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22
Intrinsic Robustness of the Price of Anarchy
 STOC'09
, 2009
"... The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium ..."
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Cited by 101 (12 self)
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The price of anarchy (POA) is a worstcase measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regretminimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a “canonical sufficient condition ” for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regretminimizing players (or “price of total anarchy”). Smoothness arguments also have automatic implications for the inefficiency of approximate and BayesianNash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomiallength bestresponse sequences. We also identify classes of games — most notably, congestion games with cost functions restricted to an arbitrary fixed set — that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worstcase upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with nonpolynomial cost functions, and the first
Welfare Guarantees for Combinatorial Auctions with Item Bidding
, 2010
"... We analyze the price of anarchy (POA) in a simple and practical nontruthful combinatorial auction when players have subadditive valuations for goods. We study the mechanism that sells every good in parallel with separate secondprice auctions. We first prove that under a standard “no overbidding ” ..."
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Cited by 39 (5 self)
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We analyze the price of anarchy (POA) in a simple and practical nontruthful combinatorial auction when players have subadditive valuations for goods. We study the mechanism that sells every good in parallel with separate secondprice auctions. We first prove that under a standard “no overbidding ” assumption, for every subadditive valuation profile, every pure Nash equilibrium has welfare at least 50 % of optimal — i.e., the POA is at most 2. For the incomplete information setting, we prove that the POA with respect to BayesNash equilibria is strictly larger than 2 — an unusual separation from the fullinformation model — and is at most 2 ln m, where m is the number of goods.
On the efficiency of equilibria in generalized second price auctions
 In EC’11
, 2011
"... The Generalized Second Price (GSP) auction is the primary auction used for monetizing the use of the Internet. It is wellknown that truthtelling is not a dominant strategy in this auction and that inefficient equilibria can arise. Edelman et al. and Varian show that an efficient equilibrium always ..."
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Cited by 35 (1 self)
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The Generalized Second Price (GSP) auction is the primary auction used for monetizing the use of the Internet. It is wellknown that truthtelling is not a dominant strategy in this auction and that inefficient equilibria can arise. Edelman et al. and Varian 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. 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 (price of anarchy) of the
The Price of Anarchy in Games of Incomplete Information
 EC'12
, 2012
"... We define smooth games of incomplete information. We prove an “extension theorem” for such games: price of anarchy bounds for pure Nash equilibria for all induced fullinformation games extend automatically, without quantitative degradation, to all mixedstrategy BayesNash equilibria with respect t ..."
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Cited by 25 (2 self)
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We define smooth games of incomplete information. We prove an “extension theorem” for such games: price of anarchy bounds for pure Nash equilibria for all induced fullinformation games extend automatically, without quantitative degradation, to all mixedstrategy BayesNash equilibria with respect to a product prior distribution over players’ preferences. We also note that, for BayesNash equilibria in games with correlated player preferences, there is no general extension theorem for smooth games. We give several applications of our definition and extension theorem. First, we show that many games of incomplete information for which the price of anarchy has been studied are smooth in our sense. Thus our extension theorem unifies much of the known work on the price of anarchy in games of incomplete information. Second, we use our extension theorem to prove new bounds on the price of anarchy of BayesNash equilibria in congestion games with incomplete information.
Simultaneous Auctions are (almost) Efficient
, 2012
"... Simultaneous item auctions are simple procedures for allocating items to bidders with potentially complex preferences over different item sets. In a simultaneous auction, every bidder submits bids on all items simultaneously. The allocation and prices are then resolved for each item separately, base ..."
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Cited by 21 (5 self)
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Simultaneous item auctions are simple procedures for allocating items to bidders with potentially complex preferences over different item sets. In a simultaneous auction, every bidder submits bids on all items simultaneously. The allocation and prices are then resolved for each item separately, based solely on the bids submitted on that item. Such procedures occur in practice (e.g. eBay) but are not truthful. We study the efficiency of Bayesian Nash equilibrium (BNE) outcomes of simultaneous first and secondprice auctions when bidders have complementfree (a.k.a. subadditive) valuations. We show that the expected social welfare of any BNE is at least 1 2 of the optimal social welfare in the case of firstprice auctions, and at least 1 4 in the case of secondprice auctions. These results improve upon the previouslyknown logarithmic bounds, which wereestablished by Hassidim et al. (2011) for firstpriceauctions and by Bhawalkar and Roughgarden (2011) for secondprice auctions. 1
On revenue in the generalized second price auction.
 In WWW.
, 2012
"... ABSTRACT 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. Most previous work of GSP focuses on envy free equilibria of the full information version of this game. Envyfree equilibri ..."
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Cited by 15 (0 self)
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ABSTRACT 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. Most previous work of GSP focuses on envy free equilibria of the full information version of this game. Envyfree equilibria are known to obtain at least the revenue of the VCG auction. Here we consider revenue in equilibria that are not envyfree, and also consider revenue in the Bayesian version of the game. We show that, at equilibrium, the GSP auction obtains at least half of the revenue of the VCG mechanism excluding the payment of a single participant. This bound is tight, and we give examples demonstrating that GSP cannot approximate the full revenue of the VCG mechanism either in the full information game, or in the Bayesian version (even if agent values are independently drawn from identical uniform distributions). We also show that the GSP revenue approximates the VCG revenue in the Bayesian game when the clickthrough rates are well separated. We also consider revenuemaximizing equilibrium of GSP in the full information model. We show that if clickthrough rates satisfy a natural convexity assumption, then the revenuemaximizing equilibrium will necessarily be envyfree. In particular, it is possible to maximize revenue and social welfare simultaneously. On the other hand, without this convexity assumption, we demonstrate that revenue may be maximized at a nonenvyfree equilibrium that generates a socially inefficient allocation.
Auctions with Unique Equilibria
, 2013
"... We study BayesNash equilibria in a large class of anonymous orderbased auctions. These include the generalized firstprice auction for allocating positions to bidders, e.g., for sponsored search. We show that when bidders ’ values are independent and identically distributed there is a unique Bayes ..."
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Cited by 7 (2 self)
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We study BayesNash equilibria in a large class of anonymous orderbased auctions. These include the generalized firstprice auction for allocating positions to bidders, e.g., for sponsored search. We show that when bidders ’ values are independent and identically distributed there is a unique Bayes Nash equilibrium; This equilibrium is symmetric and efficient. Importantly, our proof is simple and structurally revealing. This uniqueness result for the generalized firstprice auction is in stark contrast to the generalized secondprice auction where there may be no efficient equilibrium. This result suggests, e.g., that firstprice payment semantics may have advantages over secondprice payment semantics. Our results extend also to certain models of risk aversion.
Sponsored Search Auctions: An Overview of Research with emphasis on Game Theoretic Aspects
, 2010
"... We provide an overview of recent research that has been conducted on the design of sponsored search auctions. We mainly focus on game theoretic and mechanism design aspects of these auctions, and we analyze the issues associated with each of the three participating entities, i.e. the search engine, ..."
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Cited by 7 (3 self)
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We provide an overview of recent research that has been conducted on the design of sponsored search auctions. We mainly focus on game theoretic and mechanism design aspects of these auctions, and we analyze the issues associated with each of the three participating entities, i.e. the search engine, the advertisers, and the users of the search engine, as well as their resulting behavior. Regarding the search engine, we overview the various mechanisms that have been proposed including the currently used GSP mechanism. The issues that are addressed include analysis of Nash equilibria and their performance, design of alternative mechanisms and aspects of competition among search engines. We then move on to the advertisers and discuss the problem of choosing a bidding strategy, given the mechanism of the search engine. Following this, we consider the end users and we examine how user behavior may create externalities and influence the performance of the advertisers. Finally, we also overview statistical methods for estimating modeling parameters that are of interest to the three entities. In each section, we point out interesting open problems and directions for future research.