An axiomatic characterization of adaptive-liquidity market makers

Prediction markets offer contingent securities with payoffs linked to future events. The market price of a security reveals information about traders' beliefs, but prediction markets often suffer from low liquidity, which can prevent trades from occurring. One way to inject liquidity into a market is through the use of an automated market maker, an algorithmic agent willing to accept some risk in order to facilitate trades. Abernethy et al. [2013] proposed a general framework for the design of automated market makers, defining a set of axioms that a market should satisfy, and characterizing the class of market makers that satisfy these axioms. However, the liquidity of any market in their class, quantified in terms of the rate at which prices adapt to trades, is fixed a priori and does not change as the volume of trade increases. Othman and Sandholm [2011] proposed a class of liquidity-adaptive markets, but gave little guidance for how to choose a market from within this class. Combining ideas from Abernethy et al. [2013] and Othman and Sandholm [2011], we provide an axiomatic characterization of a parameterized class of automated market makers with adaptive liquidity. A primary advantage of our framework is the ability to analyze important market properties, such as its ability to aggregate information or make a profit, in terms of market parameters, which we do using techniques from convex analysis and geometry. For example, we show that the curvature of the price space can be used to manage a trade-off between information loss and profit. This analysis offers guidance for market designers who wish to choose a particular market maker to implement.