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"?
There is no simple strategy: no matter which thing the first player
picks, the second can always beat him, and viceversa. Thus, the optimum
strategy is a mix, picking each thing with a certain probability.
Determining this probabilities is a bore --you need to apply linear
programming and solve a certain system of linear equations-- but you
can find on line solvers which will do it for you.