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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
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 the efficiency of the Walrasian mechanism
 In Proceedings of the 15th ACM Conference on Economics and Computation
, 2014
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Inefficiency of Games with Social Context
 In Proc. 6th International Symposium on Algorithmic Game Theory
, 2013
"... Abstract. The study of otherregarding player behavior such as altruism and spite in games has recently received quite some attention in the algorithmic game theory literature. Already for very simple models, it has been shown that altruistic behavior can actually be harmful for society in the sens ..."
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Cited by 2 (1 self)
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Abstract. The study of otherregarding player behavior such as altruism and spite in games has recently received quite some attention in the algorithmic game theory literature. Already for very simple models, it has been shown that altruistic behavior can actually be harmful for society in the sense that the price of anarchy may increase as the players become more altruistic. In this paper, we study the severity of this phenomenon for more realistic settings in which there is a complex underlying social structure, causing the players to direct their altruistic and spiteful behavior in a refined playerspecific sense (depending, for example, on friendships that exist among the players). Our findings show that the increase in the price of anarchy is modest for congestion games and minsum scheduling games, whereas it is drastic for generalized second price auctions. 1
Simultaneous auctions are (almost) efficient (Extended Abstract)
"... Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent bids on all items simultaneously. The allocation and prices are then resolved for each item separately, bas ..."
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Simultaneous item auctions are simple and practical procedures for allocating items to bidders with potentially complex preferences. In a simultaneous auction, every bidder submits independent 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. 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. While it is known that the social welfare of every pure Nash equilibrium (NE) constitutes a constant fraction of the optimal social welfare, a pure NE rarely exists, and moreover, the full information assumption is often unrealistic. Therefore, quantifying the welfare loss in Bayesian Nash equilibria is of particular interest. Previous work established a logarithmic bound on the ratio between the social welfare of a BNE and the expected optimal social welfare in both firstprice auctions (Hassidim et al. [11]) and secondprice auctions (Bhawalkar and Roughgarden [2]), leaving a large gap between a constant and a logarithmic ratio. We introduce a new proof technique and use it to resolve both of these gaps in a unified way. Specifically, 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.
Draft Auctions
, 2014
"... We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are ..."
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We introduce draft auctions, which is a sequential auction format where at each iteration players bid for the right to buy items at a fixed price. We show that draft auctions offer an exponential improvement in social welfare at equilibrium over sequential item auctions where predetermined items are auctioned at each time step. Specifically, we show that for any subadditive valuation the social welfare at equilibrium is an O(log²(m))approximation to the optimal social welfare, where m is the number of items. We also provide tighter approximation results for several subclasses. Our welfare guarantees hold for BayesNash equilibria and for noregret learning outcomes, via the smoothmechanism framework. Of independent interest, our techniques show that in a combinatorial auction setting, efficiency guarantees of a mechanism via smoothness for a very restricted class of cardinality valuations, extend with a small degradation, to subadditive valuations, the largest complementfree class of valuations. Variants of draft auctions have been used in practice and have been experimentally shown to outperform other auctions. Our results provide a theoretical justification.
On the Inefficiency of the Uniform Price Auction
, 2013
"... We present our results on Uniform Price Auctions, one of the standard sealedbid multiunit auction formats, for selling multiple identical units of a single good to multidemand bidders. Contrary to the truthful and economically efficient multiunit Vickrey auction, the Uniform Price Auction encour ..."
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We present our results on Uniform Price Auctions, one of the standard sealedbid multiunit auction formats, for selling multiple identical units of a single good to multidemand bidders. Contrary to the truthful and economically efficient multiunit Vickrey auction, the Uniform Price Auction encourages strategic bidding and is socially inefficient in general, partly because of a ”Demand Reduction ” effect; bidders tend to bid for fewer (identical) units, so as to receive them at a lower uniform price. Despite its inefficiency, the uniform pricing rule is widely popular by its appeal to the natural anticipation, that identical items should be identically priced. Application domains of its variants include sales of U.S. Treasury notes to investors, trade exchanges over the internet facilitated by popular online brokers, allocation of radio spectrum licenses etc. In this work we study equilibria of the Uniform Price Auction for bidders with (symmetric) submodular valuation functions, over the number of units that they win. We investigate pure Nash equilibria of the auction in undominated strategies; we produce a characterization of these equilibria that allows us to prove that a fraction 1 − e−1 of the optimum social welfare is always recovered in undominated pure Nash equilibrium – and this bound is essentially tight. Subsequently, we study the auction under the incomplete information setting and prove a bound of 4 − 2 k on the economic inefficiency of (mixed) Bayes Nash equilibria that are supported by undominated strategies.
Equilibrium in Combinatorial Public Projects
"... Abstract. We study simple item bidding mechanisms for the combinatorial public project problem and explore their eciency guarantees in various wellknown solution concepts. We rst study sequential mechanisms where each agent, in sequence, reports her bid for every item in a prede ned order on the ..."
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Abstract. We study simple item bidding mechanisms for the combinatorial public project problem and explore their eciency guarantees in various wellknown solution concepts. We rst study sequential mechanisms where each agent, in sequence, reports her bid for every item in a prede ned order on the agents determined by the mechanism. We show that if agents ' valuations are unitdemand any subgame perfect equilibrium of a sequential mechanism achieves the optimal social welfare. For the simultaneous bidding equivalent of the above auction we show that for any class of bidder valuations, all Strong Nash Equilibria achieve at least a O(logn) factor of the optimal social welfare. For Pure Nash Equilibria we show that the worstcase loss in eciency is proportional to the number of agents. For public projects in which only one item is selected we show constructively that there always exists a Pure Nash Equilibrium that guarantees at least 1 2 (1 1 n) of the optimum. We also show eciency bounds for Correlated Equilibria and BayesNash Equilibria, via the recent smooth mechanism framework [26]. 1