Two French aristocrats, Chevalier Chagrin and Marquis de Renard, fight a duel. Each has a pistol loaded with one bullet. They start 10 steps apart and walk toward each other at the same pace, 1 step...

Two French aristocrats, Chevalier Chagrin and Marquis de Renard, fight a duel. Each has a pistol loaded with one bullet. They start 10 steps apart and walk toward each other at the same pace, 1 step at a time. After each step, either may fire his gun. When one shoots, the probability of scoring a hit depends on the distance. After k steps it is k5, and so it rises from 0.2 after the first step to 1 (certainty) after 5 steps, at which point they are right up against one another. If one player fires and misses while the other has yet to fire, the walk must continue even though the bullet less one now faces certain death; this rule is dictated by the code of the aristocracy. Each gets a payoff of 21 if he himself is killed and 1 if the other is killed. If neither or both are killed, each gets 0. This is a game with five sequential steps and simultaneous moves (shoot or not shoot) at each step. Find the rollback (sub game-perfect) equilibrium of this game.