Renato Paes Leme

Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction

The Generalized Second Price Auction has been the main mechanism used by search companies to auction positions for advertisements on search pages. In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full information setting, socially optimal Nash equilibria are known to exist (i.e., the Price of Stability is 1). This paper is the first to prove bounds on the price of anarchy, and to give any bounds in the Bayesian setting. Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated strategies. In the full information setting we prove a bound of 1.618 for the price of anarchy for pure Nash equilibria, and a bound of 4 for mixed Nash equilibria. We also prove a bound of 8 for the price of anarchy in the Bayesian setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the distributions used are common knowledge. Our proof exhibits a combinatorial structure of Nash equilibria and uses this structure to bound the price of anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to the Bayesian setting requires the use of novel combinatorial techniques that can be of independent interest.

GSP auctions with correlated types

The Generalized Second Price (GSP) auction is the primary method by which sponsered search advertisements are sold. We study the performance of this auction in the Bayesian setting for players with correlated types. Correlation arises very naturally in the context of sponsored search auctions, especially as a result of uncertainty inherent in the behaviour of the underlying ad allocation algorithm. We demonstrate that the Bayesian Price of Anarchy of the GSP auction is bounded by

On revenue in the generalized second price auction

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Bounding the inefficiency of outcomes in generalized second price auctions

, even when agents have arbitrarily correlated types. Our proof highlights a connection between the GSP mechanism and the concept of smoothness in games, which may be of independent interest. For the special case of uncorrelated (i.e. independent) agent types, we improve our bound to 2(1-1/e)-1 ≅ 3.16, significantly improving upon previously known bounds. Using our techniques, we obtain the same 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. Moreoever, our analysis is robust against the presence of irrational bidders and settings of asymmetric information, and our bounds degrade gracefully when agents apply strategies that form only an approximate equilibrium.

The curse of simultaneity

The Generalized Second Price (GSP) auction is the primary auction used for selling sponsored search advertisements. In this paper we consider the revenue of this auction at equilibrium. We prove that if agent values are drawn from identical regular distributions, then the GSP auction paired with an appropriate reserve price generates a constant fraction (1/6th) of the optimal revenue. In the full-information game, we show that at any Nash equilibrium of the GSP auction obtains at least half of the revenue of the VCG mechanism excluding the payment of a single participant. This bound holds also with any reserve price, and is tight. Finally, we consider the tradeoff between maximizing revenue and social welfare. We introduce a natural convexity assumption on the click-through rates and show that it implies that the revenue-maximizing equilibrium of GSP in the full information model will necessarily be envy-free. In particular, it is always possible to maximize revenue and social welfare simultaneously when click-through rates are convex. Without this convexity assumption, however, we demonstrate that revenue may be maximized at a non-envy-free equilibrium that generates a socially inefficient allocation.

Gross substitutability: An algorithmic survey

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. Edelman et al. (2007) [11] and Varian (2007) [36] show that an efficient equilibrium always exists in the full information setting. Their results, however, do not extend to the case with uncertainty, where efficient equilibria might not exist.

On the efficiency of the walrasian mechanism

Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we propose studying the sequential price of anarchy: the quality of outcomes of sequential versions of games whose simultaneous counterparts are prototypical in algorithmic game theory. We study different classes of games with high price of anarchy, and show that the subgame perfect equilibrium of their sequential version is a much more natural prediction, ruling out unreasonable equilibria, and leading to much better quality solutions. We consider three examples of such games: Cost Sharing Games, Unrelated Machine Scheduling Games and Consensus Games. For Machine Cost Sharing Games, the sequential price of anarchy is at most O(log(n)), an exponential improvement of the O(n) price of anarchy of their simultaneous counterparts. Further, the subgame perfect equilibrium can be computed by a polynomial time greedy algorithm, and is independent of the order the players arrive. For Unrelated Machine Scheduling Games we show that the sequential price of anarchy is bounded as a function of the number of jobs n and machines m (by at most O(m2n)), while in the simultaneous version the price of anarchy is unbounded even for two players and two machines. For Consensus Games we observe that the optimal outcome for generic weights is the unique equilibrium that arises in the sequential game. We also study the related Cut Games, where we show that the sequential price of anarchy is at most 4. In addition we study the complexity of finding the subgame perfect equilibrium outcome in these games.

Optimal mechanisms for selling information

Abstract The concept of gross substitute valuations was introduced by Kelso and Crawford as a sufficient conditions for the existence of Walrasian equilibria in economies with indivisible goods. The proof is algorithmic in nature: gross substitutes is exactly the condition that enables a natural price adjustment procedure – known as Walrasian tatonnement – to converge to equilibrium. The same concept was also introduced independently in other communities with different names: M ♮ -concave functions (Murota and Shioura), Matroidal and Well-Layered maps (Dress and Terhalle) and valuated matroids (Dress and Wenzel). Here we survey various definitions of gross substitutability and show their equivalence. We focus on algorithmic aspects of the various definitions. In particular, we highlight that gross substitutes are the exact class of valuations for which demand oracles can be computed via an ascending greedy algorithm. It also corresponds to a natural discrete analogue of concave functions: local maximizers correspond to global maximizers.

Efficiency Guarantees in Auctions with Budgets

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 Walrasian Mechanism that proceeds by collecting reported demands, finding clearing prices in the reported market via an ascending price tatonnement 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. We extend our results in several directions. First, our results extend to Bayes-Nash equilibria and outcomes of no regret learning via the smooth mechanism framework. We also extend our bounds to any mechanism that maximizes welfare with respect to the declared valuations and never charges agents more than their bids. Additionally, we consider other classes of valuations and bid spaces beyond those satisfying the gross substitutes conditions. Finally, we relax the no-overbidding assumption, and present bounds that are parameterized by the extent to which agents are willing to overbid.

Feature-based Dynamic Pricing

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.