won) or [-1,+1] turn non-terminal nodes into terminal leaves And, - pruning continues to apply -> Use expert knowledge and/or machine learning
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20 Building a program to play checkers Play the game: ml ml What are good strategies?
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22 The α-β algorithm
23 Suppose We want to add in heuristic evaluation function We want to specify how many ply the program will search We want to make it specific to checkers We want to order expansion of nodes by evaluation function