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Many experiments elicit subjects' prior and posterior beliefs about a randomvariable to assess how information affects one's own actions. However, beliefsare multi-dimensional objects, and experimenters often only elicit a singleresponse from subjects. In this paper, we discuss how the incentives offered byexperimenters map subjects' true belief distributions to what profit-maximizingsubjects respond in the elicitation task. In particular, we show how slightlydifferent incentives may induce subjects to report the mean, mode, or median oftheir belief distribution. If beliefs are not symmetric and unimodal, thenusing an elicitation scheme that is mismatched with the research question mayaffect both the magnitude and the sign of identified effects, or may even makeidentification impossible. As an example, we revisit Cantoni et al.'s (2019)study of whether political protests are strategic complements or substitutes.We show that they elicit modal beliefs, while modal and mean beliefs may beupdated in opposite directions following their experiment. Hence, the sign oftheir effects may change, allowing an alternative interpretation of theirresults.

Choosing The Best Incentives for Belief Elicitation with an Application to Political Protests