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 opponents piece mill three pieces formed along a line ID: 140876
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Slide1
Nine Men's Morris
Martin Boyd
Christopher HirunthanakornSlide2
Game Overview
Two player game
RULES
Players alternate turns placing pieces on the boardIf a mill is formed, player may remove an opponent's piecemill - three pieces formed along a lineAfter both players place nine pieces, players move their pieces to any free adjacent spotThe game is over when a player has less than 3 pieces or no legal moves remainSlide3
ExampleSlide4
Example (continued)Slide5
Game Classification
Determinate
Zero-sum
SymmetricPerfect InformationSequentialNormalSlide6
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 CheckersPopular variants of the game include Three Men's, Six Men's, and Twelve Men's MorrisSlide7
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?Slide8
Analyzing the Game
Searched for previous work on the game
Game States and Combinatorics
Created program with a GUIAnalyzed Five Men's Morris Created an Adaptive ProgramCreated an AIUsed python as the programming languageSlide9
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 positionDid not provide any advice on the optimal strategy or fairness of the gameSlide10
Five Men's Morris
Players have 5 pieces instead of 9
16 spots instead of 24Slide11
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)Slide12
Basic Program Structure
A
1
B
C
D
E
2
3
4
5
B2
E5Slide13
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]]Slide14
Adaptive Program
Runs the Game MANY times
Contains Matchboxes that punish a player if that player loses thus not repeating the same mistake twice.Slide15
Matchboxes
A
E
C
D
B
1
2
3
4
5
A5
C2
B4
C5
D2
C4
D3
D4
E5
MOVESSlide16
Adaptive Program
2 different Adaptive Programs written for Five Men's Morris
Opening Stage Adaptive
Contains a Matchbox for each player to select spotsSecond Stage AdaptiveContains two Matchboxes for each player to move pieces and the other for removing piecesSlide17
Adaptive Results
Opening Stage Adaptive
After 60 million runs (On the last 10 million)
72544 won by Player 144059 won by Player 29883397 end in a drawPlayer 1 has 20% advantage on win/loss
However most opening stages end in draw
Second Stage AdaptiveAfter 10 million runs still dead even
The program requires more runs to draw a conclusion.Slide18
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
2
0
2
5
9
1
6
Next State
Next Next StateSlide19
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 = +1next to opponent's piece = -1sets up 2/3 parts of a mill = +2blocks opponent's mill = +2makes a mill = +3Slide20
AI Results
Player 1 using AI
,
Player 2 playing randomlyAfter 1000 runs multiple times, Player 1 wins roughly about 70% of the timeBoth players using AIAfter 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 resultsSlide21
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 yourselfTry to force your opponent to allow you to make a millEx) player 1 takes outside corners and player 2 tries to blockIf possible, set up two potential mills next to each other so that a mill can be made by moving back and forthSlide22
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 itCome up with a more detailed strategy that will handle every situation