Learning Performance of Prediction Markets with Kelly Bettors
AbstractIn evaluating prediction markets (and other crowd-prediction mechanisms), investigators have repeatedly observed a so-called "wisdom of crowds" effect, which roughly says that the average of participants performs much better than the average participant. The market price---an average or at least aggregate of traders' beliefs---offers a better estimate than most any individual trader's opinion. In this paper, we ask a stronger question: how does the market price compare to the best trader's belief, not just the average trader. We measure the market's worst-case log regret, a notion common in machine learning theory. To arrive at a meaningful answer, we need to assume something about how traders behave. We suppose that every trader optimizes according to the Kelly criteria, a strategy that provably maximizes the compound growth of wealth over an (infinite) sequence of market interactions. We show several consequences. First, the market prediction is a wealth-weighted average of the individual participants' beliefs. Second, the market learns at the optimal rate, the market price reacts exactly as if updating according to Bayes' Law, and the market prediction has low worst-case log regret to the best individual participant. We simulate a sequence of markets where an underlying true probability exists, showing that the market converges to the true objective frequency as if updating a Beta distribution, as the theory predicts. If agents adopt a fractional Kelly criteria, a common practical variant, we show that agents behave like full-Kelly agents with beliefs weighted between their own and the market's, and that the market price converges to a time-discounted frequency. Our analysis provides a new justification for fractional Kelly betting, a strategy widely used in practice for ad-hoc reasons. Finally, we propose a method for an agent to learn her own optimal Kelly fraction.
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1201.6655.
Date of creation: Jan 2012
Date of revision:
Contact details of provider:
Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2012-02-15 (All new papers)
- NEP-CTA-2012-02-15 (Contract Theory & Applications)
- NEP-FOR-2012-02-15 (Forecasting)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Rubinstein, Mark, 1974. "An aggregation theorem for securities markets," Journal of Financial Economics, Elsevier, Elsevier, vol. 1(3), pages 225-244, September.
- Justin Wolfers & Eric Zitzewitz, 2006.
"Interpreting prediction market prices as probabilities,"
Working Paper Series, Federal Reserve Bank of San Francisco
2006-11, Federal Reserve Bank of San Francisco.
- Wolfers, Justin & Zitzewitz, Eric, 2006. "Interpreting Prediction Market Prices as Probabilities," CEPR Discussion Papers, C.E.P.R. Discussion Papers 5676, C.E.P.R. Discussion Papers.
- Justin Wolfers & Eric Zitzewitz, 2006. "Interpreting Prediction Market Prices as Probabilities," NBER Working Papers 12200, National Bureau of Economic Research, Inc.
- Wolfers, Justin & Zitzewitz, Eric, 2006. "Interpreting Prediction Market Prices as Probabilities," IZA Discussion Papers 2092, Institute for the Study of Labor (IZA).
- Grossman, Sanford J, 1981. "An Introduction to the Theory of Rational Expectations under Asymmetric Information," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 48(4), pages 541-59, October.
- Peter A. Morris, 1983. "An Axiomatic Approach to Expert Resolution," Management Science, INFORMS, INFORMS, vol. 29(1), pages 24-32, January.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators).
If references are entirely missing, you can add them using this form.