The Shooters Logic (A Three Way Duel)
Car Talk 2007.01.27


RAY: This puzzler is mathematical in nature. Imagine if you will, three gentleman, Mr. Black, Mr. Brown and Mr. White, who so detest each other that they decided to resolve their differences with pistols. It's kind of like a duel; only a three-way duel. And unlike the gunfights of the old West, where the participants would simultaneously draw their guns and shoot at each other, these three gentlemen have come up with a rather more civilized approach.

Mr. White is the worst shot of the three and hits his target one time out of three. Mr. Brown is twice as good and hits his target two times out of three. Mr. Black is deadly. He never misses. Whomever he shoots at is a goner.

To even the odds a bit, Mr. White is given first shot. Mr. Brown is next, if he's still alive. He's followed by Mr. Black, if he's still alive.

They will continue shooting like this, in this order, until two of them are dead.

HERE IS A SOLUTION: If you work backwards a little bit and assume Mr. White hits whoever he is shooting at then it is the case on the next round there will only be two folks left, him and whoever he did NOT shoot at.

So, Mr. White must not shoot at Mr. Brown because only Mr. Black will be left and he (Mr. White) will be shot. So Mr. White should shoot at Mr. Black.

If Mr. White misses then all three are left and it is Mr. Brown's turn and he will apply the same logic and will also shoot at Mr. Black.

If he hits Mr. Black then Mr. Brown gets another chance. If he misses Mr. Black then Mr. Black will shoot at Mr. Brown since he is the next best shot.

This assumes that all the shooters are rational and do not shoot back at the person who just shot at them out of anger without regard to logic.


(note:  After listening to the results on the radio this morning I find that my solution is NOT optimal. The optimal solution is for Mr. White to shoot at no one and just take a pass on his first shot.  Next, my logic below still applies and Mr. Brown is obliged to shoot at Mr. Black and if he misses then Mr. Black is obliged to shoot at Mr. Brown and in either case Mr. White still gets another shot.  If Mr. White does shoot at someone on his first shot he does decrease his chances of survival on the next round if he hits someone.  So I agree that he should 'shoot-to-miss' on his first shot.  The following illustrates the probabilities that each player survives as the rounds go by.  Since the probability that A hits anything and the probability that B hits anything are both less than one then they could wind up shooting at each other forever with the expected probability that A survives getting smaller and smaller.  Since C always hits what he shoots at then then his probability is seen to stabilize. I am working on a graphic that shows this.

Also on the web here is a page devoted to the same problem with slightly different probabilities: in that encounter the probabilities are 100%, 80% and 50%.  Dr. Chen wrote his analysis after reading Martin Gardner's description which has the same probabilities in his book Martin Gardner's Mathematical Games.


This Page Last Update Monday, February 19, 2007 11:28:11 -0500