A Dutch-Book Trap for Misspecification
Emiliano Catonini, Giacomo Lanzani
TL;DR
This paper develops Dutch-book arguments for misspecified Bayesian learning in a dynamic, information-revealing environment. It establishes that forward-consistency (Bayes updating with some likelihood) is necessary and sufficient to prevent deterministic Dutch books, while complete coherence via a lexicographic prior with the correct model blocks Dutch-booking in objective terms. The authors connect these ideas to Bayes-plausibility and CCBS concepts, showing that misspecification can harm decision-making even when beliefs update via Bayes’ rule, and they identify robustness limits when bookmakers’ models are slightly misspecified. They illustrate the results with correlation neglect in finance and Monty Hall-type problems, showing how population-level betting opportunities arise from systematic misperceptions, and they discuss the implications for population heterogeneity and the design of coherent priors in dynamic learning settings.
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.