A New Understanding of Prediction Markets Via No-Regret Learning

TL;DR

This paper reveals deep mathematical links between prediction markets, no-regret learning algorithms, and expert advice, providing new insights into market behavior and information aggregation.

Contribution

It establishes a formal equivalence between cost function based prediction markets, market scoring rules, and Follow the Regularized Leader algorithms, offering a unified theoretical framework.

Findings

Prediction markets can be interpreted as no-regret learning algorithms.

Convex cost functions correspond to Follow the Regularized Leader algorithms.

Market scoring rules are equivalent to certain prediction market algorithms.

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.