A Dutch-Book Trap for Misspecification

Abstract: We provide Dutch-book arguments against two forms of misspecified Bayesian learning. An agent progressively learns about a state and is offered a bet after every new discovery. We say the agent is Dutch-booked when they are willing to accept all bets, but their payoff is negative under each state either ex-post, or in expectation given the objective conditional probabilities of the discoveries (i.e., the correct data-generating process, DGP). Respectively, an agent cannot be Dutch-booked if and only if they update their beliefs with Bayes rule either from the previous belief, even using misspecified likelihood functions, or from one lexicographic prior, using the correct data-generating process. Under a large population interpretation of the DGP, this means that a population can suffer aggregate losses under all states when different individuals update their beliefs from different (lexicographic) priors, or using misspecified likelihoods. Thus, the Dutch-book argument offers a general characterization of the perils of misspecification.