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Nine Men's Morris

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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  1. 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 Structure A B C D E 1 2 B2 3 4 5 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 D2 4 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 • 9883397 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 0 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

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