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"?

(In reply to

Expected result by Federico Kereki)

In general, a zero-sum two-person game is not necessarily fair,
and does not necessarily result in zero expected payoff if one
player uses an optimum strategy.

However, I believe that a zero-sum two-person game is necessarily fair
and does necessarily result in a zero expected payoff if it is
"symmetrical". In a symmetrical game, like this one, the players
have the same strategies available and the same pay-off
structure. There is no way that this game favors one player over
the other, if there are two people playing who are unaware of each
others' strategies. Therefore, the optimum strategy played by
each must result in an expected zero outcome.