Weighted Congestion Games Research Articles

A Unifying Approximate Potential for Weighted Congestion Games

A Unifying Approximate Potential for Weighted Congestion Games

Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

We study the existence of approximate pure Nash equilibria (α-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree d. Previously, it was known that d-PNE always exist, whereas nonexistence was established only for small constants, namely, for 1.153-PNE. We improve significantly upon this gap, proving that such games in general do not have [Formula: see text]-PNE, which provides the first superconstant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, by deploying this technique, we are able to show that deciding whether a weighted congestion game has an [Formula: see text]-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest, and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of α-PNE, in which a certain set of players plays a specific strategy profile, is NP-hard for any [Formula: see text], even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players n. We show that n-PNE always exists, matched by an almost tight nonexistence bound of [Formula: see text], which we can again transform into an NP-completeness proof for the decision problem. Funding: Supported by the Alexander von Humboldt Foundation with funds from the German Federal Ministry of Education and Research (BMBF). D. Poças was also funded by FCT via LASIGE Research Unit, ref. UIDB/00408/2020 and ref. UIDP/00408/2020.

On the robustness of the approximate price of anarchy in generalized congestion games

One of the main results shown through Roughgarden's notions of smooth games and Robust Price of Anarchy is that, for any sum-bounded utilitarian social function, the worst-case Price of Anarchy of Coarse Correlated Equilibria coincides with that of Pure Nash Equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions, and for a broad generalization of the class of weighted congestion games with non-negative (and possibly decreasing) latency functions, the worst-case Price of Anarchy of ϵ-approximate Coarse Correlated Equilibria still coincides with that of ϵ-approximate Pure Nash Equilibria, for any ϵ≥0. As a byproduct of our proof, it also follows that such a value can always be determined by making use of the primal-dual method we introduced in a previous work.

Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses

We present a deterministic polynomial-time algorithm for computing [Formula: see text]-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most [Formula: see text]. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algorithm by Caragiannis at al. [Caragiannis I, Fanelli A, Gravin N, Skopalik A (2011) Efficient computation of approximate pure Nash equilibria in congestion games. Ostrovsky R, ed. Proc. 52nd Annual Symp. Foundations Comput. Sci. (FOCS) (IEEE Computer Society, Los Alamitos, CA), 532–541; Caragiannis I, Fanelli A, Gravin N, Skopalik A (2015) Approximate pure Nash equilibria in weighted congestion games: Existence, efficient computation, and structure. ACM Trans. Econom. Comput. 3(1):2:1–2:32.], but we utilize an approximate potential function directly on the original game instead of an exact one on a modified game. A critical component of our analysis, which is of independent interest, is the derivation of a novel bound of [Formula: see text] for the price of anarchy (PoA) of [Formula: see text]-approximate equilibria in weighted congestion games, where [Formula: see text] is the Lambert-W function. More specifically, we show that this PoA is exactly equal to [Formula: see text], where [Formula: see text] is the unique positive solution of the equation [Formula: see text]. Our upper bound is derived via a smoothness-like argument, and thus holds even for mixed Nash and correlated equilibria, whereas our lower bound is simple enough to apply even to singleton congestion games.

On approximate pure Nash equilibria in weighted congestion games with polynomial latencies

We consider weighted congestion games with polynomial latency functions of maximum degree d≥1. For these games, we investigate the existence and efficiency of approximate pure Nash equilibria which are obtained through sequences of unilateral improvement moves by the players. By exploiting a simple technique, we firstly show that these games admit an infinite set of d-approximate potential functions. This implies that there always exists a d-approximate pure Nash equilibrium which can be reached through any sequence of d-approximate improvement moves by the players. As a corollary, we also obtain that, under mild assumptions on the structure of the players' strategies, these games also admit a constant approximate potential function. Secondly, using a simple potential function argument, we are able to show that a (d+δ)-approximate pure Nash equilibrium of cost at most (d+1)/(d+δ) times the cost of an optimal state always exists, for every δ∈[0,1].

Existence and Efficiency of Equilibria for Cost-Sharing in Generalized Weighted Congestion Games

This work studies the impact of cost-sharing methods on the existence and efficiency of (pure) Nash equilibria in weighted congestion games. We also study generalized weighted congestion games , where each player may control multiple commodities. Our results are fairly general; we only require that our cost-sharing method and our set of cost functions satisfy certain natural conditions. For general weighted congestion games, we study the existence of pure Nash equilibria in the induced games, and we exhibit a separation from the standard single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees existence of pure Nash equilibria. With respect to efficiency, we present general tight bounds on the price of anarchy, which are robust and apply to general equilibrium concepts. Our analysis provides a tight bound on the price of anarchy, which depends only on the used cost-sharing method and the set of allowable cost functions. Interestingly, the same bound applies to weighted congestion games and generalized weighted congestion games. We then turn to the price of stability and prove an upper bound for the Shapley value cost-sharing method, which holds for general sets of cost functions and which is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we provide a somewhat surprising result, showing that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, for this case, the price of stability becomes as bad as the price of anarchy. Again, our bounds on the price of stability are independent on whether players are single or multi-commodity.

The Price of Stability of Weighted Congestion Games

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 $\varOmega(\Phi_d)^{d+1}$, where $\Phi_d\sim d/\ln d$ is the unique positive root of the equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $\Phi_d^{d+1}$. 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 $\varOmega((1+1/\alpha)^d/d)$ on the PoS of $\alpha$-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 $\alpha$-approximate Nash equilibria, which is sensitive to the range $W$ of the player weights and the approximation parameter $\alpha$. 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 $W$. Second, we show that for unweighted congestion games, the PoS of $\alpha$-approximate Nash equilibria is at most $(d+1)/\alpha$. Read More: https://epubs.siam.org/doi/10.1137/18M1207880

A Unifying Tool for Bounding the Quality of Non-Cooperative Solutions in Weighted Congestion Games

We present a general technique, based on a primal-dual formulation, for analyzing the quality of self-emerging solutions in weighted congestion games. With respect to traditional combinatorial approaches, the primal-dual schema has at least three advantages: first, it provides an analytic tool which can always be used to prove tight upper bounds for all the cases in which we are able to characterize exactly the polyhedron of the solutions under analysis; secondly, in each such a case the complementary slackness conditions give us an hint on how to construct matching lower bounding instances; thirdly, proofs become simpler and easy to check. For the sake of exposition, we first apply our technique to the problems of bounding the prices of anarchy and stability of exact and approximate pure Nash equilibria, as well as the approximation ratio of the solutions achieved after a one-round walk starting from the empty strategy profile, in the case of affine latency functions and we show how all the known upper bounds for these measures (and some of their generalizations) can be easily reobtained under a unified approach. Then, we use the technique to attack the more challenging setting of polynomial latency functions. In particular, we obtain the first known upper bounds on the price of stability of pure Nash equilibria and on the approximation ratio of the solutions achieved after a one-round walk starting from the empty strategy profile for unweighted players in the cases of quadratic and cubic latency functions.

Exploiting Congestion Games to Achieve Distributed Service Chaining in NFV Networks

The network function virtualization (NFV) paradigm has gained increasing interest in both academia and industry as it promises scalable and flexible network management and orchestration. In NFV networks, network services are provided as chains of different virtual network functions (VNFs), which are instantiated and executed on dedicated VNF-compliant servers. The problem of composing those chains is referred to as the service chain composition problem. In contrast to centralized solutions that suffer from scalability and privacy issues, in this paper, we leverage non-cooperative game theory to achieve a low-complexity distributed solution to the above-mentioned problem. Specifically, to account for selfish and competitive behavior of users, we formulate the service chain composition problem as an atomic weighted congestion game with unsplittable flows and player-specific cost functions. We show that the game possesses a weighted potential function and admits a Nash equilibrium (NE). We prove that the price of anarchy is upper-bounded, and also propose a distributed and privacy-preserving algorithm which provably converges toward an NE of the game in polynomial time. Finally, through extensive numerical results, we assess the performance of the proposed distributed solution to the service chain composition problem.

Restoring Pure Equilibria to Weighted Congestion Games

Congestion games model several interesting applications, including routing and network formation games, and also possess attractive theoretical properties, including the existence of and convergence of natural dynamics to a pure Nash equilibrium. Weighted variants of congestion games that rely on sharing costs proportional to players' weights do not generally have pure-strategy Nash equilibria. We propose a new way of assigning costs to players with weights in congestion games that recovers the important properties of the unweighted model. This method is derived from the Shapley value, and it always induces a game with a (weighted) potential function. For the special cases of weighted network cost-sharing and weighted routing games with Shapley value-based cost shares, we prove tight bounds on the worst-case inefficiency of equilibria. For weighted network cost-sharing games, we precisely calculate the price of stability for any given player weight vector, while for weighted routing games, we precisely calculate the price of anarchy, as a parameter of the set of allowable cost functions.

The complexity of pure equilibria in mix-weighted congestion games on parallel links

We revisit the simple class of weighted congestion games on parallel links[10], where each player has a non-negative weight and her cost on the link she chooses is the sum of the weights of all players choosing the link. We extend this class to mix-weighted congestion games on parallel links, where weights may as well be negative. For the resulting simple class, we study the complexity of deciding the existence of a pure equilibrium, where no player could unilaterally improve her cost by switching to another link.We show that even for a single negative weight, this decision problem is strongly NP-complete when the number of links is part of the input; the problem is NP-complete already for two links. When the number of links is a fixed constant, we show, through a pseudopolynomial, dynamic programming algorithm, that the problem is not strongly NP-complete unless P=NP; the algorithm works for any number of negative weights.

Approximate Pure Nash Equilibria in Weighted Congestion Games

We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ --approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ . Do such equilibria exist for small values of ρ ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ -games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5/2 + O γ for arbitrarily small γ > 0; for latency functions with maximum degree d ≥ 2, it is d 2 d + o ( d ). The running time is polynomial in the number of bits in the representation of the game and 1/ γ . As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO ( d 2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ -games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications.

Dynamic Spectrum Access in Multi-Channel Cognitive Radio Networks

In this paper, dynamic spectrum access (DSA) in multi-channel cognitive radio networks (CRNs) is studied. The two fundamental issues in DSA, spectrum sensing and spectrum sharing, for a general scenario are revisited, where the channels present different usage characteristics and the detection performance of individual secondary users (SUs) varies. First, spectrum sensing is investigated, where multiple SUs are coordinated to cooperatively sense the channels owned by the primary users (PUs) for different interests. When the PUs' interests are concerned, cooperative spectrum sensing is performed to better protect the PUs while satisfying the SUs' requirement on the expected access time. For the SUs' interests, the objective is to maximize the expected available time while keeping the interference to PUs under a predefined level. With the dynamics in the channel usage characteristics and the detection capacities, the coordination problems for the above two cases are formulated as nonlinear integer programming problems accordingly, which are proved to be NP-complete. To find the solution efficiently, for the former case, the original problem is transformed into a variant of convex bipartite matching problem by constructing a complete bipartite graph and defining proper weight vectors. Based on the problem transformation, a channel selection algorithm is proposed to compute the solution. For the latter case, the deterministic optimization problem is first transformed to an associated stochastic optimization problem, which is then solved by cross-entropy (CE) method of stochastic optimization. Then, the sharing of the available channels by SUs after sensing is modeled by a channel access game, based on the framework of weighted congestion game. An algorithm for SUs to select access channels to achieve Nash equilibrium (NE) is proposed. Simulation results are presented to validate the performance of the proposed algorithms.

Weighted Congestion Games

We characterize the Price of Anarchy (POA) 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 by-product 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 equilibria, mixed Nash equilibria, correlated equilibria, 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 large as in their more general asymmetric counterparts.

The Price of Routing Unsplittable Flow

In this paper we study the “price of for the general class of (weighted and unweighted) atomic “congestion with the sum of players' costs as the objective function. We show that for linear resource cost functions the price of anarchy is exactly $\frac{3 + \sqrt{5}}{2} \approx 2.618$ for weighted congestion games and exactly $2.5$ for unweighted congestion games. We show that for resource cost functions that are polynomials of degree $d$ the price of anarchy is $d^{\Theta(d)}$. Our results also hold for mixed strategies. In particular, these results apply to atomic routing games where the traffic demand from a source to a destination must be satisfied by choosing a single path between source and destination.

On the Existence of Pure Nash Equilibria in Weighted Congestion Games

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 continuous cost functions. We prove that a set 𝒞 of continuous functions is consistent for two-player games if and only if 𝒞 contains only monotonic functions and for all nonconstant functions c1, c2 ∈ 𝒞, there are constants a, b ∈ ℝ such that c1(x) = a c2(x) + b for all x ∈ ℝ≥0. For games with at least three players, we prove that 𝒞 is consistent if and only if exactly one of the following cases holds: (a) 𝒞 contains only affine functions; (b) 𝒞 contains only exponential functions such that c(x) = ac eϕx + bc for some ac, bc, ϕ ∈ ℝ, where ac and bc may depend on c, while ϕ must be equal for every c ∈ 𝒞. The latter characterization is even valid for three-player games. Finally, we derive several characterizations of consistency of cost functions for games with restricted strategy spaces, such as weighted network congestion games or weighted congestion games with singleton strategies.

On best response dynamics in weighted congestion games with polynomial delays

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.

The Impact of Social Ignorance on Weighted Congestion Games

We consider weighted linear congestion games, and investigate how social ignorance, namely lack of information about the presence of some players, affects the inefficiency of pure Nash equilibria (PNE) and the convergence rate of the e-Nash dynamics. To this end, we adopt the model of graphical linear congestion games with weighted players, where the individual cost and the strategy selection of each player only depends on his neighboring players in the social graph. We show that such games admit a potential function, and thus a PNE. Next, we investigate the Price of Anarchy (PoA) and the Price of Stability (PoS) of graphical linear congestion games with respect to the players’ total actual cost. Our main result is that the impact of social ignorance on the PoA and on the PoS is naturally quantified by the independence number α(G) of the social graph G. In particular, we show that the PoA grows roughly as α(G)(α(G)+2), which is essentially tight as long as α(G) does not exceed half the number of players, and that the PoS lies between α(G) and 2α(G). Moreover, we show that the e-Nash dynamics reaches an α(G)(α(G)+2)-approximate configuration in polynomial time that does not directly depend on the social graph. For unweighted graphical linear games with symmetric strategies, we show that the e-Nash dynamics reaches an e-approximate PNE in polynomial time that exceeds the corresponding time for symmetric linear games by a factor at most as large as the number of players.

Characterizing the Existence of Potential Functions in Weighted Congestion Games

Since the pioneering paper of Rosenthal a lot of work has been done in order to determine classes of games that admit a potential. First, we study the existence of potential functions for weighted congestion games. Let $\mathcal{C}$ be an arbitrary set of locally bounded functions and let $\mathcal{G}(\mathcal{C})$ be the set of weighted congestion games with cost functions in $\mathcal{C}$ . We show that every weighted congestion game $G\in\mathcal{G}(\mathcal{C})$ admits an exact potential if and only if $\mathcal{C}$ contains only affine functions. We also give a similar characterization for w-potentials with the difference that here $\mathcal{C}$ consists either of affine functions or of certain exponential functions. We finally extend our characterizations to weighted congestion games with facility-dependent demands and elastic demands, respectively.

Weighted congestion games with separable preferences

Players in a congestion game may differ from one another in their intrinsic preferences (e.g., the benefit they get from using a specific resource), their contribution to congestion, or both. In many cases of interest, intrinsic preferences and the negative effect of congestion are (additively or multiplicatively) separable. This paper considers the implications of separability for the existence of pure-strategy Nash equilibrium and the prospects of spontaneous convergence to equilibrium. It is shown that these properties may or may not be guaranteed, depending on the exact nature of player heterogeneity.