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48
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
Bayesian Algorithmic Mechanism Design
, 2010
"... The principal problem in algorithmic mechanism design is in merging the incentive constraints imposed by selfish behavior with the algorithmic constraints imposed by computational intractability. This field is motivated by the observation that the preeminent approach for designing incentive compatib ..."
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Cited by 41 (11 self)
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The principal problem in algorithmic mechanism design is in merging the incentive constraints imposed by selfish behavior with the algorithmic constraints imposed by computational intractability. This field is motivated by the observation that the preeminent approach for designing incentive compatible mechanisms, namely that of Vickrey, Clarke, and Groves; and the central approach for circumventing computational obstacles, that of approximation algorithms, are fundamentally incompatible: natural applications of the VCG approach to an approximation algorithm fails to yield an incentive compatible mechanism. We consider relaxing the desideratum of (ex post) incentive compatibility (IC) to Bayesian incentive compatibility (BIC), where truthtelling is a BayesNash equilibrium (the standard notion of incentive compatibility in economics). For welfare maximization in singleparameter agent settings, we give a general blackbox reduction that turns any approximation algorithm into a Bayesian incentive compatible mechanism with essentially the same1 approximation factor.
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
Price of Anarchy for Greedy Auctions
"... We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be singleminded. This class of problems includes combinatorial auctions, multiunit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximat ..."
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Cited by 30 (9 self)
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We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be singleminded. This class of problems includes combinatorial auctions, multiunit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximately optimize social welfare at every BayesNash equilibrium (BNE), which is the standard notion of equilibrium in settings of incomplete information. For a broad class of greedy approximation algorithms, we give a general blackbox reduction to deterministic mechanisms with almost no loss to the approximation ratio at any BNE. We also consider the special case of Nash equilibria in fullinformation games, where we obtain tightened results. This solution concept is closely related to the wellstudied price of anarchy. Furthermore, for a rich subclass of allocation problems, pure Nash equilibria are guaranteed to exist for our mechanisms. For many problems, the approximation factors we obtain at equilibrium improve upon the best known results for deterministic truthful mechanisms. In particular, we exhibit a simple deterministic mechanism for general combinatorial auctions that obtains an O(√m) approximation at every BNE.
Pure and BayesNash Price of Anarchy for Generalized Second Price Auction
"... Generalized Second Price Auction, also knows as Ad Word auctions, and its variants has been the main mechanism used by search companies to auction positions for sponsored search links. In this paper we study the social welfare of the Nash equilibria of this game. It is known that socially optimal Na ..."
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Cited by 30 (4 self)
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Generalized Second Price Auction, also knows as Ad Word auctions, and its variants has been the main mechanism used by search companies to auction positions for sponsored search links. In this paper we study the social welfare of the Nash equilibria of this game. It is known that socially optimal Nash equilibria exists (i.e., that the Price of Stability for this game is 1). This paper is the first to prove bounds on the price of anarchy. Our main result is to show that under some mild assumptions the price of anarchy is small. For pure Nash equilibria we bound the price of anarchy by 1.618, assuming all bidders are playing undominated strategies. For mixed Nash equilibria we prove a bound of 4 under the same assumption. We also extend the result to the Bayesian setting when bidders valuations are also random, and prove a bound of 8 for this case. Our proof exhibits a combinatorial structure of Nash equilibria and use 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.
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.
GSP auctions with correlated types
 In Proceedings of the 12th Annual ACM Conference on Electronic Commerce (EC
, 2011
"... 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, espec ..."
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Cited by 22 (5 self)
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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, especiallyasaresultofuncertaintyinherentinthebehaviour of the underlying ad allocation algorithm. We demonstrate that the Bayesian Price of Anarchy of the GSP auction is bounded by 4, 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 performanceofGSPatcoarsecorrelatedequilibria, whichcaptures (for example) a repeatedauction setting in which agents apply regretminimizing 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.
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
Conditional Equilibrium Outcomes via Ascending Price Processes
"... A Walrasian equilibrium in an economy with nonidentical indivisible items exists only for small classes of players’ valuations (mostly “gross substitutes” valuations), and may not generally exist even with decreasing marginal values. This paper studies a relaxed notion, “conditional equilibrium”, t ..."
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Cited by 13 (4 self)
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A Walrasian equilibrium in an economy with nonidentical indivisible items exists only for small classes of players’ valuations (mostly “gross substitutes” valuations), and may not generally exist even with decreasing marginal values. This paper studies a relaxed notion, “conditional equilibrium”, that requires individual rationality and “outward stability”, i.e., a player will not want to add items to her allocation, at given prices. While a Walrasian equilibrium outcome is unconditionally stable, a conditional equilibrium outcome is stable if players cannot choose to drop only some of their allocated items. With decreasing marginal valuations, conditional equilibrium outcomes exhibit three appealing properties: (1) An approximate version of the first welfare theorem, namely that the social welfare in any conditional equilibrium is at least half of the maximal welfare; (2) A conditional equilibrium outcome can always be obtained via a natural ascendingprice process; and (3) The second welfare theorem holds: any welfare maximizing allocation is supported by a conditional equilibrium. In particular, each of the last two properties independently implies that a conditional equilibrium always exists with decreasing marginal valuations (whereas a Walrasian