Abstract: We consider selfishrouting over a network consisting of m parallellinks through which $n$ selfish users route their traffic trying tominimize their own expected latency. We study the class of mixedstrategies in which the expected latency through each link is at mosta constant multiple of the optimum maximum latency had globalregulation been available. For the case of uniform links it is knownthat all Nash equilibria belong to this class of strategies. We areinterested in bounding the coordination ratio (or price of anarchy) ofthese strategies defined as the worst-case ratio of the maximum (overall links) expected latency over the optimum maximum latency. The loadbalancing aspect of the problem immediately implies a lower boundO(ln m ln ln m) of the coordinationratio. We give a tight (up to a multiplicative constant) upper bound.To show the upper bound, we analyze a variant of the classical ballsand bins problem, in which balls with arbitrary weights are placedinto bins according to arbitrary probability distributions. At theheart of our approach is a new probabilistic tool that we call ballfusion; this tool is used to reduce the variant of the problem whereballs bear weights to the classical version (with no weights). Ballfusion applies to more general settings such as links with arbitrarycapacities and other latency functions.

Abstract: In large scale networks users often behave selfishly trying to minimize their routing cost. Modelling this as a noncooperative game, may yield a Nash equilibrium with unboundedly poor network performance. To measure this inefficacy, the Coordination Ratio or Price of Anarchy (PoA) was introduced. It equals the ratio of the cost induced by the worst Nash equilibrium, to the corresponding one induced by the overall optimum assignment of the jobs to the network. On improving the PoA of a given network, a series of papers model this selfish behavior as a Stackelberg or Leader-Followers game.
We consider random tuples of machines, with either linear or M/M/1 latency functions, and PoA at least a tuning parameter c. We validate a variant (NLS) of the Largest Latency First (LLF) Leaderrsquos strategy on tuples with PoA ge c. NLS experimentally improves on LLF for systems with inherently high PoA, where the Leader is constrained to control low portion agr of jobs. This suggests even better performance for systems with arbitrary PoA. Also, we bounded experimentally the least Leaderrsquos portion agr0 needed to induce optimum cost. Unexpectedly, as parameter c increases the corresponding agr0 decreases, for M/M/1 latency functions. All these are implemented in an extensive Matlab toolbox.

Abstract: We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfishrouting game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + epsi, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical.
Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + epsi. Furthermore, we prove that for any arbitrary integer k > 0, it is -hard to decide whether or not any given allocation of users to links can be transformed into a pure Nash equilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is -hard to approximate the worst social cost within a multiplicative factor . The quantity is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.

Abstract: In routing games, the network performance at equilibrium can be significantly improved if we remove some edges from the network. This counterintuitive fact, widely known as Braess's paradox, gives rise to the (selfish) network design problem, where we seek to recognize routing games suffering from the paradox, and to improve the equilibrium performance by edge removal. In this work, we investigate the computational complexity and the approximability of the network design problem for non-atomic bottleneck routing games, where the individual cost of each player is the bottleneck cost of her path, and the social cost is the bottleneck cost of the network. We first show that bottleneck routing games do not suffer from Braess's paradox either if the network is series-parallel, or if we consider only subpath-optimal Nash flows. On the negative side, we prove that even for games with strictly increasing linear latencies, it is NP-hard not only to recognize instances suffering from the paradox, but also to distinguish between instances for which the Price of Anarchy (PoA) can decrease to 1 and instances for which the PoA is as large as \Omega(n^{0.121}) and cannot improve by edge removal. Thus, the network design problem for such games is NP-hard to approximate within a factor of O(n^{0.121-\eps}), for any constant \eps > 0. On the positive side, we show how to compute an almost optimal subnetwork w.r.t. the bottleneck cost of its worst Nash flow, when the worst Nash flow in the best subnetwork routes a non-negligible amount of flow on all used edges. The running time is determined by the total number of paths, and is quasipolynomial when the number of paths is quasipolynomial.

Abstract: We study extreme Nash equilibria in the context of a selfishrouting game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.
We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1+ɛ) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.
Considering pure Nash equilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is View the MathML source-hard to approximate the worst social cost within a multiplicative factor better than 2-2/(m+1).

Abstract: We study extreme Nash equilibria in the context of a selfishrouting game. Specifically, we assume a collection of n users, each employing a mixed strategy, which is a probability distribution over m parallel identical links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic on those links that minimize its expected latency cost. The social cost of a Nash equilibrium is the expectation, over all random choices of the users, of the maximum, over all links, latency through a link.We provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria. Furthermore, we show, that under a certain condition, the social cost of any Nash equilibrium is within a factor of 2h(1 + {\aa}) of that of the fully mixed Nash equilibrium, where h is the factor by which the largest user traffic deviates from the average user traffic.Considering pure Nash equilibria, we provide a PTAS to approximate the best social cost, we give an upper bound on the worst social cost and we show that it is N P-hard to approximate the worst social cost within a multiplicative factor better than 2 - 2/(m + 1).

Abstract: 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 [E. Koutsoupias, C. Papadimitriou, Worst-case equilibria, in: 16th Annual Symposium on Theoretical Aspects of Computer Science, STACS, vol. 1563, 1999, pp. 404-413; T. Roughgarden, E. Tardos, How bad is selfishrouting? in: 41st IEEE Annual Symposium of Foundations of Computer Science, FOCS, 2000, pp. 93-102]. A Leader can decrease the coordination ratio by assigning flow {\'a}r on M, and then all Followers assign selfishly the (1-{\'a})r remaining flow. This is a Stackelberg Scheduling Instance(M,r,{\'a}),0≤{\'a}≤1. It was shown [T. Roughgarden, Stackelberg scheduling strategies, in: 33rd Annual Symposium on Theory of Computing, STOC, 2001, pp. 104-113] that it is weakly NP-hard to compute the optimal Leader's strategy. For any such network M we efficiently compute the minimum portion @b"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,@a>=@b"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 {\'a}r flow can induce ≤1/{\'a} 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 [A. Kaporis, P. Spirakis, The price of optimum in stackelberg games on arbitrary single commodity networks and latency functions, in: 18th annual ACM symposium on Parallelism in Algorithms and Architectures, SPAA, 2006, pp. 19-28]. Some preliminary results have also appeared as technical report in [A.C. Kaporis, E. Politopoulou, P.G. Spirakis, The price of optimum in stackelberg games, in: Electronic Colloquium on Computational Complexity, ECCC, (056), 2005].

Abstract: We study the problem of routing traffic through a congested network. We focus on the simplest case of a network consisting of m parallel links. We assume a collection of n network users; each user employs a mixed strategy, which is a probability distribution over links, to control the shipping of its own assigned traffic. Given a capacity for each link specifying the rate at which the link processes traffic, the objective is to route traffic so that the maximum (over all links) latency is minimized. We consider both uniform and arbitrary link capacities. How much decrease in global performace is necessary due to the absence of some central authority to regulate network traffic and implement an optimal assignment of traffic to links? We investigate this fundamental question in the context of Nash equilibria for such a system, where each network user selfishly routes its traffic only on those links available to it that minimize its expected latency cost, given the network congestion caused by the other users. We use the Coordination Ratio, originally defined by Koutsoupias and Papadimitriou, as a measure of the cost of lack of coordination among the users; roughly speaking, the Coordination Ratio is the ratio of the expectation of the maximum (over all links) latency in the worst possible Nash equilibrium, over the least possible maximum latency had global regulation been available. Our chief instrument is a set of combinatorial Minimum Expected Latency Cost Equations, one per user, that characterize the Nash equilibria of this system. These are linear equations in the minimum expected latency costs, involving the user traffics, the link capacities, and the routing pattern determined by the mixed strategies. In turn, we solve these equations in the case of fully mixed strategies, where each user assigns its traffic with a strictly positive probability to every link, to derive the first existence and uniqueness results for fully mixed Nash equilibria in this setting. Through a thorough analysis and characterization of fully mixed Nash equilibria, we obtain tight upper bounds of no worse than O(ln n/ln ln n) on the Coordination Ratio for (i) the case of uniform capacities and arbitrary traffics and (ii) the case of arbitrary capacities and identical traffics.

Abstract: In this work, we study the combinatorial structure and the
computational complexity of Nash equilibria for a certain game that
models selfishrouting over a network consisting of m parallel links. We
assume a collection of n users, each employing a mixed strategy, which
is a probability distribution over links, to control the routing of its own
assigned traffic. In a Nash equilibrium, each user selfishly routes its traffic
on those links that minimize its expected latency cost, given the network
congestion caused by the other users. The social cost of a Nash equilibrium
is the expectation, over all random choices of the users, of the
maximum, over all links, latency through a link.
We embark on a systematic study of several algorithmic problems related
to the computation of Nash equilibria for the selfishrouting game we consider.
In a nutshell, these problems relate to deciding the existence of a
Nash equilibrium, constructing a Nash equilibrium with given support
characteristics, constructing the worst Nash equilibrium (the one with
maximum social cost), constructing the best Nash equilibrium (the one
with minimum social cost), or computing the social cost of a (given) Nash
equilibrium. Our work provides a comprehensive collection of efficient algorithms,
hardness results (both as NP-hardness and #P-completeness
results), and structural results for these algorithmic problems. Our results
span and contrast a wide range of assumptions on the syntax of the
Nash equilibria and on the parameters of the system.

Abstract: A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfishrouting for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium – the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou [9].
More specifically, we continue the study of the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity, methodologies to establish, and limitations of, the FMNE Conjecture.