Taxes for Linear Atomic Congestion Games

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 give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer $d$ we construct rather simple games with cost functions of degree at most $d$ which have a PoS of at least $\varOmega(Φ_d)^{d+1}$, where $Φ_d\sim d/\ln d$ is the unique positive root of equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $Φ_d^{d+1}$. Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we provide a lower bound of $\varOmega((1+1/α)^d/d)$ on the PoS of $α$-approximate Nash equilibria for singleton games. All our lower bounds hold for mixed and correlated equilibria as well. On the positive side, we give a general upper bound on the PoS of $α$-approximate Nash equilibria, which is sensitive to the range $W$ of the player weights and the approximation parameter $α$. We do this by explicitly constructing a novel approximate potential function, based on Faulhaber's formula, that generalizes Rosenthal's potential in a continuous, analytic way. From the general theorem, we deduce two interesting corollaries. First, we derive the existence of an approximate pure Nash equilibrium with PoS at most $(d+3)/2$; the equilibrium's approximation parameter ranges from $\varTheta(1)$ to $d+1$ in a smooth way with respect to $W$. Secondly, we show that for unweighted congestion games, the PoS of $α$-approximate Nash equilibria is at most $(d+1)/α$.

How can we design mechanisms to promote efficient use of shared resources? Here, we answer this question in relation to the well-studied class of atomic congestion games, used to model a variety of problems, including traffic routing. Within this context, a methodology for designing tolling mechanisms that minimize the system inefficiency (price of anarchy) exploiting solely local information is so far missing in spite of the scientific interest. In this article, we resolve this problem through a tractable linear programming formulation that applies to and beyond polynomial congestion games. When specializing our approach to the polynomial case, we obtain tight values for the optimal price of anarchy and corresponding tolls, uncovering an unexpected link with load balancing games. We also derive optimal tolling mechanisms that are constant with the congestion level, generalizing the results of Caragiannis et al. [8] to polynomial congestion games and beyond. Finally, we apply our techniques to compute the efficiency of the marginal cost mechanism. Surprisingly, optimal tolling mechanism using only local information perform closely to existing mechanism that utilize global information, e.g., Bilò and Vinci [6], while the marginal cost mechanism, known to be optimal in the continuous-flow model, has lower efficiency than that encountered levying no toll. All results are tight for pure Nash equilibria and extend to coarse correlated equilibria.

We characterize the price of anarchy in weighted congestion games, as a function of the allowable resource cost functions. Our results provide as thorough an understanding of this quantity as is already known for nonatomic and unweighted congestion games, and take the form of universal (cost function-independent) worst-case examples. One noteworthy byproduct of our proofs is the fact that weighted congestion games are “tight”, which implies that the worst-case price of anarchy with respect to pure Nash, mixed Nash, correlated, and coarse correlated equilibria are always equal (under mild conditions on the allowable cost functions). Another is the fact that, like nonatomic but unlike atomic (unweighted) congestion games, weighted congestion games with trivial structure already realize the worst-case POA, at least for polynomial cost functions.We also prove a new result about unweighted congestion games: the worst-case price of anarchy in symmetric games is, as the number of players goes to infinity, as large as in their more general asymmetric counterparts.KeywordsCost FunctionNash EquilibriumCongestion GamePure Nash EquilibriumSymmetric GameThese 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.

Almost all convergence results from each player adopting specific ``no-regret learning algorithms such as multiplicative updates or the more general mirror-descent algorithms in repeated games are only known in the more generous information model, in which each player is assumed to have access to the costs of all possible choices, even the unchosen ones, at each time step. This assumption in general may seem too strong, while a more realistic one is captured by the bandit model, in which each player at each time step is restricted to know only the cost of her currently chosen path, but not any of the unchosen ones. Can convergence still be achieved in such a more challenging bandit model? We answer this question positively. While existing bandit algorithms do not seem to work here, we develop a new family of bandit algorithms based on the mirror-descent algorithm with such a guarantee in atomic congestion games.

In this work we study weighted network congestion games with player-specific 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, 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. For several generalizations of these routing games, we show that such potential functions do not exist. We prove tight upper and lower bounds on the price of anarchy for games with polynomial latency functions. All our results on the price of anarchy translate to general congestion games.

Unlike standard congestion games, weighted congestion games and congestion games with player-specific delay functions do not necessarily possess pure Nash equilibria. It is known, however, that there exist pure equilibria for both of these variants in the case of singleton congestion games, i. e., if the players' strategy spaces contain only sets of cardinality one. In this paper, we investigate how far such a property on the players' strategy spaces guaranteeing the existence of pure equilibria can be extended. We show that both weighted and player-specific congestion games admit pure equilibria in the case of matroid congestion games, i. e., if the strategy space of each player consists of the bases of a matroid on the set of resources. We also show that the matroid property is the maximal property that guarantees pure equilibria without taking into account how the strategy spaces of different players are interweaved. In the case of player-specific congestion games, our analysis of matroid games also yields a polynomial time algorithm for computing pure equilibria.

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 study the inefficiency of equilibria for congestion games when players are (partially) altruistic. We model altruistic behavior by assuming that player i 's perceived cost is a convex combination of α i times his direct cost and α i times the social cost. Tuning the parameters α i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study primarily altruistic extensions of (atomic and nonatomic) congestion games, but also obtain some results on fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concepts. Thereto, we suitably adapt the smoothness notion introduced by Roughgarden and show that it captures the essential properties to determine the robust price of anarchy of these games. Our bounds show that for atomic congestion games and cost-sharing games, the robust price of anarchy gets worse with increasing altruism, while for valid utility games, it remains constant and is not affected by altruism. However, the increase in the price of anarchy is not a universal phenomenon: For general nonatomic congestion games with uniform altruism, the price of anarchy improves with increasing altruism. For atomic and nonatomic symmetric singleton congestion games, we derive bounds on the pure price of anarchy that improve as the average level of altruism increases. (For atomic games, we only derive such bounds when cost functions are linear.) Since the bounds are also strictly lower than the robust price of anarchy, these games exhibit natural examples in which pure Nash equilibria are more efficient than more permissive notions of equilibrium.

We study convergence of ?-Nash dynamics in congestion games when delay functions of all resources are similar. Delay functions are said to be similar if their values are within a polynomial factor at every congestion level. We show that for any ?> 0, an ?-Nash dynamics in symmetric congestion games with similar resources converge in steps polynomial in 1/? and the number of players are resources, yielding an FPTAS. Our result can be contrasted with that of Chien and Sinclair [3], which showed polynomial convergence result for symmetric congestion games where the delay functions have polynomially bounded jumps. Our assumption of similar delay functions is orthogonal to that of bounded jumps in that neither assumption implies the other. Our convergence result also hold for several natural variants of ?-Nash dynamics, including the most general polynomial liveness dynamics, where each player is given a chance to move frequently enough. We also extend our positive results to give an FPTAS for computing equilibrium in asymmetric games with similar resources, in which players share k distinct strategy spaces for any constant k. We complement our positive results by showing that computing an exact pure Nash equilibrium in symmetric congestion game with similar resources is PLS-complete. Furthermore, we show that for any ?> 0, all sequences of ?-Nash dynamics takes exponential steps to reach an approximate equilibrium in general congestion games with similar resources, as well as in symmetric congestion games with two groups of similar resources.

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.

Leadership in singleton congestion games: What is hard and what is easy

We investigate the speed of convergence of best response dynamics to approximately optimal solutions in weighted congestion games with polynomial delay functions. Awerbuch et al. (Fast convergence to nearly optimal solutions in potential games. ACM Conference on Electronic Commerce, 2008) showed that the convergence time of such dynamics to Nash equilibrium may be exponential in the number of players n even for unweighted players and linear delay functions. Nevertheless, we show that Θ(n log log W) (where W is the sum of all the players’ weights) best responses are necessary and sufficient to achieve states that approximate the optimal solution by a constant factor, under the assumption that every O(n) steps each player performs a constant (and non-null) number of best responses. For congestion games in which computing a best response is a polynomial time solvable problem, such a dynamics naturally implies a polynomial time distributed algorithm for the problem of computing the social optimum in congestion games, approximating the optimal solution by a constant factor.

Among other solution concepts, the notion of the pure Nash equilibrium plays a central role in Game Theory. Pure Nash equilibria in a game characterize situations with non-cooperative deterministic players in which no player has any incentive to unilaterally deviate from the current situation in order to achieve a higher payoff. Unfortunately, it is well known that there are games that do not have pure Nash equilibria. Furhermore, even in games where the existence of equilibria is guaranteed, their computation can be a computationally hard task. Such negative results significantly question the importance of pure Nash equilibria as solution concepts that characterize the behavior of rational players. Approximate pure Nash equilibria, which characterize situations where no player can significantly improve her payoff by unilaterally deviating from her current strategy, could serve as alternative solution concepts provided that they exist and can be computed efficiently. In this letter, we discuss recent positive algorithmic results for approximate pure Nash equilibria in congestion games.

We study the effect of combining selfishness and altruism in atomic congestion games. We allow players to be partially altruistic and partially selfish and determine the impact of this behavior on the overall system performance. Surprisingly, our results indicate that, in general, by allowing players to be (even partially) altruistic, the overall system performance deteriorates. Instead, for the class of symmetric load balancing games, a balance between selfish and altruistic behavior improves system performance to optimality.KeywordsNash EquilibriumLatency FunctionAltruistic BehaviorOptimal AssignmentCongestion GameThese 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.