  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.

Theoretical Economics