An Optimization-Based Framework for Automated Market-Making

TL;DR

This paper introduces an optimization-based framework for designing efficient securities markets over large outcome spaces, addressing computational challenges by restricting securities and using convex optimization for pricing.

Contribution

It develops a novel convex potential function approach for market design, enabling efficient pricing and market creation for complex outcome spaces.

Findings

Designed an efficient market for predicting landing locations on a sphere.

Created a new market maker for pair betting with relaxed no-arbitrage conditions.

Enabled dynamic market depth through transaction fees.

Abstract

Building on ideas from online convex optimization, we propose a general framework for the design of efficient securities markets over very large outcome spaces. The challenge here is computational. In a complete market, in which one security is offered for each outcome, the market institution can not efficiently keep track of the transaction history or calculate security prices when the outcome space is large. The natural solution is to restrict the space of securities to be much smaller than the outcome space in such a way that securities can be priced efficiently. Recent research has focused on searching for spaces of securities that can be priced efficiently by existing mechanisms designed for complete markets. While there have been some successes, much of this research has led to hardness results. In this paper, we take a drastically different approach. We start with an arbitrary space of securities with bounded payoff, and establish a framework to design markets tailored to this space. We prove that any market satisfying a set of intuitive conditions must price securities via a convex potential function and that the space of reachable prices must be precisely the convex hull of the security payoffs. We then show how the convex potential function can be defined in terms of an optimization over the convex hull of the security payoffs. The optimal solution to the optimization problem gives the security prices. Using this framework, we provide an efficient market for predicting the landing location of an object on a sphere. In addition, we show that we can relax our "no-arbitrage" condition to design a new efficient market maker for pair betting, which is known to be #P-hard to price using existing mechanisms. This relaxation also allows the market maker to charge transaction fees so that the depth of the market can be dynamically increased as the number of trades increases.