Choosing Behind the Veil: Tight Bounds for Identity-Blind Online Algorithms

In Bayesian online settings, every element is associated with a value drawn from a known underlying distribution. This distribution, representing the population from which the element is drawn, is referred to as the element’s identity. The elements arrive sequentially, with their values being revealed in an online manner. Most previous work has assumed that, upon the arrival of a new element, the online algorithm observes its value and its identity. However, practical scenarios frequently require algorithms to make decisions based solely on the element’s value, disregarding its identity. This necessity emerges either from the algorithm’s lack of knowledge about the element’s identity or in the pursuit of fairness, aiming for bias-free decisions across varying identities. We call such algorithms identity-blind algorithms, and propose the identity-blindness gap as a metric to evaluate the performance loss in online algorithms caused by identity-blindness. This gap is defined as the maximum ratio between the expected performance of an identity-blind online algorithm and an optimal online algorithm that knows the arrival order, thus also the identities. We study the identity-blindness gap in the paradigmatic prophet inequality problem, under the two common objectives of maximizing the expected value, and maximizing the probability to obtain the highest value. We provide tight bounds with respect to both objectives. For the max-expectation objective, the celebrated prophet inequality establishes a single-threshold (thus identity-blind) algorithm that gives at least 1/2 of the offline optimum, thus also an identity-blindness gap of at least 1/2. We show that this bound is tight with respect to the identity-blindness gap, even with respect to randomized algorithms. For the max-probability objective, we provide a deterministic single-threshold (thus identity-blind) algorithm that gives an identity-blindness gap of ∼ 0.562 (assuming the absence of large point masses). Moreover, we show that this bound is tight with respect to deterministic algorithms. Both results demonstrate a separation between what can be achieved with and without discrimination based on identities.