CMSC 421: Intro to Artificial Intelligence

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CMSC 421 Intro to Artificial Intelligence
October 6, 2003 Lecture 7 Games Professor
Bonnie J. Dorr TA Nate Waisbrot
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Reading
Finish Chapter 6 start Chapter 7.
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Games

• Games are a form of multiagent environment
• Why study games?
• Fun historically entertaining
• Interesting subject of study because they are
  hard
• Easy to represent and agents restricted to small
  number of actions

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Relation of Games to Search

• Search no adversary
• Solution is (heuristic) method for finding goal
• Heuristics and CSP techniques can find optimal
  solution
• Evaluation function estimate of cost from start
  to goal through given node
• Examples path planning, scheduling activities
• Games adversary
• Solution is strategy
• Time limits force an approximate solution
• Evaluation function evaluate goodness of game
  position
• Examples chess, checkers, Othello, backgammon

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Types of Games
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Game Playing

• Two players MAX and MIN
• MAX moves first and they take turns until the
  game is over
• A move is planned and executed as follows
• consider all legal moves
• evalute resulting position for each one
• pick best position and move there
• opponent moves
• repeat

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Partial Game Tree for Tic-Tac-Toe
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Minimax Algorithm
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Two-Ply Game Tree
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Two-Ply Game Tree
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Two-Ply Game Tree
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Two-Ply Game Tree
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Properties of Minimax
?
?
?
?
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How do we deal with resource limits?

• Evaluation function return an estimate of the
  expected utility of the game from a given
  position
• Alpha-beta pruning return appropriate minimax
  decision without exploring entire tree

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Heuristic Evaluation Function Tic Tac Toe
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Heuristic Evaluation Functions
Eval(s) w1f1(s) w2f2(s) wnfn(s)
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What is Alpha-Beta Cutoff?
3
3,8
MAX
3
2
-8,2
MIN
3
12
2
MAX
X
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Alpha-beta Pruning

• Replace TERMINAL-TEST
• and UTILITY in Minimax
• if TERMINAL-TEST(state) then return
  UTILITY(state)
• if CUTOFF-TEST(state) then return EVAL(state)

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Minimax with Alpha-Beta Pruning

• Set alpha-beta values for all game-tree nodes to
  -8, 8
• For each move, run DFS to generate game tree
  (DFS) to a given depth
• Back-up evaluation estimates whenever possible
• For MAX node n set alpha(n) to be max value found
  so far
• For MIN node n set beta(n) to be min value found
  so far
• Prune branches that do not change final decision
• Alpha cutoff stop searching below MIN node if
  its beta value is less than or equal to the alpha
  value of its ancestor
• Beta cutoff stop searching below MAX node if its
  alpha value is greater than or equal to the beta
  value of its ancestor

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Alpha-Beta Example
-8,8
-8, -8
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Alpha-Beta Example (continued)
3,8
-8,3
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Alpha-Beta Example (continued)
3,8
-8,3
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Alpha-Beta Example (continued)
3,8
3,3
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Alpha-Beta Example (continued)
3,8
-8,2
3,3
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Alpha-Beta Example (continued)
,
3,14
-8,2
3,3
-8,14
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Alpha-Beta Example (continued)
,
3,5
-8,2
3,3
-8,5
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Alpha-Beta Example (continued)
3,3
2,2
-8,2
3,3
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Alpha-Beta Example (continued)
3,3
2,2
-8,2
3,3
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Final Comments about Alpha-Beta Pruning

• Pruning does not affect final results
• Good move ordering improves effectiveness of
  pruning
• With perfect ordering, time complexity is
  O(bm/2)
• Definition of optimal play for MAX assumes MIN
  plays optimally maximizes worst-case outcome for
  MAX.
• But if MIN does not play optimally, MAX will do
  even better. Can be proved.