Balasubramanian Sivan

Near optimal online algorithms and fast approximation algorithms for resource allocation problems

We present algorithms for a class of resource allocation problems both in the online setting with stochastic input and in the offline setting. This class of problems contains many interesting special cases such as the Adwords problem. In the online setting we introduce a new distributional model called the adversarial stochastic input model, which is a generalization of the i.i.d model with unknown distributions, where the distributions can change over time. In this model we give a 1-O(ε) approximation algorithm for the resource allocation problem, with almost the weakest possible assumption: the ratio of the maximum amount of resource consumed by any single request to the total capacity of the resource, and the ratio of the profit contributed by any single request to the optimal profit is at most (ε2/log(1/ε)2)/(log n + log (1/ε)) where n is the number of resources available. There are instances where this ratio is #949;2/log n such that no randomized algorithm can have a competitive ratio of 1-o(ε) even in the i.i.d model. The upper bound on ratio that we require improves on the previous upper-bound for the i.i.d case by a factor of n. Our proof technique also gives a very simple proof that the greedy algorithm has a competitive ratio of 1-1/e for the Adwords problem in the i.i.d model with unknown distributions, and more generally in the adversarial stochastic input model, when there is no bound on the bid to budget ratio. All the previous proofs assume that either bids are very small compared to budgets or something very similar to this. In the offline setting we give a fast algorithm to solve very large LPs with both packing and covering constraints. We give algorithms to approximately solve (within a factor of 1+ε) the mixed packing-covering problem with O(γ m log n/ε2) oracle calls where the constraint matrix of this LP has dimension n x m, and γ is a parameter which is very similar to the ratio described for the online setting. We discuss several applications, and how our algorithms improve existing results in some of these applications.

Prior-independent mechanisms for scheduling

We study the makespan minimization problem with unrelated selfish machines under the assumption that job sizes are stochastic. We design simple truthful mechanisms that under different distributional assumptions provide constant and sublogarithmic approximations to expected makespan. Our mechanisms are prior-independent in that they do not rely on knowledge of the job size distributions. Prior-independent approximations were previously known only for the revenue maximization objective [13, 11, 26]. In contrast to our results, in prior-free settings no truthful anonymous deterministic mechanism for the makespan objective can provide a sublinear approximation [3].

Bayesian algorithmic mechanism design

This article surveys recent work with an algorithmic flavor in Bayesian mechanism design. Bayesian mechanism design involves optimization in economic settings where the designer possesses some stochastic information about the input. Recent years have witnessed huge advances in our knowledge and understanding of algorithmic techniques for Bayesian mechanism design problems. These include, for example, revenue maximization in settings where buyers have multi-dimensional preferences, optimization of non-linear objectives such as makespan, and generic reductions from mechanism design to algorithm design. However, a number of tantalizing questions remain un-solved. This article is meant to serve as an introduction to Bayesian mechanism design for a novice, as well as a starting point for a broader literature search for an experienced researcher.

Multi-parameter mechanism design and sequential posted pricing

We consider the classical mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single-parameter settings (i.e., where each agents preference is given by a single private value for being served and zero for not being served) this problem is solved. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed, and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agents preference is given by multiple values for each of the multiple services available). In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multidimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem.

On Conditional Covering Problem

The conditional covering problem (CCP) aims to locate facilities on a graph, where the vertex set represents both the demand points and the potential facility locations. The problem has a constraint that each vertex can cover only those vertices that lie within its covering radius and no vertex can cover itself. The objective of the problem is to find a set that minimizes the sum of the facility costs required to cover all the demand points. An algorithm for CCP on paths was presented by Horne and Smith (Networks 46(4):177–185, 2005). We show that their algorithm is wrong and further present a correct O(n3) algorithm for the same. We also propose an O(n2) algorithm for the CCP on paths when all vertices are assigned unit costs and further extend this algorithm to interval graphs without an increase in time complexity.

Stability of service under time-of-use pricing

We consider time-of-use pricing as a technique for matching supply and demand of temporal resources with the goal of maximizing social welfare. Relevant examples include energy, computing resources on a cloud computing platform, and charging stations for electric vehicles, among many others. A client/job in this setting has a window of time during which he needs service, and a particular value for obtaining it. We assume a stochastic model for demand, where each job materializes with some probability via an independent Bernoulli trial. Given a per-time-unit pricing of resources, any realized job will first try to get served by the cheapest available resource in its window and, failing that, will try to find service at the next cheapest available resource, and so on. Thus, the natural stochastic fluctuations in demand have the potential to lead to cascading overload events. Our main result shows that setting prices so as to optimally handle the expected demand works well: with high probability, when the actual demand is instantiated, the system is stable and the expected value of the jobs served is very close to that of the optimal offline algorithm.

Multi-Score Position Auctions

In this paper we propose a general family of position auctions used in paid search, which we call multi-score position auctions. These auctions contain the GSP auction and the GSP auction with squashing as special cases. We show experimentally that these auctions contain special cases that perform better than the GSP auction with squashing, in terms of revenue, and the number of clicks on ads. In particular, we study in detail the special case that squashes the first slot alone and show that this beats pure squashing (which squashes all slots uniformly). We study the equilibria that arise in this special case to examine both the first order and the second order effect of moving from the squashing-all-slots auction to the squash-only-the-top-slot auction. For studying the second order effect, we simulate auctions using the value-relevance correlated distribution suggested in Lahaie and Pennock [2007]. Since this distribution is derived from a study of value and relevance distributions in Yahoo! we believe the insights derived from this simulation to be valuable. For measuring the first order effect, in addition to the said simulation, we also conduct experiments using auction data from Bing over several weeks that includes a random sample of all auctions.

Price Competition, Fluctuations and Welfare Guarantees

In various markets where sellers compete in price, price oscillations are observed rather than convergence to equilibrium. Such fluctuations have been empirically observed in the retail market for gasoline, in airline pricing and in the online sale of consumer goods. Motivated by this, we study a model of price competition in which equilibria rarely exist. We seek to analyze the welfare, despite the nonexistence of equilibria, and present welfare guarantees as a function of the market power of the sellers. We first study best response dynamics in markets with sellers that provide a homogeneous good, and show that except for a modest number of initial rounds, the welfare is guaranteed to be high. We consider two variations: in the first the sellers have full information about the buyers valuation. Here we show that if there are n items available across all sellers and nmax is the maximum number of items controlled by any given seller, then the ratio of the optimal welfare to the achieved welfare will be at most log n/(n-nmax + 1))+1. As the market power of the largest seller diminishes, the welfare becomes closer to optimal. In the second variation we consider an extended model in which sellers have uncertainty about the buyers valuation. Here we similarly show that the welfare improves as the market power of the larger seller decreases, yet with a worse ratio of n/(n-nmax + 1). Our welfare bounds in both cases are essentially tight. The exponential gap in welfare between the two variations quantifies the value of accurately learning the buyers valuation in such settings. Finally, we show that extending our results to heterogeneous goods in general is not possible. Even for the simple class of k-additive valuations, there exists a setting where the welfare approximates the optimal welfare within any non-zero factor only for O(1/s) fraction of the time, where s is the number of sellers.

Testing Incentive Compatibility in Display Ad Auctions

Consider a buyer participating in a repeated auction, such as those prevalent in display advertising. How would she test whether the auction is incentive compatible? To bid effectively, she is interested in whether the auction is single-shot incentive compatible---a pure second-price auction, with fixed reserve price---and also dynamically incentive compatible---her bids are not used to set future reserve prices. In this work we develop tests based on simple bid perturbations that a buyer can use to answer these questions, with a focus on dynamic incentive compatibility. There are many potential A/B testing setups that one could use, but we find that many natural experimental designs are, in fact, flawed. For instance, we show that additive perturbations can lead to paradoxical results, where higher bids lead to lower optimal reserve prices. We precisely characterize this phenomenon and show that reserve prices are only guaranteed to be monotone for distributions satisfying the Monotone Hazard Rate (MHR) property. The experimenter must also decide how to split traffic to apply systematic perturbations. It is tempting to have this split be randomized, but we demonstrate empirically that unless the perturbations are aligned with the partitions used by the seller to compute reserve prices, the results are guaranteed to be inconclusive. We validate our results with experiments on real display auction data and show that a buyer can quantify both single-shot and dynamic incentive compatibility even under realistic conditions where only the cost of the impression is observed (as opposed to the exact reserve price). We analyze the cost of running such experiments, exposing trade-offs between test accuracy, cost, and underlying market dynamics.

Robust Repeated Auctions under Heterogeneous Buyer Behavior

We study revenue optimization in a repeated auction between a single seller anda single buyer. Traditionally, the design of repeated auctions requires strong modeling assumptions about the bidder behavior, such as it being myopic, infinite lookahead, or some specific form of learning behavior. Is it possible to design mechanisms which are simultaneously optimal against a multitude of possible buyer behaviors? We answer this question by designing a simple state-based mechanism that is simultaneously approximately optimal against a k -lookahead buyer for all k , a buyer who is a no-regret learner, and a buyer who is a policy-regret learner. Against each type of buyer our mechanism attains a constant fraction of the optimal revenue attainable against that type of buyer. We complement our positive results with almost tight impossibility results, showing that the revenue approximation tradeoffs achieved by our mechanism for different lookahead attitudes are near-optimal.