Econometrica: Jan 2021, Volume 89, Issue 1
Dynamic Belief Elicitation
Christopher P. Chambers, Nicolas S. Lambert
At an initial time, an individual forms a belief about a future random outcome. As time passes, the individual may obtain, privately or subjectively, further information, until the outcome is eventually revealed. How can a protocol be devised that induces the individual, as a strict best response, to reveal at the outset his prior assessment of both the final outcome and the information flows he anticipates and, subsequently, what information he privately receives? The protocol can provide the individual with payoffs that depend only on the outcome realization and his reports. We develop a framework to design such protocols, and apply it to construct simple elicitation mechanisms for common dynamic environments. The framework is general: we show that strategyproof protocols exist for any number of periods and large outcome sets. For these more general settings, we build a family of strategyproof protocols based on a hierarchy of choice menus, and show that any strategyproof protocol can be approximated by a protocol of this family.
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Supplement to "Dynamic Belief Elicitation"
This supplement includes additional results omitted from the main paper. In Section S.1, we provide an algorithm that computes the payoffs for a simple instanceof strategy proof protocol in the general setting of Section 4. In Section S.2, we show how to construct elicitation protocols for information structures involving potentially infinitely many time periods using menus with random deadlines. Sections S.3–S.5 are relevant to situations in which expert knowledge is solicited or evaluated for thepurpose of helping decision makers. In Section S.3, we show that, subject to regularity conditions, the knowledge of high-order beliefs elicited by the protocols we studyis sufficient to solve essentially any dynamic decision problem. In Section S.4, we argue that knowledge of these high-order beliefs is much needed when the decision environment is dynamic: we ask what decision problems can be solved using the classical methods that elicit only first-order beliefs, and show they form a degenerate class. Finally, in Section S.5, we illustrate our results in the context of simple principal-agent problems.