  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.
