Modern IT Tools and Methods. Lecture 7 - Games

Август 1, 2022

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Modern IT Tools and Methods. Lecture 7 - Games

Содержание

2. Outline Optimal decisions α-β pruning Imperfect, real-time decisions
3. Games vs. search problems "Unpredictable" opponent ? specifying a move for every possible opponent reply Time
4. Game tree (2-player, deterministic, turns)
5. Minimax Perfect play for deterministic games Idea: choose move to position with highest minimax value =
6. Minimax algorithm
7. Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time
8. α-β pruning example
9. α-β pruning example
10. α-β pruning example
11. α-β pruning example
12. α-β pruning example
13. Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruning
14. Why is it called α-β? α is the value of the best (i.e., highest-value) choice found
15. The α-β algorithm
16. The α-β algorithm
17. Resource limits Suppose we have 100 secs, explore 104 nodes/sec ? 106 nodes per move Standard
18. Evaluation functions For chess, typically linear weighted sum of features Eval(s) = w1 f1(s) + w2
19. Cutting off search MinimaxCutoff is identical to MinimaxValue except Terminal? is replaced by Cutoff? Utility is
20. Deterministic games in practice Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994.
22. Скачать презентацию

Слайд 2

Outline
Optimal decisions
α-β pruning
Imperfect, real-time decisions

Слайд 3

Games vs. search problems
"Unpredictable" opponent ? specifying a move for every

possible opponent reply
Time limits ? unlikely to find goal, must approximate

Слайд 4

Game tree (2-player, deterministic, turns)

Слайд 5

Minimax
Perfect play for deterministic games
Idea: choose move to position with highest

minimax value = best achievable payoff against best play
E.g., 2-ply game:

Слайд 6

Minimax algorithm

Слайд 7

Properties of minimax
Complete? Yes (if tree is finite)
Optimal? Yes (against an

optimal opponent)
Time complexity? O(bm)
Space complexity? O(bm) (depth-first exploration)
For chess, b ≈ 35, m ≈100 for "reasonable" games ? exact solution completely infeasible

Слайд 8

α-β pruning example

Слайд 9

α-β pruning example

Слайд 10

α-β pruning example

Слайд 11

α-β pruning example

Слайд 12

α-β pruning example

Слайд 13

Properties of α-β
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
A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

Слайд 14

Why is it called α-β?
α is the value of the best

(i.e., highest-value) choice found so far at any choice point along the path for max
If v is worse than α, max will avoid it
? prune that branch
Define β similarly for min

Слайд 15

The α-β algorithm

Слайд 16

The α-β algorithm

Слайд 17

Resource limits
Suppose we have 100 secs, explore 104 nodes/sec ? 106 nodes

per move
Standard approach:
cutoff test:
e.g., depth limit (perhaps add quiescence search)
evaluation function
= estimated desirability of position

Слайд 18

Evaluation functions
For chess, typically linear weighted sum of 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.

Слайд 19

Cutting off search
MinimaxCutoff is identical to MinimaxValue except
Terminal? is replaced by

Cutoff?
Utility is replaced by Eval
Does it work in practice?
bm = 106, 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

Слайд 20

Deterministic games in practice
Checkers: Chinook ended 40-year-reign of human world champion

Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.
Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.
Othello: human champions refuse to compete against computers, who are too good.
Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.

Modern IT Tools and Methods. Lecture 7 - Games

Содержание

OutlineOptimal decisionsα-β pruningImperfect, real-time decisions

Games vs. search problems"Unpredictable" opponent ? specifying a move for every

Game tree (2-player, deterministic, turns)

MinimaxPerfect play for deterministic games Idea: choose move to position with highest

Minimax algorithm

Properties of minimaxComplete? Yes (if tree is finite) Optimal? Yes (against an

α-β pruning example

α-β pruning example

α-β pruning example

α-β pruning example

α-β pruning example

Properties of α-βPruning does not affect final result Good move ordering improves

Why is it called α-β?α is the value of the best

The α-β algorithm

The α-β algorithm

Resource limitsSuppose we have 100 secs, explore 104 nodes/sec ? 106 nodes

Evaluation functionsFor chess, typically linear weighted sum of featuresEval(s) = w1

Cutting off searchMinimaxCutoff is identical to MinimaxValue exceptTerminal? is replaced by

Deterministic games in practiceCheckers: Chinook ended 40-year-reign of human world champion

Похожие презентации

Outline
Optimal decisions
α-β pruning
Imperfect, real-time decisions

Games vs. search problems
"Unpredictable" opponent ? specifying a move for every

Minimax
Perfect play for deterministic games
Idea: choose move to position with highest

Properties of minimax
Complete? Yes (if tree is finite)
Optimal? Yes (against an

Properties of α-β
Pruning does not affect final result
Good move ordering improves

Why is it called α-β?
α is the value of the best

Resource limits
Suppose we have 100 secs, explore 104 nodes/sec ? 106 nodes

Evaluation functions
For chess, typically linear weighted sum of features
Eval(s) = w1

Cutting off search
MinimaxCutoff is identical to MinimaxValue except
Terminal? is replaced by

Deterministic games in practice
Checkers: Chinook ended 40-year-reign of human world champion