The Two-Sided Game of Googol

José Correa, Andrés Cristi, Boris Epstein, José Soto; 23(113):1−37, 2022.

Abstract

The secretary problem or game of Googol are classic models for online selection problems. 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: maximizing the probability of selecting the maximum hidden value, and maximizing the expectation of the selected hidden value. For the former objective we obtain a simple $0.45292$-competitive algorithm. For the latter, we obtain a $0.63518$-competitive algorithm. Our main contribution is to set up a model allowing to transform probabilistic optimal stopping problems into purely combinatorial ones. For instance, we can apply our results to obtain lower bounds for the single sample prophet secretary problem.

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