Safe Contextual Pricing under Censored Feedback: Joint Inventory and Allocation with Online Conformal Guarantees

Many online decision problems are simultaneously profit-seeking and safety-constrained: an agent must minimise regret while keeping a downstream operational risk below a prescribed level, using only censored feedback. We study this tension in online contextual pricing with multi-supplier inventory and price-dependent allocation, where the operational risk is the stockout rate. The seller observes only whether each customer purchases, must commit inventory across heterogeneous suppliers before demand realises, and faces an unknown and possibly non-Lipschitz demand-noise distribution. Our algorithm CP-CPA prices by maximising an optimistic profit on a residual grid, provisions the estimated profit-optimal inventory, and raises it to an online conformally calibrated demand quantile whenever the coverage target requires more. Its regret against the unconstrained clairvoyant is $\tilde{O}(T^{2/3})$, the optimal rate under agnostic noise, plus exactly the holding cost of the safety inventory added by the conformal layer; this cost is $\tilde{O}(T^{2/3})$ when the coverage target is slack, and when it binds we prove that linear regret is unavoidable for every coverage-feasible algorithm. The realised stockout rate satisfies ${\hat{\alpha }}_T\le \alpha +O(1/(\eta T))$ deterministically, for any noise distribution and regardless of the accuracy of the demand model, and one conformal state per segment of a fixed context partition yields the same guarantee on every segment. The analysis replaces the Lipschitz and piecewise-convexity structure of prior joint pricing and allocation arguments, which fails under atomic noise, with a coupling bound on poisson-binomial tails and a monotone-demand comparator. On synthetic benchmarks with Lipschitz and atomic noise, the fitted regret exponents match the $T^{2/3}$ prediction and lie far below the distribution-free baseline; when the target binds, CP-CPA attains a stockout rate of $0.096$ with zero variance across seeds while the plug-in quantile over-provisions at higher regret. On a demand curve calibrated from the public UCI Online Retail II log, CP-CPA is the only method that meets the target, at a profit cost of $0.1\%$, and the per-segment variant never exceeds the target in any segment.