Three-Person Duel

Source

I got this problem from Rustan Leino, who got it from Johannes Kinder, who said he heard it from his brother. Rustan reformulated the setting.

I solved it and wrote up my solution.

Problem    

A particular basketball shootout game consists of a number of duels. In each duel, one player is the challenger. The challenger chooses another player to challenge, and then gets one chance to shoot the hoop. If the player makes the shot, the playing being challenged is out. If the player does not make the shot, or if the player chooses to skip his turn, then the game continues with the next duel. A player wins when only that player remains.

One day, this game is played by three players: A, B, and C. Their skill levels vary considerably: player A makes every shot, player B has a 50% chance of making a shot, and player C has a 30% chance of making a shot. Because of the difference in skill levels, they decide to let C begin, then B, then A, and so on (skipping any player who is out of the game) until there is a winner. If everyone plays to win, what strategy should each player follow? In this scenario, what probability does each player have of winning?

Solution     Reveal