Eliciting beliefs: Proper scoring rules, incentives, stakes and hedging

•We study three effects relevant to economists: Incentives, stakes and hedging.•The PSR biases are characterized for all PSR and all risk averse agents.•We conduct an experiment to confirm the empirical significance of the various biases.•The experimental results reveal complex distortions of reported beliefs.•Thus, the ability of PSR to recover beliefs in economic environments is in question.

Proper Scoring Rules (PSRs) are popular incentivized mechanisms to elicit an agent's beliefs. This paper combines theory and experiment to characterize how PSRs bias reported beliefs when (i) the PSR payments are increased, (ii) the agent has a financial stake in the event she is predicting, and (iii) the agent can hedge her prediction by taking an additional action. In contrast with previous literature, the PSR biases are characterized for all PSRs and all risk averse agents. Our results reveal complex distortions of reported beliefs, thereby raising concerns about the ability of PSRs to recover truthful beliefs in general decision-making environments.