Prophet and Secretary at the Same Time

Many online problems are studied in stochastic settings for which inputs are samples from a known distribution, given in advance, or from an unknown distribution. Such distributions model both beyond-worst-case inputs and, when given, partial foreknowledge for the online algorithm. But how robust can such algorithms be to misspecification of the given distribution? When is this detectable, and when does it matter? When can algorithms give good competitive ratios both when the input distribution is as specified, and when it is not? We consider these questions in the setting of optimal stopping, where the cases of known and unknown distributions correspond to the well-known prophet inequality and to the secretary problem, respectively. Here we ask: Can a stopping rule be competitive for the i.i.d. prophet inequality problem and the secretary problem at the same time? We constrain the Pareto frontier of simultaneous approximation ratios $(α, β)$ that a stopping rule can attain. We introduce a family of algorithms that give nontrivial joint guarantees and are optimal for the extremal i.i.d. prophet and secretary problems. We also prove impossibilities, identifying $(α, β)$ unattainable by any adaptive stopping rule. Our results hold for both $n$ fixed arrivals and for arrivals from a Poisson process with rate $n$. We work primarily in the Poisson setting, and provide reductions between the Poisson and $n$-arrival settings that may be of broader interest.