In the well known classic three way duel puzzle, 3 players Alice, Bob and Carol with different hit probabilities take turns to shoot each other in some given shooting order until only one survives.
Now they decide to try a new game. Before the game, Bob prepares three guns with hit probabilities $0\lt w\lt m\lt b$. After that, Alice will be the first one to choose her gun, Bob will be the second to choose, and Carol has whatever is leftover. Carol then determines the shooting order (i.e. who gets to shoot first, who's second and who's last). The rule is that the worst gun can skip turns, but the other 2 guns must always shoot at someone in their turns. Three players take turns to shoot in the order specified by Carol until only one survives. Everyone acts to maximize his/her surviving probability.
I have 2 questions:
- To what values should Bob set $w, m, b$? What is the maximum surviving probability for him?
- Alice and Carol still act to maximize their respective surviving probabilities, but suppose Bob now just want to have a greater surviving probability than Alice, can he achieve that? To what values should he set $w, m, b$?
(We assume Carol is friendly to Bob, that is if two different shooting orders are both optimal for her, she will choose the one that gives Bob better surviving probability.)