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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

Nine Men\'s Morris

Martin Boyd

Christopher Hirunthanakorn

game overview
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
Game Classification
  • Determinate
  • Zero-sum
  • Symmetric
  • Perfect Information
  • Sequential
  • Normal
background
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
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
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
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
Five Men\'s Morris
  • Players have 5 pieces instead of 9
  • 16 spots instead of 24
game states and combinatorics
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 structure1
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
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
Matchboxes

MOVES

A

B

C

D

E

A5

1

B4

C2

2

C4

3

C5

D2

4

D3

5

D4

E5

adaptive program1
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
adaptive results
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.
ai logic minimax and negamax
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
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
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
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
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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