 Solution

Let the Markov chain state after each toss be $(S,I)$, where $S$ is player $A_1$'s net winnings and $I$ is the next player to toss. From $(S,i)$ it moves to $(S+1,1)$ with probability $p_i$ and to $(S-1,2)$ with probability $q_i=1-p_i$. The initial state is $(0,1)$, and a return means hitting any state with $S=0$ at positive time.

From $(1,1)$, the probability of hitting zero isIndeed, in the positive-drift case the bounded harmonic solution ofuses the decaying root $q_2/p_1$ and boundary value $h_2(0)=1$, yielding $h_1(1)=q_1/p_2$; with nonpositive drift the walk hits zero surely. By the reflected argument, from $(-1,2)$ the probability isConditioning on the first toss, the required return probability isBoth formulas equal one at the boundary $p_1+p_2=1$.