1. Combinatorial Games Martin Müller

2. Contents Combinatorial game theory Thermographs Go and Amazons as combinatorial games

3. Combinatorial Games Basics Example: Domineering Simplifying games Sums of games Hot games

4. What is a Game? 2 players, Left and Right Set of positions, starting position Moves defined by rules Alternating moves Player who cannot move loses (no draws) Conway's plan: find the simplest possible definition

5. Properties of Games Complete information Perfect information No random element (no dice, coin throws, …)

6. Definition of a Game Move options of players Each move leads to a game Player who cannot move loses { A,B,C | D,E } G = { L 1,…,L n | R 1,…,R m }

7. Creating Games G = { L 1,…,L n | R 1,…,R m } Simplest possible game: { | } Next step: {{ | } | } { | { | }} {{ | } | { | }} Continue...

8. Games and Numbers Insight: some games represent a number of free moves for one player

9. Infinite Games Recursion: option leads back to game G = { A,B | C } A = { |G }

10. The Domineering Game

11. Domineering Examples

12. Inverse Game Swap all Left and Right moves Compute inverse for all options recursively G = { L 1,…,L n | R 1,…,R m }. Inverse: -G = { -R 1,…,-R m | -L 1,…,-L n } Property of inverses: -(-G) = G

13. Examples of Inverses -(0) = -({ | }) = { | } = 0 -(1) = -({0 | }) = { | -0} = { | 0} = -1 -({0|0}) = {-0 | -0} = {0|0}

14. Domineering Example Inverse of domineering position: rotate by 90˚

15. Classification of Games G > 0Left wins G < 0Right wins G = 0Second player wins G || 0First player wins

16. Classification Examples 0 = { | }First player loses { 0 | 0 }First player win { 0 | { 0 | 0 } }Left always wins {{ 0 | 0 } | 0 }Right always wins

17. Comparing Games G > HifG - H > 0 Left wins difference game G < HifG - H < 0 Right wins difference game G = HifG - H = 0 Second player wins difference game G || HifG - H || 0 First player wins difference game

18. Canonical Form of Games Loopfree games have canonical form Two operations: –Delete dominated options –Reversing reversible options Apply as long as possible End result: unique canonical form

19. Deleting Dominated Options Example: {2, -5, 6, 3 | -2, 6, 13, -8} = {6|-8} General problem: compare games Complete algorithm implemented in David Wolfe's games package

20. Sums of Games Two games, G and H Choice: play either in G or in H G+H = { G+H L, G L +H | G+H R, G R +H } Example: -5+3 = { -5+3 L, -5 L +3 | -5+3 R, -5 R +3 } = {-5+2|-4+3} = {-3|-1} = -2

21. Sum of Domineering Positions

22. Fractions Example: {0|1} + {0|1} = 1 {-1,0|1}={0|1} = 1/2

23. Hot Games First player gets extra moves Both are eager to play Example: {1|-1} The 2x2 square is hot

24. Sums of Hot Games Can be much more complex than summands Example: a = {1|-1}, b = {2|-2}, c = {3|-3}, d = {4|-4} Sums: a+b = {{3|1}|{-1|-3}} a+b+c = {{{6|4}|{2|0}}|{{0|-2}|{-4|-6}}} a+b+c+d = {{{10|8}|{6|4}}|{{4|2}|{0|-2}}} |{{{2|0}|{-2|-4}}|{{-4|-6}|{-8|-10}}}

25. Mean Mean  Average outcome Means add Examples:  4|-4) = 0  6|-4) = 1  4|{-4|-10})= -3/2  4|{-4|-20})= -4 Theorem:  a+b  a  b 

26. Temperature Measures urgency of move Sum does not become hotter Examples: temp  4|-4}) = 4 temp  4|{-4|-10})= 11/2 temp  4|{-4|-20})= 8 temp  4|{-4|-100}) = 8 temp(a+b) max(temp(a), temp(b))

27. Example a = 4|-4, b = 5|-5, c = 5 |{-4|-6} temp(a) = 4, temp(b) = 5, temp(c) = 5 temp(a + b) = 5 temp(b + c) = 1 temp(b + b) = 0

28. Leftscore and Rightscore Also called LeftStop and RightStop Minimax values of game if left (right) plays first Assumption: play stops in numbers Base points of thermograph (see next slides)

29. Thermograph

30. Thermograph (TG) Consists of left and right scaffold May coincide in a mast Leaf node: TG of numbers are masts Constructed from TG of followers –Tax right scaffold of left follower by t –Tax left scaffold of right follower by -t –Compute max (min) over all left (right) followers –Cut off above intersection of left, right, add mast

31. Sente and Gote Thermographs Three examples –Gote –One-sided sente –Double sente All examples: leftScore - rightScore = 4. Appear the same to a local minimax search But they are very different!

32. Gote Game: 4|0 leftScore 4 rightScore 0 Mean: 2 Temperature: 2

33. One-sided Sente Game: 22|4||0 leftScore 4 rightScore 0 Mean: 4 Temperature: 4

34. Double Sente Game: 12|3 || -1|-11.5 leftScore 3 rightScore -1 Mean: 0.5 Temperature: 7

35. Extensions (1) Sub-zero thermography –Problem: hard to check when game is number –extend TG to range [-1..0] –“colored ground” rule for zugzwang-like games –Can now construct TG from options in a uniform way –TG = makeTG(left-option-TGs,right-option-TGs)

36. Extensions (2) TG for games including loops –Defined by Berlekamp’s Economists’s view paper –I did the first practical algorithm and implementation –Much more complex… –Caves, hills, bent masts, backward masts,…

37. Some Wild Ko Thermographs

39. Stable and Unstable Positions Position H in game G is called stable if temperature is lower than all of its ancestors H is unstable if it has an ancestor with lower temperature H is semistable if not unstable and has ancestor of same temperature

40. Subtree of Stable Followers Root of a game tree is stable by definition Find first stable node on each line of play Go on recursively This subtree of stable followers is a (very good) small summary of the whole game

41. Mainlines and Sidelines Given G, play n copies of G optimally Let n go to infinity Some lines of play will be played more and more often –Mainlines Other lines played only finitely often –Sidelines

42. Stable Followers in Mainlines Stable mainline gote position: has two stable followers, one for each color Stable mainline one-sided sente position: –Only stable follower of one color (sente) In a “rich environment” (e.g. coupon stack), play follows mainlines.

43. Playing Sum Games Choose one subgame Choose move in that subgame Brute force algorithm: –Compute sum –Find move retaining minimax value –Problem: computing sum is slow

44. Fast Approximate Methods Goal: identify good move without computing sum Two parameters: mean and temperature Hottest games usually most urgent Refinement: Thermostrat