Nine Men's MorrisMartin Boyd

Christopher Hirunthanakorn

Game Overview• Two player game• RULES

o Players alternate turns placing pieces on the board

o If a mill is formed, player may remove an opponent's piece mill - three pieces formed along a line

o After both players place nine pieces, players move their pieces to any free adjacent spot

o The game is over when a player has less than 3 pieces or no legal moves remain

Example

Example (continued)

Game Classification

• Determinate• Zero-sum• Symmetric• Perfect Information• Sequential• Normal

Background• One of the oldest games played to date

o Game board carving from 1400 BCE found in Egypt

• Also known as Mill, Merelles, or Cowboy Checkers

• Popular variants of the game include Three Men's, Six Men's, and Twelve Men's Morris

Research Goals/Questions• Look for an optimal strategy for piece

placement• Find an optimal strategy for gameplay• Is there a winning strategy for either

player?• Is the game fair?

Analyzing the Game• Searched for previous work on the game• Game States and Combinatorics• Created program with a GUI• Analyzed Five Men's Morris • Created an Adaptive Program• Created an AI• Used python as the programming

language

Previous Publications• Ralph Gasser (Swiss computer

scientist)o Proved that perfect play in Nine Men's Morris

results in a draw and is impossible for humans to achieve

o Analyzed the midgame and endgame by going through all possible game states and labeling them a win or lose position

o Did not provide any advice on the optimal strategy or fairness of the game

Five Men's Morris• Players have 5 pieces instead of 9• 16 spots instead of 24

Game States and Combinatorics• A game state is defined as the game board and

all relevant information defining it such as Last player to move and position of last move

• Board to the right is the game state where player 1 just went but could have placed it on either side

• Used combinatorics to estimate the number of game states possible o About 1.74 * 10^11 states based on possible

combinations of placement (16*15*14*13*12*11*10*9*8*7*6)

o Can be reduced using symmetry of game states to about 7.26 * 10^8 (31+14*13*12*11*10*9*8*7*6)

Basic Program Structure

A1

B C D E

2

3

4

5

B2

E5

Basic Program Structure• Data of the Game Board is stored in 3

arrayso Basic Array

[A1,A3,A5,B2,B3,B4,C1,C2,C4,C5,D2,D3,D4,E1,E3,E5]o Mill Array

[[0,A1,A3,A5],[0,B2,B3,B4],...,[0,E1,E3,E5]] o Connection Array

[[A1,A3,C1,0],[A3,A1,A5,B3],...,[E5,C5,E3,0]]

Adaptive Program• Runs the Game MANY times• Contains Matchboxes that punish a

player if that player loses thus not repeating the same mistake twice.

Matchboxes

A EC DB1

2

3

4

5

A5

C2B4

C5D2

C4

D3D4E5

MOVES

Adaptive Program• 2 different Adaptive Programs written for Five

Men's Morris• Opening Stage Adaptive

o Contains a Matchbox for each player to select spots

• Second Stage Adaptiveo Contains two Matchboxes for each player to

move pieces and the other for removing pieces

Adaptive Results• Opening Stage Adaptive

o After 60 million runs (On the last 10 million) 72544 won by Player 1 44059 won by Player 2 9883397 end in a draw Player 1 has 20% advantage on win/loss However most opening stages end in draw

• Second Stage Adaptiveo After 10 million runs still dead eveno The program requires more runs to draw a

conclusion.

AI Logic (Minimax and Negamax)• AI is based on the game theory decision rule of

Minimax and Negamax• Both determine the worth of a game state using

a set of conditions• Efficiently searches through possible states and

presents the best one.o Negamax differs in how it eliminates certain

states that can not be achieved to increase search speeds Current State

2 0

2

5

91 6

Next State

Next Next State

AI Logic (Scoring)• Plays the game more intelligently by

choosing the best move from all possible moves for that game board

• Moves are scored based on the resultant game board o next to open connection or own piece =

+1o next to opponent's piece = -1o sets up 2/3 parts of a mill = +2o blocks opponent's mill = +2o makes a mill = +3

AI Results• Player 1 using AI, Player 2 playing

randomlyo After 1000 runs multiple times, Player 1 wins

roughly about 70% of the time• Both players using AI

o After 1000 runs multiple times, neither player has an advantage over the other (around 50% each)

• AI will require more improvements and test runs to get solid results

General Strategy• Take spots on both rings• Take spots with the most connections• Block your opponent's move in a way that

you don't trap yourself• Try to force your opponent to allow you to

make a millo Ex) player 1 takes outside corners and

player 2 tries to block• If possible, set up two potential mills next

to each other so that a mill can be made by moving back and forth

Future Work• Improve AI and adaptive learning

programs to be more efficiento Currently the Adaptive takes too long to

run through the required number of games

• Confirm the patterns found apply to Nine Men's Morris by running the programs on it

• Come up with a more detailed strategy that will handle every situation