Prophet Inequalities via the Expected Competitive Ratio

We consider prophet inequalities under general downward-closed constraints. In a prophet inequality problem, a decision-maker sees a series of online elements with values and needs to decide immediately and irrevocably whether or not to select each element upon its arrival, subject to an underlying feasibility constraint. Traditionally, the decision-maker’s expected performance has been compared to the expected performance of the prophet, i.e., the expected offline optimum. We refer to this measure as the Ratio of Expectations (or, in short, RoE). However, a major limitation of the RoE measure is that it only gives a guarantee against what the optimum would be on average, while, in theory, algorithms still might perform poorly compared to the realized ex-post optimal value. Hence, we study alternative performance measures. In particular, we suggest the Expectation of Ratio (or, in short, EoR), which is the expectation of the ratio between the value of the algorithm and the value of the prophet. This measure yields desirable guarantees, e.g., a constant EoR implies achieving a constant fraction of the ex-post offline optimum with constant probability. Moreover, in the single-choice setting, we show that the EoR is equivalent (in the worst case) to the probability of selecting the maximum, a well-studied measure in the literature. However, the probability of selecting the maximum does not generalize meaningfully to combinatorial constraints (beyond single-choice), since its direct equivalent is the probability of selecting an optimal overall set. We, thus, introduce the EoR as a cardinal variant of the probability of selecting the maximum, which extends naturally to combinatorial settings. Our main goal is to understand the relation between RoE and EoR in combinatorial settings. Specifically, we establish two reductions: For every feasibility constraint, the RoE and the EoR are at most a constant factor apart on worst-case instances. Additionally, we show that the EoR is a stronger benchmark than the RoE in that, for every instance (feasibility constraint and product distribution), the RoE is at least a constant fraction of the EoR but not vice versa. Both these reductions imply a wealth of EoR results in multiple settings where RoE results are known.