1. Martin Boyd Christopher Hirunthanakorn
Nine Men's Morris
Martin Boyd
Christopher Hirunthanakorn

2. 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

3. Example

4. Example (continued)

5. Game Classification Determinate Zero-sum Symmetric Perfect Information
Sequential
Normal

6. 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

7. 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?

8. 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

9. 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

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

11. 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)

12. Basic Program Structure1 B C D E 2 3 4 5 B2 E5

13. 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]]

14. Adaptive Program Runs the Game MANY times
Contains Matchboxes that punish a player if that player loses thus not repeating the same mistake twice.

15. Matchboxes
MOVES A B C D E A5 1 B4 C2 2 C4 3 C5 4 D2 D3 5 D4 E5

16. Adaptive Program
2 different Adaptive Programs written for Five Men's Morris
Opening Stage Adaptive
Contains a Matchbox for each player to select spots
Second Stage Adaptive
Contains two Matchboxes for each player to move pieces and the other for removing pieces

17. Adaptive Results Opening Stage Adaptive
After 60 million runs (On the last 10 million)
72544 won by Player 1
44059 won by Player 2
end in a draw
Player 1 has 20% advantage on win/loss
However most opening stages end in draw
Second Stage Adaptive
After 10 million runs still dead even
The program requires more runs to draw a conclusion.

18. 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
Next Next State 9 1 2 6

19. 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

20. 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

21. 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

22. 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