AI 中文总结
本研究推导出体育博彩中的利润-偏差恒等式，将博彩公司利润分解为抽成、价格偏移与公众倾向乘积及协方差，并通过MLB数据发现投注份额与结果的关联实为辛普森悖论。
AI 中文摘要
体育博彩持续地将资金从广大公众流向少数公司。关于这一资金流动最具影响力的解释，由Levitt（2004）提出，认为博彩公司通过偏离市场出清点定价，以利用公众可预测的偏差。该解释通过两个数字（一方赢得投注的概率及其吸引的投注额比例）计算利润，并将一方投注份额视为与结果无关。在此，我们放宽这一假设，推导出一个利润-偏差恒等式：利润是输方预期投注份额的仿射递增函数，而Levitt的表达式是独立性假设下的特例。该恒等式将博彩公司的利润分解为三个渠道：其抽成（hold）、其价格偏移与公众倾向的乘积，以及投注份额与结果之间的协方差。因此，没有价格偏移，公众倾向毫无价值；没有公众倾向，价格偏移也毫无价值。在公众信念模型下，利润驱动因素是公众的贝叶斯误差：代表性投注者选择输方的概率。我们为“金发姑娘区域”（Goldilocks Zone）提供了充要条件：在该价格下，博彩公司和投注者双方都能获利。通过对1,139场美国职业棒球大联盟比赛进行测试，我们发现投注份额与结果之间看似存在的依赖关系是一个辛普森悖论：当比赛被合并分析时该关系存在，但一旦按博彩公司所偏好的方分开分析，该关系便消失。公众强烈倾向于热门方，但我们未检测到相应的价格偏移，且实际利润与抽成无法区分。
英文摘要
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.
