arXiv subjects

Renato Paes Leme

Publications and source records attributed to Renato Paes Leme.

70 records · Page 4Linked to original sources

arXiv2016

A Field Guide to Personalized Reserve Prices

We study the question of setting and testing reserve prices in single item auctions when the bidders are not identical. At a high level, there are two generalizations of the standard second price auction: in the lazy version we first determine the winner, and then apply reserve prices; in the eager version we first discard the bidders not meeting their reserves, and then determine the winner among the rest. We show that the two versions have dramatically different properties: lazy reserves are easy to optimize, and A/B test in production, whereas eager reserves always lead to higher welfare, but their optimization is NP-complete, and naive A/B testing will lead to incorrect conclusions. Despite their different characteristics, we show that the overall revenue for the two scenarios is always within a factor of 2 of each other, even in the presence of correlated bids. Moreover, we prove that the eager auction dominates the lazy auction on revenue whenever the bidders are independent or symmetric. We complement our theoretical results with simulations on real world data that show that even suboptimally set eager reserve prices are preferred from a revenue standpoint.

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arXiv2015

Tight Bounds for Approximate Carathéodory and Beyond

We give a deterministic nearly-linear time algorithm for approximating any point inside a convex polytope with a sparse convex combination of the polytope's vertices. Our result provides a constructive proof for the Approximate Carathéodory Problem, which states that any point inside a polytope contained in the $\ell_p$ ball of radius $D$ can be approximated to within $ε$ in $\ell_p$ norm by a convex combination of only $O\left(D^2 p/ε^2\right)$ vertices of the polytope for $p \geq 2$. We also show that this bound is tight, using an argument based on anti-concentration for the binomial distribution. Along the way of establishing the upper bound, we develop a technique for minimizing norms over convex sets with complicated geometry; this is achieved by running Mirror Descent on a dual convex function obtained via Sion's Theorem. As simple extensions of our method, we then provide new algorithms for submodular function minimization and SVM training. For submodular function minimization we obtain a simplification and (provable) speed-up over Wolfe's algorithm, the method commonly found to be the fastest in practice. For SVM training, we obtain $O(1/ε^2)$ convergence for arbitrary kernels; each iteration only requires matrix-vector operations involving the kernel matrix, so we overcome the obstacle of having to explicitly store the kernel or compute its Cholesky factorization.

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arXiv2015

Core-competitive Auctions

One of the major drawbacks of the celebrated VCG auction is its low (or zero) revenue even when the agents have high value for the goods and a {\em competitive} outcome could have generated a significant revenue. A competitive outcome is one for which it is impossible for the seller and a subset of buyers to `block' the auction by defecting and negotiating an outcome with higher payoffs for themselves. This corresponds to the well-known concept of {\em core} in cooperative game theory. In particular, VCG revenue is known to be not competitive when the goods being sold have complementarities. A bottleneck here is an impossibility result showing that there is no auction that simultaneously achieves competitive prices (a core outcome) and incentive-compatibility. In this paper we try to overcome the above impossibility result by asking the following natural question: is it possible to design an incentive-compatible auction whose revenue is comparable (even if less) to a competitive outcome? Towards this, we define a notion of {\em core-competitive} auctions. We say that an incentive-compatible auction is $α$-core-competitive if its revenue is at least $1/α$ fraction of the minimum revenue of a core-outcome. We study the Text-and-Image setting. In this setting, there is an ad slot which can be filled with either a single image ad or $k$ text ads. We design an $O(\ln \ln k)$ core-competitive randomized auction and an $O(\sqrt{\ln(k)})$ competitive deterministic auction for the Text-and-Image setting. We also show that both factors are tight.

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arXiv2015

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 an equilibrium rarely exists. We seek to analyze the welfare, despite the nonexistence of an equilibrium, 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 valuation of the buyer. Here we show that if there are $n$ items available across all sellers and $n_{\max}$ is the maximum number of items controlled by any given seller, the ratio of the optimal welfare to the achieved welfare will be at most $\log(\frac{n}{n-n_{\max}+1})+1$. As the market power of the largest seller diminishes, the welfare becomes closer to optimal. In the second variation we consider an extended model where sellers have uncertainty about the buyer's valuation. Here we similarly show that the welfare improves as the market power of the largest seller decreases, yet with a worse ratio of $\frac{n}{n-n_{\max}+1}$. The exponential gap in welfare between the two variations quantifies the value of accurately learning the buyer valuation. 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.

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arXiv2014

Bounding the inefficiency of outcomes in generalized second price auctions

The Generalized Second Price (GSP) auction is the primary auction used for monetizing the use of the Internet. It is well-known that truthtelling is not a dominant strategy in this auction and that inefficient equilibria can arise. In this paper we study the space of equilibria in GSP, and quantify the efficiency loss that can arise in equilibria under a wide range of sources of uncertainty, as well as in the full information setting. The traditional Bayesian game models uncertainty in the valuations (types) of the participants. The Generalized Second Price (GSP) auction gives rise to a further form of uncertainty: the selection of quality factors resulting in uncertainty about the behavior of the underlying ad allocation algorithm. The bounds we obtain apply to both forms of uncertainty, and are robust in the sense that they apply under various perturbations of the solution concept, extending to models with information asymmetries and bounded rationality in the form of learning strategies. We present a constant bound (2.927) on the factor of the efficiency loss (\emph{price of anarchy}) of the corresponding game for the Bayesian model of partial information about other participants and about ad quality factors. For the full information setting, we prove a surprisingly low upper bound of 1.282 on the price of anarchy over pure Nash equilibria, nearly matching a lower bound of 1.259 for the case of three advertisers. Further, we do not require that the system reaches equilibrium, and give similarly low bounds also on the quality degradation for any no-regret learning outcome. Our conclusion is that the number of advertisers in the auction has almost no impact on the price of anarchy, and that the efficiency of GSP is very robust with respect to the belief and rationality assumptions imposed on the participants.

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arXiv2014

Clinching Auctions Beyond Hard Budget Constraints

Constraints on agent's ability to pay play a major role in auction design for any setting where the magnitude of financial transactions is sufficiently large. Those constraints have been traditionally modeled in mechanism design as \emph{hard budget}, i.e., mechanism is not allowed to charge agents more than a certain amount. Yet, real auction systems (such as Google AdWords) allow more sophisticated constraints on agents' ability to pay, such as \emph{average budgets}. In this work, we investigate the design of Pareto optimal and incentive compatible auctions for agents with \emph{constrained quasi-linear utilities}, which captures more realistic models of liquidity constraints that the agents may have. Our result applies to a very general class of allocation constraints known as polymatroidal environments, encompassing many settings of interest such as multi-unit auctions, matching markets, video-on-demand and advertisement systems. Our design is based Ausubel's \emph{clinching framework}. Incentive compatibility and feasibility with respect to ability-to-pay constraints are direct consequences of the clinching framework. Pareto-optimality, on the other hand, is considerably more challenging, since the no-trade condition that characterizes it depends not only on whether agents have their budgets exhausted or not, but also on prices {at} which the goods are allocated. In order to get a handle on those prices, we introduce novel concepts of dropping prices and saturation. These concepts lead to our main structural result which is a characterization of the tight sets in the clinching auction outcome and its relation to dropping prices.

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arXiv2014

Price Competition in Online Combinatorial Markets

We consider a single buyer with a combinatorial preference that would like to purchase related products and services from different vendors, where each vendor supplies exactly one product. We study the general case where subsets of products can be substitutes as well as complementary and analyze the game that is induced on the vendors, where a vendor's strategy is the price that he asks for his product. This model generalizes both Bertrand competition (where vendors are perfect substitutes) and Nash bargaining (where they are perfect complements), and captures a wide variety of scenarios that can appear in complex crowd sourcing or in automatic pricing of related products. We study the equilibria of such games and show that a pure efficient equilibrium always exists. In the case of submodular buyer preferences we fully characterize the set of pure Nash equilibria, essentially showing uniqueness. For the even more restricted "substitutes" buyer preferences we also prove uniqueness over {\em mixed} equilibria. Finally we begin the exploration of natural generalizations of our setting such as when services have costs, when there are multiple buyers or uncertainty about the the buyer's valuation, and when a single vendor supplies multiple products.

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arXiv2013

On the Efficiency of the Walrasian Mechanism

Central results in economics guarantee the existence of efficient equilibria for various classes of markets. An underlying assumption in early work is that agents are price-takers, i.e., agents honestly report their true demand in response to prices. A line of research in economics, initiated by Hurwicz (1972), is devoted to understanding how such markets perform when agents are strategic about their demands. This is captured by the \emph{Walrasian Mechanism} that proceeds by collecting reported demands, finding clearing prices in the \emph{reported} market via an ascending price tâtonnement procedure, and returns the resulting allocation. Similar mechanisms are used, for example, in the daily opening of the New York Stock Exchange and the call market for copper and gold in London. In practice, it is commonly observed that agents in such markets reduce their demand leading to behaviors resembling bargaining and to inefficient outcomes. We ask how inefficient the equilibria can be. Our main result is that the welfare of every pure Nash equilibrium of the Walrasian mechanism is at least one quarter of the optimal welfare, when players have gross substitute valuations and do not overbid. Previous analysis of the Walrasian mechanism have resorted to large market assumptions to show convergence to efficiency in the limit. Our result shows that approximate efficiency is guaranteed regardless of the size of the market.

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arXiv2013

Pricing Public Goods for Private Sale

We consider the pricing problem faced by a seller who assigns a price to a good that confers its benefits not only to its buyers, but also to other individuals around them. For example, a snow-blower is potentially useful not only to the household that buys it, but also to others on the same street. Given that the seller is constrained to selling such a (locally) public good via individual private sales, how should he set his prices given the distribution of values held by the agents? We study this problem as a two-stage game. In the first stage, the seller chooses and announces a price for the product. In the second stage, the agents (each having a private value for the good) decide simultaneously whether or not they will buy the product. In the resulting game, which can exhibit a multiplicity of equilibria, agents must strategize about whether they will themselves purchase the good to receive its benefits. In the case of a fully public good (where all agents benefit whenever any agent purchases), we describe a pricing mechanism that is approximately revenue-optimal (up to a constant factor) when values are drawn from a regular distribution. We then study settings in which the good is only "locally" public: agents are arranged in a network and share benefits only with their neighbors. We describe a pricing method that approximately maximizes revenue, in the worst case over equilibria of agent behavior, for any $d$-regular network. Finally, we show that approximately optimal prices can be found for general networks in the special case that private values are drawn from a uniform distribution. We also discuss some barriers to extending these results to general networks and regular distributions.

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arXiv2013

Efficiency Guarantees in Auctions with Budgets

In settings where players have a limited access to liquidity, represented in the form of budget constraints, efficiency maximization has proven to be a challenging goal. In particular, the social welfare cannot be approximated by a better factor then the number of players. Therefore, the literature has mainly resorted to Pareto-efficiency as a way to achieve efficiency in such settings. While successful in some important scenarios, in many settings it is known that either exactly one incentive-compatible auction that always outputs a Pareto-efficient solution, or that no truthful mechanism can always guarantee a Pareto-efficient outcome. Traditionally, impossibility results can be avoided by considering approximations. However, Pareto-efficiency is a binary property (is either satisfied or not), which does not allow for approximations. In this paper we propose a new notion of efficiency, called \emph{liquid welfare}. This is the maximum amount of revenue an omniscient seller would be able to extract from a certain instance. We explain the intuition behind this objective function and show that it can be 2-approximated by two different auctions. Moreover, we show that no truthful algorithm can guarantee an approximation factor better than 4/3 with respect to the liquid welfare, and provide a truthful auction that attains this bound in a special case. Importantly, the liquid welfare benchmark also overcomes impossibilities for some settings. While it is impossible to design Pareto-efficient auctions for multi-unit auctions where players have decreasing marginal values, we give a deterministic $O(\log n)$-approximation for the liquid welfare in this setting.

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arXiv2012

Clinching Auctions with Online Supply

Auctions for perishable goods such as internet ad inventory need to make real-time allocation and pricing decisions as the supply of the good arrives in an online manner, without knowing the entire supply in advance. These allocation and pricing decisions get complicated when buyers have some global constraints. In this work, we consider a multi-unit model where buyers have global {\em budget} constraints, and the supply arrives in an online manner. Our main contribution is to show that for this setting there is an individually-rational, incentive-compatible and Pareto-optimal auction that allocates these units and calculates prices on the fly, without knowledge of the total supply. We do so by showing that the Adaptive Clinching Auction satisfies a {\em supply-monotonicity} property. We also analyze and discuss, using examples, how the insights gained by the allocation and payment rule can be applied to design better ad allocation heuristics in practice. Finally, while our main technical result concerns multi-unit supply, we propose a formal model of online supply that captures scenarios beyond multi-unit supply and has applications to sponsored search. We conjecture that our results for multi-unit auctions can be extended to these more general models.

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arXiv2012

Polyhedral Clinching Auctions and the Adwords Polytope

A central issue in applying auction theory in practice is the problem of dealing with budget-constrained agents. A desirable goal in practice is to design incentive compatible, individually rational, and Pareto optimal auctions while respecting the budget constraints. Achieving this goal is particularly challenging in the presence of nontrivial combinatorial constraints over the set of feasible allocations. Toward this goal and motivated by AdWords auctions, we present an auction for {\em polymatroidal} environments satisfying the above properties. Our auction employs a novel clinching technique with a clean geometric description and only needs an oracle access to the submodular function defining the polymatroid. As a result, this auction not only simplifies and generalizes all previous results, it applies to several new applications including AdWords Auctions, bandwidth markets, and video on demand. In particular, our characterization of the AdWords auction as polymatroidal constraints might be of independent interest. This allows us to design the first mechanism for Ad Auctions taking into account simultaneously budgets, multiple keywords and multiple slots. We show that it is impossible to extend this result to generic polyhedral constraints. This also implies an impossibility result for multi-unit auctions with decreasing marginal utilities in the presence of budget constraints.

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arXiv2012

Signaling Schemes for Revenue Maximization

Signaling is an important topic in the study of asymmetric information in economic settings. In particular, the transparency of information available to a seller in an auction setting is a question of major interest. We introduce the study of signaling when conducting a second price auction of a probabilistic good whose actual instantiation is known to the auctioneer but not to the bidders. This framework can be used to model impressions selling in display advertising. We study the problem of computing a signaling scheme that maximizes the auctioneer's revenue in a Bayesian setting. While the general case is proved to be computationally hard, several cases of interest are shown to be polynomially solvable. In addition, we establish a tight bound on the minimum number of signals required to implement an optimal signaling scheme and show that at least half of the maximum social welfare can be preserved within such a scheme.

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arXiv2012

Optimal Mechanisms for Selling Information

The buying and selling of information is taking place at a scale unprecedented in the history of commerce, thanks to the formation of online marketplaces for user data. Data providing agencies sell user information to advertisers to allow them to match ads to viewers more effectively. In this paper we study the design of optimal mechanisms for a monopolistic data provider to sell information to a buyer, in a model where both parties have (possibly correlated) private signals about a state of the world, and the buyer uses information learned from the seller, along with his own signal, to choose an action (e.g., displaying an ad) whose payoff depends on the state of the world. We provide sufficient conditions under which there is a simple one-round protocol (i.e. a protocol where the buyer and seller each sends a single message, and there is a single money transfer) achieving optimal revenue. In these cases we present a polynomial-time algorithm that computes the optimal mechanism. Intriguingly, we show that multiple rounds of partial information disclosure (interleaved by payment to the seller) are sometimes necessary to achieve optimal revenue if the buyer is allowed to abort his interaction with the seller prematurely. We also prove some negative results about the inability of simple mechanisms for selling information to approximate more complicated ones in the worst case.

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arXiv2011

Sequential Auctions and Externalities

In many settings agents participate in multiple different auctions that are not necessarily implemented simultaneously. Future opportunities affect strategic considerations of the players in each auction, introducing externalities. Motivated by this consideration, we study a setting of a market of buyers and sellers, where each seller holds one item, bidders have combinatorial valuations and sellers hold item auctions sequentially. Our results are qualitatively different from those of simultaneous auctions, proving that simultaneity is a crucial aspect of previous work. We prove that if sellers hold sequential first price auctions then for unit-demand bidders (matching market) every subgame perfect equilibrium achieves at least half of the optimal social welfare, while for submodular bidders or when second price auctions are used, the social welfare can be arbitrarily worse than the optimal. We also show that a first price sequential auction for buying or selling a base of a matroid is always efficient, and implements the VCG outcome. An important tool in our analysis is studying first and second price auctions with externalities (bidders have valuations for each possible winner outcome), which can be of independent interest. We show that a Pure Nash Equilibrium always exists in a first price auction with externalities.

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arXiv2010

Improved Social Welfare Bounds for GSP at Equilibrium

The Generalized Second Price auction is the primary method by which sponsered search advertisements are sold. We study the performance of this auction under various equilibrium concepts. In particular, we demonstrate that the Bayesian Price of Anarchy is at most $2(1-1/e)^{-1} \approx 3.16$, significantly improving upon previously known bounds. Our techniques are intuitively straightforward and extend in a number of ways. For one, our result extends to a bound on the performance of GSP at coarse correlated equilibria, which captures (for example) a repeated-auction setting in which agents apply regret-minimizing bidding strategies. In addition, our analysis is robust against the presence of byzantine agents who cannot be assumed to participate rationally. Additionally, we present tight bounds for the social welfare obtained at pure NE for the special case of an auction for 3 slots, and discuss potential methods for extending this analysis to an arbitrary number of slots.

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