Dynamics and Contracts for an Agent with Misspecified Beliefs

TL;DR

This paper investigates how agents with misspecified beliefs behave in contracting environments, revealing convergence properties, computational challenges, and the impact of misspecification on revenue.

Contribution

It demonstrates convergence to Berk-Nash equilibrium for two actions, highlights computational complexity issues with multiple actions, and provides polynomial-time methods for revenue optimization under misspecification.

Findings

Action frequency converges to Berk-Nash equilibrium with two actions.

Convergence fails with three or more actions.

Misspecification significantly reduces optimal revenue.

Abstract

We study a single-agent contracting environment where the agent has misspecified beliefs about the outcome distributions for each chosen action. First, we show that for a myopic Bayesian learning agent with only two possible actions, the empirical frequency of the chosen actions converges to a Berk-Nash equilibrium. However, through a constructed example, we illustrate that this convergence in action frequencies fails when the agent has three or more actions. Furthermore, with multiple actions, even computing an $\varepsilon$-Berk-Nash equilibrium requires at least quasi-polynomial time under the Exponential Time Hypothesis (ETH) for the PPAD-class. This finding poses a significant challenge to the existence of simple learning dynamics that converge in action frequencies. Motivated by this challenge, we focus on the contract design problems for an agent with misspecified beliefs and two possible actions. We show that the revenue-optimal contract, under a Berk-Nash equilibrium, can be computed in polynomial time. Perhaps surprisingly, we show that even a minor degree of misspecification can result in a significant reduction in optimal revenue.