可精确求解的胜率模型中的校准-杠杆权衡
The Calibration-Leverage Tradeoff in Exactly Solvable Win-Probability Models
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中文总结 AI 辅助
该研究构建可精确求解的马尔可夫胜率模型，发现其存在校准-杠杆结构性权衡，短程得分持续性是校准偏差主因，块自举模拟器可缩小26%校准差距。
中文摘要 AI 辅助
我们研究Twenty20板球比赛中第二局追分时逐球胜率（WP），构建为一个可精确求解的马尔可夫模型：我们估计单一对象，即{0,…,6, 失球}的逐球结果分布，并通过追分图（球数、失球数、需跑分数）的非循环结构反向推导每个比赛状态下的胜率。该构造使胜率成为精确鞅，进而使杠杆（单球能多大程度改变胜率）和胜率增加值（WPA）有明确定义且可精确归因；我们用它们确认收尾击球手和死亡投球手处于最高杠杆时刻。随后我们表明模型的胜率存在系统性校准偏差，且这并非偶然。为定位误差，我们排除了尾部变薄和边际估计错误（模型的逐球结果分布在每个需跑率下与经验分布的总变差距离至多为0.02）。仅存的原因是未建模的状态依赖，我们确定了该依赖：排列零假设分解显示，短程连续得分持续性（约3-5球；局级异质性仅贡献约18%；失球反集群）。在保持边际固定的同时注入真实依赖的块自举模拟器缩小了26%的校准差距（6个随机种子重复验证），在约20球的块长时达到饱和，与测得的相关范围一致；这是证实诊断的建设性下界。结果是相对于（球数、失球数、需跑分数）状态描述的结构性权衡：精确杠杆需要鞅，鞅需要该状态下的条件逐球独立性，而该独立性正是导致胜率校准偏差的原因。无法从该状态下的同一对象同时获得可精确归因的杠杆和校准良好的胜率。
英文摘要
We study ball-by-ball win probability (WP) for second-innings run chases in Twenty20 cricket, built as an exactly solvable Markov model: we estimate a single object, the per-ball outcome distribution over {0,...,6, wicket}, and derive WP for every game state by backward induction over the acyclic (balls, wickets, runs-required) chase graph. This construction makes WP an exact martingale, which in turn makes leverage (how much a ball can swing WP) and win probability added (WPA) well-defined and exactly attributable; we use them to confirm that finishers and death bowlers occupy the highest-leverage moments. We then show the model's WP is systematically miscalibrated, and that this is not incidental. Localizing the error, we rule out tail-thinning and marginal mis-estimation (the model's per-ball outcome distribution matches the empirical one to a total variation of at most 0.02 at every required run rate). The only remaining cause is unmodelled dependence given the state, and we identify it: a permutation-null decomposition shows short-range sequential run-scoring persistence (roughly 3-5 balls; innings-level heterogeneity contributes only about 18%; wickets, if anything, anti-cluster). A block-bootstrap simulator that injects the real dependence while holding the marginals fixed closes 26% of the calibration gap (replicated over six seeds), saturating at block lengths of about 20 balls, consistent with the measured correlation range; this is a constructive lower bound that confirms the diagnosis. The result is a structural tradeoff relative to the (balls, wickets, runs) state description: exact leverage requires the martingale, the martingale requires conditional ball independence on that state, and that independence is what miscalibrates the WP. Exactly attributable leverage and well-calibrated WP cannot be obtained from the same object over this state.
