Stronger Bounds on Braess's Paradox and the Maximum Latency of Selfish Routing (original) (raw)
Related papers
Price of Anarchy in Networks with Heterogeneous Latency Functions
Mathematics of Operations Research
We address the performance of selfish network routing in multi-commodity flows where the latency or delay function on edges is dependent on the flow of individual commodities, rather than on the aggregate flow. An application of this study is the analysis of a network with differentiated traffic, i.e., in transportation networks where there are multiple types of traffic and in networks where traffic is prioritized according to type classification. We consider the inefficiency of equilibrium in this model and provide price of anarchy bounds for networks with k (types of) commodities where each link is associated with heterogeneous polynomial delays, i.e., commodity i on edge e faces delay specified by h e i (f 1 (e), f 2 (e),. .. , f k (e)) where f i (e) is the flow of the ith commodity through edge e and h e i () a polynomial delay function applicable to the ith commodity. We consider both atomic and non-atomic flows and show bounds on the price of anarchy that depend on the relative impact of each type of traffic on the edge delay where the delay functions are polynomials of degree θ, e.g., i a i f i (e) θ. The price of anarchy is unbounded for arbitrary polynomials. For networks with decomposable delay functions where the delay is the same for all commodities using the edge, i.e., delays on edge e are defined by h e (f 1 (e), f 2 (e),. .. , f k (e)), we show improved bounds on the price of anarchy, for both non-atomic and atomic flows. The results illustrate that the inefficiency of selfish routing worsens in the case of heterogeneous delays as compared to the standard delay functions that do not consider type differentiation.
Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability
Lecture Notes in Computer Science, 2005
We give the first analyses in multicommodity networks of both the worst-case severity of Braess's Paradox and the price of anarchy of selfish routing with respect to the maximum latency. Our first main result is a construction of an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both of these quantities grow exponentially with the size of the network. This construction has wide implications, and demonstrates that numerous existing analyses of selfish routing in single-commodity networks have no analogues in multicommodity networks, even in those with only two commodities. This dichotomy between single-and twocommodity networks is arguably quite unexpected, given the negligible dependence on the number of commodities of previous work on selfish routing. Our second main result is an exponential upper bound on the worst-possible severity of Braess's Paradox and on the price of anarchy for the maximum latency, which essentially matches the lower bound when the number of commodities is constant. Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability: while there is a polynomial-time algorithm with an exponential approximation ratio, subexponential approximation is unachievable in polynomial time (assuming P = N P ).
Selfish Network Creation with Non-uniform Edge Cost
ArXiv, 2017
Network creation games investigate complex networks from a game-theoretic point of view. Based on the original model by Fabrikant et al. [PODC'03] many variants have been introduced. However, almost all versions have the drawback that edges are treated uniformly, i.e. every edge has the same cost and that this common parameter heavily influences the outcomes and the analysis of these games. We propose and analyze simple and natural parameter-free network creation games with non-uniform edge cost. Our models are inspired by social networks where the cost of forming a link is proportional to the popularity of the targeted node. Besides results on the complexity of computing a best response and on various properties of the sequential versions, we show that the most general version of our model has constant Price of Anarchy. To the best of our knowledge, this is the first proof of a constant Price of Anarchy for any network creation game.
Selfish Routing in Capacitated Networks
Mathematics of Operations Research, 2004
According to Wardrop's first principle, agents in a congested network choose their routes selfishly, a behavior that is captured by the Nash equilibrium of the underlying noncooperative game. A Nash equilibrium does not optimize any global criterion per se, and so there is no apparent reason why it should be close to a solution of minimal total travel time, i.e., the system optimum. In this paper, we offer positive results on the efficiency of Nash equilibria in traffic networks. In contrast to prior work, we present results for networks with capacities and for latency functions that are nonconvex, nondifferentiable, and even discontinuous. The inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows. In this more general model, multiple Nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient. Nonetheless, our main result shows that the best equilibrium is as...
Pricing Networks with Selfish Routing
2003
such a "selfishly motivated" assignment of traffic to paths (a Nash equilibrium) does not minimize the total latency; put differently, the outcome of selfish behavior can be improved upon with coordination.
Near-optimal network design with selfish agents
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 2003
We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent's goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1 + ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65 + ε)-approximate Nash equilibrium that does not cost much more.
Upper Bounding the Price of Anarchy in Atomic Splittable Selfish Routing
Abstract. Selfish behavior of nodes of a network such as sensors of a geographically distributed sensor network, each of which owned and operated by a different stakeholder may lead to a game theoretic setting called “selfish routing”. The fact that every node strictly aims at maximizing its own utility can cause degradations of social welfare. An issue of concern would be the quantitative measure of this inefficiency.
The Price of Anarchy for Restricted Parallel Links
Parallel Processing Letters, 2006
In the model of restricted parallel links, n users must be routed on m parallel links under the restriction that the link for each user be chosen from a certain set of allowed links for the user. In a (pure) Nash equilibrium, no user may improve its own Individual Cost (latency) by unilaterally switching to another link from its set of allowed links. The Price of Anarchy is a widely adopted measure of the worst-case loss (relative to optimum) in system performance (maximum latency) incurred in a Nash equilibrium. In this work, we present a comprehensive collection of bounds on Price of Anarchy for the model of restricted parallel links and for the special case of pure Nash equilibria. Specifically, we prove: • For the case of identical users and identical links, the Price of Anarchy is [Formula: see text]. • For the case of identical users, the Price of Anarchy is [Formula: see text]. • For the case of identical links, the Price of Anarchy is [Formula: see text], which is asymptotic...
Theoretical Computer Science, 2009
Let M be a single s-t network of parallel links with load dependent latency functions shared by an infinite number of selfish users. This may yield a Nash equilibrium with unbounded Coordination Ratio [23, 43]. A Leader can decrease the coordination ratio by assigning flow αr on M , and then all Followers assign selfishly the (1 − α)r remaining flow. This is a Stackelberg Scheduling Instance (M, r, α), 0 ≤ α ≤ 1. It was shown [38] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion β M of flow r > 0 needed by a Leader to induce M 's optimum cost, as well as her optimal strategy. This shows that the optimal Leader's strategy on instances (M, r, α ≥ β M) is in P. Unfortunately, Stackelberg routing in more general nets can be arbitrarily hard. Roughgarden presented a modification of Braess's Paradox graph, such that no strategy controlling αr flow can induce ≤ 1 α times the optimum cost. However, we show that our main result also applies to any s-t net G. We take care of the Braess's graph explicitly, as a convincing example. Finally, we extend this result to k commodities. A conference version of this paper has appeared in [16]. Some preliminary results have also appeared as technical report in [18].
Network uncertainty in selfish routing
Proceedings 20th IEEE International Parallel & Distributed Processing Symposium, 2006
We study the problem of selfish routing in the presence of incomplete network information. Our model consists of a number of users who wish to route their traffic on a network of m parallel links with the objective of minimizing their latency. However, in doing so, they face the challenge of lack of precise information on the capacity of the network links. This uncertainty is modelled via a set of probability distributions over all the possibilities, one for each user. The resulting model is an amalgamation of the KP-model of [13] and the congestion games with user-specific functions of [17]. We embark on a study of Nash equilibria and the price of anarchy in this new model. In particular, we propose polynomial-time algorithms for computing some special cases of pure Nash equilibria and we show that negative results of [17], for the non-existence of pure Nash equilibria in the case of three users, do not apply to our model. Consequently, we propose an interesting open problem in this area, that of the existence of pure Nash equilibria in the general case of our model. Furthermore, we consider appropriate notions for the social cost and the price of anarchy and obtain upper bounds for the latter. With respect to fully mixed Nash equilibria, we propose a method to compute them and show that when they exist they are unique. Finally we prove that the fully mixed Nash equilibrium maximizes the social welfare.