Abstract

We explore the striking mathematical connections that exist between market scoring rules, cost function based prediction markets, and no-regret learning. We show that any cost function based prediction market can be interpreted as an algorithm for the commonly studied problem of learning from expert advice by equating trades made in the market with losses observed by the learning algorithm. If the loss of the market organizer is bounded, this bound can be used to derive an O(sqrt(T)) regret bound for the corresponding learning algorithm. We then show that the class of markets with convex cost functions exactly corresponds to the class of Follow the Regularized Leader learning algorithms, with the choice of a cost function in the market corresponding to the choice of a regularizer in the learning problem. Finally, we show an equivalence between market scoring rules and prediction markets with convex cost functions. This implies that market scoring rules can also be interpreted naturally as Follow the Regularized Leader algorithms, and may be of independent interest. These connections provide new insight into how it is that commonly studied markets, such as the Logarithmic Market Scoring Rule, can aggregate opinions into accurate estimates of the likelihood of future events.

Highlights

  • Imagine you are interested in learning an accurate estimate of the probability that the United States unemployment rate for a particular month will fall below 10%

  • Chen et al [6] were the first to formally demonstrate a connection, showing that the standard Randomized Weighted Majority regret bound [9] can be used as a starting point to rederive the well-known bound on the worst-case loss of a Logarithmic Market Scoring Rule (LMSR) marker maker. (They went on to show that PermELearn, an extension of Weighted Majority to permutation learning [20], can be used to efficiently run LMSR over combinatorial outcome spaces for betting on rankings.) As we show in Section 4, the converse is true; the Weighted Majority regret bound can be derived directly from the bound on the worst-case loss of a market maker using LMSR

  • We prove an equivalence between another common class of prediction markets, market scoring rules, and convex cost function based markets,2 which immediately implies that market scoring rules can be interpreted as Follow the Regularized Leader algorithms too

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Summary

A New Understanding of Prediction Markets Via No-Regret Learning

The Harvard community has made this article openly available. Please share how this access benefits you. A new understanding of prediction markets via no-regret learning. In 11th ACM Conference on Electronic Commerce: June 7-11, 2010, Cambridge, MA, 189-198. New York, NY: Association for Computing Machinery.

INTRODUCTION
PREDICTION MARKETS
Market Scoring Rules
Cost Function Based Markets
Positive Translation Invariance
LEARNING FROM EXPERT ADVICE
A Bound on Regret
Rederiving the Weighted Majority Bound
CONNECTIONS BETWEEN MARKET
A Representation Theorem for Convex Cost Functions
Negative Translation Invariance
Relationship Between Convex Cost Functions and Market Scoring Rules
Convex Cost Functions and FTRL
Relation to the SCPM
EXAMPLE
DISCUSSION
PROOF OF LEMMA 2
PROOF OF LEMMA 3
Findings
STABILITY OF QUAD-SCPM PRICES
Full Text
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