If I read your answer right, you're saying that the OP is really a trick question, where 10-shot and 11-shot don't matter. You never know with these puzzlemakers, it could be :-)
Personally, I suspect that the OP was intended to be a real question, where 10-shot vs. 11-shot might matter as well as who shoots first. So I interpreted it as meaning the warden puts a bullet in a chamber, spins the chambers (the cylinder, I guess), and hands the gun to the first shooter.
Therefore, I'm convinced by chicory that going first with the 11-shot is best, and going last with the 11-shot is worst, with any position with the 10-shot being intermediate. So, if I could choose first, I'd avoid the 11-shot (as my opponent would choose to shoot first). I'd either choose the 10-shot, and guarantee the intermediate chances, or choose to go first (and hope my opponent would accidentally choose the 11-shot).