  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.
