Decision rules and decision markets

Abstract

We explore settings where a principal must make a decision about which action to take to achieve a desired outcome. The principal elicits the probability of achieving the outcome by following each action from a self-interested (but decision-agnostic) expert. We prove results about the relation between the principal’s decision rule and the scoring rules that determine the expert’s payoffs. For the most natural decision rule (where the principal takes the action with highest success probability), we prove that no symmetric scoring rule, nor any of Winkler’s asymmetric scoring rules, have desirable incentive properties. We characterize the set of differentiable scoring rules with desirable incentive properties and construct one. We then consider decision markets, where the role of a single expert is replaced by multiple agents that interact by trading with an automated market maker. We prove a surprising impossibility for this setting: an agent can always benefit from exaggerating the success probability of a suboptimal action. To mitigate this, we construct automated market makers that minimize manipulability. Finally, we consider two alternative decision market designs. We prove the first, in which all outcomes live in the same probability universe, has poor incentive properties. The second, in which the experts trade in the probability of the outcome occurring unconditionally, exhibits a new kind of no-trade phenomenon.