Safe Observation Capacity for Opponent Exploitation under Showdown Censoring

Abstract

In poker-like games, folds hide private cards, so showdown data are missing not at random: per-card estimates converge to behavior conditional on reveal, and shrinking confidence sets can lose coverage. A floor-safe probe carries a line to showdown; sequence-form flow then recovers censored fold mass on reveal-certified histories. We price acquisition through safe observation capacity, the largest target reach attainable by a floor-safe plan at a given value slack. Its frontier is concave and piecewise linear, with initial slope given by the floor's shadow price. When that reach converts fully to reveal and parent flow is non-bottleneck, matching local bounds make the hands required for conditional-probability half-width $\epsilon$ inversely proportional, up to logarithms, to capacity, opponent continuation mass, and ${\epsilon }^2$. Safe Active De-censoring (SAD) combines public screening, an independent reveal batch, and robust deployment; a max-min safe audit gives positive joint reveal rate to every coordinate in a finite library-covered target set, including public-null deviations. With $10^6$ hands, SAD raises the river over-fold certified gain from $0.485$ to $0.692$ and improves both gains over public-only collection on all three deviations (Holm-adjusted paired $p\le 0.012$), while selecting no control target. On a fixed-board public twin, a disjoint audit-refit-deploy loop detects all $30$ simulation seeds and no control seed, certifying absolute value $0.655$ (95% confidence-interval half-width $0.008$). Every floor-constrained probe and response passes a floor audit.

32 pages, 5 figures. Version 2 expands the theory and empirical evaluation, adds universal public-null auditing and a disjoint audit-refit-deploy experiment, and revises the exposition. Code: https://github.com/jiaxing-guo/safe-observation-capacity