The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

Abstract

The secretary problem or the game of Googol are classic models for online selection problems that have received significant attention in the last five decades. In this paper we consider a variant of the problem and explore its connections to data-driven online selection. Specifically, we are given n cards with arbitrary non-negative numbers written on both sides. The cards are randomly placed on n consecutive positions on a table, and for each card, the visible side is also selected at random. The player sees the visible side of all cards and wants to select the card with the maximum hidden value. To this end, the player flips the first card, sees its hidden value and decides whether to pick it or drop it and continue with the next card.We study algorithms for two natural objectives. In the first one, similar to the secretary problem, the player wants to maximize the probability of selecting the maximum hidden value. We show that this can be done with probability at least 0.45292. In the second objective, similar to the prophet inequality, the player wants to maximize the expectation of the selected hidden value. Here we show a guarantee of at least 0.63518 with respect to the expected maximum hidden value.Our algorithms result from combining three basic strategies. One is to stop whenever we see a value larger than the initial n visible numbers. The second one is to stop the first time the last flipped card's value is the largest of the currently n visible numbers in the table. And the third one is similar to the latter but to stop it additionally requires that the last flipped value is larger than the value on the other side of its card.We apply our results to the prophet secretary problem with unknown distributions, but with access to a single sample from each distribution. In particular, our guarantee improves upon 1 − 1/e for this problem, which is the currently best known guarantee and only works for the i.i.d. prophet inequality with samples.

Efficient Two-Sided Markets with Limited Information

“…Secondly, in some cases, we are able to improve over previous bounds obtained with full knowledge of the distributions. Our work is close in spirit to the previous works on one-side mechanism design that obtain approximately optimal revenue with limited information about an existing distribution of bidders' values [12,18,23], and to the previous work on prophet inequalities with limited information from the distributions [2,16,17,38].…”

“…Our techniques are very different from those in prior work on revenue-maximizing one-sided mechanisms from samples [12,18,23], and also from the prophet inequalities with limited information literature [2,16,17,38].…”

“…ě E rSWpp ‹ qs p1 ´2δq, (20) where in Equation ( 16) we used the independence of v s and v b , in Inequality (17) we used Equation (13), in Inequality (18) we reorganized the terms and moved the δ part of the last term around to match the cases in the expected welfare of setting price p ‹ , and in Inequality (19) we used Equation ( 13) again and scaled one term by P pv b ă p ‹ q.…”

The Secretary Problem with Independent Sampling

“…A notable result is that for the classic prophet inequality, 2 a single sample from each distribution is enough to achieve the optimal guarantee [23]. Also for the prophet secretary problem, the variant of the prophet inequality when the elements come in random order, one sample has been proved to be quite effective [8]. Recently, Kaplan et al [17] study a model that is closest to ours.…”

Scheduling in the Secretary Model

“…A modern version considers adversarial orders but allows prior sampling [15,32]. Related models are prophet inequalities and the game of googol [14,16].…”