# Martin Boyd Christopher Hirunthanakorn

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Martin Boyd Christopher Hirunthanakorn
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

Example

Example (continued)

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 Adaptive Program 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)

Basic Program Structure
1 B C D E 2 3 4 5 B2 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]]

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

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

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.

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

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