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Intrinsic Robustness of the Price of Anarchy
"... 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 99 (11 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
Optimal mechanism design and money burning
 STOC ’08
, 2008
"... Mechanism design is now a standard tool in computer science for aligning the incentives of selfinterested agents with the objectives of a system designer. There is, however, a fundamental disconnect between the traditional application domains of mechanism design (such as auctions) and those arising ..."
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Cited by 58 (15 self)
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Mechanism design is now a standard tool in computer science for aligning the incentives of selfinterested agents with the objectives of a system designer. There is, however, a fundamental disconnect between the traditional application domains of mechanism design (such as auctions) and those arising in computer science (such as networks): while monetary transfers (i.e., payments) are essential for most of the known positive results in mechanism design, they are undesirable or even technologically infeasible in many computer systems. Classical impossibility results imply that the reach of mechanisms without transfers is severely limited. Computer systems typically do have the ability to reduce service quality—routing systems can drop or delay traffic, scheduling protocols can delay the release of jobs, and computational payment schemes can require computational payments from users (e.g., in spamfighting systems). Service degradation is tantamount to requiring that users burn money, and such “payments ” can be used to influence the preferences of the agents at a cost of degrading the social surplus. We develop a framework for the design and analysis of moneyburning mechanisms to maximize the residual surplus— the total value of the chosen outcome minus the payments required. Our primary contributions are the following. • We define a general template for priorfree optimal mechanism design that explicitly connects Bayesian optimal mechanism design, the dominant paradigm in economics, with worstcase analysis. In particular, we establish a general and principled way to identify appropriate performance benchmarks in priorfree mechanism design. • For general singleparameter agent settings, we char
Coordination mechanisms for selfish scheduling
 THEORETICAL COMPUTER SCIENCE
, 2009
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Designing networks with good equilibria
 In SODA ’08
, 2007
"... In a network with selfish users, designing and deploying a protocol determines the rules of the game by which end users interact with each other and with the network. We study the problem of designing a protocol to optimize the equilibrium behavior of the induced network game. We consider network co ..."
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Cited by 36 (4 self)
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In a network with selfish users, designing and deploying a protocol determines the rules of the game by which end users interact with each other and with the network. We study the problem of designing a protocol to optimize the equilibrium behavior of the induced network game. We consider network costsharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge costsharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal costsharing protocols for undirected and directed graphs, singlesink and multicommodity networks, different classes of costsharing methods, and different measures of the inefficiency of equilibria. One of our main technical tools is a complete characterization of the uniform costsharing protocols—protocols that are designed without foreknowledge of or assumptions on the network in which they will be deployed. We use this characterization result to identify the optimal uniform protocol in several scenarios: for example, the Shapley protocol is optimal in directed graphs, while the optimal protocol in undirected graphs, a simple priority scheme, has exponentially smaller worstcase price of anarchy than the Shapley protocol. We also provide several matching upper and lower bounds on the bestpossible performance of nonuniform costsharing protocols.
(Almost) optimal coordination mechanisms for unrelated maching scheduling
 IN 18TH ACMSIAM SYMP. ON DISCRETE ALGORITHMS (SODA
, 2008
"... We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and ..."
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Cited by 35 (6 self)
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We investigate the influence of different algorithmic choices on the approximation ratio in selfish scheduling. Our goal is to design local policies that minimize the inefficiency of resulting equilibria. In particular, we design optimal coordination mechanisms for unrelated machine scheduling, and improve the known approximation ratio from Θ(m) to Θ(log m), where m is the number of machines. A local policy for each machine orders the set of jobs assigned to it only based on parameters of those jobs. A strongly local policy only uses the processing time of jobs on the the same machine. We prove that the approximation ratio of any set of strongly local ordering policies in equilibria is at least Ω(m). In particular, it implies that the approximation ratio of a greedy shortestfirst algorithm for machine scheduling is at least Ω(m). This closes the gap between the known lower and upper bounds for this problem, and answers an open question raised by Ibarra and Kim [16], and Davis and Jaffe [10]. We then design a local ordering policy with the approximation ratio of Θ(log m) in equilibria, and prove that this policy is optimal among all local ordering policies. This policy orders the jobs in the nondecreasing order of their inefficiency, i.e, the ratio between the processing time on that machine over the minimum processing time. Finally, we show that best responses of players for the inefficiencybased policy may not converge to a pure Nash equilibrium, and present a Θ(log² m) policy for which we can prove fast convergence of best responses to pure Nash equilibria.
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 21 (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
Designing Network Protocols for Good Equilibria
, 2009
"... Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network costsharing games, where the set ..."
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Cited by 15 (1 self)
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Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network costsharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge costsharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal costsharing protocols for undirected and directed graphs, singlesink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the costsharing protocols that only induce network games with purestrategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs, and that simple priority protocols are essentially optimal in undirected graphs.
Inner product spaces for minsum coordination mechanisms
 In STOC
, 2011
"... We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obta ..."
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Cited by 13 (3 self)
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We study policies aiming to minimize the weighted sum of completion times of jobs in the context of coordination mechanisms for selfish scheduling problems. Our goal is to design local policies that achieve a good price of anarchy in the resulting equilibria for unrelated machine scheduling. To obtain the approximation bounds, we introduce a new technique that while conceptually simple, seems to be quite powerful. The method entails mapping strategy vectors into a carefully chosen inner product space; costs are shown to correspond to the norm in this space, and the Nash condition also has a simple description. With this structure in place, we are able to prove a number of results, as follows. First, we consider Smith’s Rule, which orders the jobs on a machine in ascending processing time to weight ratio, and show that it achieves an approximation ratio of 4. We also demonstrate that this is the best possible for deterministic nonpreemptive strongly local policies. Since Smith’s Rule is always optimal for a given fixed assignment, this may seem unsurprising, but we then show that better approximation ratios can be obtained if either preemption or randomization is allowed.
Scheduling selfish tasks: about the performance of truthful algorithms
"... This paper deals with problems which fall into the domain of selfish scheduling: a protocol is in charge of building a schedule for a set of tasks without directly knowing their length. The protocol gets these informations from agents who control the tasks. The aim of each agent is to minimize the c ..."
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Cited by 10 (7 self)
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This paper deals with problems which fall into the domain of selfish scheduling: a protocol is in charge of building a schedule for a set of tasks without directly knowing their length. The protocol gets these informations from agents who control the tasks. The aim of each agent is to minimize the completion time of her task while the protocol tries to minimize the maximal completion time. When an agent reports the length of her task, she is aware of what the others bid and also of the protocol’s algorithm. Then, an agent can bid a false value in order to optimize her individual objective function. With erroneous information, even the most efficient algorithm may produce unreasonable solutions. An algorithm is truthful if it prevents the selfish agents from lying about the length of their task. The central question in this paper is: “How efficient a truthful algorithm can be? We study the problem of scheduling selfish tasks on parallel identical machines. This question has been raised by Christodoulou et al [8] in a distributed system, but it is also relevant in centrally controlled systems. Without considering side payments, our goal is to give a picture of the performance under the condition of truthfulness.
Selfish task allocation
, 2003
"... Recently there has been a lot of interest in problems at the intersection of Game Theory, Economics, and Computer Science. For example, there are interesting developments concerning algorithms for equilibria and cost sharing, algorithmic mechanism design, and the efficiency of systems with selfish ..."
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Cited by 10 (1 self)
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Recently there has been a lot of interest in problems at the intersection of Game Theory, Economics, and Computer Science. For example, there are interesting developments concerning algorithms for equilibria and cost sharing, algorithmic mechanism design, and the efficiency of systems with selfish users [17]. In this note, I will focus on the last area and in particular on the price of anarchy of scheduling or task allocation. I will discuss the issues of this area, mention the central results, and suggest some open problems. This is definitely not a review paper: My aim is to stimulate the reader, not to provide complete coverage of the area. The presentation favors simplicity to preciseness, examples to formal presentation, and intuition to rigor.