Fair Odds for Noisy Probabilities

We suggest that one individual holds multiple degrees of belief about an outcome, given the evidence. We then investigate the implications of such noisy probabilities for a buyer and a seller of binary options and find the odds agreed upon to ensure zero-expectation betting, differ from those consistent with the relative frequency of outcomes. More precisely, the buyer and the seller agree to odds that are higher (lower) than the reciprocal of their averaged unbiased probabilities when this average indicates the outcome is more (less) likely to occur than chance. The favorite-longshot bias thereby emerges to establish the foundation of an equitable market. As corollaries, our work suggests the old-established way of revealing someone's degree of belief through wagers may be more problematic than previously thought, and implies that betting markets cannot generally promise to support rational decisions.