A random-turn game is a graph game which is parameterized by a ratio $p$. In each turn, we throw a coin with bias $p$ to determine which player moves the token. Formally, random-turn games are a special case of stochastic games. We are interested in finding, for each vertex, the "value" of the game, which is the optimal probability with which Player 1 can guarantee winning. Stated as a decision problem: decide whether the value is greater than 0.5. The problem is known to be in NP and coNP (again, a special case of stochastic games), but not known to be in P nor to be as hard as stochastic games (or any other problem in NP and coNP).
My interest in random-turn games stems from the fact that they are equivalent to bidding games: a solution to a random-turn game can be used to construct optimal bidding strategies in a bidding game. Moreover, random-turn games have been extensively studied in the combinatorics community.