Safe Probing of Censored Showdown Records: What an Agent Can Learn and How to Act on It

In poker-like imperfect-information games every hand leaves a public record of betting and payoffs, yet an opponent's private cards appear only at showdown. Weak hands fold, so showdown records systematically under-represent the hands an opponent model needs. Safe exploitation rests on the sequence-form exploitability certificate, which bounds what a response can lose against any opponent. It certifies safety but does not quantify observation: how much a probe may collect inside a safety budget, or what the censored records identify. Even the full terminal record can leave the opponent plan only partially identified. This paper couples the two questions and deploys against the residual ambiguity. A linear observation-rate objective composed with the certificate yields a *capacity linear program* (LP-C), whose optimum is the maximum target-showdown mass extractable inside a bounded exploitability budget under a nominal opponent model. The sharp identified set of opponent plans is a moment polytope: the opponent's *plan* may remain partially identified, while the probe's own *value* is point-identified from observable payoffs. Deploying robustly over the identified set is a single LP whose optimum certifies a one-sided finite-sample lower bound on deployed value at nominal coverage. Stacking probes refines the identified set and its guarantees. Under a fixed total sample budget, a dual-certificate analysis bounds the width of any deployment whose payoff vector lies in the certificate row space. It decomposes the effect of adding a probe into a weight term and a radius term. On the exhaustively enumerated Leduc Hold'em tree, the cell-polytope certificate over two probes attains a mean margin over the game value of $0.0895$ against $0.0000$ for a certificate built only from the two probes' mean payoffs. Over three probes, empirical–Bernstein intervals lift it to $0.0822$ from $0.0062$ under a Hoeffding box. The constructions apply to any finite perfect-recall two-player zero-sum game with hero-observable terminal cells.