Aggregating predictions via sequential mini-trading

Prediction markets which trade on contracts representing unknown future outcomes are designed specifically to aggregate expert predictions via the market price. While there are some existing machine learning interpretations for the market price and connections to Bayesian updating under the equilibrium analysis of such markets, there is less of an understanding of what the instantaneous price in sequentially traded markets means. In this paper we show that the prices generated in sequentially traded prediction markets are stochastic approximations to the price given by an equilibrium analysis. We do so by showing the equilibrium price is a solution to a stochastic optimisation problem which is solved by stochastic mirror descent (SMD) by a class of sequential pricing mechanisms. This connection leads us to propose a scheme called mini-trading" which introduces a parameter related to the learning rate in SMD. We prove several properties of this scheme and show that it can improve the stability of prices in sequentially traded prediction markets.