Combinatorial games played randomly: Chomp and nim

In this note, we investigate combinatorial games where both players move randomly (each turn, independently selecting a legal move uniformly at random). In this model, we provide closed-form expressions for the expected number of turns in a game of Chomp with any starting condition. We also derive and prove formulas for the win probabilities for any game of Chomp with at most two rows. Additionally, we completely analyze the game of nim under random play by finding the expected number of turns and win probabilities from any starting position.