Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation

In practice, scoring rules elicit good probability estimates from individuals, while betting markets elicit good consensus estimates from groups. Market scoring rules combine these features, eliciting estimates from individuals or groups, with groups costing no more than individuals. Regarding a bet on one event given another event, only logarithmic versions preserve the probability of the given event. Logarithmic versions also preserve the conditional probabilities of other events, and so preserve conditional independence relations. Given logarithmic rules that elicit relative probabilities of base event pairs, it costs no more to elicit estimates on all combinations of these base events. Introduction In theory, probability elicitation is hard. For expected utility maximizers, choices are jointly determined by utilities and probabilities, and so without additional constraints on utilities or probabilities subjective probabilities cannot be separated from event-dependent utilities (Kadane & Winkler, 1988). Some sophisticated approaches can in theory overcome this problem (Jaffray & Karni, 1999; Hanson, 2002b). Yet in practice, less sophisticated uses of scoring rules and related methods are widely and successfully used to elicit informative event probabilities from individuals in weather forecasting (Murphy & Winkler, 1984), economic forecasting (O’Carroll, 1977), risk analysis (DeWispelare, Herren, & Clemen, 1995), and the engineering of intelligent computer systems (Druzdzel & van der Gaag, 1995). Furthermore, simple conversational statements are often taken as reliable belief elicitations. Given an ability to elicit probabilities, in theory it should be easy to induce individuals to aggregate their information into common estimates.1 Given a finite state space, Bayesians with a common prior who repeatedly state their beliefs, and who have common knowledge about previous statements, must eventually achieve common knowledge of future statements (Geanakoplos & Polemarchakis, 1982), and thus common estimates (Aumann, 1976). Along the way, Bayesians ∗For their comments, I thank Chris Hibbert and David Pennock. †[email protected] http://hanson.gmu.edu 703-993-2326 FAX: 704-993-2323 MSN 1D3, Carow Hall, Fairfax VA 22030 This can be true even if directly combining probability distributions is more difficult (Genest & Zidek, 1986).