The apportionment problem constitutes a fundamental problem in democratic societies: How to distribute a fixed number of seats among a set of states in proportion to the states’ populations? This—seemingly simple—task has led to a rich literature and has become well known in the context of the US House of Representatives. In this paper, we connect the design of monotone apportionment methods to classic problems from discrete geometry and combinatorial optimization and explore the extent to which randomization can enhance proportionality.

We study the prophet secretary problem, a well-studied variant of the classic prophet inequality, where values are drawn from independent known distributions but arrive in uniformly random order. Upon seeing a value at each step, the decision-maker has to either select it and stop or irrevocably discard it. Traditionally, the chosen benchmark is the expected reward of the prophet, who knows all the values in advance and can always select the maximum one. In this work, we study the prophet secretary problem against a less pessimistic but equally well-motivated benchmark; the online optimal. Here, the main goal is to find polynomial-time algorithms that guarantee near-optimal expected reward. As a warm-up, we present a quasi-polynomial time approximation scheme (QPTAS) achieving a (1 − ε)-approximation in O(npoly log n· f(ε)) time through careful discretization and non-trivial bundling processes. Using the toolbox developed for the QPTAS, coupled with a novel frontloading technique that enables us to reduce the number of decisions we need to make, we are able to remove the dependence on n in the exponent and obtain a polynomial time approximation scheme (PTAS) for this problem.

In the secretary problem we are faced with an online sequence of elements with values. Upon seeing an element we have to make an irrevocable take-it-or-leave-it decision. The goal is to maximize the probability of picking the element of maximum value. The most classic version of the problem is that in which the elements arrive in random order and their values are arbitrary. Here, the optimal algorithm picks the maximum value with probability at least 1/e. However, by varying the available information, new interesting problems arise. For instance, in the full information variant of the secretary problem the values are i.i.d. samples from a known distribution. Naturally, the best possible success probability increases and turns out to be approximately 0.58. Also, the case in which the arrival order is adversarial instead of random leads to interesting variants that have been considered in the literature. In this paper we study both the random order and adversarial order secretary problems with an additional twist. The values are arbitrary, but before starting the online sequence we independently sample each element with a fixed probability p. The sampled elements become our information or history set and the game is played over the remaining elements. We call these problems the random order secretary problem with p-sampling (ROSp for short) and the adversarial order secretary problem with p-sampling (AOSp for short). Our main result is to obtain best possible algorithms for both problems and all values of p. As p grows to 1 the obtained guarantees converge to the optimal guarantees in the full information case. In the adversarial order setting, the best possible algorithm turns out to be a simple fixed threshold algorithm in which the optimal threshold is a function of p only. Therefore, even knowledge of the total number of elements is unnecessary. Proving that this algorithm is optimal involves a novel technique, which boils down to analyzing a related game in a conflict graph over binary sequences. In the random order setting we prove that the best possible algorithm is characterized by a fixed sequence of time thresholds, dictating at which point in time we should start accepting a value that is both a maximum of the online sequence and has a given ranking within the sampled elements. Surprisingly, this sequence of time thresholds arises from a separable and convex optimization problem whose solution is independent of p.

We study the problem of multi-dimensional revenue maximization when selling m items to a buyer that has additive valuations for them, drawn from a (possibly correlated) prior distribution. Unlike traditional Bayesian auction design, we assume that the seller has a very restricted knowledge of this prior: they only know the mean $$\mu _j$$ and an upper bound $$\sigma _j$$ on the standard deviation of each item’s marginal distribution. Our goal is to design mechanisms that achieve good revenue against an ideal optimal auction that has full knowledge of the distribution in advance. Informally, our main contribution is a tight quantification of the interplay between the dispersity of the priors and the aforementioned robust approximation ratio. Furthermore, this can be achieved by very simple selling mechanisms. More precisely, we show that selling the items via separate price lotteries achieves an $$O(\log r)$$ approximation ratio where $$r=\max _j(\sigma _j/\mu _j)$$ is the maximum coefficient of variation across the items. If forced to restrict ourselves to deterministic mechanisms, this guarantee degrades to $$O(r^2)$$ . Assuming independence of the item valuations, these ratios can be further improved by pricing the full bundle. For the case of identical means and variances, in particular, we get a guarantee of $$O(\log (r/m))$$ which converges to optimality as the number of items grows large. We demonstrate the optimality of the above mechanisms by providing matching lower bounds. Our tight analysis for the deterministic case resolves an open gap from the work of Azar and Micali [ITCS’13].