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Explanations, Hints, and Suggestions for Assignment #2
[
Revised 03/18/08
]
These notes help explain the problems and oﬀer suggestions for how to proceed if you’re
stuck. The suggestions give one particular way of solving the problems; feel free to ignore
them and solve the problems your own way.
1 Problem 1
Start by doing the ﬁrst part, looking for Bob’s violations of probability axioms and theorems
(Bob’s quotients are nonprobabilistic in more than one way). On the second part, if you’re
having trouble ﬁnding a Dutch Book against Bob, try the following:
•
Look back at the lecture notes where Branden proved the Dutch Book Theorem. See
how he constructed books against agents with probability violations like Bob’s. Try to
build an analogous book against Bob.
•
If you’re really stuck, try working backwards. Imagine you already had a book of bets
against Bob’s quotients. Let the stake of the ﬁrst bet be
s
1
, the stake of the second bet
be
s
2
, etc. Write a table of possible ways the bets could come out, then calculate Bob’s
total payoﬀ for each possible outcome using the formulas from Lecture 11, Slide 12.
Since Bob must lose on every possible outcome if your bets are to constitute a Dutch
Book, each total payoﬀ must be a negative number. Thus you can use your table to
write some inequalities that have to be true for this to be a Book against Bob. Once
you have a set of inequalities, either solve them for the
s
n
s using algebra or use trial
anderror to ﬁnd some
s
n
values that satisfy all the inequalities. Finally, translate the
stake values back into actual bets using the formulas from Lecture 11, Slide 12.
On the third part of Problem 1, we’re not looking for anything like a rigorous proof. Just
give a reasonable argument for why you think your answer is correct.
2 Problem 2.1
2.1 The Framework
A credence function
q(X)
assigns a real number between 0 and 1 (inclusive) to every propo
sition in some set
B
. We can also imagine a “world function”
w(X)
that assigns 1 to every
true proposition and 0 to every false proposition. It seems reasonable that the most accu
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This note was uploaded on 08/01/2008 for the course PHIL 148 taught by Professor Fitelson during the Spring '08 term at University of California, Berkeley.
 Spring '08
 FITELSON

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