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72
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
Mdpop: Faithful distributed implementation of efficient social choice problems
 In AAMAS’06  Autonomous Agents and Multiagent Systems
, 2006
"... In the efficient social choice problem, the goal is to assign values, subject to side constraints, to a set of variables to maximize the total utility across a population of agents, where each agent has private information about its utility function. In this paper we model the social choice problem ..."
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Cited by 48 (17 self)
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In the efficient social choice problem, the goal is to assign values, subject to side constraints, to a set of variables to maximize the total utility across a population of agents, where each agent has private information about its utility function. In this paper we model the social choice problem as a distributed constraint optimization problem (DCOP), in which each agent can communicate with other agents that share an interest in one or more variables. Whereas existing DCOP algorithms can be easily manipulated by an agent, either by misreporting private information or deviating from the algorithm, we introduce MDPOP, the first DCOP algorithm that provides a faithful distributed implementation for efficient social choice. This provides a concrete example of how the methods of mechanism design can be unified with those of distributed optimization. Faithfulness ensures that no agent can benefit by unilaterally deviating from any aspect of the protocol, neither informationrevelation, computation, nor communication, and whatever the private information of other agents. We allow for payments by agents to a central bank, which is the only central authority that we require. To achieve faithfulness, we carefully integrate the VickreyClarkeGroves (VCG) mechanism with the DPOP algorithm, such that each agent is only asked to perform computation, report
Strong equilibrium in cost sharing connection games
 Proc. 8th ACM Conference on Electronic Commerce, 84–92
, 2007
"... In this work we study cost sharing connection games, where each player has a source and sink he would like to connect, and the cost of the edges is either shared equally (fair connection games) or in an arbitrary way (general connection games). We study the graph topologies that guarantee the existe ..."
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Cited by 47 (6 self)
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In this work we study cost sharing connection games, where each player has a source and sink he would like to connect, and the cost of the edges is either shared equally (fair connection games) or in an arbitrary way (general connection games). We study the graph topologies that guarantee the existence of a strong equilibrium (where no coalition can improve the cost of each of its members) regardless of the specific costs on the edges. Our main existence results are the following: (1) For a single source and sink we show that there is always a strong equilibrium (both for fair and general connection games). (2) For a single source multiple sinks we show that for a series parallel graph a strong equilibrium always exists (both for fair and general connection games). (3) For multi source and sink we show that an extension parallel graph always admits a strong equilibrium in fair connection games. As for the quality of the strong equilibrium we show that in any fair connection games the cost of a strong equilibrium is Θ(log n) from the optimal solution, where n is the number of players. (This should be contrasted with the Ω(n) price of anarchy for the same setting.) For single source general connection games and single source single sink fair connection games, we show that a strong equilibrium is always an optimal solution.
On the Value of Coordination in Network Design
"... We study network design games where n selfinterested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy o ..."
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Cited by 36 (0 self)
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We study network design games where n selfinterested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as n. Therefore, recent research has focused on evaluating the price of stability, i.e. the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of Ω(log W / log log W) for weighted games, where W is the total weight of all the agents. This almost matches the known upper bound of O(log W). Our results imply that, for most settings, the worstcase performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs.
Basic Network Creation Games
, 2010
"... We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes, by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they ..."
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Cited by 22 (1 self)
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We study a natural network creation game, in which each node locally tries to minimize its local diameter or its local average distance to other nodes, by swapping one incident edge at a time. The central question is what structure the resulting equilibrium graphs have, in particular, how well they globally minimize diameter. For the localaveragedistance version, we prove an upper bound of 2 O( √ lg n), a lower bound of 3, a tight bound of exactly 2 for trees, and give evidence of a general polylogarithmic upper bound. For the localdiameter version, we prove a lower bound of Ω ( √ n), and a tight upper bound of 3 for trees. All of our upper bounds apply equally well to previously extensively studied network creation games, both in terms of the diameter metric described above and the previously studied price of anarchy (which are related by constant factors). In surprising contrast, our model has no parameter α for the link creation cost, so our results automatically apply for all values of α without additional effort; furthermore, equilibrium can be checked in polynomial time in our model, unlike previous models. Our perspective enables simpler and more general proofs that get at the heart of network creation games.
Improved Equilibria via Public Service Advertising
"... Many natural games have both high and low cost Nash equilibria: their Price of Anarchy is high and yet their Price of Stability is low. In such cases, one could hope to move behavior from a high cost equilibrium to a low cost one by a “public service advertising campaign ” encouraging players to fol ..."
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Cited by 22 (7 self)
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Many natural games have both high and low cost Nash equilibria: their Price of Anarchy is high and yet their Price of Stability is low. In such cases, one could hope to move behavior from a high cost equilibrium to a low cost one by a “public service advertising campaign ” encouraging players to follow the lowcost equilibrium, and if every player follows the advice then we are done. However, the assumption that everyone follows instructions is unrealistic. A more natural assumption is that some players will follow them, while other players will not. In this paper we consider the question of to what extent can such an advertising campaign cause behavior to switch from a bad equilibrium to a good one even if only a fraction of people actually follow the given advice, and do so only temporarily. Unlike
Circumventing the Price of Anarchy: Leading Dynamics to Good Behavior
"... Many natural games can have a dramatic difference between the quality of their best and worst Nash equilibria, even in pure strategies. Yet, nearly all work to date on dynamics shows only convergence to some equilibrium, especially within a polynomial number of steps. In this work we study how age ..."
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Cited by 14 (6 self)
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Many natural games can have a dramatic difference between the quality of their best and worst Nash equilibria, even in pure strategies. Yet, nearly all work to date on dynamics shows only convergence to some equilibrium, especially within a polynomial number of steps. In this work we study how agents with some knowledge of the game might be able to quickly (within a polynomial number of steps) find their way to states of quality close to the best equilibrium. We consider two natural learning models in which players choose between greedy behavior and following a proposed good but untrusted strategy and analyze two important classes of games in this context, fair costsharing and consensus games. Both games have extremely high Price of Anarchy and yet we show that behavior in these models can efficiently reach lowcost states.
On the existence of pure Nash equilibria in weighted congestion games.
 Proc. 37rd Internat. Colloquium on Automata, Languages and Programming,
, 2010
"... Abstract We study the existence of pure Nash equilibria in weighted congestion games. Let C denote a set of cost functions. We say that C is consistent if every weighted congestion game with cost functions in C possesses a pure Nash equilibrium. Our main contribution is a complete characterization ..."
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Abstract We study the existence of pure Nash equilibria in weighted congestion games. Let C denote a set of cost functions. We say that C is consistent if every weighted congestion game with cost functions in C possesses a pure Nash equilibrium. Our main contribution is a complete characterization of consistency of cost functions. Specifically, we prove that a nonempty set C of twice continuously differentiable functions is consistent for twoplayer games if and only if C contains only monotonic functions and for all c 1 , c 2 ∈ C, there are constants a, b ∈ R such that c 1 (x) = a c 2 (x) + b for all x ∈ R ≥0 . For games with at least 3 players, we prove that C is consistent if and only if exactly one of the following cases hold: (a) C contains only affine functions; (b) C contains only exponential functions such that c(x) = a c e φ x + b c for some a c , b c , φ ∈ R, where a c and b c may depend on c, while φ must be equal for every c ∈ C. The latter characterization is even valid for 3player games, thus, closing the gap to 2player games considered above. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as games with singleton strategies or weighted network congestion games.
Quantifying Inefficiency in CostSharing Mechanisms
, 2009
"... In a costsharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A costsharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a co ..."
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Cited by 12 (1 self)
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In a costsharing problem, several participants with unknown preferences vie to receive some good or service, and each possible outcome has a known cost. A costsharing mechanism is a protocol that decides which participants are allocated a good and at what prices. Three desirable properties of a costsharing mechanism are: incentivecompatibility, meaning that participants are motivated to bid their true private value for receiving the good; budgetbalance, meaning that the mechanism recovers its incurred cost with the prices charged; and economic efficiency, meaning that the cost incurred and the value to the participants are traded off in an optimal way. These three goals have been known to be mutually incompatible for thirty years. Nearly all the work on costsharing mechanism design by the economics and computer science communities has focused on achieving two of these goals while completely ignoring the third. We introduce novel measures for quantifying efficiency loss in costsharing mechanisms and prove simultaneous approximate budgetbalance and approximate efficiency guarantees for mechanisms for a wide range of costsharing problems, including all submodular and Steiner tree problems. Our key technical tool is an exact characterization of worstcase efficiency loss in Moulin mechanisms, the dominant paradigm in costsharing mechanism design.
Strong Price of Anarchy for Machine Load Balancing
"... As defined by Aumann in 1959, a strong equilibrium is a Nash equilibrium that is resilient to deviations by coalitions. We give tight bounds on the strong price of anarchy for load balancing on related machines. We also give tight bounds for kstrong equilibria, where the size of a deviating coali ..."
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Cited by 11 (0 self)
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As defined by Aumann in 1959, a strong equilibrium is a Nash equilibrium that is resilient to deviations by coalitions. We give tight bounds on the strong price of anarchy for load balancing on related machines. We also give tight bounds for kstrong equilibria, where the size of a deviating coalition is at most k.