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"?

A simple solution to the first part is

Fire: 20%
Life: 20%
Water: 20%
Earth: 20%
Cold: 20%
Air: 0

The program's opponent will always pick the same element as long as your probability set is weak to any of them.  With this set, no matter what the opponent picks, his chance of victory is 50%.  If he picks air, he wins 40% and program wins 60%. 

Thus, you will win 50% of the time with this probability set.  I don't see how you could win more than that. 

The second part is a lot more complex.  If you try the same thing, your opponent will take air every time and win 60%, leaving you with 40%. 

By fiddling with numbers (not the perfect solution since I'm not using a computer), I came up with a 47.06% (40/85) chance of victory with...

Fire: 26.087%  (30/115)
Life: 30.435%  (35/115)
Water: 17.391% (20/115)
Earth: 17.391% (20/115)
Cold: 8.696% (10/115)
Air: 0

Opponent win % with each:
Fire: 52.94%
Life: 50%
Water: 52.63%
Earth: 42.1%
Cold: 52.38%
Air:  43.48%

Therefore against this set, he will choose fire every time, and you will win 47.06%.  I'm guessing this is near the actual solution but precision here might be impossible for a human. 
I don't think there is a way to get 50% here.

Posted by Matt on 2005-06-17 04:59:57