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iCAMP: Game Simulation and Analysis Analysis of the Game “Poison” Sarah Eichhorn University of California Irvine iCAMP Overview Meeting, 5/5/10

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Rules of Poison Two players alternate turns There are 10 objects Each turn a player must take either 1 or 2 objects The player to take the last object loses and is “poisoned”

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Questions: Is Poison a fair game? What is the best strategy for each player? Are there interesting variants of the game?

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“ If you can't solve a problem, then there is an easier problem you can solve: find it.” - George Polya “How to Solve It”

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# of ObjectsWinner 1 2 3 4 5 6 7 8 9 10

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# of ObjectsWinner 1Player 2 2 3 4 5 6 7 8 9 10

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# of ObjectsWinner 1Player 2 2Player 1 3 4 5 6 7 8 9 10

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# of ObjectsWinner 1Player 2 2Player 1 3 4 5 6 7 8 9 10

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or

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# of ObjectsWinner 1Player 2 2Player 1 3 4Player 2 5 6 7 8 9 10

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# of ObjectsWinner 1Player 2 2Player 1 3 4Player 2 5Player 1 6 7Player 2 8Player 1 9 10Player 2

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Poison is an unfair game! With 10 objects, Player 2 should be able to win regardless of what Player 1 does. Could we predict who should win with more objects?

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Modular Arithmetic Modular arithmetic basically only keeps track of the remainder when dividing by a given integer Ex.) 4=1(mod3) 5=2(mod3) 2=2(mod3) 3=0(mod3) 67=1(mod3)

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# of ObjectsWinner 1Player 2 2Player 1 3 4Player 2 5Player 1 6 7Player 2 8Player 1 9 10Player 2

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Player 2 wins if the # of objects=1(mod3) Otherwise, Player 1 wins Ex.) Poison with 2009 objects 2009/3=669.66667 2009=2(mod3) Player 1 should have a winning strategy

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Player 2 wins!

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Winning Strategy If you are Player 2 when there are 10 objects, you would like to continue having it be your turn when # of objects=1(mod3) Therefore, your strategy is always to do the opposite of what your opponent does ie.) If the other player takes 1, you take 2. If the other player takes 2, you take 1.

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Game Variants The variants of Poison are often called Nim Variants: Vary the number of objects Vary number of objects allowed to take per turn Nim Heaps – Piles of objects, on your turn you can remove any number of objects from a single pile, including the whole pile.

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Game Theory and Adaptive Learning Can we write a program to train a computer to figure out the best strategy to play a game? Idea: - Start by picking moves randomly - Let the computer play game many times - Reward the moves that lead to a win and punish those leading to loss by weighting the “move” options

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Conclusions You are now a Poison grand master and can always win (provided you can talk the other person to going first!) Computational game theory is an exciting area of mathematics with many interesting applications

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