# Nine Men's Morris - PowerPoint PPT Presentation

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Nine Men's Morris. Martin Boyd Christopher Hirunthanakorn. Game Overview. Two player game RULES Players alternate turns placing pieces on the board If a mill is formed, player may remove an opponent's piece mill - three pieces formed along a line

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Nine Men's Morris

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## Nine Men's Morris

Martin Boyd

Christopher Hirunthanakorn

### Game Overview

• Two player game

• RULES

• Players alternate turns placing pieces on the board

• If a mill is formed, player may remove an opponent's piece

• mill - three pieces formed along a line

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

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

### Game Classification

• Determinate

• Zero-sum

• Symmetric

• Perfect Information

• Sequential

• Normal

### Background

• One of the oldest games played to date

• 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 AI

• Used python as the programming language

### Previous Publications

• Ralph Gasser (Swiss computer scientist)

• Proved that perfect play in Nine Men's Morris results in a draw and is impossible for humans to achieve

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

• 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

• About 1.74 * 10^11 states based on possible combinations of placement (16*15*14*13*12*11*10*9*8*7*6)

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

A

B

C

D

E

1

2

B2

3

4

5

E5

### Basic Program Structure

• Data of the Game Board is stored in 3 arrays

• Basic Array

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

• Mill Array

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

• Connection Array

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

• Runs the Game MANY times

• Contains Matchboxes that punish a player if that player loses thus not repeating the same mistake twice.

### Matchboxes

MOVES

A

B

C

D

E

A5

1

B4

C2

2

C4

3

C5

D2

4

D3

5

D4

E5

• 2 different Adaptive Programs written for Five Men's Morris

• Contains a Matchbox for each player to select spots

• Contains two Matchboxes for each player to move pieces and the other for removing pieces

• 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

• After 10 million runs still dead even

• 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.

• Negamax differs in how it eliminates certain states that can not be achieved to increase search speeds

Current State

Next State

2

5

0

Next Next State

9

1

2

6

### 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

• next to open connection or own piece = +1

• next to opponent's piece = -1

• sets up 2/3 parts of a mill = +2

• blocks opponent's mill = +2

• makes a mill = +3

### AI Results

• Player 1 using AI, Player 2 playing randomly

• After 1000 runs multiple times, Player 1 wins roughly about 70% of the time

• Both players using AI

• 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 mill

• 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 efficient

• 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