tag:blogger.com,1999:blog-3891434218564545511.post6389777815957630739..comments2024-03-28T19:56:42.305-05:00Comments on Alexander Pruss's Blog: Lebesgue sums previsions don't always lead to Dutch Books for inconsistent credencesAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3891434218564545511.post-90106059413107241792020-01-24T03:34:02.842-06:002020-01-24T03:34:02.842-06:00Here is another way of looking at it.
For Dutch B...Here is another way of looking at it.<br /><br />For Dutch Books as applied in the post, the only relevant feature of a prevision (for given credences) is its “acceptance set”, i.e. the set of wagers assigned non-negative value. Previsions with the same acceptance set will be subject (or not) to the same Dutch Books, however else they may differ.<br /><br />In the 2-point case, the acceptance set for credences α and β (your notation) is the same as for the consistent credences α/(α+β) and β/(α+β). Hence no Dutch Books.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-53387634045256374392020-01-23T09:16:25.946-06:002020-01-23T09:16:25.946-06:00That all looks correct.
In fact, Lebesgue sum pre...That all looks correct.<br /><br />In fact, Lebesgue sum prevision always gives failures of domination in cases of inconsistency.<br /><br />I was just surprised that it doesn't always give a dutch book.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-47321143138872244442020-01-23T01:32:40.493-06:002020-01-23T01:32:40.493-06:00It is good to avoid Dutch books, but it is not eno...It is good to avoid Dutch books, but it is not enough. You also want to avoid choosing dominated options. With inconsistent credences, the value assigned by the Lebesgue sum prevision is discontinuous in the neighbourhood of wagers with equal payoffs on two or more outcomes. This leads to cases in which the Lebesgue sum prevision will assign a higher value to a wager with lower payoffs than another, whatever the outcome. (Note that level set method avoids these issues.)<br /><br />Here is an example. Suppose you have inconsistent credences Cr(a) = Cr(b) = 0.2, Cr(a or b) = 1. Wager#1 pays 1 on a and o.9 on b. This is given value 0.38. Wager#2 pays 0.5 on a or b. This has value 0.5. So, based on Lebesgue sum prevision, you would choose Wager#2. But Wager#2 pays less than Wager#1 whatever the outcome. This is clearly the wrong choice.<br /><br />As you show, there are no Dutch books if Ω has two points. But if Ω has three or more points, Dutch books are possible. An example follows.<br /><br />Suppose Ω has three points. Call them a, b and c. Suppose your credences are Cr(a) = Cr(b) = Cr(c) = 1/3 and (inconsistently) Cr(a or b) = Cr(b or c) =Cr(c or a) = 1/2. Then you would accept the wager with payoffs 1.8 on a and -1 on b or c. (Expectation is 1.8/3 – 1/2 = +0.1) You would also accept each of the two similar wagers with the payoffs cyclically permuted. The three wagers together make a Dutch book. (Whatever the outcome, you win one bet and lose two with net payoff 1.8 – 2*1 = -0.2.)IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.com