An Axiomatic Study of Scoring Rule Markets

TL;DR

This paper provides an axiomatic framework for understanding scoring rule markets (SRMs), characterizing when they behave like traditional markets and introducing new mechanisms for eliciting complex statistical predictions.

Contribution

It introduces an axiomatic classification of SRMs, characterizes cost-function markets as the only fully compliant type, and proposes a novel share-based market for ratios of expectations.

Findings

Cost-function markets uniquely satisfy all axioms for finite outcomes.

Many SRMs satisfy weaker axioms, broadening market design options.

A new share-based mechanism for ratios of expectations is proposed.

Abstract

Prediction markets are well-studied in the case where predictions are probabilities or expectations of future random variables. In 2008, Lambert, et al. proposed a generalization, which we call "scoring rule markets" (SRMs), in which traders predict the value of arbitrary statistics of the random variables, provided these statistics can be elicited by a scoring rule. Surprisingly, despite active recent work on prediction markets, there has not yet been any investigation into the properties of more general SRMs. To initiate such a study, we ask the following question: in what sense are SRMs "markets"? We classify SRMs according to several axioms that capture potentially desirable qualities of a market, such as the ability to freely exchange goods (contracts) for money. Not all SRMs satisfy our axioms: once a contract is purchased in any market for prediction the median of some variable, there will not necessarily be any way to sell that contract back, even in a very weak sense. Our main result is a characterization showing that slight generalizations of cost-function-based markets are the only markets to satisfy all of our axioms for finite-outcome random variables. Nonetheless, we find that several SRMs satisfy weaker versions of our axioms, including a novel share-based market mechanism for ratios of expected values.