Search: Games

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Search Games Adversarial Search

• Artificial Intelligence
• CMSC 25000
• January 29, 2002

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Agenda

• Game search characteristics
• Minimax procedure
• Adversarial Search
• Alpha-beta pruning
• If its bad, we dont need to know HOW awful!
• Game search specialties
• Progressive deepening
• Singular extensions

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Games as Search

• Nodes Board Positions
• Each ply (depth 1) Move
• Special feature
• Two players, adversial
• Static evaluation function
• Instantaneous assessment of board configuration
• NOT perfect (maybe not even very good)

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Minimax Lookahead

• Modeling adversarial players
• Maximizer positive values
• Minimizer negative values
• Decisions depend on choices of other player
• Look forward to some limit
• Static evaluate at limit
• Propagate up via minimax

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Minimax Procedure

• If at limit of search, compute static value
• Relative to player
• If minimizing level, do minimax
• Report minimum
• If maximizing level, do minimax
• Report maximum

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Minimax Example
2
MAX
MIN
2
1
MAX
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Alpha-Beta Pruning

• Alpha-beta principle If you know its bad, dont
  waste time finding out HOW bad
• May eliminate some static evaluations
• May eliminate some node expansions
• Similar to branch bound

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Simple Alpha-Beta Example
2
gt2
MAX
MIN
lt1
2
MAX
1
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Alpha-Beta Detailed Example
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Alpha-Beta Procedure
If levelTOP_LEVEL, alpha NEGMAX beta
POSMAX If (reached Search-limit), compute
return static value of current If level is
minimizing level, While more children to explore
AND alpha lt beta ab alpha-beta(child) if
(ab lt beta), then beta ab Report beta If level
is maximizing level, While more children to
explore AND alpha lt beta ab
alpha-beta(child) if (ab gt alpha), then alpha
ab Report alpha
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Alpha-Beta Pruning Analysis

• Worst case
• Bad ordering Alpha-beta prunes NO nodes
• Best case
• Assume cooperative oracle orders nodes
• Best value on left
• If an opponent has some response that makes move
  bad no matter what the moving player does, then
  the move is bad.
• Implies check move where opposing player has
  choice, check all own moves

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Optimal Alpha-Beta Ordering
1
4
3
2
7
5
6
10
8
9
13
11
12
35
36
37
38
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40
32
33
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29
30
31
26
27
28
23
24
25
14
15
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21
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Optimal Ordering Alpha-Beta

• Significant reduction of work
• 11 of 27 static evaluations
• Lower bound on of static evaluations
• if d is even, s 2bd/2-1
• if d is odd, s b(d1)/2b(d-1)/2-1
• Upper bound on of static evaluations
• bd
• Reality somewhere between the two
• Typically closer to best than worst

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Heuristic Game Search

• Handling time pressure
• Focus search
• Be reasonably sure best option found is likely
  to be a good option.
• Progressive deepening
• Always having a good move ready
• Singular extensions
• Follow out stand-out moves

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Progressive Deepening

• Problem Timed turns
• Limited depth
• If too conservative, too shallow
• If too generous, wont finish
• Solution
• Always have a (reasonably good) move ready
• Search at progressively greater depths
• 1,2,3,4,5..

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Progressive Deepening

• Question Arent we wasting a lot of work?
• E.g. cost of intermediate depths
• Answer (surprisingly) No!
• Assume cost of static evaluations dominates
• Last ply (depth d) Cost bd
• Preceding plies b0 b1b(d-1)
• (bd - 1)/(b -1)
• Ratio of last ply cost/all preceding b - 1
• For large branching factors, prior work small
  relative to final ply

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Singular Extensions

• Problem Explore to some depth, but things change
  a lot in next ply
• False sense of security
• aka horizon effect
• Solution Singular extensions
• If static value stands out, follow it out
• Typically, forced moves
• E.g. follow out captures

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Additional Pruning Heuristics

• Tapered search
• Keep more branches for higher ranked children
• Rank nodes cheaply
• Rule out moves that look bad
• Problem
• Heuristic May be misleading
• Could miss good moves

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Summary

• Game search
• Key features Alternating, adversarial moves
• Minimax search Models adversarial game
• Alpha-beta pruning
• If a branch is bad, dont need to see how bad!
• Exclude branch once know cant change value
• Can significantly reduce number of evaluations
• Heuristics Search under pressure
• Progressive deepening Singular extensions