A probabilistic two-pile game

We consider a game with two piles in which two players take turns adding a or b chips, randomly and independently, to their respective piles. Here a, b are not necessarily positive. The player who collects at least n chips first wins the game. We derive general formulas for p_n, the probability of the second player winning the game by collecting n chips first, and give the calculation for the cases {a, b} = {−1, 1} and {−1, 2}. The latter case was considered by Wong and Xu. At the end, we derive a general formula for p_n1,n₂, the probability of the second player winning the game by collecting n₂ chips before the first player collects n₁ chips.