The parimutuel Kelly probability scoring rule

Kilgour and Gerchak (2004) 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.