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"?

Note how many others each element beats:

Life, Water, and Fire each beat 3 others (upper tier)

Air, Cold, and Earth each beat 2 others (lower tier)

So the first three should outrank the other 3

Air only beats Cold and Earth (both lower tier)
Cold beats Life and Water (both higher tier)
Earth beats Cold and Fire (one of each tier)

I would rank Air 6th, Earth 5th, and Cold 4th.

Each of the top tier beats one of the other top tier and two of the lower tier, but;
Fire beats 4th and 6th rank
Life beats 5th and 6th rank
Water beats 5th and 6th rank

So I would rank Fire 1st

Life beats Water, so Life is 2nd and Water 3rd

My ranking (best to worst):

Fire
Life
Water
Cold
Earth
Air

Is this the only possible ranking?  I don't know.

Where to go from here?  I don't know but I'd make the computer pick the higher ranks more often.  Maybe triangular 6/21, 5/21, etc.

Posted by Jer on 2005-06-16 20:10:33