This game is similar to "rock, paper, scissors" in that two players independently pick one of the six things, and if one thing somehow "beats" the other, then that player wins. If both players pick the same thing, they repeat until someone wins.

Life grows on Earth.
Water douses Fire.
Air resists Cold.
Life drinks Water.
Fire consumes Air.
Cold freezes Water.
Earth smothers Fire.
Life breathes Air.
Fire and Earth both warm Cold.
Air and Water both erode Earth.
Fire and Cold both destroy Life.
Water displaces Air.

A program that plays this game has a single set of probabilities for picking each of the six things. Assuming that the program's opponent knows what these probabilities are, what probabilities will give the program the best chances of winning?

What if the rules of the game are changed so that "Water displaces Air" is replaced with "Air ripples Water"?

Fire (3)
beats: Life, Air, Cold
loses: Water, Earth
So: 3+3+2+2-3-2=5

Life (3)
beats: Air, Water, Earth
loses: Fire, Cold
So: 3+2+3+2-3-2=1

Earth (2)
beats: Fire, Cold
loses: Air, Water, Life
So: 2+3+2-2-3-3=-1

Water (3)
beats: Earth, Fire, Air
loses: Cold, Life
So: 3+2+3+2-2-3=5

Air (2)
beats: Earth, Cold
loses: Fire, Life, Water
So: 2+2+2-3-3-3=-3

Cold (2)
beats: Life, Water
loses: Fire, Earth, Air
So: 2+3+3-3-2-2=1

Therefore:

5: Fire and Water
1: Cold and Life
-1: Earth
-3: Air

But after the rule switch:

Water (2)
beats: Earth, Fire
loses: Cold, Life, Air
So: 2+2+3-2-3-3=-1

Air (3)
beats: Earth, Cold, Water
loses: Fire, Life
So: 3+2+2+2-3-3=3

Therefore:

5: Fire
3: Air
1: Cold, Life, and Water
-1: Earth

So before the ruleswitch, the best pick is either Fire or Water. Each has a 60% chance of winning, and overall, they come out on top. After the ruleswitch, Fire stands alone.

Posted by Cougar Draven on 2005-06-20 02:14:54