Let there initially be $m$ players in a game of rock-paper-scissors. Everyone uniformly outputs one of rock, paper, and scissors. Rock wins against scissors, scissors win against paper, and paper wins against rock.
There are three cases for each game:
Case 1, only two of the possible outputs are played: There is a group of people with the winning output over the group with the losing output. This group will stay in play while the losing group leaves.
Case 2, there is at least one of each output: Everyone stays and another game is played with the players still in the game.
Case 3, everyone outputs the same thing: Everyone stays and another game is played with the players still in the game.
What is the expected number of games until only one person is still in play?
I tried a few small cases and figured that using a function $f(x)$ where $x$ is the number of people left and $f(x)$ is the expected number of games until only one person is still in play is a possible way of figuring out the solution, but I'm not sure how to generalize this to a case of $m$ players. Does anyone have any ideas?