Pivot Mechanisms in Probability Revelation

Abstract

The Groves mechanism and k^th price auctions are well-known examples of pivot mechanisms. In this paper an analogous pivot mechanism is defined for probability revelation and then the Bayesian equilibria are characterized for the three pivot mechanisms. The main result is that in Bayesian games with these pivot mechanisms, equilibria must satisfy a simple fixed point condition. The result does not require signal ordering properties and thus generalizes and simplifies results by Milgrom and others. When the fixed point is unique there is "no regret." The result also holds for games less structured than Bayesian games (where the common knowledge and consistency assumptions are relaxed). The pivot mechanism in probability revelation is shown to generalize and characterize proper scoring rules. The characterization yields an optimization of research incentives for proper scoring rules and suggests that under some conditions the new mechanisms, which are pivot mechanisms but not proper scoring rules, outperform proper scoring rules.