Sequential Selection with Expirations

Motivated by applications where impatience is pervasive and evaluation times are uncertain, we study a selection model where options may expire at an unknown point in time and evaluation times are stochastic. Initially, the decision-maker (DM) has access to $n$ options with known non-negative values: these options have unknown stochastic evaluation and expiration times with known distributional information, which we assume to be independent. When the DM is free, we can select an available option that occupies the DM for an unknown amount of time and collect its value. The objective is to maximize the expected total value obtained from options selected by the DM. Natural formulations of this problem suffer from the curse of dimensionality. In fact, this problem is NP-hard even in the deterministic case. Hence, we focus on efficiently computable approximation algorithms that can provide high expected reward compared to the optimal expected value. Towards this end, we first provide a compact linear programming (LP) relaxation that gives an upper bound on the expected value obtained by the optimal policy. Then we design a polynomial-time algorithm that is nearly a $(1/2)\cdot (1-1/e)$-approximation to the optimal LP value (so also to the optimal expected value). We next shift our focus to the case of independent and identically distributed (i.i.d.) evaluation times. In this case, we show that the greedy policy that always selects the highest-valued option whenever the DM is free obtains a $1/2$-approximation to the optimal expected value. Our approaches extend effortlessly, and we demonstrate their flexibility by providing approximations to natural extensions of our problem. Finally, we evaluate our LP-based policies and the greedy policy empirically on synthetic and real datasets.