Buuut, the whole point in game theory is that if you miss intentionally, you have no guarantee the other players will do the same and honor the “miss intentionally agreement”.

A history of past moves from other players help in analysing their reputation and knowing if they can be trusted, but the first player to shoot has no history. Assuming the worst, the optimal move is to shoot. Then the optimal move for the second player, seeing the first player aggressive history, is to also shoot.

The good news is that in real life, “civilized people” have the implicit reputation of not being aggressive, the “assuming the worst” doesn´t happen, and peaceful agreements can often be made.

In regards to the shooting truel with Charlie being the first shot, I think it not only behoove him to purposely miss, which you said, but to never shoot at Adam. The reason is that there’s no guarantee that Adam and Bob are completely rational, and being shot at would probably put Adam on the defensive and want to kill Charlie even if he’s not a big threat. Same would go with Bob, except Bob is not a guaranteed hit, so shooting and missing Bob first might work.

I wonder if Charlie should just obviously shoot so badly that both opponents have no suspicion as to his bad shooting (shooting very wide) or if he should show them both he refuses to shoot first by shooting into the air or putting his gun away. No point in wasting ammo after all, especially if it’s limited. Though this might draw suspicion.

Also, I’m terrible at poker, but often manage to knock out a few players because I’m not a big threat, and have managed to win a game…sometimes people let me get chips because they know they’ll get it back from me.

Anyway, I’ve been working on truels ever since I heard the word 11 days ago. I eventually found your site and noticed the percentages for the two weaker shooters differed from those given in the problem I was given. This motivated me to generalize the computer program I wrote which simulated the situation (it now accepts the shooters’ percentages from the keyboard). I only addressed the case that the first and weakest shooter intentionally misses on his first shot. The mathematics which analytically solves this situation and a computer program which simulates it may be seen if you follow the thread at

http://tobee-interpres.blogspot.com/2009/08/zee-da-gud-tobee-da-badd-and-luna-da.html

The simulated probability that the weakest shooter will survive matches up well with the analytically determined probability.

It was fun thinking about this problem and, like most computer programs, took many revisions to get it “just right.”