  In Bayesian online settings, every element has a value that is drawn from a
known underlying distribution, which we refer to as the element's identity. The
elements arrive sequentially. Upon the arrival of an element, its value is
revealed, and the decision maker needs to decide, immediately and irrevocably,
whether to accept it or not. While most previous work has assumed that the
decision maker, upon observing the element's value, also becomes aware of its
identity -- namely, its distribution -- practical scenarios frequently demand
that decisions be made based solely on the element's value, without considering
its identity. This necessity arises either from the algorithm's ignorance of
the element's identity or due to the pursuit of fairness. We call such
algorithms identity-blind algorithms, and propose the identity-blindness gap as
a metric to evaluate the performance loss 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 objectives of maximizing the expected value, and
maximizing the probability to obtain the highest value. For the max-expectation
objective, the celebrated prophet inequality establishes a single-threshold
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. For
the max-probability objective, while the competitive ratio is tightly 1/e, we
provide a deterministic single-threshold algorithm that gives an
identity-blindness gap of $\sim 0.562$ under the assumption that there are no
large point masses. Moreover, we show that this bound is tight with respect to
deterministic algorithms.
