Rational Degree of Belief Ceases to Be Additive When the Dutch Book Argument Is Transported to a New Environment

Abstract

The strength of a person’s beliefs can be measured by the buying and selling prices they offer on contingent promissory notes. Consider a promissory note contingent on a proposition; it pays off one unit of money if the proposition is true and nothing otherwise. The more strongly a person believes the proposition, the higher the minimum price would be at which they would sell it. The same would apply to the maximum purchase price. The well-known Dutch Book Argument claims that, if the person’s beliefs are rational, their buying/selling prices should combine additively, meaning that the price of a promissory note contingent on the disjunction of two incompatible propositions should be the sum of the prices of the promissory notes contingent on the individual incompatible propositions. This paper shows that the essence of the Dutch Book Argument is that rational belief is additive because money is additive. It is proved that, if the structure of the Dutch Book Argument is kept, but a nonadditive resource is substituted for money, then rational belief will follow a nonadditive combining rule. It is also shown how rational buying/selling prices behave when the pay-off amount of a contingent note changes.

Keywords

• Dutch Book Argument
• partial belief
• rationality
• consistency
• coherence
• additivity