Price Updating in Combinatorial Prediction Markets with Bayesian Networks

TL;DR

This paper extends the use of Bayesian networks to efficiently update prices in combinatorial prediction markets, providing a complete characterization of structure-preserving securities and linking market mechanisms with probabilistic belief aggregation.

Contribution

It generalizes previous work by employing Bayesian networks in broader combinatorial markets and characterizes all structure-preserving securities for decomposable networks.

Findings

Complete characterization of structure-preserving securities

Decomposability of Bayesian network graphs is necessary for certain securities

Discussion of approximation techniques for non-structure-preserving securities

Abstract

To overcome the #P-hardness of computing/updating prices in logarithm market scoring rule-based (LMSR-based) combinatorial prediction markets, Chen et al. [5] recently used a simple Bayesian network to represent the prices of securities in combinatorial predictionmarkets for tournaments, and showed that two types of popular securities are structure preserving. In this paper, we significantly extend this idea by employing Bayesian networks in general combinatorial prediction markets. We reveal a very natural connection between LMSR-based combinatorial prediction markets and probabilistic belief aggregation,which leads to a complete characterization of all structure preserving securities for decomposable network structures. Notably, the main results by Chen et al. [5] are corollaries of our characterization. We then prove that in order for a very basic set of securities to be structure preserving, the graph of the Bayesian network must be decomposable. We also discuss some approximation techniques for securities that are not structure preserving.