Secretary Problems: Weights and Discounts

The classical secretary problem studies the problem of selecting online an element (a “secretary”) with maximum value in a randomly ordered sequence. The difficulty lies in the fact that an element must be either selected or discarded upon its arrival, and this decision is irrevocable. Constant-competitive algorithms are known for the classical secretary problems and several variants. We study the following two extensions of the secretary problem: In the discounted secretary problem, there is a time-dependent “discount” factor $d(t)$, and the benefit derived from selecting an element/secretary e at time t is $d(t)v(e)$. For this problem with arbitrary (not necessarily decreasing) functions $d(t)$, we show a constant-competitive algorithm when the expected optimum is known in advance. With no prior knowledge, we exhibit a lower bound and give a nearly matching $O(log n)$-competitive algorithm. In the weighted secretary problem, up to K secretaries can be selected; when a secretary is selected (s)he must be irrevocably assigned to one of K positions, with position k having weight $w(k)$, and assigning object/secretary e to position k has benefit $w(k)v(e)$. The goal is to select secretaries and assign them to positions to maximize the sum of $w(k)v(e_k)$ where $e_k$ is the secretary assigned to position k. We give constant-competitive algorithms for this problem. Most of these results can also be extended to the matroid secretary case for a large family of matroids with a constant-factor loss, and an O(log rank) loss for general matroids. These results are based on a reduction from various matroids to partition matroids which present a unified approach to many of the upper bounds of Babaioff et al. These problems have connections to online mechanism design. All our algorithms are monotone, and hence lead to truthful mechanisms for the corresponding online auction problems

Matroids, secretary problems, and online mechanisms

We study a generalization of the classical secretary problem which we call the "matroid secretary problem". In this problem, the elements of a matroid are presented to an online algorithm in random order. When an element a

A Knapsack Secretary Problem with Applications

We consider situations in which a decision-maker with a fixed budget faces a sequence of options, each with a cost and a value, and must select a subset of them online so as to maximize the total value. Such situations arise in many contexts, e.g., hiring workers, scheduling jobs, and bidding in sponsored search auctions. This problem, often called the online knapsack problem, is known to be inapproximable. Therefore, we make the enabling assumption that elements arrive in a random order. Hence our problem can be thought of as a weighted version of the classical secretary problem, which we call the knapsack secretary problem. Using the random-order assumption, we design a constant-competitive algorithm for arbitrary weights and values, as well as a e-competitive algorithm for the special case when all weights are equal (i.e., the multiple-choice secretary problem). In contrast to previous work on online knapsack problems, we do not assume any knowledge regarding the distribution of weights and values beyond the fact that the order is random

Online Auctions and Generalized Secretary Problems

We present generalized secretary problems as a framework for online auctions. Elements, such as potential employees or customers, arrive one by one online. After observing the value derived from an element, but without knowing the values of future elements, the algorithm has to make an irrevocable decision whether to retain the element as part of a solution, or reject it. The way in which the secretary framework diers from traditional online algorithms is that the elements arrive in uniformly random order. Many natural online auction scenarios can be cast as generalized secretary problems, by impos- ing natural restrictions on the feasible sets. For many such settings, we present surprisingly strong constant factor guarantees on the expected value of solutions obtained by online algorithms. The framework is also easily augmented to take into account time-discounted revenue and incentive compatibility. We give an overview of recent results and future research directions

Coalition formation and price of anarchy in Cournot oligopolies

Non-cooperative game theory purports that economic agents behave with little regard towards the negative externalities they impose on each other. Such behaviors generally lead to inefficient outcomes where the social welfare is bounded away from its optimal value. However, in practice, self-interested individuals explore the possibility of circumventing such negative externalities by forming coalitions. What sort of coalitions should we expect to arise? How do they affect the social welfare? We study these questions in the setting of Cournot markets, one of the most prevalent models of firm competition. Our model of coalition formation has two dynamic aspects. First, agents choose strategically how to update the current coalition partition. Furthermore, coalitions compete repeatedly between themselves trying to minimize their long-term regret. We prove tight bounds on the social welfare, which are significantly higher than that of the Nash equilibria of the original game. Furthermore, this improvement in performance is robust across different supply-demand curves and depends only on the size of the market

A Combinatorial Allocation Mechanism for Banner Advertisement with Penalties

Most current banner advertising is sold through negotiation thereby incurring large transaction costs and possibly sub-optimal allocations. We propose a new automated system for selling banner advertising. In this system, each advertiser specifies a collection of host webpages which are relevant to his product, a desired total quantity of impressions on these pages, and a maximum per-impression price. The system selects a subset of advertisers as winners and maps each winner to a set of impressions on pages within his desired collection. The distinguishing feature of our system as opposed to current combinatorial allocation mechanisms is that, mimicking the current negotiation system, we guarantee that winners receive at least as many advertising opportunities as they requested or else receive ample compensation in the form of a monetary payment by the host. Such guarantees are essential in markets like banner advertising where a major goal of the advertising campaign is developing brand recognition. As we show, the problem of selecting a feasible subset of advertisers with maximum total value is inapproximable. We thus present two greedy heuristics and discuss theoretical techniques to measure their performances. Our first algorithm iteratively selects advertisers and corresponding sets of impressions which contribute maximum marginal per-impression profit to the current solution. We prove a bi-criteria approximation for this algorithm, showing that it generates approximately as much value as the optimum algorithm on a slightly harder problem. However, this algorithm might perform poorly on instances in which the value of the optimum solution is quite large, a clearly undesirable failure mode. Hence, we present an adaptive greedy algorithm which again iteratively selects advertisers with maximum marginal per-impression profit, but additionally reassigns impressions at each iteration. For this algorithm, we prove a structural approximation result, a newly defined framework for evaluating heuristics. We thereby prove that this algorithm has a better performance guarantee than the simple greedy algorithm

The role of compatibility in the diffusion of technologies through social networks

In many settings, competing technologies -- for example, operating systems, instant messenger systems, or document formats -- can be seen adopting a limited amount of compatibility with one another; in other words, the difficulty in using multiple technologies is balanced somewhere between the two extremes of impossibility and effortless interoperability. There are a range of reasons why this phenomenon occurs, many of which -- based on legal, social, or business considerations -- seem to defy concise mathematical models. Despite this, we show that the advantages of limited compatibility can arise in a very simple model of diffusion in social networks, thus offering a basic explanation for this phenomenon in purely strategic terms. Our approach builds on work on the diffusion of innovations in the economics literature, which seeks to model how a new technology A might spread through a social network of individuals who are currently users of technology B. We consider several ways of capturing the compatibility of A and B, focusing primarily on a model in which users can choose to adopt A, adopt B, or -- at an extra cost -- adopt both A and B. We characterize how the ability of A to spread depends on both its quality relative to B, and also this additional cost of adopting both, and find some surprising non-monotonicity properties in the dependence on these parameters: in some cases, for one technology to survive the introduction of another, the cost of adopting both technologies must be balanced within a narrow, intermediate range. We also extend the framework to the case of multiple technologies, where we find that a simple model captures the phenomenon of two firms adopting a limited "strategic alliance" to defend against a new, third technology