HexMoHex

experiments

MoHex: Computer Hex World Champion

Arneson Hayward Henderson

Comp SciU Alberta

Edmonton AB Canada

NSERC/iCORE/UofA GAMES/Schaeffer/Müller

4 Aug 2010

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

Hex

properties: n×n, no draw, monotonic, 1pw (swap)

inferior cells

connections

books on Hex by Cameron BrowneHex Strategy: Making the Right ConnectionsConnection Games: Variations on a Theme

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

Hex v Go

stones on a gridconnected groups

subgame decompositionwide branching

connect territorywin/loss score

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

inferior cells: dead/captured

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

inferior cells: reversible/dominated

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

connections: VC/SC

+

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

connections: mustplay

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

alpha-beta players

1950 Shannon (Hex) circuit saddle point1950 Shannon (Bird Cage) circuit voltage drop2000 Van Rijswijck Queenbee 2-distance2000 Anshelevich Hexy circuit + virt’l conn’n2003 Melis Six2003 H . . . Mongoose2008 AHH Wolve

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex basic framework

MCTS: tree traversal, game simulation, tree updateMoHex

tree traversal/update: AMAF on, UCT offgame simulation: bridge pattern

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

unused simulation pattern

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: lock-free

MCTS: lock-free parallelization

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: tree knowledge

MCTS: tree knowledge

b

b

b

b b

b b

b

b b b

b

b b

b

b

b b b

1. node hits visit threshold: apply ICE/VCE

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: tree knowledge

MCTS: tree knowledge . . .

b

b

b

b b

b b

b

b b b

b

b b

b

2. prune inferior/non-mustplay children

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: tree knowledge

MCTS: tree knowledge . . .

b

b

b

b b

b b b

3. remove other children’s subtrees

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: tree knowledge

tree knowledge: typical position, ICE OFF . . .

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

MoHex enhancements: tree knowledge

tree knowledge: typical position, ICE ON . . .

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

exp’s: scaling

45

50

55

60

65

70

1 2 3 4 5 6 7 8

Per

cent

age

win

s

Number threads / time multiplier

single threadedlock-free

locked

locked/lock-free/time-scaled-1-thread v 1-thread 1s/moveArneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

exp’s: bridge/AMAF

win % Elo gainbridge v — 64.7% ± 1.4% 105

AMAF + bridge v bridge 73.9% ± 1.3% 181

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

exp’s: tree knowledge threshold

50

55

60

65

70

75

0 100 200 300 400 500 600 700 800 900 1000

Per

cent

age

win

s

Knowledge threshold multiplied by number of threads

1 thread2 thread4 thread8 thread

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

exp’s: opening book

30

35

40

45

50

55

60

65

70

75

80

1 2 3 4 5

Per

cent

age

win

s

MoHex with 25k * 2^(x-1) Simulations / Move

no book10k book40k book

160k book

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

tournaments

a

a

b

b

c

c

d

d

e

e

f

f

g

g

h

h

i

i

j

j

k

k1

1

2

2

3

3

4

4

5

5

6

6

7

7

8

8

9

9

10

10

11

111

23

4

56

78

910

1112

1314

1516

1718

1920

2122

2324

2526

2728

29

3031

32

3334

35

36

3738

39

4041

4243

44

4546

47

4849

50

51

52

535455

565758

5960

6162

63

6465

6667

68

2009: Wolve v MoHex (white)

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

tournaments

a

a

b

b

c

c

d

d

e

e

f

f

g

g

h

h

i

i

j

j

k

k1

1

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2

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3

4

4

5

5

6

6

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8

9

9

10

10

11

11

1

2

34

56

78

910

1112

1314

15

1617

18

19

20

2122

23

24

25

26

27

2829

30

31

32

3334

35

36 37

38

39 40

41

42

4344

4546

4748

49

50

51

5253

54

55

5657

58

5960

6162

2009: Six v MoHex (white)

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

tournaments

a

a

b

b

c

c

d

d

e

e

f

f

g

g

h

h

i

i

j

j

k

k1

1

2

2

3

3

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4

5

5

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6

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7

8

8

9

9

10

10

11

111

2

3

4

56

7

8

910

1112

1314

15

16

1718

19

20

2122

23

24

2526

2728

2930

3132

33

34

3536

37 38

3940

41 42

43

2009: Yopt v MoHex (white)

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

tournaments

Olympiad gold silver bronze2009 MoHex Wolve Six Yopt

6-0 3-3 2-4 1-52008 Wolve MoHex Six Yopt

9-3 8-4 4-8 3-9

242-game rounds solver/book off2009 win %

MoHex v Six 76.6 ± 3.6MoHex v Wolve 49.2 ± 3.2

Arneson Hayward Henderson MoHex: Computer Hex World Champion

HexMoHex

experiments

conclusion

other gamesYHavannahGo

MoHex todosimulations: connection AMAFtree: solver init’n

MoHex summaryno big surprises

Arneson Hayward Henderson MoHex: Computer Hex World Champion