Suppose that a game is played on an $n \times n$ board as follows. There are two players, Player 1 has (only) $Q$ queens and Player 2 has only $K$ knights. Suppose that $Q, K \leq n/3$. The game is played as per usual in chess, with Player 1 going first. The objective is to capture all of the other players' pieces. Now we fix the initial configuration by placing the queens arbitrarily on the left most column, then place all of the knights on the right most column so that no knight and queen are initially on the same row. Now suppose that both players play with no strategy; that is, each player randomly chooses a legal move each turn with all legal moves having equal probability, and illegal moves (defined as a move that will cause the moving piece to be captured in the next move) with probability zero. Consider the random variable $X$ that satisfies $X = k$ if on the $k$th move Player 1 wins the game, and $X = -k$ if on the $k$th move Player 2 wins the game. Then what can we say about $X$? For example, does it have finite mean or variance? Is it possible to give an explicit distribution? Are there any existing probabilistic models that model its behaviour?

## 1 Answer

It is standard to show that $X$ has exponential decay: for any position, there is a fixed positive probability that the game will terminate within the next (say) 10n steps. In particular, all moments of $X$ are finite.