Combinatorial Games Martin Mller Contents Combinatorial game theory

Combinatorial Games Martin Müller

Contents • Combinatorial game theory • Thermographs • Go and Amazons as combinatorial games

Combinatorial Games • Basics • Example: Domineering • Simplifying games • Sums of games • Hot games

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

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

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

Creating Games G = { L 1, …, Ln | R 1, …, Rm } • Simplest possible game: {|} • Next step: {{ | }} {{ | } | { | }} • Continue. . .

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

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

The Domineering Game

Domineering Examples

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

Domineering Example • Inverse of domineering position: rotate by 90˚

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

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

Comparing Games • G > H if G-H>0 Left wins difference game • G < H if G-H<0 Right wins difference game • G = H if G-H=0 Second player wins difference game • G || H if G - H || 0 First player wins difference game

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

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

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

Sum of Domineering Positions

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

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

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}}}

Mean • Mean m • Average outcome • Means add Examples: m(4|-4) m(6|-4) m(4|{-4|-10}) m(4|{-4|-20}) Theorem: m(a+b) = m(a) + m(b) =0 =1 = -3/2 = -4

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

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

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

Thermograph

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

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

Gote • Game: 4|0 • left. Score 4 • right. Score 0 • Mean: 2 • Temperature: 2

One-sided Sente • Game: 22|4||0 • left. Score 4 • right. Score 0 • Mean: 4 • Temperature: 4

Double Sente • Game: 12|3 || -1|-11. 5 • left. Score 3 • right. Score -1 • Mean: 0. 5 • Temperature: 7

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 = make. TG(left-option-TGs, right-option-TGs)

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, …

Some Wild Ko Thermographs

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

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

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

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.

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

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