A Dutch-Book Trap for Misspecification

TL;DR

This paper demonstrates that Bayesian agents updating beliefs via Bayes' rule are immune to Dutch-booking, even with misspecification, and explores how certain financial and game scenarios relate to Dutch-book strategies.

Contribution

It establishes conditions under which Bayesian updating prevents Dutch-booking despite model misspecification and links Dutch books to financial instruments and game problems.

Findings

Bayesian updating with Bayes' rule prevents Dutch-booking under misspecification.

Dutch books can be constructed using financial instruments and Monty Hall scenarios.

Agents with lexicographic priors can avoid Dutch-booking even with misspecified models.

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.