iCAMP: Game Simulation and Analysis
Analysis of the Game “Poison”
Sarah Eichhorn
University 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
1
2
3
4
5
6
7
8
9
10
Winner
# of Objects
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
# of Objects
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
Player 1
# of Objects
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
Player 1
Player 1
or
# of Objects
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
Player 1
Player 1
Player 2
# of Objects
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
Player 1
Player 1
Player 2
Player 1
Player 1
Player 2
Player 1
Player 1
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
1
2
3
4
5
6
7
8
9
10
Winner
Player 2
Player 1
Player 1
Player 2
Player 1
Player 1
Player 2
Player 1
Player 1
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