Nash equilibria in singleton congestion games : Symmetric case (original) (raw)
A formula for Nash equilibria in monotone singleton congestion games
2011
This paper provides a simple formula describing all Nash equilibria in symmetric monotone singleton congestion games. Our approach also yields a new and short proof establishing the existence of a Nash equilibrium in this kind of congestion games without invoking the potential function or the nite improvement property.
Nash equilibria in nonsymmetric singleton congestion games with exact partition
2011
We define a new class of games, which we qualify as congestion games with exact partition. These games constitute a subfamily of singleton congestion games for which the players are restricted to choose only one strategy, but they each possess their own utility function. The aim of this paper is to develop a method leading to an easier identification of
Nonsymmetric singleton congestion games: case of two resources
2011
In this note we study the existence of Nash equilibria in nonsymmetric finite congestion games, complementing the results obtained by Milchtaich on monotone-decreasing congestion games. More specifically, we examine the case of two resources and we propose a simple method describing all Nash equilibria in this kind of congestion games. Additionally, we give a new and short proof establishing the existence of a Nash equilibrium in this type of games without invoking the potential function or the finite improvement property. (H. Smaoui), abderrahmane.ziad@unicaen.fr (A.Ziad). Example 2 . Let N = {1, 2, 3, 4, 5} and R = {a, b}. Suppose that the players' preferences are given by the following weak ordering:
On the inefficiency of equilibria in congestion games
2005
We present a short geometric proof of the price of anarchy and price of stability results that have recently been established in a series of papers on selfish routing. This novel proof also facilitates two types of new results: On the one hand, we give pseudoapproximation results that depend on the class of allowable cost functions. On the other hand, we offer improved bounds on the inefficiency of Nash equilibria for situations in which the equilibrium travel times are within reasonable limits of the free-flow travel times, a scenario that captures empirical observations in vehicular traffic networks. Our results actually hold in the more general context of congestion games, which provide the framework in which we describe this work.
Congestion Games and Potentials Reconsidered
International Game Theory Review, 1999
In congestion games, players use facilities from a common pool. The benefit that a player derives from using a facility depends, possibly among other things, on the number of users of this facility. The paper gives an easy alternative proof of the isomorphism between exact potential games and the set of congestion games introduced by Rosenthal (1973). It clarifies the relations between existing models on congestion games, and studies a class of congestion games where the sets of Nash equilibria, strong Nash equilibria and potential-maximising strategies coincide. Particular emphasis is on the computation of potential-maximising strategies.
Bounding the inefficiency of equilibria in nonatomic congestion games
Games and Economic Behavior, 2004
Equilibria in noncooperative games are typically inefficient, as illustrated by the Prisoner's Dilemma. In this paper, we quantify this inefficiency by comparing the payoffs of equilibria to the payoffs of a "best possible" outcome. We study a nonatomic version of the congestion games defined by Rosenthal , and identify games in which equilibria are approximately optimal in the sense that no other outcome achieves a significantly larger total payoff to the players-games in which optimization by individuals approximately optimizes the social good, in spite of the lack of coordination between players. Our results extend previous work on traffic routing games . * We thank Lou Billera for encouraging us to explore generalizations of our previous work on traffic routing, and Robert Rosenthal for introducing us to his congestion games. The first author thanks Amir Ronen for comments on a previous paper [16] that facilitated a generalization of the results therein to the nonatomic congestion games of this paper.
Solution-based congestion games
In this paper we develop the theory of potential of cooperative games for semivalues, characterize congestion models that are defined by semivalues, and suggest an application of these results to combinatorial auctions, which may explain the success of the Iowa electronic market. and they have been analyzed by several researchers from various additional fields, in particular computer science 2 , communication networks 3 , and economics/game theory. 4
On the Performance of Approximate Equilibria in Congestion Games
Algorithmica, 2011
We study the performance of approximate Nash equilibria for congestion games with polynomial latency functions. We consider how much the price of anarchy worsens and how much the price of stability improves as a function of the approximation factor ε. We give tight bounds for the price of anarchy of atomic and non-atomic congestion games and for the price of
On the Inefficiency Ratio of Stable Equilibria in Congestion Games
Lecture Notes in Computer Science, 2009
Price of anarchy and price of stability are the primary notions for measuring the efficiency (i.e. the social welfare) of the outcome of a game. Both of these notions focus on extreme cases: one is defined as the inefficiency ratio of the worst-case equilibrium and the other as the best one. Therefore, studying these notions often results in discovering equilibria that are not necessarily the most likely outcomes of the dynamics of selfish and non-coordinating agents. The current paper studies the inefficiency of the equilibria that are most stable in the presence of noise. In particular, we study two variations of non-cooperative games: atomic congestion games and selfish load balancing. The noisy best-response dynamics in these games keeps the joint action profile around a particular set of equilibria that minimize the potential function. The inefficiency ratio in the neighborhood of these "stable" equilibria is much better than the price of anarchy. Furthermore, the dynamics reaches these equilibria in polynomial time. Our observations show that in the game environments where a small noise is present, the system as a whole works better than what a pessimist may predict. They also suggest that in congestion games, introducing a small noise in the payoff of the agents may improve the social welfare.
On Singleton Congestion Games with Resilience Against Collusion
Lecture Notes in Computer Science, 2021
We study the subclass of singleton congestion games with identical and increasing cost functions, i.e., each agent tries to utilize from the least crowded resource in her accessible subset of resources. Our main contribution is a novel approach for proving the existence of equilibrium outcomes that are resilient to weakly improving deviations: (i) by singletons (Nash equilibria), (ii) by the grand coalition (Pareto efficiency), and (iii) by coalitions with respect to an a priori given partition coalition structure (partition equilibria). To the best of our knowledge, this is the strongest existence guarantee in the literature of congestion games that is resilient to weakly improving deviations by coalitions.
Lecture Notes in Computer Science, 2009
We introduce a new class of games, asynchronous congestion games (ACGs). In an ACG, each player has a task that can be carried out by any element of a set of resources, and each resource executes its assigned tasks in a random order. Each player's aim is to minimize his expected cost which is the sum of two terms -the sum of the fixed costs over the set of his utilized resources and the expected cost of his task execution. The cost of a player's task execution is determined by the earliest time his task is completed, and thus it might be beneficial for him to assign his task to several resources. We prove the existence of pure strategy Nash equilibria in ACGs. Moreover, we present a polynomial time algorithm for finding such an equilibrium in a given ACG.
Exact Price of Anarchy for Polynomial Congestion Games
SIAM Journal on Computing, 2011
We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion games.
Congestion network problems and related games
European Journal of Operational Research, 2006
This paper analyzes network problems with congestion effects from a cooperative game theoretic perspective. It is shown that for network problems with convex congestion costs, the corresponding games have a non-empty core. If congestion costs are concave, then the corresponding game has not necessarily core elements, but it is derived that, contrary to the convex congestion situation, there always exist optimal tree networks. Extensions of these results to a class of relaxed network problems and associated games are derived.
Congestion Models and Weighted Bayesian Potential Games
1997
Games associated to congestion situations a la Rosenthal (1973) have pure Nash equilibria. This result implicitly relies on the existence of a potential function. In this paper we will provide a characterization of potential games in terms of coordination games and dummy games. Secondly, w e extend Rosenthal's congestion model to an incomplete information setting, and show that the related Bayesian games are potential games and therefore have pure Bayesian equilibria.
Congestion Games with Player-Specific Constants
Lecture Notes in Computer Science
We consider a special case of weighted congestion games with playerspecific latency functions where each player uses for each particular resource a fixed (non-decreasing) delay function together with a player-specific constant. For each particular resource, the resource-specific delay function and the playerspecific constant (for that resource) are composed by means of a group operation (such as addition or multiplication) into a player-specific latency function. We assume that the underlying group is a totally ordered abelian group. In this way, we obtain the class of weighted congestion games with player-specific constants; we observe that this class is contained in the new intuitive class of dominance weighted congestion games. We obtain the following results: Games on parallel links:-Every unweighted congestion game has a generalized ordinal potential.-There is a weighted congestion game with 3 players on 3 parallel links that does not have the Finite Best-Improvement Property.-There is a particular best-improvement cycle for general weighted congestion games with player-specific latency functions and 3 players whose outlaw implies the existence of a pure Nash equilibrium. This cycle is indeed outlawed for dominance weighted congestion games with 3 players-and hence for weighted congestion games with player-specific constants and 3 players. Network congestion games: For unweighted symmetric network congestion games with player-specific additive constants, it is PLS-complete to find a pure Nash equilibrium. Arbitrary (non-network) congestion games: Every weighted congestion game with linear delay functions and player-specific additive constants has a weighted potential.
2006
We consider congestion pricing as a mechanism for sharing bandwidth in communication networks, and model the interaction among the users as a game. We propose a decentralized algorithm for the users that is based on the history of the price process, where user response to congestion prices is analogous to “fictitious play” in game theory, and show that this results in convergence to the unique Wardrop equilibrium. We further show that the Wardrop equilibrium coincides with the welfare maximizing capacity allo-
Congestion games with failures
Discrete Applied Mathematics, 2011
We introduce a new class of games, congestion games with failures (CGFs), which extends the class of congestion games to allow for facility failures. In a CGF agents share a common set of facilities (service providers), where each service provider (SP) may fail with some known probability. For reliability reasons, an agent may choose a subset of the SPs in order to try and perform his task. The cost of an agent for utilizing any SP is an agent-specific function of the total number of agents using this SP. A main feature of this setting is that the cost for an agent for successful completion of his task is the minimum of the costs of his successful attempts. We show that although congestion games with failures do not admit a potential function, and thus are not isomorphic to classic congestion games, they always possess a pure-strategy Nash equilibrium. Moreover, an efficient algorithm for the construction of pure-strategy Nash equilibrium profile is presented. We also show that the SPs' congestion experienced in different Nash equilibria is (almost) unique. For the subclass of symmetric CGFs we give a characterization of best and worst Nash equilibria, present algorithms for their construction, and compare the social disutilities of the agents at these points.
Nash equilibrium problems with congestion costs and shared constraints
Decision and Control, 2009 …, 2009
Generalized Nash equilibria (GNE) represent extensions of the Nash solution concept when agents have shared strategy sets. This generalization is particularly relevant when agents compete in a networked setting. In this paper, we consider such a setting and focus on a congestion game in which agents contend with shared network constraints. We make two sets of contributions: (1) Under two types of congestion cost functions, we prove the existence of the primal generalized Nash equilibrium. The results are provided without a compactness assumption on the constraint set and are shown to hold when the mappings associated with the resulting variational inequality are non-monotone. Under further assumptions, the local and global uniqueness of the primal and primal-dual generalized Nash equilibrium is also provided. (2) We provide two distributed schemes for obtaining such equilibria: a dual and a primal-dual algorithm. Convergence of both algorithms is analyzed and preliminary numerical evidence is presented with the aid of an example.
Computers & Mathematics with Applications, 2008
We derive several bounds for the price of anarchy of the noncooperative congestion games with elastic demands and asymmetric linear or nonlinear cost functions. The bounds established depend on a constant from the cost functions as well as the ratio between user benefit and social surplus at Nash equilibrium. The results can be viewed a generalization of that of Chau and Sim [C.K. Chau, K.M. Sim, The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands, Operations Research Letters 31 (2003) 327-334] for the symmetric case, or a generalization of Perakis [G. Perakis, The price of anarchy when costs are nonseparable and asymmetric, Lecture Notes in Computer Science 3064 (2004) 46-58] to the elastic demand.