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49
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
Coordination mechanisms
 PROCEEDINGS OF THE 31ST INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING, IN: LECTURE NOTES IN COMPUTER SCIENCE
, 2004
"... We introduce the notion of coordination mechanisms to improve the performance in systems with independent selfish and noncolluding agents. The quality of a coordination mechanism is measured by its price of anarchy—the worstcase performance of a Nash equilibrium over the (centrally controlled) soc ..."
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Cited by 57 (5 self)
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We introduce the notion of coordination mechanisms to improve the performance in systems with independent selfish and noncolluding agents. The quality of a coordination mechanism is measured by its price of anarchy—the worstcase performance of a Nash equilibrium over the (centrally controlled) social optimum. We give upper and lower bounds for the price of anarchy for selfish task allocation and congestion games.
Tight bounds for selfish and greedy load balancing
 ICALP 2006. LNCS
, 2006
"... Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it ..."
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Cited by 43 (6 self)
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Abstract. We study the load balancing problem in the context of a set of clients each wishing to run a job on a server selected among a subset of permissible servers for the particular client. We consider two different scenarios. In selfish load balancing, each client is selfish in the sense that it selects to run its job to the server among its permissible servers having the smallest latency given the assignments of the jobs of other clients to servers. In online load balancing, clients appear online and, when a client appears, it has to make an irrevocable decision and assign its job to one of its permissible servers. Here, we assume that the clients aim to optimize some global criterion but in an online fashion. A natural local optimization criterion that can be used by each client when making its decision is to assign its job to that server that gives the minimum increase of the global objective. This gives rise to greedy online solutions. The aim of this paper is to determine how much the quality of load balancing is affected by selfishness and greediness. We characterize almost completely the impact of selfishness and greediness in load balancing by presenting new and improved, tight or almost tight bounds on the price of anarchy and price of stability of selfish load balancing as well as on the competitiveness of the greedy algorithm for online load balancing when the objective is to minimize the total latency of all clients on servers with linear latency functions. 1
Nash Equilibria in Discrete Routing Games with Convex Latency Functions
, 2004
"... In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary nondecreasing, nonconstant and convex latency function φ. In a Nash equilibrium, each user alone is ..."
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Cited by 38 (11 self)
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In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary nondecreasing, nonconstant and convex latency function φ. In a Nash equilibrium, each user alone is minimizing her (Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users ’ (Expected) Individual Costs. The Price of Anarchy is the worstcase ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with nonzero probability. We obtain: For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization. For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function φ(x) = x d, the Price of Anarchy is the Bell number of order d + 1. For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with nonnegative coefficients and degree d, this yields an upper bound of d + 1. For 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.
The price of anarchy for polynomial social cost
 IN PROC. MFCS
, 2004
"... In this work, we consider an interesting variant of the wellstudied KP model [18] for selfish routing that reflects some influence from the much older Wardrop model [31]. In the new model, user traffics are still unsplittable, while social cost is now the expectation of the sum, over all links, of ..."
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Cited by 24 (8 self)
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In this work, we consider an interesting variant of the wellstudied KP model [18] for selfish routing that reflects some influence from the much older Wardrop model [31]. In the new model, user traffics are still unsplittable, while social cost is now the expectation of the sum, over all links, of a certain polynomial evaluated at the total latency incurred by all users choosing the link; we call it polynomial social cost. The polynomials that we consider have nonnegative coefficients. We are interested in evaluating Nash equilibria in this model, and we use the Price of Anarchy as our evaluation measure. We prove the Fully Mixed Nash Equilibrium Conjecture for identical users and two links, and establish an approximate version of the conjecture for arbitrary many links. Moreover, we give upper bounds on the Price of Anarchy.
On the Complexity of PureStrategy Nash Equilibria in Congestion and LocalEffect Games
 In Proc. of the 2nd Int. Workshop on Internet and Network Economics (WINE
, 2006
"... doi 10.1287/moor.1080.0322 ..."
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Efficient coordination mechanisms for unrelated machine scheduling
 In: Proc. AMCSIAM SODA
, 2009
"... We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game ..."
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Cited by 20 (1 self)
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We present three new coordination mechanisms for scheduling n selfish jobs on m unrelated machines. A coordination mechanism aims to mitigate the impact of selfishness of jobs on the efficiency of schedules by defining a local scheduling policy on each machine. The scheduling policies induce a game among the jobs and each job prefers to be scheduled on a machine so that its completion time is minimum given the assignments of the other jobs. We consider the maximum completion time among all jobs as the measure of the efficiency of schedules. The approximation ratio of a coordination mechanism quantifies the efficiency of pure Nash equilibria (price of anarchy) of the induced game. Our mechanisms are deterministic, local, and preemptive in the sense that the scheduling policy does not necessarily process
Local Smoothness and the Price of Anarchy in Atomic Splittable Congestion Games
"... We resolve the worstcase price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest nontrivial special case of affine cost functions. We make two distinct contributions. On the upperbound side, we def ..."
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Cited by 20 (3 self)
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We resolve the worstcase price of anarchy (POA) of atomic splittable congestion games. Prior to this work, no tight bounds on the POA in such games were known, even for the simplest nontrivial special case of affine cost functions. We make two distinct contributions. On the upperbound side, we define the framework of “local smoothness”, which refines the standard smoothness framework for games with convex strategy sets. While standard smoothness arguments cannot establish tight bounds on the POA in atomic splittable congestion games, we prove that local smoothness arguments can. Further, we prove that every POA bound derived via local smoothness applies automatically to every correlated equilibrium of the game. Unlike standard smoothness arguments, bounds proved using local smoothness do not always apply to the coarse correlated equilibria of the game. Our second contribution is a very general lower bound: for every set L that satisfies mild technical conditions, the worstcase POA of pure Nash equilibria in atomic splittable congestion games with cost functions in L is exactly the smallest upper bound provable using local smoothness arguments. In particular, the worstcase POA of pure Nash equilibria, mixed Nash equilibria, and correlated equilibria coincide in such games. 1
Routing (Un) Splittable Flow in Games with PlayerSpecific Linear Latency Functions
 In Proceedings of the 33rd International Colloquium on Automata, Languages, and Programming (ICALP’06), LNCS 4051
, 2006
"... Abstract. In this work we study weighted network congestion games with playerspecific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows: – For routing games ..."
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Cited by 19 (1 self)
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Abstract. In this work we study weighted network congestion games with playerspecific latency functions where selfish players wish to route their traffic through a shared network. We consider both the case of splittable and unsplittable traffic. Our main findings are as follows: – For routing games on parallel links with linear latency functions without a constant term we introduce two new potential functions for unsplittable and for splittable traffic respectively. We use these functions to derive results on the convergence to pure Nash equilibria and the computation of equilibria. We also show for several generalizations of these routing games that such potential functions do not exist. – We prove upper and lower bounds on the price of anarchy for games with linear latency functions. For the case of unsplittable traffic the upper and lower bound are asymptotically tight. 1