Nondistortionary belief elicitation

Abstract

A researcher wants to ask a decision-maker about a belief related to a choice the decision-maker made; examples include eliciting confidence or cognitive uncertainty. When can the researcher provide incentives for the decision-maker to report her belief truthfully without distorting her choice? We identify necessary and sufficient conditions for nondistortionary elicitation and fully characterize all incentivizable questions in three canonical classes of problems. For these problems, we show how to elicit beliefs using variants of the Becker-DeGroot-Marschak mechanism.

Figures (6)

• Figure 1: Summary of main results. Nodes of the adjacency graph are actions, with edges indicating adjacencies.
• Figure 2: Adjacency graph for Example \ref{['ex: Multiple choice test']} with $\Omega=\{0,1\}$ and $I=3$.
• Figure 3: Problems with three states. Each triangle depicts the simplex of beliefs. The black line segments illustrate the partition of the simplex according to which action is $u$-optimal. Blue line segments represent sets of beliefs on which $\mathsf{X}(a^*)$ is constant for the $u$-optimal action $a^*$. In the left triangle, because the blue lines form a triangle, adjacency considerations do not rule out incentivizability of $X$. In the right triangle, $X$ is not incentivizable if the payment for elicitation depends only on the value of $X$ and the reported belief.
• Figure 4: Small cycles of type (3) and type (2)
• Figure 5: Cycles with free transitions in case I.

Theorems & Definitions (54)

• Definition 1
• Definition 2
• Definition 3
• Proposition 1
• Corollary 1
• Lemma 1: Adjacency Lemma
• Example 1: Second-order beliefs
• Theorem 1
• Example 2
• Proposition 2