An Axiomatic Characterization of Wagering Mechanisms

We construct a budget-balanced wagering mechanism that flexibly extracts information about event probabilities, as well as the mean, median, and other statistics from a group of individuals whose beliefs are immutable to the actions of others. We show how our mechanism, called the Brier betting mechanism, arises naturally from a modied parimutuel betting market. We prove that it is essentially the unique wagering mechanism that is anonymous,  proportional, sybilproof, and homogeneous. While the Brier betting mechanism is designed for individuals with immutable beliefs, we nd that it continues to perform well even for Bayesian individuals who learn from the actions of others.