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Quick on the Draw: Hamilton and Burr

Warning: This is a difficult problem set.  Print the questions, read them, and think about them.  Leave yourself enough time to get this done by the due date.

Early in the 19th century (7/11/1804) Alexander Hamilton and Aaron Burr fought a duel atop the bluffs known as The Palisades, overlooking Manhattan on the New Jersey side of the Hudson River.  We use their duel and a little poetic license to construct a problem.  In class you will be asked to provide a business rivalry that has similar structure.

Last Name

First Name

Social Security Number

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

Paces apart 10 8 6 4 2
Probability of a hit .2 .4 .6 .8 1

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. An example web site is the PBS page.

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

2. Will either of them wait until they are two paces apart to fire?
Yes No 

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

Yes No

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

Yes No

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

Yes No

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

Yes No