How to Ask for Belief Statistics without Distortion?

2026-02-11

AI Summary

This study addresses the problem that traditional incentive mechanisms in behavioral experiments often distort participants’ optimal decisions in the primary task, thereby biasing reported beliefs. To resolve this, the authors propose the Counterfactual Scoring Rule (CSR), which truthfully elicits statistical properties of participants’ beliefs without interfering with their primary-task behavior. Using graph-theoretic methods, they characterize the joint alignment conditions necessary for distortion-free incentive compatibility across multiple statistics and establish sufficiency via a refined Becker–DeGroot–Marschak mechanism. The resulting theoretical framework accommodates general task payoff structures and arbitrary belief elicitation questions, enabling effective and incentive-compatible elicitation of both single and multiple belief statistics.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Decision/Utility Theory

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Belief elicitation is ubiquitous in experiments but can distort behavior in the main tasks. We study when, and how, an experimenter can ask for a series of action-dependent belief statistics after a subject chooses an action, while incentivize truthful reports without distorting the subject's optimal action in the main experimental tasks. We first propose a novel mechanism called the Counterfactual Scoring Rule (CSR), which achieves such nondistortionary elicitation of any single belief statistic by decomposing it into supplemental action-independent statistics. In contrast, when eliciting a fixed set of belief statistics without such decomposition, we show that robust nondistortionary elicitation is achievable if and only if the questions satisfy a joint alignment condition with the task payoff. The necessity of joint alignment is established through a graph theoretical approach, while its sufficiency follows from invoking an adaptation of the Becker-DeGroot-Marschak mechanism. Our characterization applies to experiments with general task-payoff structures and belief elicitation questions.

Problem

• belief elicitation
• nondistortionary elicitation
• action-dependent beliefs
• experimental design
• incentive compatibility

Innovation

• Counterfactual Scoring Rule
• belief elicitation
• nondistortionary mechanism
• joint alignment condition
• Becker-DeGroot-Marschak mechanism