I am done with implementing the second strategy for the duel problem but, now, I think that I may have gotten the question wrong.

When it says Jason has to miss intentionally on his first shot, does this mean his win rate would be 0%? The reason I think his win rate will be zero is that he's the first one to take a shot in the game and he's dueling with Lenny first. So, if his misses his shot on Lenny intentionally, then he's basically dead as Lenny does not miss at all.

 

Recall that Jason is the lowest priority target. If he misses his first shot every time, this ensures that Kyle will be shooting at Lenny, and there's a 50 percent chance Lenny will be dead before having a chance to shoot. Should Lenny die before firing, the duel is now between Kyle and Jason. There's a 33 percent chance that Jason kills Kyle before Kyle can shoot at him; if he misses, there's a 50 percent chance that Kyle will also miss, giving Jason another chance. The other potential scenario is that Kyle misses and Lenny gets a turn. In that case, Kyle is guaranteed to die, after which Jason has a 33 percent chance to kill Lenny before Lenny fires. I think you're misunderstanding the turn order and target priorities here. Jason will miss his first shot, then Kyle fires, then Lenny fires, so Lenny will not have an immediate chance to shoot Jason. Beyond that, Lenny will aim at Kyle first, as Kyle has a better chance of hitting him than Jason would; in all scenarios, Jason lives long enough to fire at least once, and by the time he does, there is only one other target; if he hits them, he wins the duel.