iCAMP: Game Simulation and Analysis

Analysis of the Game “Poison”

Sarah EichhornUniversity of California Irvine

iCAMP Overview Meeting, 5/5/10

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”

Questions:

• Is Poison a fair game?

• What is the best strategy for each player?

• Are there interesting variants of the game?

“ If you can't solve a problem, then there is an easier problem you can solve: find it.”

- George Polya

“How to Solve It”

# of Objects Winner

1

2

3

4

5

6

7

8

9

10

# of Objects Winner

1 Player 2

2

3

4

5

6

7

8

9

10

# of Objects Winner

1 Player 2

2 Player 1

3

4

5

6

7

8

9

10

# of Objects Winner

1 Player 2

2 Player 1

3 Player 1

4

5

6

7

8

9

10

or

# of Objects Winner

1 Player 2

2 Player 1

3 Player 1

4 Player 2

5

6

7

8

9

10

# of Objects Winner

1 Player 2

2 Player 1

3 Player 1

4 Player 2

5 Player 1

6 Player 1

7 Player 2

8 Player 1

9 Player 1

10 Player 2

• 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?

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)

# of Objects Winner

1 Player 2

2 Player 1

3 Player 1

4 Player 2

5 Player 1

6 Player 1

7 Player 2

8 Player 1

9 Player 1

10 Player 2

• 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

Player 2 wins!

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.

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.

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

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