Indeed. Obviously Gelman is accomplished in the field, so it's a bit surprising. But it's easy to get confused on these issues, for anyone apparently. There is

Indeed. Obviously Gelman is accomplished in the field, so it's a bit surprising. But it's easy to get confused on these issues, for anyone apparently. There is a part towards the end of the paper where he suggests that the coin flip is really a "point distribution" at .5, while the outcome of the fight is something else. I think the intuition that he was going for was that if the fight was repeated N times, the outcome likely would not be N/2 wins for each. This suggests that there is "something" about the problem that could probably be known, and that is different from 1/2. But obviously that "something" is not our prior in the scenario given before the fight occurs. Determining the prior should be as simple as: Are both outcomes possible? Yes. Do you have any reason to expect one over the other? No. Well, then you have your prior. I think this "something" is where the idea of taking a uniform distribution on [0,1] comes from -- the idea that there is some "true" probability of the outcome of the fight, but we just don't know what it is, rather than realizing that no such thing exists. Really it seems like re-reading Jaynes should just clear the whole thing up.