Duel Ok, in this classic gunfighting duel, let's assume for just a second that we add just one little obvious bit of twist onto the scheme: -Each gunfighter clearly would take the action that best ensures his survival each round. Given that the solution for best survival chance of the first gunfighter (you) is to shoot into the air, the second gunfighter (Mr. 50%), is posed with an interesting dilemma.  It's clear that he can shoot at the third(100%), and risk his own demise as either he is shot in return if he misses (he's a clear threat), or being shot 33.33% of the time by the first gunslinger.  Clearly, his survival rate within the next round if he shoots at the third and hits is 66.67%.  If he misses, his chances of surviving apparently drop to 50%...  the third gunslinger has clearly not decided who to shoot at yet.  Assuming that the second(50%) gunslinger shoots at the firrst, the third(100%) shooter MUST choose to engage the 50% shooter to ensure his own survival (since you, as the first shooter, are shooting into the air, you are no longer a threat). But wait a second... If the second (50%) shooter declines to shoot at the third (100%)) gunslinger by ALSO firing into the air, the chances of the third gunslinger dying in a given round drop to 0%!!!  Obviously, changing the status quo in the third gunslinger's mind is quite out of the question.  He obviously can't shoot at either of the first two gunslingers, as the successful killing of one of the two forces the other to pull down his or her guns and shoot back at the senior gunslinger, as a one-on-one duel between either of the two with Mr.100% clearly ends in thier death if they decline to fire again! So the second gunslinger as well, fires straight up into the air.  The third, realizing that his survival chances as well are maximized at 100% if he doesn't break the status quo, holsters his guns and walks off into the sunset (or invites you two into a nearby saloon for a drink). Clearly, pacifism is the optimal solution in this scenario.  Shooting into the air creates a clear 100% survival ratio each round for yourself as the shooter, given that shooter #2 and #3 play by the same rules of logic that you use when you vent your anger at the empty sky. Addendum : Clearly, the ammunition limitation eventually pulls the stalemate out of the fire, but you can just as easily decline to shoot as shoot into the air, which would perpetuate the stalemate. Stephen Hoffman Wednesday, July 24, 2002 "Each gunfighter clearly would take the action that best ensures his survival each round." If you are going to assume this, then obviously calling off the gun fight is the best chance for survival.  Everyone has a 100% chance of surviving if nobody shoots.  I don't even have to go through the numbers to prove that to myself. "where or who should you shoot at in round 1?" What is my objective?  Am I just to assume that I want to live?  Am I to assume that I want both the other fighters dead?  How badly do I want them dead?  Equally? See, I wouldn't be in a gun fight in the first place unless I wanted one or more of the other fighters dead at least enough that I'm willing to risk my own life to make the attempt. Suppose I really want to see sloppy Joe 50% bite it and I don't care so much about 100% dead eye Dick.  If so, then I'm going to take my shot at sloppy Joe.  Sure, if I hit, I'm dead with the next shot, but I will die with the satisfaction of knowing that ol' Joe is pushing daisies.  If I don't shoot him, then I know he's going to shoot at dead eye Dick.  If he misses Dick then Dick will get him for sure.  If Joe nails Dick, then I'll get another shot at Joe before he can turn on me. So how badly do I want to see Joe die?  If I want it bad enough, then I'll shoot at him even though it raises the certainty of my own demise. William Frantz Friday, July 26, 2002 It's not really a "Duel" if there are three people. As the problem was originally stated by Silverman, it's a "truel". Wei-Hwa Huang Thursday, August 15, 2002   Fog Creek Home