Design and Analysis of a Synthetic Prediction Market using Dynamic Convex Sets

TL;DR

This paper introduces a synthetic prediction market model based on convex semi-algebraic sets and sigmoid transformations, capable of approximating binary functions and modeling data distributions, with convergence analysis and empirical validation.

Contribution

It proposes a novel prediction market framework using convex geometry and sigmoid functions, with theoretical analysis and an evolutionary training algorithm.

Findings

Market can approximate binary functions arbitrarily closely

Conditions for market convergence are established

Market models data distributions effectively, comparable to standard methods

Abstract

We present a synthetic prediction market whose agent purchase logic is defined using a sigmoid transformation of a convex semi-algebraic set defined in feature space. Asset prices are determined by a logarithmic scoring market rule. Time varying asset prices affect the structure of the semi-algebraic sets leading to time-varying agent purchase rules. We show that under certain assumptions on the underlying geometry, the resulting synthetic prediction market can be used to arbitrarily closely approximate a binary function defined on a set of input data. We also provide sufficient conditions for market convergence and show that under certain instances markets can exhibit limit cycles in asset spot price. We provide an evolutionary algorithm for training agent parameters to allow a market to model the distribution of a given data set and illustrate the market approximation using two open source data sets. Results are compared to standard machine learning methods.