The asymptotic price of anarchy for k-uniform congestion games

We consider the atomic version of congestion games with affine cost functions, and analyze the quality of worst case Nash equilibria when the strategy spaces of the players are the set of bases of a k-uniform matroid. In this setting, for some parameter k, each player is to choose k out of a finite set of resources, and the cost of a player for choosing a resource depends affine linearly on the number of players choosing the same resource. Earlier work shows that the price of anarchy for this class of games is larger than 1.34 but at most 2.15. We determine a tight bound on the asymptotic price of anarchy equal to ≈1.35188. Here, asymptotic refers to the fact that the bound holds for all instances with sufficiently many players. In particular, the asymptotic price of anarchy is bounded away from 4/3. Our analysis also yields an upper bound on the price of anarchy <1.4131, for all instances

The Value of Information in Selfish Routing

Path selection by selfish agents has traditionally been studied by comparing social optima and equilibria in the Wardrop model, i.e., by investigating the Price of Anarchy in selfish routing. In this work, we refine and extend the traditional selfish-routing model in order to answer questions that arise in emerging path-aware Internet architectures. The model enables us to characterize the impact of different degrees of congestion information that users possess. Furthermore, it allows us to analytically quantify the impact of selfish routing, not only on users, but also on network operators. Based on our model, we show that the cost of selfish routing depends on the network topology, the perspective (users versus network operators), and the information that users have. Surprisingly, we show analytically and empirically that less information tends to lower the Price of Anarchy, almost to the optimum. Our results hence suggest that selfish routing has modest social cost even without the dissemination of path-load information.Comment: 27th International Colloquium on Structural Information and Communication Complexity (SIROCCO 2020

Nash Social Welfare in Selfish and Online Load Balancing

In load balancing problems there is a set of clients, each wishing to select a resource from a set of permissible ones, in order to execute a certain task. Each resource has a latency function, which depends on its workload, and a client's cost is the completion time of her chosen resource. Two fundamental variants of load balancing problems are {\em selfish load balancing} (aka. {\em load balancing games}), where clients are non-cooperative selfish players aimed at minimizing their own cost solely, and {\em online load balancing}, where clients appear online and have to be irrevocably assigned to a resource without any knowledge about future requests. We revisit both selfish and online load balancing under the objective of minimizing the {\em Nash Social Welfare}, i.e., the geometric mean of the clients' costs. To the best of our knowledge, despite being a celebrated welfare estimator in many social contexts, the Nash Social Welfare has not been considered so far as a benchmarking quality measure in load balancing problems. We provide tight bounds on the price of anarchy of pure Nash equilibria and on the competitive ratio of the greedy algorithm under very general latency functions, including polynomial ones. For this particular class, we also prove that the greedy strategy is optimal as it matches the performance of any possible online algorithm

Optimal routing of traffic flows with length restrictions in networks with congestion

When traffic flows are routed through a road network it is desirable to minimize the total road usage. Since a route guidance system can only recommend paths to the drivers, special care has to be taken not to route them over paths they perceive as too long. This leads in a simplified model to a nonlinear multicommodity flow problem with constraints on the available paths. In this article an algorithm for this problem is given, which combines the convex combinations algorithm by Frank and Wolfe with column generation and algorithms for the constrained shortest path problem. Computational results stemming from a cooperation with DaimlerChrysler are presented

Efficiency of equilibria in uniform matroid congestion games

Network routing games, and more generally congestion games play a central role in algorithmic game theory, comparable to the role of the traveling salesman problem in combinatorial optimization. It is known that the price of anarchy is independent of the network topology for non-atomic congestion games. In other words, it is independent of the structure of the strategy spaces of the players, and for affine cost functions it equals 4/3. In this paper, we show that the dependence of the price of anarchy on the network topology is considerably more intricate for atomic congestion games. More specifically, we consider congestion games with affine cost functions where the strategy spaces of players are symmetric and equal to the set of bases of a k-uniform matroid. In this setting, we show that the price of anarchy is strictly larger than the price of anarchy for singleton strategy spaces where the latter is 4/3. As our main result we show that the price of anarchy can be bounded from above by 28/13. This constitutes a substantial improvement over the price of anarchy bound 5/2, which is known to be tight for arbitrary network routing games with affine cost functions