A Dutch-Book Trap for Misspecification

Abstract

We provide Dutch-book arguments against misspecified Bayesian learning. An agent progressively learns about a state and is offered a bet after every discovery. We say the agent is deterministically Dutch-booked when they would accept all bets, but their payoff is ex-post negative under each state. More generally, we say that the agent is Dutch-booked when they would accept all bets, but their expected payoff under each fundamental state is negative. With this, the agent cannot be deterministically Dutch-booked if and only if they update their beliefs using Bayes' rule, even with misspecified likelihoods. The agent cannot be Dutch booked if and only if they update their beliefs using Bayes' rule with a lexicographic prior and using the correct data-generating process. We show that offers of financial instruments and behavior in Monty Hall problems can be viewed as Dutch books that extract a sure expected gain from a misspecified population.