Adversarial Search: Game Playing

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Adversarial Search Game Playing

• Reading Chess paper

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Agenda

• Review of game-playing approaches
• A how-to example Checkers
• Your homework

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• 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
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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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MAX
MIN
-1
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MAX
MIN
-1
-1
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MAX
MIN
-1
-1
-1
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MAX
MIN
-1
-1
-1
1
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MAX
MIN
-1
1
1
-1
1
-1
1
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MAX
MIN
-1
1
1
-1
1
-1
1
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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
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Which values are necessary?
X
X
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• ? 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
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? 3
? gt ß so no need to look further
? 3
? 2
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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

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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
• -gt Use expert knowledge and/or machine learning

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Building a program to play checkers

• Play the game http//www.darkfish.com/checkers/Ch
  eckers.html
• What are good strategies? http//dqsoft.com/checke
  rstips.html

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The a-ß algorithm
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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