Upping the Stakes and the Preface Paradox

Abstract

The preface paradox, first introduced by David Makinson (Analysis 25:205–207, 1961), presents a plausible scenario where an agent is evidentially certain of each of a set of propositions without being evidentially certain of the conjunction of the set of propositions. Given reasonable assumptions about the nature of evidential certainty, this appears to be a straightforward contradiction. We solve the paradox by appeal to stake size sensitivity, which is the claim that evidential probability is sensitive to stake size. The argument is that because the informational content in the conjunction is greater than the sum of the informational content of the conjuncts, the stake size in the conjunction is higher than the sum of the stake sizes in the conjuncts. We present a theory of evidential probability that identifies knowledge with value and allows for coherent stake-sensitive beliefs. An agent’s beliefs are represented two-dimensionally as a bid-ask spread, which gives a bid price and an ask price for bets at each stake size. The bid-ask spread gets wider when there is less valuable evidence relative to the stake size and narrower when there is more valuable evidence according to a simple formula. The bid-ask spread can represent the uncertainty in the first-order probabilistic judgement. According to the theory, it can be coherent to be evidentially certain at low stakes, but less than certain at high stakes, and therefore there is no contradiction in the preface. The theory not only solves the paradox but also gives a good model of decisions under risk that overcomes many of the problems associated with classic expected utility theory.