Russell,
I'm not suggesting that an omniscient player would not win over time as a
result of its superior knowledge.
I am suggesting that a non-omniscient player need not necessarily be
bilked in the sense meant by De Finetti; that is, it needn't be forced to
lose automatically due to dutch books made against it.
To illustrate a dutch book:
Say you believe in life on Mars with p=.1 and in intelligent life on Mars
with p=.01.
To De Finetti (and Ramsey), this is the same as saying you would pay 10
cents for a ticket worth $1 if there is life on Mars, and 1 cent for a
ticket worth $1 if there is intelligent life on Mars. Also you believe
these are fair bets such that you would be willing to take either side of
either transaction.
You are coherent here in the De Finetti sense no matter how right or wrong
you may be about the probabilities of life on Mars. No dutch books can be
made against you. No bookie can bilk you.
Would you consider instead valuing the tickets such that the first is
worth 1 cent and the second is worth 10 cents? No, you would not, because
in that case you would be incoherent: someone (an omniscient bookie or
otherwise) could exploit your incoherency by buying from you the first
ticket and selling you the second, locking in a profit of at least 9 cents
at your expense no matter what is true about life on Mars.
That is a dutch book.
Can entities with limited knowledge and resources be coherent in the sense
described, thus avoiding being bilked by omniscient bookies seeking to
make dutch books? I don't see why not.
-gts
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