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101
Routing without regret: On convergence to nash equilibria of regretminimizing algorithms in routing games
 In PODC
, 2006
"... Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging envi ..."
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Cited by 58 (7 self)
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Abstract There has been substantial work developing simple, efficient noregret algorithms for a wideclass of repeated decisionmaking problems including online routing. These are adaptive strategies an individual can use that give strong guarantees on performance even in adversariallychanging environments. There has also been substantial work on analyzing properties of Nash equilibria in routing games. In this paper, we consider the question: if each player in a routing game uses a noregret strategy, will behavior converge to a Nash equilibrium? In general games the answer to this question is known to be no in a strong sense, but routing games havesubstantially more structure. In this paper we show that in the Wardrop setting of multicommodity flow and infinitesimalagents, behavior will approach Nash equilibrium (formally, on most days, the cost of the flow will be close to the cost of the cheapest paths possible given that flow) at a rate that dependspolynomially on the players ' regret bounds and the maximum slope of any latency function. We also show that priceofanarchy results may be applied to these approximate equilibria, and alsoconsider the finitesize (noninfinitesimal) loadbalancing model of Azar [2].
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
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.
Multiplicative Updates Outperform Generic NoRegret . . .
, 2009
"... We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the wellknown multiplicativeweights learning algorithm results in convergence to ..."
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Cited by 29 (8 self)
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We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the wellknown multiplicativeweights learning algorithm results in convergence to pure equilibria. Our results show that natural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the loadbalancing game introduced by Koutsoupias and Papadimitriou, which has superconstant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined gametheoretically, but we show that this property holds whenever a stability criterion from the theory of dynamical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribution). We further extend our results to show that players can use algorithms with different (sufficiently small) learning rates, i.e. they can trade off convergence speed and long term average regret differently.
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.
Mechanism design in large games: Incentives and privacy. arXiv preprint arXiv:1207.4084,
, 2013
"... ABSTRACT We study the problem of implementing equilibria of complete information games in settings of incomplete information, and address this problem using "recommender mechanisms." A recommender mechanism is one that does not have the power to enforce outcomes or to force participation, ..."
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Cited by 20 (12 self)
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ABSTRACT We study the problem of implementing equilibria of complete information games in settings of incomplete information, and address this problem using "recommender mechanisms." A recommender mechanism is one that does not have the power to enforce outcomes or to force participation, rather it only has the power to suggestion outcomes on the basis of voluntary participation. We show that despite these restrictions, recommender mechanisms can implement equilibria of complete information games in settings of incomplete information under the condition that the game is largei.e. that there are a large number of players, and any player's action affects any other's payoff by at most a small amount. Our result follows from a novel application of differential privacy. We show that any algorithm that computes a correlated equilibrium of a complete information game while satisfying a variant of differential privacywhich we call joint differential privacycan be used as a recommender mechanism while satisfying our desired incentive properties. Our main technical result is an algorithm for computing a * We gratefully acknowledge the support of NSF Grant CCF1101389. We thank Nabil AlNajjar, Eduardo Azevdeo, Eric Budish, Tymofiy Mylovanov, Andy Postlewaite, Al Roth and Tim Roughgarden for helpful comments and discussions. correlated equilibrium of a large game while satisfying joint differential privacy. Although our recommender mechanisms are designed to satisfy gametheoretic properties, our solution ends up satisfying a strong privacy property as well. No group of players can learn "much" about the type of any player outside the group from the recommendations of the mechanism, even if these players collude in an arbitrary way. As such, our algorithm is able to implement equilibria of complete information games, without revealing information about the realized types.
Distributed Welfare Games
"... We consider a variation of the resource allocation problem. In the traditional problem, there is a global planner who would like to assign a set of players to a set of resources so as to maximize welfare. We consider the situation where the global planner does not have the authority to assign player ..."
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Cited by 20 (7 self)
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We consider a variation of the resource allocation problem. In the traditional problem, there is a global planner who would like to assign a set of players to a set of resources so as to maximize welfare. We consider the situation where the global planner does not have the authority to assign players to resources; rather, players are selfinterested. The question that emerges is how can the global planner entice the players to settle on a desirable allocation with respect to the global welfare? To study this question, we focus on a class of games that we refer to as distributed welfare games. Within this context, we investigate how the global planner should distribute the welfare to the players. We measure the efficacy of a distribution rule in two ways: (i) Does a pure Nash equilibrium exist? (ii) How does the welfare associated with a pure Nash equilibrium compare to the global welfare associated with the optimal allocation? In this paper we explore the applicability of cost sharing methodologies for distributing welfare in such resource allocation problems. We demonstrate that obtaining desirable distribution rules, such as distribution rules that are budget balanced and guarantee the existence of a pure Nash equilibrium, often comes at a significant informational and computational cost. In light of this, we derive a systematic procedure for designing desirable distribution rules with a minimal informational and computational cost for a special class of distributed welfare games. Furthermore, we derive a bound on the price of anarchy for distributed welfare games in a variety of settings. Lastly, we highlight the implications of these results using the problem of sensor coverage.