Level-strategyproof Belief Aggregation Mechanisms

In the problem of aggregating experts' probabilistic predictions over an ordered set of outcomes, we introduce the axiom of level-strategy\-proofness (level-SP) and prove that it is a natural notion with several applications. Moreover, it is a robust concept as it implies incentive compatibility in a rich domain of single-peakedness over the space of cumulative distribution functions (CDFs). This contrasts with the literature which assumes single-peaked preferences over the space of probability distributions. Our main results are: (1) a reduction of our problem to the aggregation of CDFs; (2) the axiomatic characterization of level-SP probability aggregation functions with and without the addition of other axioms; (3) impossibility results which provide bounds for our characterization; (4) the axiomatic characterization of two new and practical level-SP methods: the proportional-cumulative method and the middlemost-cumulative method; and (5) the application of proportional-cumulative to extend approval voting, majority rule, and majority judgment methods to situations where voters/experts are uncertain about how to grade the candidates/alternatives to be ranked.\footnote{We are grateful to Thomas Boyer-Kassem, Roger Cooke, Aris Filos-Ratsikas, Herv\'e Moulin, Clemens Puppe and some anonymous EC2021 referees for their helpful comments and suggestions.} \keywords{Probability Aggregation Functions \and ordered Set of Alternatives \and Level Strategy-Proofness \and Proportional-Cumulative \and Middlemost-Cumulative}