1. Chapter 6 :
Game Search
게임 탐색
(Adversarial Search)

2. 게임 탐색의 특성 - Exhaustive search is almost impossible.
==> mostly too many branching factor and too many depths.
(e.g., 바둑: (18 * 18)! ), 체스:DeepBlue ?
- 정적평가점수(Static evaluation score) ==> board quality
- maximizing player ==> hoping to win (me)
minimizing player ==> hoping to lose (enemy)
- Game tree ==> is a semantic tree with node (board configuration)
and branch (moves).
original
board state
new
board state
new
board state

3. Minimax Game Search
Idea: take maximum score at maximizing level (my turn).
take minimum score at minimizing level (your turn).
maximizing level 2 7 1 8 2 7 1 8 2 1
minimizing level
maximizing level
상대는?
나는? 2 2 7 1 8
“this move
gurantees
best”

4. 최소최대 탐색 예 평가함수 값을 최대화 시키려는 최대화자 A의 탐색 최소화자 B의 의도를 고려한 A의 선택 [c1] [c2]
f=0.8
[c2]
f=0.3
[c3]
f=-0.2 A
최소화자(B) 단계
[c1]
f=0.8
[c2]
f=0.3
[c3]
f=-0.2
[c11]
f=0.9
[c12]
f=0.1
[c13]
f=-0.6
[c21]
f=0.1
[c22]
f=-0.7
[c31]
f=-0.1
[c32]
f=-0.3

5. Minimax Algorithm Function MINIMAX( state ) returns an action
inputs: state, current state in game
v = MAX-VALUE (state)
return the action corresponding with value v
Function MAX-VALUE( state ) returns a utility value
if TERMINAL-TEST( state ) then return UTILITY( state )
v = -
for a, s in SUCCESSORS( state ) do
v = MAX( v, MIN-VALUE( s ) )
return v
Function MIN-VALUE( state ) returns a utility value
v = 
v = MIN( v, MAX-VALUE( s ) )

6. Minimax Example
MAX node
MIN node 3 12 8 2 4 6 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

7. Minimax Example
MAX node
MIN node 3 12 8 2 4 6 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

8. Minimax Example
MAX node
MIN node 3 2 2 3 12 8 2 4 6 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

9. Minimax Example
MAX node 3
MIN node 3 2 2 3 12 8 2 4 6 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

10. Tic-Tac-Toe
Tic-tac-toe, also called noughts and crosses (in the British Commonwealth countries) and X's and O's in the Republic of Ireland, is a pencil-and-paper game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The X player usually goes first. The player who succeeds in placing three respective marks in a horizontal, vertical, or diagonal row wins the game.
The following example game is won by the first player, X:

13. Save
time

14. Game tree (2-player)
How do we search this tree to find the optimal move? 14

15. Applying MiniMax to tic-tac-toe
The static heuristic evaluation function

17. a-b Pruning Idea: 탐색 공간을 줄인다! (mini-max  지수적으로 폭발)
a-b principle: “if you have an idea that is surely bad,
do not take time to see how truly bad it is.” =2
>=2
Max
>=2
a-cut
Min =2 =2
<=1 2 7 2 7 2 7 1

18. 최대화 노드에서 가능한 최소의 값(알파 )과 최소화의 노드에서 가능한 최대의 값(베타 )를 사용한 게임 탐색법
알파베타 가지치기
최대화 노드에서 가능한 최소의 값(알파 )과 최소화의 노드에서 가능한 최대의 값(베타 )를 사용한 게임 탐색법
기본적으로 DFS로 탐색 진행
[c0]
=0.2
[c1]
f=0.2
[c2]
f= -0.1
a-cut
[c11]
f=0.2
[c12]
f=0.7
[c21]
f= -0.1
[c22]
[c23]
C21의 평가값 -0.1이 C2에 올려지면
나머지 노드들(C22, C23)을
더 이상 탐색할 필요가 없음

19. Tic-Tac-Toe Example with Alpha-Beta Pruning
Backup Values

20. a-b Procedure a never decrease (initially - infinite) -∞
b never increase (initially infinite) +∞
- Search rule:
1. a-cutoff ==> cut when below any minimizing node
that have a b <= a (ancestor).
2, b-cutoff ==> cut when below any maximizing node
that have a a >= b (ancestor).

21. Example
max
a-cut
min
max
b-cut
min 90 89
100 99 60 59 75 74

22. Alpha-Beta Pruning Algorithm
Function ALPHA-BETA( state ) returns an action
inputs: state, current state in game
v = MAX-VALUE (state, -, +)
return the action corresponding with value v
Function MAX-VALUE( state, ,  ) returns a utility value
, the value of the best alternative for MAX along the path to state
, the value of the best alternative for MIN along the path to state
if TERMINAL-TEST( state ) then return UTILITY( state )
v = -
for a, s in SUCCESSORS( state ) do
v = MAX( v, MIN-VALUE( s, ,  ) )
if v >=  then return v
 = MAX(, v )
return v

23. Alpha-Beta Pruning Algorithm
Function MIN-VALUE( state, ,  ) returns a utility value
inputs: state, current state in game
, the value of the best alternative for MAX along the path to state
, the value of the best alternative for MIN along the path to state
if TERMINAL-TEST( state ) then return UTILITY( state )
v = +
for a, s in SUCCESSORS( state ) do
v = MIN( v, MAX-VALUE( s, ,  ) )
if v <=  then return v
 = MIN( , v )
return v

24. Alpha-Beta Pruning Example
, , initial values
=−
 =+
, , passed to kids
=−
 =+
The nodes are “MAX nodes” The nodes are “MIN nodes”

25. Alpha-Beta Pruning Example
=−
 =+
=−
 =3
MIN updates , based on kids 3
The nodes are “MAX nodes” The nodes are “MIN nodes” 25 25

26. Alpha-Beta Pruning Example
[-, + ]
[-, 3] 3
The nodes are “MAX nodes” The nodes are “MIN nodes”

27. Alpha-Beta Pruning Example
[-, + ]
[-, 3]
=−
 =3
MIN updates , based on kids.
No change. 3 12
The nodes are “MAX nodes” The nodes are “MIN nodes”

28. Alpha-Beta Pruning Example
[3, +]
MAX updates , based on kids.
[-, 3]
3 is returned
as node value. 3 3 12 8
The nodes are “MAX nodes” The nodes are “MIN nodes”

29. Alpha-Beta Pruning Example
[3, +]
[-, 3] 3 12 8
The nodes are “MAX nodes” The nodes are “MIN nodes”

30. Alpha-Beta Pruning Example
[3, +]
, , passed to kids
[-, 3]
[3,+]
=3
 =+ 3 12 8
The nodes are “MAX nodes” The nodes are “MIN nodes”

31. Alpha-Beta Pruning Example
[3, +]
MIN updates ,
based on kids.
[-, 3]
[3, 2]
=3
 =2 X X
 ≥ ,
so prune. 3 12 8 2
The nodes are “MAX nodes” The nodes are “MIN nodes” 31 31

32. Alpha-Beta Pruning Example
MAX updates , based on kids.
No change.
[3, +]
2 is returned
as node value.
[-, 3]
[3, 2] X X 3 12 8 2
The nodes are “MAX nodes” The nodes are “MIN nodes” 32 32

33. Alpha-Beta Pruning Example
[3,+]
, , passed to kids
[-, 3]
[3, 2]
=3
 =+ X X 3 12 8 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

34. Alpha-Beta Pruning Example
[3, +]
MIN updates ,
based on kids.
[-, 3]
[3, 2]
[3, 14]
=3
 =14 X X 3 12 8 2 14
The nodes are “MAX nodes” The nodes are “MIN nodes” 34 34

35. Alpha-Beta Pruning Example
[3, +]
MIN updates ,
based on kids.
[-, 3]
[3, 2]
[3, 5]
=3
 =5 X X 3 12 8 2 14 5
The nodes are “MAX nodes” The nodes are “MIN nodes”

36. Alpha-Beta Pruning Example
[3, +]
MIN updates ,
based on kids.
[-, 3]
[3, 2]
[3, 2]
=3
 =2 X X 3 12 8 2 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

37. Alpha-Beta Pruning Example
[3, +]
2 is returned
as node value.
[-, 3]
[3, 2]
[3, 2] X X 3 12 8 2 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes”

38. Alpha-Beta Pruning Example
MAX updates , based on kids.
No change.
[3, +]
[-, 3]
[3, 2]
[3, 2] X X 3 12 8 2 14 5 2
The nodes are “MAX nodes” The nodes are “MIN nodes” 38 38

40. Example
-which nodes can be pruned? 3 4 1 2 7 8 5 6
Max
Min 40

41. Answer to Example 3 4 1 2 7 8 5 6 41 -which nodes can be pruned? Max
Min
Answer: NONE! Because the most favorable nodes for both are explored last (i.e., in the diagram, are on the right-hand side). 41

42. Second Example (the exact mirror image of the first example)
-which nodes can be pruned? 4 3 6 5 8 7 2 1 42

43. Answer to Second Example (the exact mirror image of the first example)
-which nodes can be pruned? 6 5 8 7 2 1 3 4
Min
Max
Answer: LOTS! Because the most favorable nodes for both are explored first (i.e., in the diagram, are on the left-hand side). 43

46. 점진적 심화방법
Time limits  unlikely to find goal, must approximate

47. Iterative (Progressive) Deepening :점진적 심화방법
In real games, there is usually a time limit T on making a move
How do we take this into account?
using alpha-beta we cannot use “partial” results with any confidence unless the full breadth of the tree has been searched
So, we could be conservative and set a conservative depth-limit which guarantees that we will find a move in time < T
disadvantage is that we may finish early, could do more search
In practice, iterative deepening search (IDS) is used
IDS runs depth-first search with an increasing depth-limit
when the clock runs out we use the solution found at the previous depth limit

48. 점진적 심화방법

49. Iterative deepening search l =049

50. Iterative deepening search l =150

51. Iterative deepening search l =251

52. Iterative deepening search l =352

54. Heuristic Continuation: fight horizon effect