We have the following duel problem: https://mathoverflow.net/questions/75318/the-duel-problem (You can read about it here). We have $P:\frac12, \frac23, \frac34, 1$, Q: $\frac14, \frac13, \frac12, 1$. Player $1$ has $2$ shoots, Player $2$ has only $1$ shoot. The problem is simple to understand. At first position, both players can decide if they want to shoot the other, they have $\frac14$ chance for that both. If they decide not to shoot, then they can step $1$ further, therefore they chances increases(you can see in $P,Q$). Players don't know if the other player tried to shoot them, but missed.
The question is the optimal strategy for shooting for both Player $1$ and Player $2$. How should I consider that problem? If Player $1$ wins, he gets paid , if Player $2$ wins, he gets paid, if they both survive, or both die, nobody gets anything.
Writing down all cases would be pretty hard, is that the only way, or is there something more "professional" for such problems? Thanks!