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20
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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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.
The robust price of anarchy of altruistic games
 In Proc. 7th Workshop on Internet and Network Economics (WINE
, 2011
"... We study the inefficiency of equilibria for several classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i’s perceived cost is a convex combination of 1−αi times his direct cost and αi times the social cost. Tuning the parameters αi allows sm ..."
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Cited by 14 (3 self)
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We study the inefficiency of equilibria for several classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i’s perceived cost is a convex combination of 1−αi times his direct cost and αi times the social cost. Tuning the parameters αi allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of costsharing games, utility games, and linear congestion games. Our main contribution is an adaptation of Roughgarden’s smoothness notion to altruistic extensions of games. We show that this extension captures the essential properties to determine the robust price of anarchy of these games, and use it to derive mostly tight bounds. For congestion games and costsharing games, the worstcase robust price of anarchy increases with increasing altruism, while for utility games, it remains constant and is not affected by altruism. However, the increase in the price of anarchy is not a universal phenomenon: for symmetric singleton linear congestion games, the pure price of anarchy decreases both under increasing uniform altruism and as the fraction of entirely altruistic individuals increases.
Coevolutionary opinion formation games.
 In STOC,
, 2013
"... ABSTRACT We present gametheoretic models of opinion formation in social networks where opinions themselves coevolve with friendships. In these models, nodes form their opinions by maximizing agreements with friends weighted by the strength of the relationships, which in turn depend on difference ..."
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ABSTRACT We present gametheoretic models of opinion formation in social networks where opinions themselves coevolve with friendships. In these models, nodes form their opinions by maximizing agreements with friends weighted by the strength of the relationships, which in turn depend on difference in opinion with the respective friends. We define a social cost of this process by generalizing recent work of Bindel et al., FOCS 2011. We tightly bound the price of anarchy of the resulting dynamics via local smoothness arguments, and characterize it as a function of how much nodes value their own (intrinsic) opinion, as well as how strongly they weigh links to friends with whom they agree more.
N.K.: Smooth inequalities and equilibrium inefficiency in scheduling games
, 2012
"... We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. In an egalitarian objective, the social cost would be the maximal job completion ..."
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We study coordination mechanisms for Scheduling Games (with unrelated machines). In these games, each job represents a player, who needs to choose a machine for its execution, and intends to complete earliest possible. In an egalitarian objective, the social cost would be the maximal job completion time, i.e. the makespan of the schedule. In an utilitarian objective, the social cost would be the average completion time. Instead of studying one of those objectives, we focus on the more general class of `knorm (for some parameter k) on job completion times as social cost. This permits to balance overall quality of service and fairness. In this setting, a coordination mechanism is a fixed policy, which specifies how jobs assigned to a same machine will be scheduled. This policy is known to the players and influences therefore their behavior. Our goal is to design scheduling policies that always admit a pure Nash equilibrium and guarantee a small price of anarchy for the `knorm social cost. We consider policies with different amount of knowledge about jobs: nonclairvoyant (not depending on the job processing times), stronglylocal (where the schedule of machine i depends only on processing times for this machine i and jobs j assigned to i) and local (where the schedule of machine i depends only on processing
Barriers to nearoptimal equilibria
 IN PROCEEDINGS OF THE 55TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS
"... This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worstcase quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each moti ..."
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This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worstcase quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games; welfaremaximization in combinatorial auctions with “blackbox” bidder valuations; and welfaremaximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worstcase price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players’ utility functions. For this reason, our lower bounds are particularly significant for problems of game design — ranging from the design of simple combinatorial auctions to the existence of effective tolls for routing networks — where the goal is to design a game that has only nearoptimal equilibria. For example, our results imply that the simultaneous firstprice auction format is optimal among all “simple combinatorial auctions” in several settings.
LPbased Covering Games with Low Price of Anarchy
"... We present a new class of vertex cover and set cover games. The price of anarchyboundsmatchthebestknownconstant factorapproximationguaranteesfor the centralized optimization problems for linear and also for submodular costs – in contrast to all previously studied covering games, wherethe price of an ..."
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We present a new class of vertex cover and set cover games. The price of anarchyboundsmatchthebestknownconstant factorapproximationguaranteesfor the centralized optimization problems for linear and also for submodular costs – in contrast to all previously studied covering games, wherethe price of anarchy cannot be bounded by a constant (e.g. [6, 7, 11, 5, 2]). In particular, we describe a vertex covergamewithapriceofanarchyof2. Therulesofthegamescapturethestructure of the linear programmingrelaxations of theunderlyingoptimization problems, and our bounds are established by analyzing these relaxations. Furthermore, for linear costs we exhibit linear time best response dynamics that converge to these almost optimal Nash equilibria. These dynamics mimic the classical greedy approximation algorithm of BarYehuda and Even [3]. 1
Game Couplings: Learning Dynamics and Applications
"... Modern engineering systems (such as the Internet) consist of multiple coupled subsystems. Such subsystems are designed with local (possibly conflicting) goals, with little or no knowledge of the implementation details of other subsystems. Despite the ubiquitous nature of such systems very little is ..."
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Modern engineering systems (such as the Internet) consist of multiple coupled subsystems. Such subsystems are designed with local (possibly conflicting) goals, with little or no knowledge of the implementation details of other subsystems. Despite the ubiquitous nature of such systems very little is formally known about their properties and global dynamics. We investigate such distributed systems by introducing a novel gametheoretic construct, that we call gamecoupling. Game coupling intuitively allows us to stitch together the payoff structures of subgames. In order to study efficiency issues, we extend the price of anarchy approach (a major focus of gametheoretical multiagent systems [22]) to this setting, where we now care about the performance of each individual subsystem as well as the global performance. Such concerns give rise to a new notion of equilibrium, as well as a new learning paradigm. We prove matching welfare guarantees for both, both for individual subsystems as well as for the global system, using a generalization of the (λ, µ)smoothness framework [19]. In the second part of the paper, we work on understanding conditions that allow for wellstructured couplings. More generally, we examine when do game couplings preserve or enhance desirable properties of the original games, such as convergence of best response dynamics and low price of anarchy.