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PeerInfinity comments on Open Thread: April 2010 - Less Wrong
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I remember seeing an LW post about why it's cheating to always guess 50%, but I haven't found the link to that post yet... I think the basic idea was that you could technically be perfectly calibrated by always guessing 50%, but that's like always claiming that you don't know anything at all. It also means that you're never updating your probabilities. It also makes you easily exploitable, since you'll always assume that your probability of winning any gamble is 50%. Oh, and then there are the times when you'll give different probabilities for the same event, if the question is worded in different ways.
Your probability of winning any two-sided bet is 50%, as long as you pick which side of the bet you take at random. A "rational ignoramus" who always had minimum confidence wouldn't accept any arrangement where the opponent got to pick which side of the bet to take.
That implies a very easy Dutch-book:
Create a lottery with three possible outcomes (a), (b), and (c) - for example, (a) 1, (b) 2, 3, or 4, and (c) 5 or 6 on a six-sided die. (Note that the probabilities are not equal - I have no need of that stipulation.)
Ask "what are the odds that (a) will happen?" In response to the proposed even-odds, bet against (a).
Ask "what are the odds that (b) will happen?" In response to the proposed even-odds, bet against (b).
Ask "what are the odds that (c) will happen?" In response to the proposed even-odds, bet against (c).
Collect on two bets out of three, regardless of outcome.