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Burr and Hamilton each have a loaded gun with one bullet. They start out ten paces apart and walk toward each other one step at a time. At the outset either may choose to fire. If there is not a hit, then each takes one step closer, when again they may choose to fire. The probability of scoring a hit increases as they get closer together.   When, say, Burr shoots at a distance of ten paces there is .2 chance of a hit, similarly for Hamilton. The probability of either combatant scoring a hit is summarized as

If Burr fires and misses while Hamilton has yet to fire, the walk must continue even though Burr now faces certain death; the reverse is also true.  Each faces a payoff of -1 if he himself is killed and 1 if the opponent is killed.  If neither or both is killed then each gets 0.

1. Who won the historic duel? Enter the last name here: You can find out the answer by doing a search on the WWW. Burr

Before proceeding you should try to draw the game tree for the fire-hold decision at ten paces.  See the tree at the bottom.

2. Will either of them wait until they are two paces apart to fire?
Yes No The only way for the game to get to this point is if both hold all the way along.  If either fires and hits the target then the game is over.  If either fires and misses then the game is over de facto.  If they both miss then they are out of ammunition.  If just one fires and misses then he is a dead man. If neither has fired until they are two paces apart then the expected payoff for Burr is 0, similarly for Hamilton.  Neither would wait this long.

3. Will either of them wait until they are four paces apart to fire?  

Yes No

No, Waiting this long means that the probability of death has risen above 1/2.

4. Will either of them wait until they are six paces apart to fire? 

Yes No

 Yes. When they are six paces apart they will both fire.  The probability of scoring a hit has risen above 1/2.

5. Will either of them fire when they are eight paces apart?

Yes No

When they are this far apart the probability of a miss, and sure death is .60.

6. Will either of them fire at the full ten paces?

Yes No

 We can construct a table of payoffs for Burr at ten paces.  His dominant strategy is to Hold.

You could also draw the extensive form of the game. Note that the game tree continues only if they both hold initially. If they both hold initially then the tree repeats itself from the Hold-Hold branch, but with revised probabilities.