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"?
If both apply their optimum (random) strategies the outcome need not be
zero; it can be any value. (Who says the game must be fair?)
Speaking from memory, it could be proven that there always exists a
pair of optimum strategies (one for each player) meaning that if any
one plays otherwise, the outcome would be worse.
I also remember you had to do many linear algebra calculations to solve
a simplex problem... but I don't remember the details any more!