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. We show how our mechanism, called Brier betting mechanism, arises naturally from a modified parimutuel betting market. We prove that it is essentially the unique wagering mechanism that is anonymous, proportional, sybilproof, and homogeneous. In a Bayesian setting, we find that a slight bias away from truthful reporting may arise under asymmetric information, through the correlation between the total wealth wagered and the event outcome. The bias is driven towards zero as the fraction of any individual’s wealth compared to the group converges towards zero. ∗Correspondence to: Nicolas Lambert, Knight Management Center, 655 Knight Way, Stanford, CA 94305, USA. Part of this work was conducted while Chen, Lambert, Reeves and Vaughan were working at Yahoo! Research, New York. We are grateful to Alina Beygelzimer, Francis Bloch, Drew Fudenberg, Alfred Galichon, Antoine Lallour, Michael Ostrovsky, Marco Ottaviani, Eduardo Perez, Bethany Soule, Eugene Vorobeychik, Robert Wilson and the participants of the ninth ACM Conference of Electronic Commerce for helpful discussions. This material is based upon work supported by Google Research and the National Science Foundation under grants No. CCF-1101209, IIS-1054911, and CCF-0953516. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies. A preliminary version circulated under the title “Self-Financed Wagering Mechanisms for Forecasting” and was awarded an ACM Conference on Electronic Commerce Best Paper Award. †Stanford University, Graduate School of Business. Email: [email protected]. ‡Yahoo! Research New York. Email: [email protected]. §UCLA, Department of Computer Science. Email: [email protected] ¶Harvard University, School of Engineering and Applied Sciences. Email: [email protected]. ‖Beeminder LLC. Email: [email protected] ∗∗Stanford University, Department of Computer Science. Email: [email protected]. ††Yahoo! Research New York. Email: [email protected].