Abstract

We provide the first concrete algorithm for combining market makers and limit orders in a prediction market with continuous trade. Our mechanism is general enough to handle both bundle orders and arbitrary securities defined over combinatorial outcome spaces. We define the notion of an approximately fair trading path, a path in security space along which no order executes at a price more than a fixed tolerance above its limit, and every order executes when its market price falls more than a fixed tolerance below its limit. We show that under a certain supermodularity condition, a fair trading path exists for which the endpoint is efficient, but that under very general conditions, reaching an efficient endpoint via a fair trading path is not possible. We develop an algorithm for operating a continuous market maker with limit orders that respects the fairness conditions in the general case in which the supermodularity condition may not hold. We conduct simulations of our algorithm using real combinatorial predictions made during the 2008 U.S. Presidential election and evaluate it against a natural baseline according to trading volume, social welfare, and violations of the two fairness conditions.