Isoelastic Agents and Wealth Updates in Machine Learning Markets

Recently, prediction markets have shown considerable promise for developing flexible mechanisms for machine learning. In this paper, agents with isoelastic utilities are considered. It is shown that the costs associated with homogeneous markets of agents with isoelastic utilities produce equilibrium prices corresponding to alpha-mixtures, with a particular form of mixing component relating to each agent's wealth. We also demonstrate that wealth accumulation for logarithmic and other isoelastic agents (through payoffs on prediction of training targets) can implement both Bayesian model updates and mixture weight updates by imposing different market payoff structures. An iterative algorithm is given for market equilibrium computation. We demonstrate that inhomogeneous markets of agents with isoelastic utilities outperform state of the art aggregate classifiers such as random forests, as well as single classifiers (neural networks, decision trees) on a number of machine learning benchmarks, and show that isoelastic combination methods are generally better than their logarithmic counterparts.Comment: Appears in Proceedings of the 29th International Conference on Machine Learning (ICML 2012

Proportional Dynamics in Exchange Economies

We study the Proportional Response dynamic in exchange economies, where each player starts with some amount of money and a good. Every day, the players bring one unit of their good and submit bids on goods they like, each good gets allocated in proportion to the bid amounts, and each seller collects the bids received. Then every player updates the bids proportionally to the contribution of each good in their utility. This dynamic models a process of learning how to bid and has been studied in a series of papers on Fisher and production markets, but not in exchange economies. Our main results are as follows: - For linear utilities, the dynamic converges to market equilibrium utilities and allocations, while the bids and prices may cycle. We give a combinatorial characterization of limit cycles for prices and bids. - We introduce a lazy version of the dynamic, where players may save money for later, and show this converges in everything: utilities, allocations, and prices. - For CES utilities in the substitute range $[0,1)$, the dynamic converges for all parameters. This answers an open question about exchange economies with linear utilities, where tatonnement does not converge to market equilibria, and no natural process leading to equilibria was known. We also note that proportional response is a process where the players exchange goods throughout time (in out-of-equilibrium states), while tatonnement only explains how exchange happens in the limit.Comment: 25 pages, 6 figure

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