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Jarrar © 2014 1
Dr. Mustafa Jarrar
Sina Institute, University of Birzeit
mjarrar@birzeit.edu
www.jarrar.info
Mustafa Jarra...
Jarrar © 2014 2
Watch this lecture and download the slides fromWatch this lecture and download the slides from
http://jarr...
Jarrar © 2014 3
Can you plan ahead with these games
Jarrar © 2014 4
Game Tree (2-player, deterministic, turns)
Calculated by utility function,
depends on the game.
Last state...
Jarrar © 2014 5
Your Moves
My Moves
My Moves
Your Moves
Your Moves
My Moves My Moves
My Moves
• Two players take turns mak...
Jarrar © 2014 6
Computer Games
• Playing games can be seen as a Search Problem
• Multiplayer games as multi-agent environm...
Jarrar © 2014 7
Minimax
• Create a utility function
– Evaluation of board/game state to determine how strong the
position ...
Jarrar © 2014 8
Minimax Tree
Max
Min
Max
Min
You are Max and your enemy is Min.You are Max and your enemy is Min.
You play...
Jarrar © 2014 9
Minimax Tree Evaluation
• Assign utility values to leaves
– Sometimes called “board evaluation function”
–...
Jarrar © 2014 10
Minimax Tree
Max
Min
Max
Min
100
-24-8-14-73-100-5-70-12-470412-3212823
Terminal nodes: values calculated...
Jarrar © 2014 11
Minimax Tree Evaluation
For the MAX player
1. Generate the game as deep as time permits
2. Apply the eval...
Jarrar © 2014 12
Minimax Tree
-24-8-14-73-100-5-70-12-470412-3212823
Max
Min
Max
Min
Terminal nodes: values calculated fro...
Jarrar © 2014 13
Minimax Tree
100
-24-8-14-73-100-5-70-12-470412-3212823
28 -3 12 70 -4 -73 -14 -8
Max
Min
Max
Min
Other n...
Jarrar © 2014 14
Minimax Tree
100
-24-8-14-73-100-5-70-12-470412-3212823
21 -3 12 70 -4 -73 -14 -8
-3 -4 -73
Max
Min
Max
M...
Jarrar © 2014 15
Minimax Tree
100
-24-8-14-73-100-5-70-12-470412-3212823
21 -3 12 70 -4 -73 -14 -8
-3 -4 -73
-3
Max
Min
Ma...
Jarrar © 2014 16
Minimax Tree
100
-24-8-14-73-100-5-70-12-470412-3212823
21 -3 12 70 -4 -73 -14 -8
-3 -4 -73
-3
Max
Min
Ma...
Jarrar © 2014 17
MiniMax Example-2
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
Terminal nodes: values calculated from the utility ...
Jarrar © 2014 18
MiniMax Example-2
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3
Other nodes: va...
Jarrar © 2014 19
MiniMax Example-2
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3
7 6 5 5 6 4
Max...
Jarrar © 2014 20
MiniMax Example-2
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3
7 6 5 5 6 4
5 3...
Jarrar © 2014 21
MiniMax Example-2
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3
7 6 5 5 6 4
5 3...
Jarrar © 2014 22
MiniMax Example-2
Max
Max
Min
Min
4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3
4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3...
Jarrar © 2014 23
Properties of MiniMax
Complete? Yes (if tree is finite)
Space complexity? A complete evaluation takes spa...
Jarrar © 2014 24
Alpha-Beta Pruning Algorithm
Jarrar © 2014 25
Pruning the Minimax Tree
• Since we have limited time available, we want to avoid
unnecessary computation...
Jarrar © 2014 26
α Cut Example
10021 -3 12 70 -4 -73 -14
-3 -4
-3
-73
Max
Max
Min
Jarrar © 2014 27
α Cut Example
• Depth first search along path 1
21
Max
Max
Min
Jarrar © 2014 28
α Cut Example
• 21 is minimum so far (second level)
• Can’t evaluate yet at top level
21
21
Max
Max
Min
Jarrar © 2014 29
α Cut Example
• -3 is minimum so far (second level)
• -3 is maximum so far (top level)
21 -3
-3
-3Max
Max...
Jarrar © 2014 30
α Cut Example
• 12 is minimum so far (second level)
• -3 is still maximum (can’t use second node yet)
21 ...
Jarrar © 2014 31
α Cut Example
• -70 is now minimum so far (second level)
• -3 is still maximum (can’t use second node yet...
Jarrar © 2014 32
α Cut Example
• Since second level node will never be > -70, it
will never be chosen by the previous leve...
Jarrar © 2014 33
α Cut Example
• Evaluation at second level is again -73
10021 -3 12 -70 -4 -73
-3
-3
-70 -73
Max
Max
Min
Jarrar © 2014 34
α Cut Example
• Again, can apply α cut since the second level
node will never be > -73, and thus will nev...
Jarrar © 2014 35
α Cut Example
• As a result, we evaluated the Max node without
evaluating several of the possible paths
1...
Jarrar © 2014 36
β Cuts
• Similar idea to α cuts, but the other way around
• If the current minimum is less than the
succe...
Jarrar © 2014 37
β Cut Example
• Some subtrees at second level already have
values > min from previous, so we can stop
eva...
Jarrar © 2014 38
Alpha-Beta Example 2
[-∞, +∞]
– we assume a depth-first, left-to-right search as basic strategy
– the ran...
Jarrar © 2014 39
Alpha-Beta Example 2
Max
Min[-∞, 7]
[-∞, +∞]
α best choice for Max ?
β best choice for Min 7
7
Jarrar © 2014 40
Alpha-Beta Example 2
Max
Min[-∞, 6]
[-∞, +∞]
α best choice for Max ?
β best choice for Min 6
7 6
Jarrar © 2014 41
Alpha-Beta Example 2
Max
Min5
[5, +∞]
α best choice for Max 5
β best choice for Min 5
7 6 5
– Min obtains...
Jarrar © 2014 42
Alpha-Beta Example 2
Max
Min[-∞, 5]
[5, +∞]
α best choice for Max 5
β best choice for Min 3
7 6 5
– Min c...
Jarrar © 2014 43
Alpha-Beta Example 2
Max
Min[-∞, 5]
[5, +∞]
α best choice for Max 5
β best choice for Min 3
7 6 5
– Min k...
Jarrar © 2014 44
Alpha-Beta Example 2
Max
Min[-∞, 5]
[5, +∞]
α best choice for Max 5
β best choice for Min 3
7 6 5
– Min e...
Jarrar © 2014 45
Alpha-Beta Example 2
Max
Min[-∞, 5]
[5, +∞]
α best choice for Max 5
β best choice for Min 3
7 6 5
– Min i...
Jarrar © 2014 46
Alpha-Beta Example 2
Max
Min[-∞, 5]
5
α best choice for Max 5
β best choice for Min 3
7 6 5
– Min could c...
Jarrar © 2014 47
max
min
max
min
Exercise
Jarrar © 2014 48
max
min
max
min
10 9 14 2 4
10 14 4
10 4
10
Exercise (Solution)
Jarrar © 2014 49
α-β Pruning
• Pruning by these cuts does not affect final result
– May allow you to go much deeper in tre...
Jarrar © 2014 50
α-β Pruning
• Can store information along an entire path, not
just at most recent levels!
• Keep along th...
Jarrar © 2014 51
Pruning at MAX node
α is possibly updated by the MAX of successors evaluated so far
• If the value that w...
Jarrar © 2014 52
Pruning at MIN node
β is possibly updated by the MIN of successors evaluated so far
• If the value that w...
Jarrar © 2014 53
Idea of α-β Pruning
• We know β on this path is 21
• So, when we get max=70, we
know this will never be u...
Jarrar © 2014 54
Why is it called α-β?
• α is the value of the best (i.e., highest-
value) choice found so far at any
choi...
Jarrar © 2014 55
Imperfect Decisions
• Complete search is impractical for most games
• Alternative: search the tree only t...
Jarrar © 2014 56
Utility Evaluation Function
• Very game-specific
• Take into account knowledge about game
• “Stupid” util...
Jarrar © 2014 57
Utility Evaluation
• Need to assign a numerical value to the state
– Could assign a more complex utility ...
Jarrar © 2014 58
Utility Evaluation
• The individual factors are combined by some function
• Usually a linear weighted com...
Jarrar © 2014 59
Evaluation Functions
• If you had a perfect utility evaluation function, what
would it mean about the min...
Jarrar © 2014 60
Evaluation Functions for Ordering
• As mentioned earlier, order of branch evaluation can make
a big diffe...
Jarrar © 2014 61
The following are extra Examples
(Self Study)
Jarrar © 2014 62
Example: Tic-Tac-Toe (evaluation function)
• Simple evaluation function
E(s) = (rx + cx + dx) - (ro + co ...
Jarrar © 2014 63
E(s12)
8
- 6
= 2
E(s13)
8
- 5
= 3
E(s14)
8
- 6
= 2
E(s15)
8
- 4
= 4
E(s16)
8
- 6
= 2
E(s17)
8
- 5
= 3
E(s...
Jarrar © 2014 64
E(s2:16)
5
- 6
= -1
E(s2:15)
5
-6
= -1
E(s28)
5
- 5
= 0
E(s27)
6
- 5
= 1
E(s2:48)
5
- 4
= 1
E(s2:47)
6
- ...
Jarrar © 2014 65
31
Checkers Case Study
• Initial board configuration
– Black single on 20
single on 21
king on 31
– Red s...
Jarrar © 2014 66
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -8 -8
1
20
->
16
21-...
Jarrar © 2014 67
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 68
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 69
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 70
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 71
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 72
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 73
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->
16
21-...
Jarrar © 2014 74
Checkers Alpha-Beta Example
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8...
Jarrar © 2014 75
22->31
1
1
1 1 1 2
2
6
6
1
1
1 1 1 1 1
1
1 1 1 1 6
6
0
0
0 0 -4
-4
-4 -8
-8
-8 -8
-8
-8
1 0 -4 -8
1
20
->...
Jarrar © 2014 76
References
[1][1] S. Russell and P. Norvig: Artificial Intelligence: A Modern ApproachS. Russell and P. N...
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Jarrar: Games

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Lecture video by Mustafa Jarrar at Birzeit University, Palestine.
See the course webpage at: http://jarrar-courses.blogspot.com/2012/04/aai-spring-jan-may-2012.html and http://www.jarrar.info
The lecture covers: Games

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  1. 1. Jarrar © 2014 1 Dr. Mustafa Jarrar Sina Institute, University of Birzeit mjarrar@birzeit.edu www.jarrar.info Mustafa Jarrar: Lecture Notes on Games, Birzeit University, Palestine Fall Semester, 2014 Artificial Intelligence Chapter 6Chapter 6 Games (adversarial search problems)
  2. 2. Jarrar © 2014 2 Watch this lecture and download the slides fromWatch this lecture and download the slides from http://jarrar-courses.blogspot.com/2011/11/artificial-intelligence-fall-2011.html Most information based on Chapter 5 of [1] Reference: Mustafa Jarrar: Lecture Notes on Conceptual Analyses University of Birzeit, Palestine, 2015 Online Course Watch Download Slides Mustafa Jarrar Birzeit University, Palestine mjarrar@birzeit.edu www.jarrar.info Mustafa Jarrar: Online Courses – Watch and Download Slides
  3. 3. Jarrar © 2014 3 Can you plan ahead with these games
  4. 4. Jarrar © 2014 4 Game Tree (2-player, deterministic, turns) Calculated by utility function, depends on the game. Last state, game is over How to see the game as a treeHow to see the game as a tree Image from [2]
  5. 5. Jarrar © 2014 5 Your Moves My Moves My Moves Your Moves Your Moves My Moves My Moves My Moves • Two players take turns making moves • Board state fully known, deterministic evaluation of moves • One player wins by defeating the other (or else there is a tie) • Want a strategy to win, assuming the other person plays as well as possible Two-Person Perfect Information Deterministic Game
  6. 6. Jarrar © 2014 6 Computer Games • Playing games can be seen as a Search Problem • Multiplayer games as multi-agent environments. • Agents' goals are in conflict. • Mostly deterministic and fully observable environments. • Some games are not trivial search problems, thus needs AI techniques, e.g. Chess has an average branching factor of 35, and games often go to 50 moves by each player, so the search tree has about 35100 or 10154 nodes. • Finding optimal move: choosing a good move with time limits. • Heuristic evaluation functions allow us to approximate the true utility of a state without doing a complete search.
  7. 7. Jarrar © 2014 7 Minimax • Create a utility function – Evaluation of board/game state to determine how strong the position of player 1 is. – Player 1 wants to maximize the utility function – Player 2 wants to minimize the utility function • Minimax Tree – Generate a new level for each move – Levels alternate between “max” (player 1 moves) and “min” (player 2 moves)
  8. 8. Jarrar © 2014 8 Minimax Tree Max Min Max Min You are Max and your enemy is Min.You are Max and your enemy is Min. You play with your enemy in this way.You play with your enemy in this way.
  9. 9. Jarrar © 2014 9 Minimax Tree Evaluation • Assign utility values to leaves – Sometimes called “board evaluation function” – If leaf is a “final” state, assign the maximum or minimum possible utility value (depending on who would win). – If leaf is not a “final” state, must use some other heuristic, specific to the game, to evaluate how good/bad the state is at that point
  10. 10. Jarrar © 2014 10 Minimax Tree Max Min Max Min 100 -24-8-14-73-100-5-70-12-470412-3212823 Terminal nodes: values calculated from the utility function, evaluates how good/bad the state is at this point
  11. 11. Jarrar © 2014 11 Minimax Tree Evaluation For the MAX player 1. Generate the game as deep as time permits 2. Apply the evaluation function to the leaf states 3. Back-up values • At MIN assign minimum payoff move • At MAX assign maximum payoff move 1. At root, MAX chooses the operator that led to the highest payoff
  12. 12. Jarrar © 2014 12 Minimax Tree -24-8-14-73-100-5-70-12-470412-3212823 Max Min Max Min Terminal nodes: values calculated from the utility function
  13. 13. Jarrar © 2014 13 Minimax Tree 100 -24-8-14-73-100-5-70-12-470412-3212823 28 -3 12 70 -4 -73 -14 -8 Max Min Max Min Other nodes: values calculated via minimax algorithm
  14. 14. Jarrar © 2014 14 Minimax Tree 100 -24-8-14-73-100-5-70-12-470412-3212823 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 Max Min Max Min
  15. 15. Jarrar © 2014 15 Minimax Tree 100 -24-8-14-73-100-5-70-12-470412-3212823 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min
  16. 16. Jarrar © 2014 16 Minimax Tree 100 -24-8-14-73-100-5-70-12-470412-3212823 21 -3 12 70 -4 -73 -14 -8 -3 -4 -73 -3 Max Min Max Min The best next move for Max
  17. 17. Jarrar © 2014 17 MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 Terminal nodes: values calculated from the utility function Max Max Min Min Based on [3]
  18. 18. Jarrar © 2014 18 MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 Other nodes: values calculated via minimax algorithm Max Max Min Min
  19. 19. Jarrar © 2014 19 MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 Max Max Min Min
  20. 20. Jarrar © 2014 20 MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 Max Max Min Min
  21. 21. Jarrar © 2014 21 MiniMax Example-2 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 5 Max Max Min Min
  22. 22. Jarrar © 2014 22 MiniMax Example-2 Max Max Min Min 4 7 9 6 9 8 8 5 6 7 5 2 3 2 5 4 9 3 4 7 6 2 6 3 4 5 1 2 5 4 1 2 6 3 4 3 7 6 5 5 6 4 5 3 4 5 moves by Max and countermoves by Min
  23. 23. Jarrar © 2014 23 Properties of MiniMax Complete? Yes (if tree is finite) Space complexity? A complete evaluation takes space bm (depth-first exploration) For chess, b ≈ 35, m ≈100 for "reasonable" games  exact solution completely infeasible, since it’s too big Instead, we limit the depth based on various factors, including time available. » • Optimal? Yes (against an optimal opponent) • Time complexity? A complete evaluation
  24. 24. Jarrar © 2014 24 Alpha-Beta Pruning Algorithm
  25. 25. Jarrar © 2014 25 Pruning the Minimax Tree • Since we have limited time available, we want to avoid unnecessary computation in the minimax tree. • Pruning: ways of determining that certain branches will not be useful. αα CutsCuts • If the current max value is greater than the successor’s min value, don’t explore that min subtree any more.
  26. 26. Jarrar © 2014 26 α Cut Example 10021 -3 12 70 -4 -73 -14 -3 -4 -3 -73 Max Max Min
  27. 27. Jarrar © 2014 27 α Cut Example • Depth first search along path 1 21 Max Max Min
  28. 28. Jarrar © 2014 28 α Cut Example • 21 is minimum so far (second level) • Can’t evaluate yet at top level 21 21 Max Max Min
  29. 29. Jarrar © 2014 29 α Cut Example • -3 is minimum so far (second level) • -3 is maximum so far (top level) 21 -3 -3 -3Max Max Min
  30. 30. Jarrar © 2014 30 α Cut Example • 12 is minimum so far (second level) • -3 is still maximum (can’t use second node yet) 21 -3 12 -3 -3 12 Max Max Min
  31. 31. Jarrar © 2014 31 α Cut Example • -70 is now minimum so far (second level) • -3 is still maximum (can’t use second node yet) 21 -3 12 -70 -3 -3 -70 Max Max Min
  32. 32. Jarrar © 2014 32 α Cut Example • Since second level node will never be > -70, it will never be chosen by the previous level • We can stop exploring that node 21 -3 12 -70 -3 -3 -70 Max Max Min
  33. 33. Jarrar © 2014 33 α Cut Example • Evaluation at second level is again -73 10021 -3 12 -70 -4 -73 -3 -3 -70 -73 Max Max Min
  34. 34. Jarrar © 2014 34 α Cut Example • Again, can apply α cut since the second level node will never be > -73, and thus will never be chosen by the previous level 10021 -3 12 -70 -4 -73 -3 -3 -70 -73 Max Max Min
  35. 35. Jarrar © 2014 35 α Cut Example • As a result, we evaluated the Max node without evaluating several of the possible paths 10021 -3 12 -70 -4 -73 -14 -3 -3 -70 -73 Max Max Min
  36. 36. Jarrar © 2014 36 β Cuts • Similar idea to α cuts, but the other way around • If the current minimum is less than the successor’s max value, don’t look down that max tree any more
  37. 37. Jarrar © 2014 37 β Cut Example • Some subtrees at second level already have values > min from previous, so we can stop evaluating them. 10021 -3 12 70 -4 73 -14 Min Min Max 21 21 70 73
  38. 38. Jarrar © 2014 38 Alpha-Beta Example 2 [-∞, +∞] – we assume a depth-first, left-to-right search as basic strategy – the range of the possible values for each node are indicated • initially [-∞, +∞] • from Max’s or Min’s perspective • these local values reflect the values of the sub-trees in that node; the global values α and β are the best overall choices so far for Max or Min [-∞, +∞] α best choice for Max ? β best choice for Min ? Max Min
  39. 39. Jarrar © 2014 39 Alpha-Beta Example 2 Max Min[-∞, 7] [-∞, +∞] α best choice for Max ? β best choice for Min 7 7
  40. 40. Jarrar © 2014 40 Alpha-Beta Example 2 Max Min[-∞, 6] [-∞, +∞] α best choice for Max ? β best choice for Min 6 7 6
  41. 41. Jarrar © 2014 41 Alpha-Beta Example 2 Max Min5 [5, +∞] α best choice for Max 5 β best choice for Min 5 7 6 5 – Min obtains the third value from a successor node – this is the last value from this sub-tree, and the exact value is known – Max now has a value for its first successor node, but hopes that something better might still come
  42. 42. Jarrar © 2014 42 Alpha-Beta Example 2 Max Min[-∞, 5] [5, +∞] α best choice for Max 5 β best choice for Min 3 7 6 5 – Min continues with the next sub-tree, and gets a better value – Max has a better choice from its perspective, however, and will not consider a move in the sub-tree currently explored by min – Initially [-∞, +∞] [-∞,3] 3
  43. 43. Jarrar © 2014 43 Alpha-Beta Example 2 Max Min[-∞, 5] [5, +∞] α best choice for Max 5 β best choice for Min 3 7 6 5 – Min knows that Max won’t consider a move to this sub-tree, and abandons it – this is a case of pruning, indicated by [-∞,3] 3
  44. 44. Jarrar © 2014 44 Alpha-Beta Example 2 Max Min[-∞, 5] [5, +∞] α best choice for Max 5 β best choice for Min 3 7 6 5 – Min explores the next sub-tree, and finds a value that is worse than the other nodes at this level – if Min is not able to find something lower, then Max will choose this branch, so Min must explore more successor nodes [-∞,3] 3 [-∞,6] 6
  45. 45. Jarrar © 2014 45 Alpha-Beta Example 2 Max Min[-∞, 5] [5, +∞] α best choice for Max 5 β best choice for Min 3 7 6 5 – Min is lucky, and finds a value that is the same as the current worst value at this level – Max can choose this branch, or the other branch with the same value [-∞,3] 3 [-∞,5] 6 5
  46. 46. Jarrar © 2014 46 Alpha-Beta Example 2 Max Min[-∞, 5] 5 α best choice for Max 5 β best choice for Min 3 7 6 5 – Min could continue searching this sub-tree to see if there is a value that is less than the current worst alternative in order to give Max as few choices as possible – this depends on the specific implementation – Max knows the best value for its sub-tree [-∞,3] 3 [-∞,5] 6 5
  47. 47. Jarrar © 2014 47 max min max min Exercise
  48. 48. Jarrar © 2014 48 max min max min 10 9 14 2 4 10 14 4 10 4 10 Exercise (Solution)
  49. 49. Jarrar © 2014 49 α-β Pruning • Pruning by these cuts does not affect final result – May allow you to go much deeper in tree • “Good” ordering of moves can make this pruning much more efficient – Evaluating “best” branch first yields better likelihood of pruning later branches – Perfect ordering reduces time to bm/2 instead of O(bd ) – i.e. doubles the depth you can search to!
  50. 50. Jarrar © 2014 50 α-β Pruning • Can store information along an entire path, not just at most recent levels! • Keep along the path: α: best MAX value found on this path (initialize to most negative utility value) β: best MIN value found on this path (initialize to most positive utility value)
  51. 51. Jarrar © 2014 51 Pruning at MAX node α is possibly updated by the MAX of successors evaluated so far • If the value that would be returned is ever > β, then stop work on this branch • If all children are evaluated without pruning, return the MAX of their values
  52. 52. Jarrar © 2014 52 Pruning at MIN node β is possibly updated by the MIN of successors evaluated so far • If the value that would be returned is ever < α, then stop work on this branch • If all children are evaluated without pruning, return the MIN of their values
  53. 53. Jarrar © 2014 53 Idea of α-β Pruning • We know β on this path is 21 • So, when we get max=70, we know this will never be used, so we can stop here 100 21 -3 12 70 -4 21 21 70
  54. 54. Jarrar © 2014 54 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
  55. 55. Jarrar © 2014 55 Imperfect Decisions • Complete search is impractical for most games • Alternative: search the tree only to a certain depth – Requires a cutoff-test to determine where to stop • Replaces the terminal test • The nodes at that level effectively become terminal leave nodes – Uses a heuristics-based evaluation function to estimate the expected utility of the game from those leave nodes.
  56. 56. Jarrar © 2014 56 Utility Evaluation Function • Very game-specific • Take into account knowledge about game • “Stupid” utility – 1 if player 1 wins – -1 if player 0 wins – 0 if tie (or unknown) – Only works if we can evaluate complete tree – But, should form a basis for other evaluations
  57. 57. Jarrar © 2014 57 Utility Evaluation • Need to assign a numerical value to the state – Could assign a more complex utility value, but then the min/max determination becomes trickier. • Typically assign numerical values to lots of individual factors: – a = # player 1’s pieces - # player 2’s pieces – b = 1 if player 1 has queen and player 2 does not, -1 if the opposite, or 0 if the same – c = 2 if player 1 has 2-rook advantage, 1 if a 1-rook advantage, etc.
  58. 58. Jarrar © 2014 58 Utility Evaluation • The individual factors are combined by some function • Usually a linear weighted combination is used: – u = αa + βb + χc – Different ways to combine are also possible • Notice: quality of utility function is based on: – What features are evaluated – How those features are scored – How the scores are weighted/combined • Absolute utility value doesn’t matter – relative value does.
  59. 59. Jarrar © 2014 59 Evaluation Functions • If you had a perfect utility evaluation function, what would it mean about the minimax tree? You would never have to evaluate more than one level deep! • Typically, you can’t create such perfect utility evaluations, though.
  60. 60. Jarrar © 2014 60 Evaluation Functions for Ordering • As mentioned earlier, order of branch evaluation can make a big difference in how well you can prune • A good evaluation function might help you order your available moves: – Perform one move only – Evaluate board at that level – Recursively evaluate branches in order from best first move to worst first move (or vice-versa if at a MIN node)
  61. 61. Jarrar © 2014 61 The following are extra Examples (Self Study)
  62. 62. Jarrar © 2014 62 Example: Tic-Tac-Toe (evaluation function) • Simple evaluation function E(s) = (rx + cx + dx) - (ro + co + do) where r,c,d are the numbers of row, column and diagonal lines still available; x and o are the pieces of the two players. • 1-ply lookahead – start at the top of the tree – evaluate all 9 choices for player 1 – pick the maximum E-value • 2-ply lookahead – also looks at the opponents possible move • assuming that the opponents picks the minimum E-value
  63. 63. Jarrar © 2014 63 E(s12) 8 - 6 = 2 E(s13) 8 - 5 = 3 E(s14) 8 - 6 = 2 E(s15) 8 - 4 = 4 E(s16) 8 - 6 = 2 E(s17) 8 - 5 = 3 E(s18) 8 - 6 = 2 E(s19) 8 - 5 = 3 Tic-Tac-Toe 1-Ply X X X X X X X X X E(s11) 8 - 5 = 3 E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4 Based on [3]
  64. 64. Jarrar © 2014 64 E(s2:16) 5 - 6 = -1 E(s2:15) 5 -6 = -1 E(s28) 5 - 5 = 0 E(s27) 6 - 5 = 1 E(s2:48) 5 - 4 = 1 E(s2:47) 6 - 4 = 2 E(s2:13) 5 - 6 = -1 E(s2:9) 5 - 6 = -1 E(s2:10) 5 -6 = -1 E(s2:11) 5 - 6 = -1 E(s2:12) 5 - 6 = -1 E(s2:14) 5 - 6 = -1 E(s25) 6 - 5 = 1 E(s21) 6 - 5 = 1 E(s22) 5 - 5 = 0 E(s23) 6 - 5 = 1 E(s24) 4 - 5 = -1 E(s26) 5 - 5 = 0 E(s1:6) 8 - 6 = 2 E(s1:7) 8 - 5 = 3 E(s1:8) 8 - 6 = 2 E(s1:9) 8 - 5 = 3 E(s1:5) 8 - 4 = 4 E(s1:3) 8 - 5 = 3 E(s1:2) 8 - 6 = 2 E(s1:1) 8 - 5 = 3 E(s2:45) 6 - 4 = 2 Tic-Tac-Toe 2-Ply X X X X X X X X X E(s1:4) 8 - 6 = 2 X O X O X O E(s2:41) 5 - 4 = 1 E(s2:42) 6 - 4 = 2 E(s2:43) 5 - 4 = 1 E(s2:44) 6 - 4 = 2 E(s2:46) 5 - 4 = 1 O X O X O X O X X O X O X O X O XX O X OO X X O X O X O X O X O X O X OX O X O O E(s0) = Max{E(s11), E(s1n)} = Max{2,3,4} = 4
  65. 65. Jarrar © 2014 65 31 Checkers Case Study • Initial board configuration – Black single on 20 single on 21 king on 31 – Red single on 23 king on 22 – Evaluation function E(s) = (5 x1 + x2) - (5r1 + r2) where x1 = black king advantage, x2 = black single advantage, r1 = red king advantage, r2 = red single advantage 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 Based on [4]
  66. 66. Jarrar © 2014 66 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -8 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->27 16->11 31 -> 27 31->24 22->13 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 MAX MAX MIN Checkers MiniMax Example
  67. 67. Jarrar © 2014 67 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->27 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 6 MAX MAX MIN Checkers Alpha-Beta Example
  68. 68. Jarrar © 2014 68 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->27 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 1 MAX MAX MIN Checkers Alpha-Beta Example
  69. 69. Jarrar © 2014 69 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 1 β− cutoff: no need to examine further branches MAX MAX MIN Checkers Alpha-Beta Example
  70. 70. Jarrar © 2014 70 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 1 MAX MAX MIN Checkers Alpha-Beta Example
  71. 71. Jarrar © 2014 71 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 1 β− cutoff: no need to examine further branches MAX MAX MIN Checkers Alpha-Beta Example
  72. 72. Jarrar © 2014 72 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 1 MAX MAX MIN Checkers Alpha-Beta Example
  73. 73. Jarrar © 2014 73 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->13 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β 0 MAX MAX MIN Checkers Alpha-Beta Example
  74. 74. Jarrar © 2014 74 Checkers Alpha-Beta Example 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 22->31 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β -4 α− cutoff: no need to examine further branches MAX MAX MIN
  75. 75. Jarrar © 2014 75 22->31 1 1 1 1 1 2 2 6 6 1 1 1 1 1 1 1 1 1 1 1 1 6 6 0 0 0 0 -4 -4 -4 -8 -8 -8 -8 -8 -8 1 0 -4 -8 1 20 -> 16 21->17 31 -> 26 31 -> 27 22 -> 17 22 -> 18 22->25 22->26 23->26 23 -> 27 21 -> 14 16 -> 11 31->27 16->11 31->27 31->22 16->11 31 -> 27 31->24 22->18 23 -> 30 23 -> 32 20->16 31->27 31->26 21->17 20->16 21->17 20->16 20->16 21->17 31 1 2 3 4 865 9 10 11 12 161413 17 18 19 20 242221 25 26 27 28 323029 7 15 23 α 1 β -8 MAX MAX MIN Checkers Alpha-Beta Example
  76. 76. Jarrar © 2014 76 References [1][1] S. Russell and P. Norvig: Artificial Intelligence: A Modern ApproachS. Russell and P. Norvig: Artificial Intelligence: A Modern Approach Prentice Hall, 2003, Second EditionPrentice Hall, 2003, Second Edition [2][2] Nilufer Onden: Lecture Notes on Artificial IntelligenceNilufer Onden: Lecture Notes on Artificial Intelligence http://www.cs.mtu.edu/~nilufer/classes/cs4811/2014-spring/lecture-slides/cs4811-ch05-adversarial-http://www.cs.mtu.edu/~nilufer/classes/cs4811/2014-spring/lecture-slides/cs4811-ch05-adversarial- search.pdfsearch.pdf [3][3] Samy Abu Nasser: Lecture Notes on Artificial IntelligenceSamy Abu Nasser: Lecture Notes on Artificial Intelligence http://up.edu.ps/ocw/repositories/academic/up/bs/it/ITLS4213/022009/data/ITLS4213.101_11042009.ppthttp://up.edu.ps/ocw/repositories/academic/up/bs/it/ITLS4213/022009/data/ITLS4213.101_11042009.ppt [4][4] Franz Kurfess: Lecture Notes on Artificial IntelligenceFranz Kurfess: Lecture Notes on Artificial Intelligence http://users.csc.calpoly.edu/~fkurfess/Courses/Artificial-Intelligence/F09/Slides/3-Search.ppthttp://users.csc.calpoly.edu/~fkurfess/Courses/Artificial-Intelligence/F09/Slides/3-Search.ppt
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