assignment_2_hints - Explanations, Hints, and Suggestions...

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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 non-probabilistic 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- and-error 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.

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assignment_2_hints - Explanations, Hints, and Suggestions...

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