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"?
For the first problem, the first player should pick (C,F,W,E,A,L) with
odds (1/3, 1/9, 1/3, 1/9, 1/9, 0) and the second player should use odds
(1/5, 1/5, 1/5, 1/5, 0, 1/5).
For the second problem, the first
player should use odds (1/5, 1/5, 1/5, 1/5, 0, 1/5) and the second
player (1/9, 1/3, 1/3, 0, 1/9, 1/9).
If both players play with these probabilities, the expected outcome is 0 -- no one will win.