Routing (un-) splittable flow in games with player-specific affine latency functions

We study congestion games where players aim to access a set of resources. Each player has a set of possible strategies and each resource has a function associating the latency it incurs to the players using it. Players are non–cooperative and each wishes to follow strategies that minimize her own latency with no regard to the global optimum. Previous work has studied the impact of this selfish behavior to system performance. In this paper, we study the question of how much the performance can be improved if players are forced to pay taxes for using resources. Our objective is to extend the original game so that selfish behavior does not deteriorate performance. We consider atomic congestion games with linear latency functions and present both negative and positive results. Our negative results show that optimal system performance cannot be achieved even in very simple games. On the positive side, we show that there are ways to assign taxes that can improve the performance of linear congestion games by forcing players to follow strategies where the total latency suffered is within a factor of 2 of the minimum possible; this result is shown to be tight. Furthermore, even in cases where in the absence of taxes the system behavior may be very poor, we show that the total disutility of players (latency plus taxes) is not much larger than the optimal total latency. Besides existential results, we show how to compute taxes in time polynomial in the size of the game by solving convex quadratic programs. Similar questions have been extensively studied in the model of non-atomic congestion games. To the best of our knowledge, this is the first study of the efficiency of taxes in atomic congestion games.KeywordsNash EquilibriumCongestion GameIdentical MachinePure Nash EquilibriumConvex Quadratic ProgramThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We investigate the speed of convergence of congestion games with linear latency functions under best response dynamics. Namely, we estimate the social performance achieved after a limited number of rounds, during each of which every player performs one best response move. In particular, we show that the price of anarchy achieved after krounds, defined as the highest possible ratio among the total latency cost, that is the sum of all players latencies, and the minimum possible cost, is $O(\sqrt[2^{k-1}] {n})$, where nis the number of players. For constant values of ksuch a bound asymptotically matches the $\Omega(\sqrt[2^{k-1}] {n}/k)$ lower bound that we determine as a refinement of the one in [7]. As a consequence, we prove that order of loglognrounds are not only necessary, but also sufficient to achieve a constant price of anarchy, i.e. comparable to the one at Nash equilibrium. This result is particularly relevant, as reaching an equilibrium may require a number of rounds exponential in n. We then provide a new lower bound of $\Omega(\sqrt[2^k-1] {n})$ for load balancing games, that is congestion games in which every strategy consists of a single resource, thus showing that a number of rounds proportional to loglognis necessary and sufficient also under such a restriction. Our results thus solve the important left open question of the polynomial speed of convergence of linear congestion games to constant factor solutions.

We study the inefficiency of equilibrium outcomes in Bottleneck Congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We analyze the (strong) Price of Anarchy of these games for a natural load balancing social cost objective, i.e., minimize the maximum latency of a facility. In our studies, we focus on Bottleneck Congestion games with linear latency functions. These games still constitute a rich class of games and generalize, for example, Load Balancing games with identical or uniformly related machines (with or without restricted assignments). We derive upper and (asymptotically) matching lower bounds on the (strong) Price of Anarchy of these games. We also derive more refined bounds for several special cases of these games, including the cases of identical player weights, identical latency functions and symmetric strategy sets. Further, we provide lower bounds on the Price of Anarchy for k-strong equilibria.

We study taxes in the well-known game theoretic traffic model due to Wardrop. Given a network and a subset of edges, on which we can impose taxes, the problem is to find taxes inducing an equilibrium flow of minimal network-wide latency cost. If all edges are taxable, then marginal cost pricing is known to induce the socially optimal flow for arbitrary multi-commodity networks. In contrast, if only a strict subset of edges is taxable, we show NP-hardness of finding optimal taxes for general networks with linear latency functions and two commodities. On the positive side, for single-commodity networks with parallel links and linear latency function, we provide a polynomial time algorithm for finding optimal taxes.

Nash equilibria in discrete routing games with convex latency functions

Stackelberg thresholds in network routing games or the value of altruism

We study the degradation in network performance caused by the selfish behavior of noncooperative network users. We consider a directed network in which each edge possesses a latency function describing the common latency incurred by all traffic on the edge as a function of the edge congestion. Given a rate of traffic between each pair of nodes in the network, we aspire toward an assignment of traffic to paths in which the sum of all travel times (the total latency) is minimized; however, in many settings network users are free to route their traffic in a selfish manner, without regard to the total latency. We therefore assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a selfishly motivated assignment of traffic to paths (a Nash equilibrium) will not minimize the total latency; hence, selfish behavior carries the cost of decreased network performance. We quantify this degradation in network performance via the price of anarchy, defined as the worst possible ratio between the total latency of a Nash equilibrium and of a minimum-latency routing of the traffic.In this paper, we show that the underlying network topology plays no role in the determination of the price of anarchy. Specifically, we show that under weak hypotheses on the class of allowable edge latency functions, the worst-case ratio between the total latency of a Nash equilibrium and of a minimum-latency routing for any multicommodity flow network is achieved by a single-commodity instance on a set of parallel links. In the special case where the class of allowable latency functions includes all of the constant functions, we prove that a network with only two parallel links suffices to achieve the worst-possible ratio. Informally, these results imply that the inefficiency inherent in a flow at Nash equilibrium stems from the inability of selfish users to discern which of two competing routes is superior and not from the topological complexity arising from the diverse intersections of many paths belonging to different commoditie.Our proof techniques also give powerful methods for computing the price of anarchy with respect to an arbitrary class of latency functions. We apply these methods to function classes that have been well studied in the literature (such as degree-bounded polynomials and functions of the form $\ell(x) = (u—x)—1} that are used to model edges with capacity u), thereby achieving the first tight analyses of the price of anarchy for significant classes of latency functions outside the class of linear functions.

As increasing concern for environmental sustainability urges to bring attention to green-aware multi-agent systems, we put forward a game-theoretic model in which agents compete for the usage of power-consuming resources and are charged a cost proportional to their fair share of the power consumption. Using the widely adopted cube-root rule for CMOS-based devices, our model becomes a congestion game in which two distinct parts coexist, namely, congestion games with polynomial latency functions and fair cost-sharing games. The interplay between these two components is governed by two resource-specific constants regulating the static and dynamic power consumption of each resource. Our findings show that, despite these games being highly inefficient in the general case (a super-constant price of stability), performance at equilibrium significantly improves (a constant price of anarchy) when the ratio between the static and dynamic power consumption of each resource remains bounded by a constant. This suggests that, in uncoordinated green-aware multi-agent systems, technology plays a fundamental role in shaping the efficiency of stable solutions.

We study Nash and strong equilibria in weighted and unweighted bottleneck games. In such a game every (weighted) player chooses a subset of a given set of resources as her strategy. The cost of a resource depends on the total weight of players choosing it and the personal cost every player tries to minimize is the cost of the most expensive resource in her strategy, the bottleneck value. To derive efficient algorithms for finding equilibria in unweighted games, we generalize a transformation of a bottleneck game into a congestion game with exponential cost functions introduced by Caragiannis et al. (2005). For weighted routing games we show that Greedy methods give Nash equilibria in extension-parallel and series-parallel graphs. Furthermore, we show that the strong Price of Anarchy can be arbitrarily high for special cases and give tight bounds depending on the topology of the graph, the number and weights of the users and the degree of the polynomial latency functions. Additionally we investigate the existence of equilibria in generalized bottleneck games, where players aim to minimize not only the bottleneck value, but also the second most expensive resource in their strategy and so on.

We present an extension to nonatomic congestion games (NCG). An NCG models a large number of players depending on a set of resources (e.g., network links) in certain combinations (e.g., paths or multicast trees) called strategies. The rate of consumption ZeS specifies how aggressively resource e is consumed when used via strategy S, but it also effects how strongly the resource's latency is experienced by the players. Our extension allows essentially unrelated factors CeS and ReS instead of ZeS. Factor CeS is the actual rate of consumption, whereas ReS expresses the amplification of the resource latency of e for players choosing strategy S, or, in other words, the relevance of resource e for strategy S. We call the extended model nonatomic consumption‐relevance congestion games (NCRCG). NCRCGs exhibit new phenomena, including multiple Nash equilibria of different social cost and—even from a worst‐case point of view—a dependence of the price of anarchy on structural parameters not limited to the class of resource latency functions used. We prove almost tight lower, upper, and bicriteria bounds for the price of anarchy for polynomial latency functions with nonnegative coefficients. We conjecture that the lower bound is the best possible. © 2013 Wiley Periodicals, Inc. NETWORKS, 2013

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2009 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Equilibria of Atomic Flow Games are not UniqueUmang Bhaskar, Lisa Fleischer, Darrell Hoy, and Chien-Chung HuangUmang Bhaskar, Lisa Fleischer, Darrell Hoy, and Chien-Chung Huangpp.748 - 757Chapter DOI:https://doi.org/10.1137/1.9781611973068.82PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In routing games with infinitesimal players, it follows from well-known convexity arguments that equilibria exist and are unique (up to induced delays, and under weak assumptions on delay functions). In routing games with players that control large amounts of flow, uniqueness has been demonstrated only in limited cases: in 2-terminal, nearly-parallel graphs; when all players control exactly the same amount of flow; when latency functions are polynomials of degree at most three. In this work, we answer an open question posed by Cominetti, Correa, and Stier-Moses (ICALP 2006) and show that there may be multiple equilibria in atomic player routing games. We demonstrate this multiplicity via two specific examples. In addition, we show our examples are topologically minimal by giving a complete characterization of the class of network topologies for which unique equilibria exist. Our proofs and examples are based on a novel characterization of these topologies in terms of sets of circulations. Previous chapter Next chapter RelatedDetails Published:2009ISBN:978-0-89871-680-1eISBN:978-1-61197-306-8 https://doi.org/10.1137/1.9781611973068Book Series Name:ProceedingsBook Code:PR132Book Pages:xviii + 1288

One of the most widely used solution concepts for strategic games is the concept of Nash equilibrium. A Nash equilibrium is a state in which no player can improve its objective by unilaterally changing its strategy. A Nash equilibrium is called pure if all players choose a pure strategy, and mixed if players choose probability distributions over strategies. Of special interest are fully mixed Nash equilibria where each player chooses each strategy with non-zero probability. Rosenthal [12] introduced a special class of strategic games, now widely known as congestion games. Here, the strategy set of each player is a subset of the power set of given resources. The players share a private objective function, defined as the sum (over their chosen resources) of functions in the number of players sharing this resource. In his seminal work, Rosenthal showed with help of a potential function that congestion games (in sharp contrast to general strategic games) always admit at least one pure Nash equilibrium. An extension to congestion games are weighted congestion games, in which the players have weights and thus different influence on the congestion of the resources. In (weighted) network congestion games the strategy sets of the players correspond to paths in a network. In order to measure the degradation of social welfare due to the selfish behavior of the players, Koutsoupias and Papadimitriou [8] introduced a global objective function, usually coined as social cost. They defined the price of anarchy, also called coordination ratio, as the worst-case ratio between the value of social cost in a Nash equilibrium and that of some social optimum. Thus the coordination ratio measures the extent to which non-cooperation approximates cooperation. As a starting point for studying the coordination ratio, Koutsoupias and Papadimitriou considered a very simple weighted network congestion game, now known as KP-model. Here, the network consists of a single source and a single destination which are connected by parallel links. The load on a link is the total weight of players assigned to this link. Associated with each link is a capacity representing the rate at which the link processes load. Each of the players selfishly routes from the source to the destination by choosing a probability distribution over the links. The private objective function of a player is defined as its expected latency. In the KP-model the social cost is defined as the expected maximum latency on a link, where the expectation is taken over all random choices of the players. In this paper, we give a thorough survey on the most exciting results on finite (weighted) congestion games and on special classes which are related to the KP-model. In particular, we review the findings on the existence and computational complexity of pure Nash equilibria. Furthermore, we discuss results on the price of anarchy. Last but not least, we survey known facts on fully mixed Nash equilibria.

We look at the scenario of having to route a continuous rate of traffic from a source node to a sink node in a network, where the objective is to maximize throughput. This is of interest, e.g., for providers of streaming content in communication networks. The overall path latency, which was relevant in other non-cooperative network routing games such as the classic Wardrop model, is of lesser concern here.To that end, we define bottleneck games with splittable traffic where the throughput on a path is inversely proportional to the maximum latency of an edge on that very path—the bottleneck latency. Therefore, we define a Wardrop equilibrium as a traffic distribution where this bottleneck latency is at minimum on all used paths. As a measure for the overall system well-being—called social cost—we take the weighted sum of the bottleneck latencies of all paths.Our main findings are as follows: First, we prove social cost of Wardrop equilibria on series parallel graphs to be unique. Even more, for any graph whose subgraph induced by all simple start-destination paths is not series parallel, there exist games having equilibria with different social cost. For the price of stability, we give an independence result with regard to the network topology. Finally, our main result is giving a new exact price of stability for Wardrop/bottleneck games on parallel links with M/M/1 latency functions. This result is at the same time the exact price of stability for bottleneck games on general graphs.KeywordsNash EquilibriumSocial CostSink NodeLatency FunctionMaximum LatencyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study rerouting policies in a dynamic round-based variant of a well known game theoretic traffic model due to Wardrop. Previous analyses (mostly in the context of selfish routing) based on Wardrop's model focus mostly on the static analysis of equilibria. In this paper, we ask the question whether the population of agents responsible for routing the traffic can jointly compute or better learn a Wardrop equilibrium efficiently. The rerouting policies that we study are of the following kind. In each round, each agent samples an alternative routing path and compares the latency on this path with its current latency. If the agent observes that it can improve its latency then it switches with some probability depending on the possible improvement to the better path.We can show various positive results based on a rerouting policy using an adaptive sampling rule that implicitly amplifies paths that carry a large amount of traffic in the Wardrop equilibrium. For general asymmetric games, we show that a simple replication protocol in which agents adopt strategies of more successful agents reaches a certain kind of bicriteria equilibrium within a time bound that is independent of the size and the structure of the network but only depends on a parameter of the latency functions, that we call the relative slope. For symmetric games, this result has an intuitive interpretation: Replication approximately satisfies almost everyone very quickly.In order to achieve convergence to a Wardrop equilibrium besides replication one also needs an exploration component discovering possibly unused strategies. We present a sampling based replication-exploration protocol and analyze its convergence time for symmetric games. For example, if the latency functions are defined by positive polynomials in coefficient representation, the convergence time is polynomial in the representation length of the latency functions. To the best of our knowledge, all previous results on the speed of convergence towards Wardrop equilibria, even when restricted to linear latency functions, were pseudopolynomial.In addition to the upper bounds on the speed of convergence, we can also present a lower bound demonstrating the necessity of adaptive sampling by showing that static sampling methods result in a slowdown that is exponential in the size of the network. A further lower bound illustrates that the relative slope is, in fact, the relevant parameter that determines the speed of convergence.

We investigate the efficiency of some Stackelberg strategies in congestion games with affine latency functions. A Stackelberg strategy is an algorithm that chooses a subset of players and assigns them a prescribed strategy with the purpose of mitigating the detrimental effect that the selfish behavior of the remaining uncoordinated players may cause to the overall performance of the system. The efficiency of a Stackelberg strategy is measured in terms of the price of anarchy of the pure Nash equilibria they induce. Three Stackelberg strategies, namely Largest Latency First, Cover and Scale, were already considered in the literature and non-tight upper and lower bounds on their price of anarchy were given. We reconsider these strategies and provide the exact bound on the price of anarchy of both Largest Latency First and Cover and a better upper bound on the price of anarchy of Scale.