Information, Rewards, and Quasi-Utilities

A set of mutually exclusive and exhaustive outcomes are assumed to have in some sense ‘true’ probabilities p i (i = 1,2,...n) and a forecaster estimates these probabilities as q i . After the outcome is known, the forecaster’s client pays him a fee f i (q 1,..., q n; π1,..., π n ) = f i (q; π) where π denotes the client’s estimates of the probabilities. If this fee has the property that its ‘true’ expectation is maximized when q = p it is called an ‘accuracy incentive’. (When p denotes the forecaster’s true attainable subjective probabilities, the fee is also an ‘honesty incentive’.) A brief historical survey of the problem of finding accuracy incentives is given, together with a recommendation that a forecaster should not be the same person as a valuer of the outcomes. One form of accuracy incentive (which depends on a parameter β ≥ 1) is analogous to a generalized surprise index. It has the merit of being ‘splitative’, that is, it is unaffected if an outcome is arbitrarily split up into more than one outcome. It is conjectured that there are no other splitative accuracy incentives, but β remains to be chosen. When β = 1 the fee has an additive property that seems fairly desirable.