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Dynamics and Contracts for an Agent with Misspecified Beliefs

Yingkai Li, Argyris Oikonomou

TL;DR

This paper analyzes contract design when an agent holds misspecified beliefs about outcome distributions. It establishes that with two actions, a myopic Bayesian learner's action frequencies converge to a Berk-Nash equilibrium, while with three or more actions convergence may fail and computing approximate Berk-Nash equilibria becomes computationally hard under ETH/PPAD. For two-action settings, the authors provide a polynomial-time procedure to compute revenue-maximizing contracts under Berk-Nash equilibrium by exploiting posterior sparsity and solving linear programs. They also show that even small degrees of misspecification can cause substantial revenue losses for the principal, underscoring robustness concerns in contract design. Overall, the work highlights tractable contract design with binary actions, delineates computational barriers in richer action spaces, and motivates robust approaches to misspecification in economic design.

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.

Dynamics and Contracts for an Agent with Misspecified Beliefs

TL;DR

This paper analyzes contract design when an agent holds misspecified beliefs about outcome distributions. It establishes that with two actions, a myopic Bayesian learner's action frequencies converge to a Berk-Nash equilibrium, while with three or more actions convergence may fail and computing approximate Berk-Nash equilibria becomes computationally hard under ETH/PPAD. For two-action settings, the authors provide a polynomial-time procedure to compute revenue-maximizing contracts under Berk-Nash equilibrium by exploiting posterior sparsity and solving linear programs. They also show that even small degrees of misspecification can cause substantial revenue losses for the principal, underscoring robustness concerns in contract design. Overall, the work highlights tractable contract design with binary actions, delineates computational barriers in richer action spaces, and motivates robust approaches to misspecification in economic design.

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 -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.
Paper Structure (22 sections, 24 theorems, 94 equations, 3 figures)

This paper contains 22 sections, 24 theorems, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Utilities
  5. Contracts
  6. Subjective Beliefs and Misspecification
  7. Berk-Nash Equilibrium
  8. Bayesian Learning Agent
  9. Convergence to Berk-Nash for a Learning Agent with Two Actions
  10. Computational Challenges for Berk-Nash Equilibria
  11. Optimal Contracts Under Misspecification
  12. Polynomial Time Algorithms for Computing Optimal Contracts with Two Actions
  13. Revenue Losses from Misspecification
  14. Conclusions and Discussions
  15. Missing Proofs for Convergence with Binary Actions
  16. ...and 7 more sections

Key Result

Proposition 1

For any contract $P:R\rightarrow \mathbb{R}_+$, there exists at least one Berk-Nash equilibrium.

Figures (3)

  • Figure 1: Quadratic program for Finding a Berk-Nash equilibrium that maximized the utility of the principal when the posterior (and actions resp.) of the agent has support $\mathcal{B}^*$ ($A^*$ resp.).
  • Figure 2: LP for finding feasible distributions over actions that support a Berk-Nash equilibrium over a set of beliefs $\mathcal{B}^*\subseteq \mathcal{B}$.
  • Figure 3: LP for computing a Berk-Nash equilibrium that maximized the utility of the principal when the posterior (and actions resp.) of the agent has support $\mathcal{B}^*$ ($A^*$ resp.).

Theorems & Definitions (35)

  • Proposition 1: esponda2016berk
  • Definition 1: $\epsilon$-Berk-Nash equilibrium
  • Theorem 1
  • Lemma 1: esponda2021asymptotic
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Definition 2: General-sum Game
  • Definition 3: End-of-the-Line Problem and PPAD Class
  • ...and 25 more