Combinatorial Games

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 simplestpossible definition

Properties of Games

• Complete information

• Perfect information

• No random element (no dice, coin throws, …)

Definition of a Game

• Move options of players• Each move leads to a game• Player who cannot move loses

A B C D E

{ A,B,C | D,E }

G = { L1,…,Ln | R1,…,Rm }

Creating Games

G = { L1,…,Ln | R1,…,Rm }

• Simplest possible game:{ | }

• Next step:{{ | } | }{ | { | }}

{{ | } | { | }}• Continue...

Games and Numbers

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

0 = { | }

1 = { 0 | }- 1 = { | 0 }

- 2 = { | - 1 }2 = { 1 | }

Infinite Games

• Recursion: option leads back to game

G = { A,B | C }A = { |G }

A B C

G

The Domineering Game

R

L

Domineering Examples

Inverse Game

• Swap all Left and Right moves• Compute inverse for all options recursively

G = { L1,…,Ln | R1,…,Rm }. • Inverse:

-G = { -R1,…,-Rm | -L1,…,-Ln }• Property of inverses:

-(-G) = G

Examples of Inverses

-(0) = -({ | }) = { | } = 0-(1) = -({0 | }) = { | -0} = { | 0} = -1-({0|0}) = {-0 | -0} = {0|0}

Domineering Example

• Inverse of domineering position: rotate by 90˚

G -G

90˚

Classification of Games

G > 0 Left winsG < 0 Right winsG = 0 Second player winsG || 0 First player wins

Classification Examples

0 = { | } First player loses

{ 0 | 0 } First player win

{ 0 | { 0 | 0 } } Left always wins

{{ 0 | 0 } | 0 } Right always wins

Comparing Games

• G > H if G - H > 0Left wins difference game

• G < H if G - H < 0Right wins difference game

• G = H if G - H = 0Second player wins difference game

• G || H if G - H || 0First 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+3L, -5L+3 | -5+3R, -5R+3 }

= {-5+2|-4+3} = {-3|-1} = -2

Sum of Domineering Positions

Fractions

• Example: {0|1} + {0|1} = 1

-10 1

{-1,0|1}={0|1} = 1/2

Hot Games

• First player gets extra moves• Both are eager to play

• Example: {1|-1}

The 2x2 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 • Average outcome• Means add

Examples: 4|-4) = 06|-4) = 14|{-4|-10}) = -3/24|{-4|-20}) = -4

Theorem: a+bab

Temperature

• Measures urgency of move

• Sum does not become hotter

Examples:temp4|-4}) = 4temp4|{-4|-10}) = 11/2temp4|{-4|-20}) = 8temp4|{-4|-100}) = 8

temp(a+b) 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 LeftStop and RightStop

• Minimax values of game if left (right) plays first

• Assumption: play stops in numbers

• Base points of thermograph (see next slides)

Thermograph

t

score

Left

scaffold

Right

scaffold

mean

temperature

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: leftScore - rightScore = 4. • Appear the same to a local minimax search• But they are very different!

Gote• Game: 4|0

• leftScore 4

• rightScore 0

• Mean: 2

• Temperature: 2

a

One-sided Sente

a

• Game: 22|4||0

• leftScore 4

• rightScore 0

• Mean: 4

• Temperature: 4

Double Sente

a b

• Game: 12|3 || -1|-11.5

• leftScore 3

• rightScore -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 = makeTG(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