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, so it rises from 0.2 after the first step to 1 (certainty) after 5 steps, at which point the two are right up against each other. If one player fires and misses while the other has yet to fire, the walk must continue even though the bulletless one now faces certain death; this rule is dictated by the code of the aristocracy. Each player gets a payoff of 1 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 dont shoot) at each step. Find the rollback (subgame-perfect) equilibrium of this game.

Hint: Begin at step 5, when the duelists are right up against each other. Set up the two-by-two table for the simultaneous-move game at this step, and find the Nash equilibrium. Now move back to step 4, where the probability of scoring a hit is 4/5, or 0.8, for each. Set up the two-by-two table for the simultaneous-move game at this step, correctly specifying in the appropriate cell what happens in the future.

For example, if one shoots and misses, but the other does not shoot, then the other will wait until step 5 and score a sure hit. If neither shoots, then the game will go to the next step, for which you have already found the equilibrium.

Questions:

1. Using all this information, find the payoffs in the two-by-two table of step 4, and find the Nash equilibrium at this step. Work backward in the same way through the rest of the steps to find the Nash equilibrium of the full game.

2. Describe an example of business competition that is similar in structure to the duel in Exercise.