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Manfred comments on Open Thread, November 8 - 14, 2013 - Less Wrong Discussion
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The difference between them becomes apparent once they start betting on other things, like the number of tails in a series of 10 coinflips. The question is: what is special about betting on heads vs. tails of a fair coin that doesn't allow Alice to do any better than Bob?
A fair coin is maximally entropic. There is no skill that will let you do anything with sheer chaos.
I think it is better to say that bet on offer is fair. It is not a property of just the coin, but also of the bet. We do not notice that there is a choice of bet because it is even odds (which corresponds to max ent), but for any weighted coin there is a corresponding fair bet.
Fair bets do have lots of special properties, but we would have the same situation if a correct choice of tails paid 1 and a correct choice of heads paid 2: Alice and Bob would both always bet H. (except in the 1/1000 chance that we start with 10 Ts and Alice updates wrongly; but the asymptotics are the same)