I think this is the key. Given that the game is winner-take-all, maximising your expected value isn't necessarily the right way to go. To take an extreme exampl

I think this is the key. Given that the game is winner-take-all, maximising your expected value isn't necessarily the right way to go. To take an extreme example, suppose I had a magic strategy that gave me a value of 1 every few throws and a lousy number most other times. That might give me a good expected value, but I'd still lose most rounds. So what's the right objective? Trying to beat your opponent's median score, so that you win more than half the time? Edit: I think you're on the right track, but I'm not sure I believe that you will lose more than half the time if you land below your opponent's expected value. I think that assumes their distribution of scores is non-skewed, which is not obvious to me.