Abstract
We consider here an important family of conditional bets, those that proceed to settlement if and only if some agreed evidence is received that a condition has been met. Despite an opinion widespread in the literature, we observe that when the evidence is strong enough to generate certainty as to whether the condition has been met or not, using traditional conditional probabilities for such bets will NOT preserve a gambler from having a synchronic Dutch Book imposed upon him. On the contrary (I show) the gambler can be Dutch-Booked if his betting ratios ever depart from a rather different probability, one that involves the probability of the agreed evidence being provided. We note furthermore that this same alternative probability assessment is necessary if the evidence is weaker (i.e. if it fails to provide knowledge whether or not the condition has been met.) By contrast, some of the (rather different) probability assessments proposed by Jeffrey, precisely for such situations, still expose the gambler to a Dutch-Book.
Highlights
My core task below is to refute a common belief about conditional probabilities, the claim that such probability assessments preserve a gambler from having a Dutch-Book imposed upon him when he makes an important type of conditional bet, one that proceeds to settlement if and only if the participating gamblers find out that some agreed condition has been met
The discussion below, is set in the standard DutchBook context: i.e. two gamblers (Peter and Mary say) are betting against each other on the outcomes of some chance-process, with Peter setting the odds through his assessments of the probability p(Z) of each outcome Z, and Mary setting the bets
It is well-known that if Peter’s odds are “coherent”, Mary cannot offer Peter a book composed of simple bets that will result in a net gain for Mary no matter which of the possible outcomes Z eventuates
Summary
We consider here an important family of conditional bets, those that proceed to settlement if and only if some agreed evidence is received that a condition has been met. Despite an opinion widespread in the literature, we observe that when the evidence is strong enough to generate certainty as to whether the condition has been met or not, using traditional conditional probabilities for such bets will not preserve a gambler from having a synchronic Dutch Book imposed upon him. On the contrary (I show) the gambler can be Dutch-Booked if his betting ratios ever depart from a rather different probability, one that involves the probability of the agreed evidence being provided. Some of the (rather different) probability assessments proposed by Jeffrey, precisely for such situations, still expose the gambler to a Dutch-Book
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