[Paper Review] Dynamic Information Provision: Rewarding the Past and Guiding the Future

Ian Ball|arXiv (Cornell University)|Mar 16, 2023

Game Theory and Applications10 citations

TL;DR

The paper characterizes the sender’s optimal dynamic information policy in a long-term(sender-receiver) relationship, showing that delayed reporting with a bias-variance tradeoff can implement the Pareto frontier, featuring a transition phase and a stationary phase.

ABSTRACT

I study the optimal provision of information in a long-term relationship between a sender and a receiver. The sender observes a persistent, evolving state and commits to send signals over time to the receiver, who sequentially chooses public actions that affect the welfare of both players. I solve for the sender's optimal policy in closed form: the sender reports the value of the state with a delay that shrinks over time and eventually vanishes. Even when the receiver knows the current state, the sender retains leverage by threatening to conceal the future evolution of the state.

Motivation & Objective

• Motivate how information can substitute for money to influence behavior in a long-term relationship.
• Model a sender who observes a persistent evolving state and commits to signals over time.
• Characterize the optimal information policy and show how it balances bias and precision over time.

Proposed method

• Model in continuous time a sender-receiver pair with a persistent state following a diffusion process.
• Define action bias and posterior variance as core statistics of a decision rule.
• Prove Bayes plausibility of variance paths and show that any Bayes-plausible path can be induced by delayed reporting.
• Solve for the optimal bias and variance paths using Lagrangian relaxation and dynamic programming.
• Provide a two-period example to illustrate the bias-precision tradeoff and extend to the general continuous-time solution.
• Establish an obedience-based reduction to deterministic bias-variance paths and characterize transition and stationary phases.

Experimental results

Research questions

• RQ1 What is the optimal dynamic information policy a sender should commit to when the receiver acts over time and the state evolves persistently?
• RQ2 How do bias and information precision trade off over time to maximize the sender’s payoff?
• RQ3 Under what conditions can the sender implement first-best decisions, and when is a gradual, delayed reporting policy necessary?
• RQ4 How does the policy change when transitioning from partial to full information about the current state?

Key findings

• The sender’s optimal policy is a delayed reporting scheme where the action recommendation reveals the state realization from an earlier time and uses a bias that evolves over time.
• Two phases emerge: a transition phase where the receiver’s uncertainty is reduced while the bias narrows the gap to the sender’s preferred action, followed by a stationary phase with perfect state information and a fixed bias.
• If the sender’s bias is small relative to state volatility, the first-best rule can be induced; otherwise, a bias-precision tradeoff governs the optimal policy.
• Bayes-plausible variance paths are necessary and sufficient for inducibility; deterministic (path) bias and variance suffice to trace the Pareto frontier.
• Obedience constraints bind when variance is positive, linking the rate of bias to the reduction in posterior variance via a differential equation (r-2κ)b^2 = 2κv + σ^2 − v′.
• The optimal policy features a transition time T where full disclosure occurs and a stationary phase thereafter with a fixed bias and zero variance.