1. Game Playing
Fifth Lecture
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2. Contents: Minimax algorithm Resource limits - pruning
Games of chance
Example: Tic Tac Toe
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3. Types of games
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4. Game tree (2-player, deterministic, turns)
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5. Game Trees
You learn about how programs can play games, such as checkers and chess.
p = d = 2
The ply of a game tree, p = d, where d is the depth of a tree.
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6. Game Trees (cont.) A game tree is a representation
That is a semantic tree
In which
Nodes denote board configurations
Branches denote moves
With writers that
Establish that a node is for the maximizer or for the minimizer
Connect a board configuration with a board-configuration description
With readers that
Determine whether the node is for the maximizer of minimizer
Produce a board configuration’s description
Exhaustive Search Is Impossible!
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7. Minimax Perfect play for deterministic, perfect-information games
Idea: choose move to position with highest minimax value
= best achievable payoff against best play
Evaluations
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8. Minimax algorithm
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9. Properties of minimax
Complete？ Yes, if tree is finite (chess has specific rules for this)
Optimal？ Yes, against an optimal opponent. Otherwise??
Time？ complexity O(b^m)
Space？ complexity O(bm) (depth-first exploration)
For chess, b  35, m  100 for “reasonable” games
=> exact solution completely infeasible
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10. Resource limits Suppose we have 100 seconds, explore 10^4 nodes/second
=>10^6 nodes per move
Standard approach:
cutoff test, e.g., depth limit
evaluation function = estimated desirability of position
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11. Evaluation functions For chess, typically linear weighted sum features
Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s)
e.g., w1 = 9 with
f1(s) = (number of white queens) - (number of black queens), etc.
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12. Exact values don't matter for deterministic games
Behaviour is preserved under any monotonic transformation of Eval
Only the order matters:
payoff in deterministic games acts as an ordinal utility function.
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13. Cutting off search MinimaxCutoff is identical to MinimaxValue except
1. Terminal? is replaced by Cutoff?
2. Utility is replaced by Eval
Does it work in practice?
b^m = 10^6, b=35 => m=4
4-ply lookahead is a hopeless chess player!
4-ply  human novice
8-ply  typical PC, human master
12-ply  Deep Blue, Kasparov
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14. - pruning example The Principle:
If you have an idea that is surely bad, do not take time to see how truly awful it is.
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15. Another example
The step number of the process
Max
Min
Max
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16. Properties of - Pruning does not affect final result
Good move ordering improves effectiveness of pruning
With “perfect ordering,” time complexity = O(bm/2)
=> doubles depth of search
=> can easily reach depth 8 and play good chess
A simple example of the value of reasoning about which
computations are relevant (a form of meta reasoning)
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17. Why is it called - ?
is the best value (to max) found so far off the current path
If V is worse than , max will avoid it
=> prune that branch
Define similarly for min
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18. An Example of Game Tree
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19. Example: Tic Tac Toe
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20. Example: The Minimax Procedure Applied to Playing Tic-Tac-Toe
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21. Two-ply minimax applied to the opening move of tic-tac- toe
Min’s move
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22. Two-ply minimax applied to X’s second move of tic-tac- toe
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23. Two-ply minimax applied to X’s move near end game
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24. Summary Games are fun to work on! (and dangerous)
They illustrate several important points about AI
perfection is unattainable => must approximate
good idea to think about what to think about
uncertainty constrains the assignment of values to states
Games are to AI as grand prix racing is to automobile design
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25. End of Lecture 5
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