Given n random variables X1β,β¦,Xnβ taken from known distributions,
a gambler observes their realizations in this order, and needs to select one of
them, immediately after it is being observed, so that its value is as high as
possible. The classical prophet inequality shows a strategy that guarantees a
value at least half (in expectation) of that an omniscience prophet that picks
the maximum, and this ratio is tight.
Esfandiari, Hajiaghayi, Liaghat, and Monemizadeh introduced a variant of the
prophet inequality, the prophet secretary problem in [1]. The difference being
that that the realizations arrive at a random permutation order, and not an
adversarial order. Esfandiari et al. gave a simple 1β1/eβ0.632
competitive algorithm for the problem. This was later improved in a surprising
result by Azar, Chiplunkar and Kaplan [2] into a 1β1/e+1/400β0.634
competitive algorithm. In a subsequent result, Correa, Saona, and Ziliotto [3]
took a systematic approach, introducing blind strategies, and gave an improved
0.669 competitive algorithm. Since then, there has been no improvements on
the lower bounds. Meanwhile, current upper bounds show that no algorithm can
achieve a competitive ratio better than 0.7235 [4].
In this paper, we give a 0.6724-competitive algorithm for the prophet
secretary problem. The algorithm follows blind strategies introduced by [3] but
has a technical difference. We do this by re-interpretting the blind
strategies, framing them as Poissonization strategies. We break the non-iid
random variables into iid shards and argue about the competitive ratio in terms
of events on shards. This gives significantly simpler and direct proofs, in
addition to a tighter analysis on the competitive ratio. The analysis might be
of independent interest for similar problems such as the prophet inequality
with order-selectionComment: 15 page