Optimal Betting Odds Against Insider Traders

This paper examines the optimal pricing decision of a bookmaker facing a group of gamblers, some of whom have superior information to the bookmaker. The central result is that the normalized betting odds diverge systematically from the true possibilities of winning. Specifically, it is optimal for the bookmaker to follow a "square root rule" in which the ratio of posted prices is given by the square root of the ratio of winning probabilities. One consequence of this rule is that the betting odds understate the winning chances of the favorites and exaggerate the winning chances of the longshots. This is precisely the bias observed in several empirical investigations of betting markets.