The Parimutuel Kelly Probability Scoring Rule

Kilgour and Gerchak (Kilgour, D. M., Y. Gerchak. 2004. Elicitation of probabilities using competitive scoring rules. Decision Anal.2 108--113) introduce a competitive probability scoring rule designed to reward forecasters for their accuracy relative to rival forecasters. The rule proposed is both proper (in the usual sense of scoring rules) and self-financing in that forecasters share a preset reward pool that can be fixed at net zero. This paper elaborates on the Kilgour-Gerchak forecasting competition by demonstrating its possible analogy with forecasters making “Kelly bets” in an exogenous fixed-odds betting market. The construction of forecasters as Kelly (log utility) bettors leads to a modification of the Kilgour-Gerchak rule, called the parimutuel Kelly competitive scoring rule. Unlike the Kilgour-Gerchak score, the parimutuel Kelly score is not proper. Its appeal, however, is that it emulates a prediction market formed within a cohort of forecasters, as if they engage in a closed betting tournament where each attempts to accumulate maximum possible wealth over the long run and ruin all others. Within such a competition, near honesty is a Nash equilibrium forecasting strategy under realistic conditions.