# An Axiomatic Study of Scoring Rule Markets

arXiv: https://arxiv.org/abs/1709.10065 · HTML: https://mlsanity.com/paper/1709.10065/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2017.

## Built on by (most cited first, 2 of 2)

- [Optimal Automated Market Makers: Differentiable Economics and Strong Duality](https://mlsanity.com/paper/2402.09129/) (2024)
- [Smooth Quadratic Prediction Markets](https://mlsanity.com/paper/2505.02959/) (2025)

## Abstract (arXiv)

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.