Moshe Babaioff

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 randomorder. 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 differs 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 imposing 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.

Concurrent auctions across the supply chain

With the recent technological feasibility of electronic commerce over the Internet, much attention has been given to the design of electronic markets for various types of electronically-tradable goods. Such markets, however, will normally need to function in some relationship with markets for other related goods, usually those downstream or upstream in the supply chain. Thus, for example, an electronic market for rubber tires for trucks, will likely need to be strongly influenced by the rubber market as well as by the truck market.In this paper we design protocols for exchange of information between a sequence of markets along a single supply chain. These protocols allow each of these markets to function separately, while the information exchanged guarantees efficient global behavior across the supply chain. Each market form a link in the supply chain operates as a double auction, where the bids on one side of the double auction come from bidders in the corresponding segment of the industry, and the bids on the other side are synthetically generated by the protocol to express the combined information from all other links in the chain. The double auctions in each of the markets can be of several types, and we study several variants of incentive compatible double auctions, comparing them in terms of their efficiency and of the market revenue.

A Simple and Approximately Optimal Mechanism for an Additive Buyer

We consider a monopolist seller with n heterogeneousitems, facing a single buyer. The buyer hasa value for each item drawn independently according to(non-identical) distributions, and his value for a set ofitems is additive. The seller aims to maximize his revenue.It is known that an optimal mechanism in this setting maybe quite complex, requiring randomization [19] and menusof infinite size [15]. Hart and Nisan [17] have initiated astudy of two very simple pricing schemes for this setting:item pricing, in which each item is priced at its monopolyreserve; and bundle pricing, in which the entire set ofitems is priced and sold as one bundle. Hart and Nisan [17]have shown that neither scheme can guarantee more thana vanishingly small fraction of the optimal revenue. Insharp contrast, we show that for any distributions, thebetter of item and bundle pricing is a constant-factorapproximation to the optimal revenue. We further discussextensions to multiple buyers and to valuations that arecorrelated across items.

Truthful mechanisms with implicit payment computation

It is widely believed that computing payments needed to induce truthful bidding is somehow harder than simply computing the allocation. We show that the opposite is true for single-parameter domains: creating a randomized truthful mechanism is essentially as easy as a single call to a monotone allocation function. Our main result is a general procedure to take a monotone allocation rule and transform it (via a black-box reduction) into a randomized mechanism that is truthful in expectation and individually rational for every realization. Moreover, the mechanism implements the same outcome as the original allocation rule with probability arbitrarily close to 1, and requires evaluating that allocation rule only once. Because our reduction is simple, versatile, and general, it has many applications to mechanism design problems in which re-evaluating the allocation function is either burdensome or informationally impossible. Applying our result to the multi-armed bandit problem, we obtain truthful randomized mechanisms whose regret matches the information-theoretic lower bound up to logarithmic factors, even though prior work showed this is impossible for truthful deterministic mechanisms. We also present applications to offline mechanism design, showing that randomization can circumvent a communication complexity lower bound for deterministic payments computation, and that it can also be used to create truthful shortest path auctions that approximate the welfare of the VCG allocation arbitrarily well, while having the same running time complexity as Dijkstras algorithm.

Characterizing Truthful Multi-armed Bandit Mechanisms

We consider a multiround auction setting motivated by pay-per-click auctions for Internet advertising. In each round the auctioneer selects an advertiser and shows her ad, which is then either clicked or not. An advertiser derives value from clicks; the value of a click is her private information. Initially, neither the auctioneer nor the advertisers have any information about the likelihood of clicks on the advertisements. The auctioneers goal is to design a (dominant strategies) truthful mechanism that (approximately) maximizes the social welfare. If the advertisers bid their true private values, our problem is equivalent to the multi-armed bandit problem, and thus can be viewed as a strategic version of the latter. In particular, for both problems the quality of an algorithm can be characterized by regret, the difference in social welfare between the algorithm and the benchmark which always selects the same “best” advertisement. We investigate how the design of multi-armed bandit algorithms is affecte...

Selling ad campaigns: online algorithms with cancellations

We study online pricing problems in markets with cancellations, i.e., markets in which prior allocation decisions can be revoked, but at a cost. In our model, a seller receives requests online and chooses which requests to accept, subject to constraints on the subsets of requests which may be accepted simultaneously. A request, once accepted, can be canceled at a cost which is a fixed fraction of the request value. This scenario models a market for web advertising campaigns, in which the buyback cost represents the cost of canceling an existing contract. We analyze a simple constant-competitive algorithm for a single-item auction in this model, and we prove that its competitive ratio is optimal among deterministic algorithms, but that the competitive ratio can be improved using a randomized algorithm. We then model ad campaigns using knapsack domains, in which the requests differ in size as well as in value. We give a deterministic online algorithm that achieves a bi-criterion approximation in which both approximation factors approach 1 as the buyback factor and the size of the maximum request approach 0. We show that the bi-criterion approximation is unavoidable for deterministic algorithms, but that a randomized algorithm is capable of achieving a constant competitive ratio. Finally, we discuss an extension of our randomized algorithm to matroid domains (in which the sets of simultaneously satisfiable requests constitute the independent sets of a matroid) as well as present results for domains in which the buyback factor of different requests varies.

Single-value combinatorial auctions and algorithmic implementation in undominated strategies

In this article, we are interested in general techniques for designing mechanisms that approximate the social welfare in the presence of selfish rational behavior. We demonstrate our results in the setting of Combinatorial Auctions (CA). Our first result is a general deterministic technique to decouple the algorithmic allocation problem from the strategic aspects, by a procedure that converts any algorithm to a dominant-strategy ascending mechanism. This technique works for any single value domain, in which each agent has the same value for each desired outcome, and this value is the only private information. In particular, for “single-value CAs”, where each player desires any one of several different bundles but has the same value for each of them, our technique converts any approximation algorithm to a dominant strategy mechanism that almost preserves the original approximation ratio. Our second result provides the first computationally efficient deterministic mechanism for the case of single-value multi-minded bidders (with private value and private desired bundles). The mechanism achieves an approximation to the social welfare which is close to the best possible in polynomial time (unless P=NP). This mechanism is an algorithmic implementation in undominated strategies, a notion that we define and justify, and is of independent interest.

Mechanisms for a spatially distributed market

We consider the problem of a spatially distributed market with strategic agents. In this problem a single good is traded in a set of independent markets, where shipment between markets is possible but incurs a cost. The problem has previously been studied in the non-strategic case, inwhich it can be analyzed and solved as a min-cost-flow problem. We considerthe case where buyers and sellers are strategic. Our first result gives adouble characterization of the VCG prices, first as distances in acertain residue graph and second as the minimal (for buyers) and maximal (forsellers) equilibrium prices. This provides a computationally efficient, individually rational and incentive compatible welfare maximizing mechanism. This mechanism is, necessarily, not budget balanced and we provide alsoa budget-balanced mechanism (which is also computationally efficient,incentive compatible, and individually rational) that achieves highwelfare. Some of our results extend to the cases where buyers andsellers have arbitrary convex demand and supply functions and to the case where transportation is controlled by strategic agents as well.

Computationally-feasible truthful auctions for convex bundles

In many economic settings, convex figures on the plane are for sale. For example, one might want to sell advertising space on a newspaper page. Selfish agents must be motivated to report their true values for the figures as well as to report the true figures. Moreover, an approximation algorithm should be used for guaranteeing a reasonable solution for the underlying NP-complete problem. We present truthful mechanisms that guarantee a certain fraction of the social welfare, as a function of a measure on the geometric diversity of the shapes. We give the first approximation algorithm for packing arbitrary weighted compact convex figures. We use this algorithm, and variants of existing algorithms, to create polynomial-time truthful mechanisms that approximate the social welfare. We show that each mechanism achieves the best approximation over all the mechanisms of its kind. We also study different models of information and a discrete model, where players bid for sets of predefined building blocks.