Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation

Prediction markets


Market scoring rules combine simple scoring rules and betting markets to elicit common probability estimates efficiently. Individuals report probabilities for events and can change their reports, getting paid accordingly. This method induces common estimates without requiring matching bets, like traditional markets. Market scoring rules are cost-effective and preserve local and conditional probabilities, making them a valuable tool for probability estimation.

Market scoring rules, as a method for probability estimation, operate by allowing individuals to report their probability estimates for various events. These rules are unique in that they permit anyone to alter the official report, and they are compensated based on this new report, provided they are willing to compensate the last person who reported according to their estimate. This essentially enables individuals to make infinitesimal fair bets at the odds reflected in the most recent report without the need to locate a counterpart willing to take the opposite side of the bet, as is customary in traditional betting markets. By doing so, market scoring rules effectively combine the advantages of both simple scoring rules and traditional betting markets, encouraging each individual to provide their probability estimate while also fostering the emergence of common estimates among participants. This approach is not only cost-effective, particularly when using logarithmic rules, but also preserves local inferences and conditional independence relations, making it a robust and efficient method for eliciting and aggregating probability estimates from individuals.

Article summary:

“Simple scoring rules are regularly used to elicit probability estimates from individuals, and more sophisticated versions are available if needed to overcome risk-aversion and state-dependent utility.

In theory, repeatedly eliciting and announcing individual estimates should be sufficient to induce common estimates, but in practice estimate differences remain. In practice, however, standard betting markets do elicit common estimates that seem to aggregate individual information well, even though participation seems irrational and requires a coordination of trading activity. With a simple scoring rule, a person reports a probability for each event, and gets paid depending on that report and the actual event. Market scoring rules are scoring rules where anyone can change the official report, and be paid according to that new report, as long as they are willing to pay the last person reporting according to their report. This in effect lets anyone make any infinitesimal fair bet at the odds in the last report, with no need to find another person willing to make a matching bet, as in ordinary betting markets. Market scoring rules thus combine the advantages of both simple scoring rules and betting markets, inducing each individual to make an estimate and inducing common estimates.

Market scoring rules cost no more to implement than simple scoring rules. The cost does depend on the number of base events for which probability estimates are invited, but for logarithmic rules there is no additional cost to elicit estimates on all combinations of these base events. The logarithmic rule is also unique in making very local inferences from the trades it sees; regarding a bet on one event given another event, only a logarithmic rule preserves the probability of the given event. Logarithmic versions also preserve the conditional probabilities of further events, and so preserve conditional independence relations.”