How to Ask for Belief Statistics without Distortion?
Yi-Chun Chen, Ruoyu Wang, Xinhan Zhang
TL;DR
The paper addresses distortion in belief elicitation by introducing a nondistortionary framework that preserves the DM’s task choice while extracting beliefs. It develops the Counterfactual Scoring Rule (CSR) to elicit a single statistic independently of action and extends to multi-question settings via joint alignment, enabling non-distorting incentives through a Becker–DeGroot–Marschak style mechanism. A graph-theoretic analysis, using adjacency graphs and Kirchhoff’s law, establishes necessary and sufficient conditions for robust incentivizability, with extensions to product and multi-block graphs and a comparison to prior work PS25. The proposed approach offers a practical blueprint for designing belief-elicitation questions in experiments with arbitrary payoffs and state spaces, significantly reducing distortions in observed behavior and enabling reliable inference from belief reports.
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.