The profit-bias identity in sports betting: bookmaker profit as the public's prediction error

Sports betting moves money continuously from a large public to a small number of firms. The most influential account of that flow, due to Levitt (2004), holds that books price away from the market-clearing point to exploit predictable public biases. It computes profit from two numbers (the probability that a side wins the proposition and the fraction of handle it attracts), treating the share on a side as independent of the outcome. Here we relax that assumption and derive a profit-bias identity: profit is affine and increasing in the expected share of handle on the losing side, with Levitt's expression as the special case of independence. The identity resolves the book's margin into exactly three channels: its hold, the product of its price shading and the public's lean, and the covariance between bet share and outcome. A lean is thus worthless without shading, and shading is worthless without a lean. Under a public-belief model, the profit driver is the public's Bayes error: the probability of a representative bettor selecting the losing side. We provide necessary and sufficient conditions for a "Goldilocks Zone": prices at which book and bettor both profit. Testing these predictions on 1,139 Major League Baseball games, we find that the apparent dependence between bet share and outcome is a Simpson's paradox: present when games are pooled, but absent once they are separated by which side the book favored. The public leans heavily toward favorites, but we detect no matching shading, and the realized margin is indistinguishable from the hold.