Abernethy et al. (2011) showed that Blackwell approachability and no-regret learning are equivalent, in the sense that any algorithm that solves a specific Blackwell approachability instance can be converted to a sublinear regret algorithm for a specific no-regret learning instance, and vice versa. In this paper, we study a more fine-grained form of such reductions, and ask when this translation between problems preserves not only a sublinear rate of convergence, but also preserves the optimal rate of convergence. That is, in which cases does it suffice to find the optimal regret bound for a no-regret learning instance in order to find the optimal rate of convergence for a corresponding approachability instance? We show that the reduction of Abernethy et al. (2011) (and of other subsequent work) does not preserve rates: their reduction may reduce a $d$-dimensional approachability instance $\mathcal{I}_1$ with optimal convergence rate $R_1$ to a no-regret learning instance $\mathcal{I}_2$ with optimal regret-per-round of $R_2$, with $R_{2}/R_{1}$ arbitrarily large (in particular, it is possible that $R_1 = 0$ and $R_{2} > 0$). On the other hand, we show that it is possible to tightly reduce any approachability instance to an instance of a generalized form of regret minimization we call \emph{improper $\phi$-regret minimization} (a variant of the $\phi$-regret minimization of Gordon et al. (2008) where the transformation functions may map actions outside of the action set). Finally, we characterize when linear transformations suffice to reduce improper $\phi$-regret minimization problems to standard classes of regret minimization problems (such as external regret minimization and proper $\phi$-regret minimization) in a rate preserving manner. We prove that some improper $\phi$-regret minimization instances cannot be reduced to either subclass of instance in this way, suggesting that approachability can capture some problems that cannot be easily phrased in the standard language of online learning.

Swap regret is a notion that has proven itself to be central to the study of general-sum normal-form games, with swap-regret minimization leading to convergence to the set of correlated equilibria and guaranteeing non-manipulability against a self-interested opponent. However, the situation for more general classes of games - such as Bayesian games and extensive-form games - is less clear-cut, with multiple candidate definitions for swap-regret but no known efficiently minimizable variant of swap regret that implies analogous non-manipulability guarantees.

We study the social efficiency of bilateral trade between a seller and a buyer. In the classical Bayesian setting, the celebrated Myerson-Satterthwaite impossibility theorem states that no Bayesian incentive-compatible, individually rational, and budget-balanced mechanism can achieve full efficiency. As a counterpoint, Deng, Mao, Sivan, and Wang (STOC 2022) show that if pricing power is delegated to the right person (either the seller or the buyer), the resulting mechanism can guarantee at least a constant fraction of the ideal (yet unattainable) gains from trade.

In many real-world applications such as data centers and cloud computing, the systems often consist of multiple frontends (routers) that receive job requests and backends (servers) that process these jobs. Efficient resource management is becoming increasingly important given the growing demand for serving machine learning inference queries, which incur high latencies and require expensive computational resources.

Budget pacing is a popular service that has been offered by major internet advertising platforms since their inception. In the past few years, autobidding products that provide real-time bidding as a service to advertisers have seen a prominent rise in adoption. A popular autobidding stategy is value maximization subject to return-on-spend (ROS) constraints. For historical or business reasons, the systems that govern these two services, namely budget pacing and ROS pacing, are not necessarily always a single unified and coordinated entity that optimizes a global objective subject to both constraints. The purpose of this work is to theoretically and empirically compare algorithms with different degrees of coordination between these two pacing systems. In particular, we compare (a) a fully-decoupled sequential algorithm; (b) a minimally-coupled min-pacing algorithm; (c) a fully-coupled dual-based algorithm. Our main contribution is to theoretically analyze the min-pacing algorithm and show that it attains similar guarantees to the fully-coupled canonical dual-based algorithm. On the other hand, we show that the sequential algorithm, even though appealing by virtue of being fully decoupled, could badly violate the constraints. We validate our theoretical findings empirically by showing that the min-pacing algorithm performs almost as well as the canonical dual-based algorithm on a semi-synthetic dataset that was generated from a large online advertising platform's auction data.

Learning to bid in repeated first-price auctions is a fundamental problem at the interface of game theory and machine learning, which has seen a recent surge in interest due to the transition of display advertising to first-price auctions. In this work, we propose a novel concave formulation for pure-strategy bidding in first-price auctions, and use it to analyze natural Gradient-Ascent-based algorithms for this problem. Importantly, our analysis goes beyond regret, which was the typical focus of past work, and also accounts for the strategic backdrop of online-advertising markets where bidding algorithms are deployed---we provide the first guarantees of strategic-robustness and incentive-compatibility for Gradient Ascent.

Blackwell’s celebrated approachability theory provides a general framework for a variety of learning problems, including regret minimization. However, Blackwell’s proof and implicit algorithm measure approachability using the $\ell_2$ (Euclidean) distance. We argue that in many applications such as regret minimization, it is more useful to study approachability under other distance metrics, most commonly the $\ell_\infty$-metric. But, the time and space complexity of the algorithms designed for $\ell_\infty$-approachability depend on the dimension of the space of the vectorial payoffs, which is often prohibitively large. Thus, we present a framework for converting high-dimensional $\ell_\infty$-approachability problems to low-dimensional \emph{pseudonorm} approachability problems, thereby resolving such issues. We first show that the $\ell_\infty$-distance between the average payoff and the approachability set can be equivalently defined as a \emph{pseudodistance} between a lower-dimensional average vector payoff and a new convex set we define. Next, we develop an algorithmic theory of pseudonorm approachability, analogous to previous work on approachability for $\ell_2$ and other norms, showing that it can be achieved via online linear optimization (OLO) over a convex set given by the Fenchel dual of the unit pseudonorm ball. We then use that to show, modulo mild normalization assumptions, that there exists an $\ell_\infty$-approachability algorithm whose convergence is independent of the dimension of the original vectorial payoff. We further show that that algorithm admits a polynomial-time complexity, assuming that the original $\ell_\infty$-distance can be computed efficiently. We also give an $\ell_\infty$-approachability algorithm whose convergence is logarithmic in that dimension using an FTRL algorithm with a maximum-entropy regularizer. Finally, we illustrate the benefits of our framework by applying it to several problems in regret minimization.

Major Internet advertising platforms offer budget pacing tools as a standard service for advertisers to manage their ad campaigns. Given the inherent non-stationarity in an advertiser’s value and also competing advertisers’ values over time, a commonly used approach is to learn a target expenditure plan that specifies a target spend as a function of time, and then run a controller that tracks this plan. This raises the question: how many historical samples are required to learn a good expenditure plan? We study this question by considering an advertiser repeatedly participating in $T$ second-price auctions, where the tuple of her value and the highest competing bid is drawn from an unknown time-varying distribution. The advertiser seeks to maximize her total utility subject to her budget constraint. Prior work has shown the sufficiency of $T\log T$ samples per distribution to achieve the optimal $O(\sqrt{T})$-regret. We dramatically improve this state-of-the-art and show that just one sample per distribution is enough to achieve the near-optimal $\tilde O(\sqrt{T})$-regret, while still being robust to noise in the sampling distributions.

Myerson-regularity (or simply regularity) is a standard condition in Economics that was originally introduced by Myerson in his seminar paper on optimal auctions [Myerson, 1981]. A regular distribution is a distribution with CDF F such that the revenue curve in quantile space R(q) = q · F−1(1 − q) is concave.

A patient seller aims to sell a good to an impatient buyer (i.e., one who discounts utility over time). The buyer will remain in the market for a period of time T, and her private value is drawn from a publicly known distribution. What is the revenue-optimal pricing-curve (sequence of (price, time) pairs) for the seller? Is randomization of help here? Is the revenue-optimal pricing-curve computable in polynomial time? We answer these questions in this paper. We give an efficient algorithm for computing the revenue-optimal pricing curve. We show that pricing curves, that post a price at each point of time and let the buyer pick her utility maximizing time to buy, are revenue-optimal among a much broader class of sequential lottery mechanisms: namely, mechanisms that allow the seller to post a menu of lotteries at each point of time cannot get any higher revenue than pricing curves. We also show that the even broader class of mechanisms that allow the menu of lotteries to be adaptively set, can earn strictly higher revenue than that of pricing curves, and the revenue gap can be as big as the support size of the buyer's value distribution. * The full version of the paper can be accessed at https://arxiv.org/abs/2106.02149.

The common way to optimize auction and pricing systems is to set aside a small fraction of the traffic to run experiments. This leads to the question: how can we learn the most with the smallest amount of data? For truthful auctions, this is the sample complexity problem. For posted price auctions, we no longer have access to samples. Instead, the algorithm is allowed to choose a price pt; then for a fresh sample vt ~ D we learn the sign st = sign(pt — vt) ∈ {-1, +1}. How many pricing queries are needed to estimate a given parameter of the underlying distribution? We give tight upper and lower bounds on the number of pricing queries required to find an approximately optimal reserve price for general, regular and MHR distributions. Interestingly, for regular distributions, the pricing query and sample complexities match. But for general and MHR distributions, we show a strict separation between them. All known results on sample complexity for revenue optimization follow from a variant of using the optimal reserve price of the empirical distribution. In the pricing query complexity setting, we show that learning the entire distribution within an error of ε in Levy distance requires strictly more pricing queries than to estimate the reserve. Instead, our algorithm uses a new property we identify called relative flatness to quickly zoom into the right region of the distribution to get the optimal pricing query complexity. * The full version of the paper can be accessed at https://arxiv.org/abs/2111.03158

We study repeated two-player games where one of the players, the learner, employs a no-regret learning strategy, while the other, the optimizer, is a rational utility maximizer. We consider general Bayesian games, where the payoffs of both the optimizer and the learner could depend on the type, which is drawn from a publicly known distribution, but revealed privately to the learner. We address the following questions: (a) what is the bare minimum that the optimizer can guarantee to obtain regardless of the no-regret learning algorithm employed by the learner? (b) are there learning algorithms that cap the optimizer payoff at this minimum? (c) can these algorithms be implemented efficiently? While building this theory of optimizer-learner interactions, we define a new combinatorial notion of regret called polytope swap regret, that could be of independent interest in other settings.

We study the classic online bipartite matching problem with a twist: offline nodes are reusable any number of times. Every offline node i becomes available d steps after it was assigned to. Nothing better than a 0.5-approximation, obtained by the trivial deterministic greedy algorithm, was known for this problem. We give the first approximation factor beating 0.5, namely a 0.505 approximation, by suitably adapting and interpreting the powerful technique of Online Correlated Selection.

We study bilateral trade between two strategic agents. The celebrated result of Myerson and Satterthwaite states that in general, no incentive-compatible, individually rational and weakly budget balanced mechanism can be efficient. I.e., no mechanism with these properties can guarantee a trade whenever buyer value exceeds seller cost. Given this, a natural question is whether there exists a mechanism with these properties that guarantees a constant fraction of the first-best gains-from-trade, namely a constant fraction of the gains-from-trade attainable whenever buyer’s value weakly exceeds seller’s cost. In this work, we positively resolve this long-standing open question on constant-factor approximation, mentioned in several previous works, using a simple mechanism that obtains a 1/8.23 ≈ 0.121 fraction of the first-best.

First-price auctions have many desirable properties, including uniquely possessing some, like credibility. However, first-price auctions are also inherently non-truthful, and non-truthfulness may result in instability and inefficiencies. Given these pros and cons, we seek to quantify the extent to which first-price auctions are susceptible to manipulation.

In the Learning to Price setting, a seller posts prices over time with the goal of maximizing revenue while learning the buyer’s valuation. This problem is very well understood when values are stationary (fixed or iid). Here we study the problem where the buyer’s value is a moving target, i.e., they change over time either by a stochastic process or adversarially with bounded variation. In either case, we provide matching upper and lower bounds on the optimal revenue loss. Since the target is moving, any information learned soon becomes out-dated, which forces the algorithms to keep switching between exploring and exploiting phases.

Product Design is an important problem in marketing research where a firm tries to learn what features of a product are more valuable to consumers. We study this problem from the viewpoint of online learning: a firm repeatedly interacts with a buyer by choosing a product configuration as well as a price and observing the buyer's purchasing decision. The goal of the firm is to maximize revenue throughout the course of $T$ rounds by learning the buyer's preferences. We study both the case of a set of discrete products and the case of a continuous set of allowable product features. In both cases we provide nearly tight upper and lower regret bounds.

We study dynamic mechanisms for optimizing revenue in repeated auctions, that are robust to heterogeneous forward-looking and learning behavior of the buyers. Typically it is assumed that the buyers are either all myopic or are all infinite lookahead, and that buyers understand and trust the mechanism. These assumptions raise the following question: is it possible to design approximately revenue optimal mechanisms when the buyer pool is heterogeneous? We provide this fresh perspective on the problem by considering a heterogeneous population of buyers with an unknown mixture of k-lookahead buyers, myopic buyers, no-regret-learners and no-policy-regret learners. Facing this population, we design a simple state-based mechanism that achieves a constant fraction of the optimal achievable revenue.

Bidding dashboards are used in online marketplaces to aid a bidder in computing good bidding strategies, particularly when the auction used by the marketplace is constrained to have the winners-pay-bid payment format. A dashboard predicts the outcome a bidder can expect to get at each possible bid. To convince a bidder to best respond to the information published in a dashboard, a dashboard mechanism should ensure either (a) that best responding maximizes the bidder's utility (a weaker requirement) or (b) that the mechanism implements the outcome published in the dashboard (a stronger requirement that subsumes (a)). Recent work by Hartline et al. EC'19 formalized the notion of dashboard mechanisms and designed winners-pay-bid mechanisms that guaranteed epsilon-optimal utility (an epsilon-approximate version of (a)), but not (b). I.e., the mechanism could end up implementing arbitrarily different outcomes from what was promised. While this guarantee is sufficient from a purely technical perspective, it is far from enough in the real world: it is hard to convince bidders to best respond to information which could be arbitrarily inaccurate, regardless of the theoretical promise of near-optimality. In this paper we study guaranteed dashboard mechanisms, namely, ones that are guaranteed to implement what they publish, and obtain good welfare. We study this question in a repeated auction setting for general single-dimensional valuations and give tight characterizations of the loss in welfare as a function of natural parameters upper bounding the difference in valuation profile across the rounds. In particular, we give three different characterizations, bounding the loss in welfare in terms of the 0 norm, 1 norm and infinite norm of difference in valuation profile across rounds. All the characterizations generalize at least up to matroid feasibility constraints, and the infinite norm characterization extends to general downward-closed feasibility constraints. We bring to bear different techniques for each of these characterizations, including connections to differential privacy and online convex optimizations.

We introduce a new decomposition technique for random variables that maps a generic instance of the prophet inequalities problem to a new instance where all but a constant number of variables have a tractable structure that we refer to as (ε, δ)-smallness. Using this technique, we make progress on several outstanding problems in the area: We show that, even in the case of non-identical distributions, it is possible to achieve (arbitrarily close to) the optimal approximation ratio of β ~0.745 when the items arrive in a random order (this version is commonly known as prophet secretary) as long as we are allowed to remove a small constant number of distributions. We show that forfrequent instances (where each distribution reoccurs some number of times) and random arrival order, it is possible to achieve the optimal approximation ratio of β (improving over the previous best-known bound of 0.738). We give a new, simpler proof of Kertz's optimal approximation guarantee of β ~0.745 for prophet inequalities with i.i.d. distributions. The proof is primal-dual and simultaneously produces upper and lower bounds. Using this decomposition in combination with a novel convex programming formulation, we construct the first (in parallel work with[1]) an Efficient PTAS (EPTAS) for the Optimal Ordering problem.

We consider a setting in which bidders participate in multiple auctions run by different sellers, and optimize their bids for the aggregate auction. We analyze this setting by formulating a game between sellers, where a seller’s strategy is to pick an auction to run. Our analysis aims to shed light on the recent change in the Display Ads market landscape: here, ad exchanges (sellers) were mostly running second-price auctions earlier and over time they switched to variants of the first-price auction, culminating in Google’s Ad Exchange moving to a first-price auction in 2019. Our model and results offer an explanation for why the first-price auction occurs as a natural equilibrium in such competitive markets.

How should a player who repeatedly plays a game against a no-regret learner strategize to maximize his utility? We study this question and show that under some mild assumptions, the player can always guarantee himself a utility of at least what he would get in a Stackelberg equilibrium. When the no-regret learner has only two actions, we show that the player cannot get any higher utility than the Stackelberg equilibrium utility. But when the no-regret learner has more than two actions and plays a mean-based no-regret strategy, we show that the player can get strictly higher than the Stackelberg equilibrium utility. We construct the optimal game-play for the player against a mean-based no-regret learner who has three actions. When the no-regret learner's strategy also guarantees him a no-swap regret, we show that the player cannot get anything higher than a Stackelberg equilibrium utility.

We study the problem of how to repeatedly sell to a buyer running a no-regret, mean-based algorithm. Previous work [Braverman et al., 2018] shows that it is possible to design effective mechanisms in such a setting that extract almost all of the economic surplus, but these mechanisms require the buyer's values each round to be drawn independently and identically from a fixed distribution. In this work, we do away with this assumption and consider the prior-free setting where the buyer's value each round is chosen adversarially (possibly adaptively). We show that even in this prior-free setting, it is possible to extract a $(1-\varepsilon)$-approximation of the full economic surplus for any $\varepsilon > 0$. The number of options offered to a buyer in any round scales independently of the number of rounds $T$ and polynomially in $\varepsilon$. We show that this is optimal up to a polynomial factor; any mechanism achieving this approximation factor, even when values are drawn stochastically, requires at least $\Omega(1/\varepsilon)$ options. Finally, we examine what is possible when we constrain our mechanism to a natural auction format where overbidding is dominated. Braverman et al. [2018] show that even when values are drawn from a known stochastic distribution supported on $[1/H, 1]$, it is impossible in general to extract more than $O(\log\log H / \log H)$ of the economic surplus. We show how to achieve the same approximation factor in the prior-independent setting (where the distribution is unknown to the seller), and an approximation factor of $O(1 / \log H)$ in the prior-free setting (where the values are chosen adversarially).

We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an unknown distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a 1−ϵ fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor ϵ approaches 0 when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the tradeoff we obtain here that determines how fast ϵ approaches 0, is near optimal: We give a nearly matching lower bound showing that the tradeoff cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the adversarial stochastic input model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a 1−ϵ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. Further, if the algorithm is given one number per distribution, namely the optimal profit possible for each of the adversary’s distribution, then we design an algorithm that achieves a 1−ϵ fraction of the weighted average of the optimal profit of each distribution the adversary picks. In the offline setting we give a fast algorithm to solve very large linear programs (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 /δ)/ϵ 2 ) oracle calls where the constraint matrix of this LP has dimension n × m , the success probability of the algorithm is 1−δ, and γ quantifies how significant a single request is when compared to the sum total of all requests. We discuss implications of our results to several special cases including online combinatorial auctions, network routing, and the adwords problem.

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.

We study a basic auction design problem with online supply. There are two unit-demand bidders and two types of items. The first item type will arrive first for sure, and the second item type may or may not arrive. The auctioneer has to decide the allocation of an item immediately after each item arrives, but is allowed to compute payments after knowing how many items arrived. For this problem we show that there is no deterministic truthful and individually rational mechanism that, even with unbounded computational resources, gets any finite approximation factor to the optimal social welfare.

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.

We study the fundamental problem of prediction with expert advice and develop regret lower bounds for a large family of algorithms for this problem. We develop simple adversarial primitives, that lend themselves to various combinations leading to sharp lower bounds for many algorithmic families. We use these primitives to show that the classic Multiplicative Weights Algorithm (MWA) has a regret of (T*ln(k)/2)^{0.5} (where T is the time horizon and k is the number of experts), there by completely closing the gap between upper and lower bounds. We further show a regret lower bound of (2/3)* (T*ln(k)/2)^{0.5} for a much more general family of algorithms than MWA, where the learning rate can be arbitrarily varied over time, or even picked from arbitrary distributions over time. We also use our primitives to construct adversaries in the geometric horizon setting for MWA to precisely characterize the regret at 0.391/(\delta)^{0.5} for the case of 2 experts and a lower bound of (1/2)*(ln(k)/(2*\delta))^{0.5}, for the case of arbitrary number of experts k (here \delta is the probability that the game ends in any given round).

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.

We consider a pricing problem where a buyer is interested in purchasing/using a good, such as an app or music or software, repeatedly over time. The consumer discovers his value for the good only as he uses it, and the value evolves with each use. Optimizing for the seller's revenue in such dynamic settings is a complex problem and requires assumptions about how the buyer behaves before learning his future value(s), and in particular, how he reacts to risk. We explore the performance of a class of pricing mechanisms that are extremely simple for both the buyer and the seller to use: the buyer reacts to prices myopically without worrying about how his value evolves in the future; the seller needs to optimize for revenue over a space of only two parameters, and can do so without knowing the buyer's risk profile or fine details of the value evolution process. We present simple-versus-optimal type results, namely that under certain assumptions, simple pricing mechanisms of the above form are approximately optimal regardless of the buyer's risk profile. Our results assume that the buyer's value per usage evolves as a martingale. For our main result, we consider pricing mechanisms in which the seller offers the product for free for a certain number of uses, and then charges an appropriate fixed price per usage. We assume that the buyer responds by buying the product for as long as his value exceeds the fixed price. Importantly, the buyer does not need to know anything about how his future value will evolve, only how much he wants to use the product right now. Regardless of the buyers' initial value, our pricing captures as revenue a constant fraction of the total value that the buyers accumulate in expectation over time.

We study the classical problem of prediction with expert advice in the adversarial setting with a geometric stopping time. In 1965, Cover gave the optimal algorithm for the case of 2 experts. In this paper, we design the optimal algorithm, adversary and regret for the case of 3 experts. Further, we show that the optimal algorithm for 2 and 3 experts is a probability matching algorithm (analogous to Thompson sampling) against a particular randomized adversary. Remarkably, our proof shows that the probability matching algorithm is not only optimal against this particular randomized adversary, but also minimax optimal. Our analysis develops upper and lower bounds simultaneously, analogous to the primal-dual method. Our analysis of the optimal adversary goes through delicate asymptotics of the random walk of a particle between multiple walls. We use the connection we develop to random walks to derive an improved algorithm and regret bound for the case of 4 experts, and, provide a general framework for designing the optimal algorithm and adversary for an arbitrary number of experts.

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.

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 buyer's 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 buyer's 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 buyer's valuation in such settings.

A special case of Myerson's classic result describes the revenue-optimal equilibrium when a seller offers a single item to a buyer. We study a natural repeated sales extension of this model: a seller offers to sell a single fresh copy of an item to the same buyer every day via a posted price. The buyer's value for the item is unknown to the seller but is drawn initially from a publicly known distribution F and remains the same throughout. One key aspect of this game is revelation of the buyer's type through his actions: while the seller might try to learn this value to extract more revenue, the buyer is motivated to hide it to induce lower prices. If the seller is able to commit to future prices, then it is known that the best he can do is extract the Myerson optimal revenue each day. In a more realistic scenario, the seller is unable to commit and must play a perfect Bayesian equilibrium. It is known that not committing to future prices does not help the seller. Thus extracting Myerson optimal revenue each day is a natural upper bound and revenue benchmark in a setting without commitment. We study this setting without commitment and find several suprises. First, if the horizon is fixed, previous work showed that an equilibrium always exists, and all equilibria yield a very low revenue, often times only a constant amount of revenue. This is unintuitive and a far cry from the linearly growing benchmark of obtaining Myerson optimal revenue each day. Our first result shows that this is because the buyer strategies in these equilibria are necessarily unnatural. We restrict to a natural class of buyer strategies, which we call threshold strategies, and show that pure strategy threshold equilibria rarely exist. This offers an explanation for the non-prevalence of bizarre outcomes predicted by previous results. Second, if the seller can commit not to raise prices upon purchase, while still retaining the possibility of lowering prices in future, we recover the natural threshold equilibria by showing that they exist for a large class of distributions including the power law family of distributions. As an example, if the distribution F is uniform in [0,1], the seller can extract revenue of order in n rounds as opposed to the constant revenue obtainable when he is unable to make any commitments. Finally, we consider the infinite horizon game with partial commitment, where both the seller and the buyer discount the future utility by a factor of 1 – δ ∊ [0,1). When the value distribution is uniform in [0, 1], there exists a threshold equilibrium with expected revenue at least of the Myerson optimal revenue benchmark. Under some mild assumptions, this equilibrium is also unique.

We study the design of Bayesian incentive compatible mechanisms in single parameter domains, for the objective of optimizing social efficiency as measured by social cost. In the problems we consider, a group of participants compete to receive service from a mechanism that can provide such services at a cost. The mechanism wishes to choose which agents to serve in order to maximize social efficiency, but is not willing to suffer an expected loss: the agents' payments should cover the cost of service in expectation.

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].

In this paper we consider the adwords problem in the unknown distribution model. We consider the case where the budget to bid ratio k is at least 2, and give improved competitive ratios. Earlier results had competitive ratios better than 1-1/e only for "large enough" k, while our competitive ratio increases continuously with k. For k=2 the competitive ratio we get is 0.729 and it is 0.9 for k=16. We also improve the asymptotic competitive ratio for large k from 1 - O(√log n/k) to 1 - O(√1/k), thus removing any dependence on n, the number of advertisers. This ratio is optimal, even with known distributions. That is, even if an algorithm is tailored to the distribution, it cannot get a competitive ratio of 1 - o(√1/k), whereas our algorithm does not depend on the distribution. The algorithm is rather simple, it computes a score for every advertiser based on his original budget, the remaining budget and the remaining number of steps in the algorithm and assigns a query to the advertiser with the highest bid plus his score. The analysis is based on a "hybrid argument" that considers algorithms that are part actual, part hypothetical, to prove that our (actual) algorithm is better than a completely hypothetical algorithm whose performance is easy to analyze.

Truthfulness is fragile and demanding. It is oftentimes computationally harder than solving the original problem. Even worse, truthfulness can be utterly destroyed by small uncertainties in a mechanism's outcome. One obstacle is that truthful payments depend on outcomes other than the one realized, such as the lengths of non-shortest-paths in a shortest-path auction. Single-call mechanisms are a powerful tool that circumvents this obstacle --- they implicitly charge truthful payments, guaranteeing truthfulness in expectation using only the outcome realized by the mechanism. The cost of such truthfulness is a trade-off between the expected quality of the outcome and the risk of large payments.

We study the design and approximation of optimal crowdsourcing contests. Crowdsourcing contests can be modeled as all-pay auctions because entrants must exert effort up-front to enter. Unlike all-pay auctions where a usual design objective would be to maximize revenue, in crowdsourcing contests, the principal only benefits from the submission with the highest quality. We give a theory for optimal crowdsourcing contests that mirrors the theory of optimal auction design: the optimal crowdsourcing contest is a virtual valuation optimizer (the virtual valuation function depends on the distribution of contestant skills and the number of contestants). We also compare crowdsourcing contests with more conventional means of procurement. In this comparison, crowdsourcing contests are relatively disadvantaged because the effort of losing contestants is wasted. Nonetheless, we show that crowdsourcing contests are 2-approximations to conventional methods for a large family of “regular” distributions, and 4-approximations, otherwise.

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.

We investigate the power of randomness in the context of a fundamental Bayesian optimal mechanism design problem - a single seller aims to maximize expected revenue by allocating multiple kinds of resources to "unit-demand" agents with preferences drawn from a known distribution. When the agents' preferences are single-dimensional Myerson's seminal work [14] shows that randomness offers no benefit - the optimal mechanism is always deterministic. In the multi-dimensional case, where each agent's preferences are given by different values for each of the available services, Briest et al.[6] recently showed that the gap between the expected revenue obtained by an optimal randomized mechanism and an optimal deterministic mechanism can be unbounded even when a single agent is offered only 4 services. However, this large gap is attained through unnatural instances where values of the agent for different services are correlated in a specific way. We show that when the agent's values involve no correlation or a specific kind of positive correlation, the benefit of randomness is only a small constant factor (4 and 8 respectively). Our model of positively correlated values (that we call the common base value model) is a natural model for unit-demand agents and items that are substitutes. Our results extend to multiple agent settings as well.

We study the classic 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 agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25].