Data-Driven Sequential Search

We consider a sequential search problem where the distribution of alternative values is unknown. In our data-driven setting, feasible policies are based solely on the history of explored alternative values. We seek to identify a policy that maximizes the worst-case ratio of expected reward compared to an oracle (referred to as \emph{Pandora}) with full knowledge of the value distribution. We design static policies that commit to a prespecified number of explorations. We show that these policies guarantee a competitive ratio of at least $1/e \approx 37$% of the Pandora benchmark for any arbitrary value distribution. Our approach involves studying nature's problem to select a distribution to counter a policy and identifying worst-case distributions. Moreover, we study how the structure of the unknown value distribution influences achievable performance guarantees by considering a setting where feasible distributions belong to the class of monotone hazard rate distributions, where we improve our guarantee to $(e/(e+1))^2 \approx 53$% of the Pandora benchmark. We show that static policies are especially effective against smaller classes of unknown distributions, guaranteeing at least $e/(e+1) \approx 73$% against exponential distributions with an unknown rate and $9/(8(4-\sqrt{7})) \approx 83$% against uniform distributions with an unknown maximum. In the latter case, we show that static policies achieve the best possible performance among all feasible policies, including the dynamic ones. Finally, we derive performance limits for all feasible policies to further highlight the efficiency and robustness of our static policies for data-driven search problems.