I am really confused about how to think about this question. It was presented as a challenge by a peer.
Two people seek to kill a duck at a location $Y$ meters from their origin. They walk from $x=0$ to $x=Y$ together. At any time, one of the two may pull out their gun and shoot at the duck, however, the probability that person A hits is $P_{A}(x)$ and the probability that person B hits is $P_{B}(x)$. It is also known that $P_A(0)=P_B(0)=0$ and $P_A(Y)=P_B(Y)=1$ and both functions are increasing functions.
What is the optimal strategy for each player?