Elicitation of probabilities using competitive scoring rules

Several forecasters predict the probability of an event, and then make or receive payments contingent on their predictions and on whether the event actually occurs. The payment functions generalize the concept of scoring rule to a competitive setting. We allow for exogenously determined subsidies to each forecaster, and require that the scheme be anonymous, neutral, and truth-inducing. By centering each forecaster's payment at the average payment to all other forecasters, we construct competitive scoring rules that reward the better predictors. Applications include multiparty betting and fixed-budget surveys to determine subjects' truthful probability assessments. We discuss when forecasters would voluntarily participate in such a competition, and relate our results to the scoring rules proposed by De Finetti for eliciting probabilities.