Prediction Markets for Machine Learning: Equilibrium Behaviour through Sequential Markets

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 thesis I show that the prices generated in sequentially traded prediction markets are stochastic approximations to the price given by an equilibrium analysis. This is done by showing that 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 to proposing a scheme called “mini-trading” which introduces a parameter related to the learning rate in SMD. I prove several properties of this scheme and show that it can improve the stability of prices in sequentially traded prediction markets. Also I analyse two popular trading models (namely the Maximum Expected Utility model and the Risk-measure model) in respect to an assumption on the class of traders I required to interpret sequential markets as SMD. I derive a sufficient condition for when the Maximum Expected Utility traders satisfy this assumption, but show that risk-measure based traders naturally satisfy this assumption for the type of markets I consider. Then I show that the “regret” of mini-trading markets (with respect to equilibrium markets) depend on the mini-trade parameter. Finally I attempt to compare the wealth updates of traders in sequential markets to the wealth updates in equilibrium markets, since this would help to extend the interpretation of equilibrium markets as performing Bayesian updates to sequential markets. For this I present preliminary results