Sunday, June 21, 2009

A truel

From Fermat´s Last Theorem, by Simon Singh,

"A truel is similar to a duel, except there are three participants rather than two. One morning Mr Black, Mr Grey and Mr White decide to resolve a conflict by truelling with pistols until only one of them survives. Mr Black is the worst shot hitting his target on average only one time in three. Mr Grey is a better shot, hitting his target two times out of three. Mr White is the best shot, hitting his target every time. To make the truel fairer Mr Black is allowed to shoot first, followed by Mr Grey (if he is still alive), followed by Mr White (if he is still alive) and round again until only one of them is alive. The question is this: Where should Mr Black aim his first shot?"

If you've not seen this question before, please feel free to drop me a comment with your answer and a short description of why.

This kind of question is part of the branch of maths which has been called Game Theory, a term coined by John von Neumann in 1944. This became fundamental in the study of military strategy, particularly during the Cold War. People of my generation will be familiar with the 1983 film War Games, in which a young Matthew Broderick attempts to teach game theory to a computer which is about to start a global nuclear war, using his Commodore Vic 20.


David said...

Mike, My intuitive answer is that Mr. Black should try to take out Mr. White with his first shot, but I'm going to try to work through some outcomes later to see if I can get more insight.

Regards from Hastings, UK, your old pal


GodBotherer said...

I've run through all the options and possible outcomes. If Black aims first at Grey then death is 8x more likely than victory. If Black first aims at White then death is around 3x more likely than victory. But my analysis takes no account of judgement: I have assumed that given a choice of two targets, the shooter has no preference (i.e. he might as well toss a coin to choose his target). I fear that taking the shooters' judgement into account will not be a simple matter... I may try just tweaking the frequencies at such 'choices', so use 0.4 and 0.6 instead of 0.5 and 0.5, and see which way it moves the outcomes.

Mike Smith said...

Seems like good reasoning from this chap. But unfortunately not the correct answer yet.

John DeLuca said...

BLACK SHOOTS: I think that Black should shoot straight up in the air. If Black kills Grey, he is certainly killed by White and if Black kills White, he has a 2 in 3 chance of being killed by Grey.

GREY SHOOTS: This leaves Grey fearing White's perfect aim. Grey would then be forced take his shot at White. Otherwise, If Grey kills White (p=0.666), then Black can attempt to kill Grey (p=0.333).

WHITE SHOOTS: If White is still alive, then White still has to shoot at Grey since Grey is the greater threat. In this case, White kills Grey (p=1.000).

BLACK SHOOTS: White (p=0.333) or Grey (p=0.666) survive to this point, so now Black has a chance (p=0.333) of killing the survivor.

This scenario has Black with the best chance of survival, then Grey, then White.

Mike Smith said...

An excellent piece of reasoning. Thankyou John