The Calibration-Leverage Tradeoff in Exactly Solvable Win-Probability Models

Abstract

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.