一个方程分配竞技团队运动中的上场时间：规则手册设定算术，联赛设定习惯
One equation allocates playing time in competitive team sports: the rulebook sets the arithmetic and the league sets the habit
中文总结 AI 辅助
本研究通过八种团队运动的数据发现一个统一方程分配上场时间，区分规则固定与联赛选择，并验证联赛偏好由换人时间价格决定，规则为几何，文化为物质。
中文摘要 AI 辅助
每项允许球队替换球员的团队运动都会对替换进行配给，而配给决定了谁上场以及上场多长时间。在八种运动项目中（其中三种涵盖男女两个性别），我们测量了上场时间的分配方式，并发现其背后存在一个方程，该方程在单向替换规则的运动项目中是完备的，其各项将规则手册所固定的内容与竞赛所选择的内容区分开来。在稀缺条件下，每个联赛的行为都相似：在四大洲，负荷较重的首发球员以相同的速率被保留在场上。在自由条件下，每个联赛表达出各自的偏好，而联赛偏离程度的大小，几乎以一对一的比例由其自身的换人时间价格来定价（λ = 0.87，95% 置信区间为 0.54 至 1.21，涉及十二个联赛），这一规律在四次样本外预测中得到证实，并在一次预测中被拒绝。联赛按偏好排序的顺序在规则变更后依然保持，而俱乐部财富并不能为其定价。规则是竞赛的几何学；文化是其物质。
英文摘要
Every team sport that lets a team replace players rations the replacements, and the ration settles who plays and for how long. Across eight codes, three in both sexes, we measure how playing time is allocated and find one equation behind it, complete in the one-way codes, its terms separating what the rulebook fixes from what a competition chooses. Under scarcity every league behaves alike: loaded starters are kept on at the same rate on four continents. Under freedom each expresses a preference, and the size of a league's departure is priced near one-for-one by what its own withdrawal minute prices (lambda = 0.87, 95% interval 0.54 to 1.21, twelve leagues), a law confirmed in four out-of-sample predictions and refused in one. The ordering of leagues by appetite persists across a rule change, and club wealth does not price it. Rules are the geometry of a competition; culture is its material.
