A Decomposition of Forecast Error in Prediction Markets

We introduce and analyze sources of error in prediction market forecasts in order to characterize and bound the difference between a security’s price and its ground truth value. We consider cost-function-based prediction markets in which an automated market maker adjusts security prices according to the history of trade. We decompose the forecasting error into four components: \emph{sampling error}, occurring because traders only possess noisy estimates of ground truth; \emph{risk-aversion effect}, arising because traders reveal beliefs only through self-interested trade; \emph{market-maker bias}, resulting from the use of a particular market maker (i.e., cost function) to facilitate trade; and finally, \emph{convergence error}, arising because, at any point in time, market prices may still be in flux. Our goal is to understand the tradeoffs between these error components, and how they are influenced by design decisions such as the functional form of the cost function and the amount of liquidity in the market. We specifically consider a model in which traders have exponential utility and exponential-family beliefs drawn with an independent noise relative to ground truth. In this setting, sampling error and risk-aversion effect vanish as the number of traders grows, but there is a tradeoff between the other two components: decreasing the market maker’s liquidity results in smaller market-maker bias, but may also slow down convergence. We provide both upper and lower bounds on market-maker bias and convergence error, and demonstrate via numerical simulations that these bounds are tight. Our results yield new insights into the question of how to set the market’s liquidity parameter, and into the extent to which markets that enforce coherent prices across securities produce better predictions than markets that price securities independently.