A seminal result of [Fleischer et al., 2004], [Karakostas and Kolliopoulos, 2004] and [Yang and Huang, 2004] states that system optimal multi-commodity static network flows are always implementable as tolled Wardrop equilibrium flows even if users have heterogeneous value-of-time sensitivities. Their proof uses LP-duality to characterize the general implementability of network flows by tolls. For the much more complex setting of dynamic flows, [Graf et al., 2025] identified necessary and sufficient conditions for a dynamic s-d flow to be implementable as a tolled dynamic equilibrium. They used the machinery of (infinite-dimensional) strong duality to obtain their characterizations. Their work, however, does not answer the question of whether system optimal dynamic network flows are implementable by tolls.

We consider dynamic network flows and study the following question: Which dynamic edge flows can be implemented as tolled dynamic equilibrium flows? We study this question for the “heterogeneous-user” model, where the flow particles are partitioned into populations having different valuations of travel time and money spent. As our main result, we give the first characterization of this type of implementability showing that for single-source single-destination networks and heterogeneous users, a dynamic edge flow is implementable by tolls if and only if the induced subgraph of the edge flow contains no cycle of positive length containing the destination. For the proof of this result we make several technical contributions: We formulate a novel infinite dimensional optimization problem, where the goal is to minimize the weighted travel times with respect to the fixed network loading induced by the given edge flow. Using the recently introduced concept of parameterized network loadings (cf. [23]), we prove existence of optimal solutions, strong duality, and a characterization of special optimal solutions for which an inequality is tight. These results are then all used for the proof of the above mentioned main characterization.

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We study a market game with n ≥ 2 players competing over m ≥ 1 divisible resources of different finite capacities. Resources are traded via the proportional sharing mechanism, where players are price-anticipating, meaning that they can influence the prices with their bids. Additionally, each player has an initial endowment of the resources which are sold at market prices. Although the players’ total profit functions may be discontinuous in the bids, we prove existence and uniqueness of pure Nash equilibria of the resulting market game. Then, we study a discrete dynamic arising from repeatedly taking the (unique) equilibrium resource allocation as initial endowments for the next market game. We prove that the total utility value of the dynamic converges to either an optimal allocation value (maximizing total utility over the allocation space) or to a restricted optimal allocation value, where the restriction is defined by fixing some tight resources which are exclusively allocated to a single player. As a corollary, it follows that for strictly concave utility functions, the aggregated allocation vector of the dynamic converges to the unique (possibly restricted) optimal aggregated allocation, and for linear utility functions, we even get convergence of the dynamic to a (possibly restricted) optimal solution in the (non-aggregated) original allocation space.

In this article, we study the dynamic traffic assignment problem using the general path-delay-operator form as proposed by Friesz et al. [1989] and augment this model with side-constraints. Our contribution consists of four types of results:

We study a dynamic traffic assignment model, where agents base their instantaneous routing decisions on real-time delay predictions. We formulate a mathematically concise model and define dynamic prediction equilibrium (DPE) in which no agent can at any point during their journey improve their predicted travel time by switching to a different route. We demonstrate the versatility of our framework by showing that it subsumes the well-known full information and instantaneous information models, in addition to admitting further realistic predictors as special cases. We then proceed to derive properties of the predictors that ensure a dynamic prediction equilibrium exists. Additionally, we define $\varepsilon$-approximate DPE wherein no agent can improve their predicted travel time by more than $\varepsilon$ and provide further conditions of the predictors under which such an approximate equilibrium can be computed. Finally, we complement our theoretical analysis by an experimental study, in which we systematically compare the induced average travel times of different predictors, including two machine-learning based models trained on data gained from previously computed approximate equilibrium flows, both on synthetic and real world road networks.

We study a dynamic traffic assignment model, where agents base their instantaneous routing decisions on real-time delay predictions. We formulate a mathematically concise model and derive properties of the predictors that ensure a dynamic prediction equilibrium exists. We demonstrate the versatility of our framework by showing that it subsumes the well-known full information and instantaneous information models, in addition to admitting further realistic predictors as special cases. We complement our theoretical analysis by an experimental study, in which we systematically compare the induced average travel times of different predictors, including a machine-learning model trained on data gained from previously computed equilibrium flows, both on a synthetic and a real road network.

We study a multi-leader single-follower congestion game where multiple users (leaders) choose one resource out of a set of resources and, after observing the realized loads, an adversary (single-follower) attacks the resources with maximum loads causing additional costs for the leaders. For the resulting strategic game among the leaders, we show that pure Nash equilibria fail to exist and therefore, we consider approximate equilibria instead. As our first main result, we show that the existence of a K-approximate equilibrium can always be guaranteed, where K (approximately equal to 1.1974) is the unique solution of a cubic polynomial equation. To this end, we give a polynomial time combinatorial algorithm which computes a K-approximate equilibrium. The factor K is tight, meaning that there is an instance that does not admit an A-approximate equilibrium for any A < K. Thus A = K is the smallest possible value of A such that the existence of an A-approximate equilibrium can be guaranteed for any instance of the considered game. Secondly, we focus on approximate equilibria of a given fixed instance. We show how to compute efficiently a best approximate equilibrium, that is, with smallest possible A among all A-approximate equilibria of the given instance.

We initiate the study of dynamic traffic assignment for electrical vehicles addressing the specific challenges such as range limitations and the possibility of battery recharge at predefined charging locations. We pose the dynamic equilibrium problem within the deterministic queueing model of Vickrey and as our main result, we establish the existence of an energy-feasible dynamic equilibrium. There are three key modeling-ingredients for obtaining this existence result: * We introduce a walk-based definition of dynamic traffic flows which allows for cyclic routing behavior as a result of recharging events en route. * We use abstract convex feasibility sets in an appropriate function space to model the energy-feasibility of used walks. * We introduce the concept of capacitated dynamic equilibrium walk-flows which generalize the former unrestricted dynamic equilibrium path-flows. Viewed in this framework, we show the existence of an energy-feasible dynamic equilibrium by applying an infinite dimensional variational inequality, which in turn requires a careful analysis of continuity properties of the network loading as a result of injecting flow into walks. We complement our theoretical results by a computational study in which we design a fixed-point algorithm computing energy-feasible dynamic equilibria. We apply the algorithm to standard real-world instances from the traffic assignment community illustrating the complex interplay of resulting travel times, energy consumption and prices paid at equilibrium.

This paper studies the existence of pure Nash equilibria in resource graph games, a general class of strategic games succinctly representing the players’ private costs. These games are defined relative to a finite set of resources and the strategy set of each player corresponds to a set of subsets of resources. The cost of a resource is an arbitrary function of the load vector of a certain subset of resources. As our main result, we give complete characterizations of the cost functions guaranteeing the existence of pure Nash equilibria for weighted and unweighted players, respectively. For unweighted players, pure Nash equilibria are guaranteed to exist for any choice of the players’ strategy space if and only if the cost of each resource is an arbitrary function of the load of the resource itself and linear in the load of all other resources where the linear coefficients of mutual influence of different resources are symmetric. This implies in particular that for any other cost structure there is a resource graph game that does not have a pure Nash equilibrium. For weighted games where players have intrinsic weights and the cost of each resource depends on the aggregated weight of its users, pure Nash equilibria are guaranteed to exist if and only if the cost of a resource is linear in all resource loads, and the linear factors of mutual influence are symmetric, or there is no interaction among resources and the cost is an exponential function of the local resource load. We further discuss the computational complexity of pure Nash equilibria in resource graph games showing that for unweighted games where pure Nash equilibria are guaranteed to exist, it is coNP-complete to decide for a given strategy profile whether it is a pure Nash equilibrium. For general resource graph games, we prove that the decision whether a pure Nash equilibrium exists is Σ p 2 -complete.

In a seminal paper, Chen, Roughgarden, and Valiant [ SIAM J. Comput., 39 (5) (2010), pp. 1799--1832] studied cost sharing protocols for network design with the objective to implement a low-cost Steiner forest as a Nash equilibrium of an induced cost-sharing game. One of the most intriguing open problems to date is to understand the power of separable cost sharing protocols in order to induce low-cost Steiner forests. In this work, we focus on undirected networks and analyze topological properties of the underlying graph so that an optimal Steiner forest can be implemented as a Nash equilibrium (by some separable cost sharing protocol) independent of the edge costs. We term a graph efficient if the above stated property holds. As our main result, we give a complete characterization of efficient undirected graphs for two-player network design games: an undirected graph is efficient if and only if it does not contain (at least) one out of few forbidden subgraphs. Our characterization implies that several graph classes are efficient: generalized series-parallel graphs, fan and wheel graphs, and graphs with small cycles.

We study uniqueness of Nash equilibria in atomic splittable congestion games and derive a uniqueness result based on polymatroid theory: when the strategy space of every player is a bidirectional flow polymatroid, then equilibria are unique. Bidirectional flow polymatroids are introduced as a subclass of polymatroids possessing certain exchange properties. We show that important cases such as base orderable matroids can be recovered as a special case of bidirectional flow polymatroids. On the other hand we show that matroidal set systems are in some sense necessary to guarantee uniqueness of equilibria: for every atomic splittable congestion game with at least three players and non-matroidal set systems per player, there is an isomorphic game having multiple equilibria. Our results leave a gap between base orderable matroids and general matroids for which we do not know whether equilibria are unique.

In competitive packet routing games, packets are routed selfishly through a network and scheduling policies at edges determine which packages are forwarded first if there is not enough capacity on an edge to forward all packages at once. We analyze the impact of priority lists on the worst-case quality of pure Nash equilibria. A priority list is an ordered list of players that may or may not depend on the edge. Whenever the number of packets entering an edge exceeds the inflow capacity, packets are processed in list order. We derive several new bounds on the price of anarchy and stability for global and local priority policies. We also consider the question of the complexity of computing an optimal priority list. It turns out that even for very restricted cases, i.e., for routing on a tree, the computation of an optimal priority list is APX-hard.

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived.

Bottleneck routing games are a well-studied model to investigate the impact of selfish behavior in communication networks. In this model, each user selects a path in a network for routing their fixed demand. The disutility of a used only depends on the most congested link visited. We extend this model by allowing users to continuously vary the demand rate at which data is sent along the chosen path. As our main result we establish tight conditions for the existence of pure strategy Nash equilibria. These 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.

Reducing traffic congestion via toll pricing has been a central topic in the operations research and transportation literature and, recently, it has been implemented in several cities all over the world. Since, in practice, it is not feasible to impose tolls on every edge of a given traffic network, we study the resulting mathematical problem of computing tolls on a predefined subset of edges of the network so as to minimize the total travel time of the induced equilibrium flow. We first present an analytical study for the special case of parallel edge networks highlighting the intrinsic complexity and nonconvexity of the resulting optimization problem. We then present algorithms for general networks for which we systematically test the solution quality for large‐scale network instances. Finally, we discuss the related optimization problem of computing tolls subject to a cardinality constraint on the number of edges that have tolls. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(3), 262–285 2015

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived.

We consider the problem of designing cost sharing protocols to minimize the price of anarchy and stability for a class of scheduling games. Here, we are given a set of players, each associated with a job of certain non-negative weight. Any job fits on any machine, and the cost of a machine is a non-decreasing function of the total load on the machine. We assume that the private cost of a player is determined by a cost sharing protocol. We consider four natural design restrictions for feasible protocols: stability, budget balance, separability, and uniformity. While budget balance is self-explanatory, the stability requirement asks for the existence of pure-strategy Nash equilibria. Separability requires that the resulting cost shares only depend on the set of players on a machine. Uniformity additionally requires that the cost shares on a machine are instance-independent, that is, they remain the same even if new machines are added to or removed from the instance. We call a cost sharing protocol basic, if it satisfies only stability and budget balance. Separable and uniform cost sharing protocols additionally satisfy separability and uniformity, respectively. For n-player games we show that among all basic and separable cost sharing protocols, there is an optimal protocol with price of anarchy and stability of precisely the n-th harmonic number. For uniform protocols we present a strong lower bound showing that the price of anarchy is unbounded. Moreover, we obtain several results for special cases in which either the cost functions are restricted, or the job sizes are restricted. As a byproduct of our analysis, we obtain a complete characterization of outcomes that can be enforced as a pure-strategy Nash equilibrium by basic and separable cost sharing protocols.

We initiate the study of congestion games with variable demands where the (variable) demand has to be assigned to exactly one subset of resources. The players' incentives to use higher demands are stimulated by non-decreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of non-cooperative games captures many elements of real-world applications, it has not been studied in this generality, to our knowledge, in the past.

In communication networks, resource assignment problems appear in several different settings. These problems are often modeled by a maximum weight matching problem in bipartite graphs and efficient matching algorithms are well known. In several applications, the corresponding matching problem has to be solved many times in a row as the underlying system operates in a time-slotted fashion and the edge weights change over time. However, changing the assignments can come with a certain cost for reconfiguration that depends on the number of changed edges between subsequent assignments. In order to control the cost of reconfiguration, we propose the k-constrained bipartite matching problem for bipartite graphs, which seeks an optimal matching that realizes at most k changes from a previous matching. We provide fast approximation algorithms with provable guarantees for this problem. Furthermore, to cope with the sequential nature of assignment problems, we introduce an online variant of the k-constrained matching problem and derive online algorithms that are based on our approximation algorithms for the k-constrained bipartite matching problem. Finally, we establish the applicability of our model and our algorithms in the context of OFDMA wireless networks finding a significant performance improvement for the proposed algorithms.

We present a framework for designing delay- independent end-to-end congestion control algorithms, where each end-user may have a different utility function. We only require that utility functions are strictly increasing. In this framework, we design an algorithm that maximizes the minimum utility value in the network, that is, the resulting resource allocation is utility max-min fair. To achieve this, we first extend the congestion control algorithm EMKC proposed by Zhang et al. [1], which aims at max-min fair bandwidth allocation. Our extension (xMKC) allows for arbitrary rate allocations in the steady state. We investigate xMKC analytically and prove local asymptotic stability with heterogeneous time-varying feedback delays in multi-link networks and global asymptotic stability with homogeneous time-varying feedback delays in single-link networks. Then, we propose uMKC (Utility Max-Min Fair Kelly Control), which achieves utility max-min fairness in its steady state. Based on the analysis of xMKC, we establish stability results for uMKC in the presence of time-varying feedback delays. Finally, we evaluate the performance of uMKC using NS-2 simulations [2].

The inter-domain resource exchange (iREX) architecture uses economic market mechanisms to automate the ad-hoc negotiation and deployment of end to end inter-domain quality of service policy among resource consumer and resource provider . In this paper, we explore iREX's network load distribution by comparing its performance to a lower bound for network congestion in two ways. We first present an analytical model of iREX in terms of an online algorithm and analyze its efficiency via competitive analysis. Our main result shows that the efficiency loss of iREX with respect to monetary cost is upper-bounded by a factor of 8 K/2 K+1, where K s the number of deployments, provided affine linear price functions are used. When the price functions are used to model congestion in the network, this result implies upper bounds on the efficiency loss of iREX with respect to network congestion. We then complement the analytical model with a numerical study using simulations.with optimal solutions derived from unsplittable and splittable multi-commodity flow optimization models. Our numerical results show that for nominal to high traffic loads of 40% or more, iREX deviates a maximum of about 20% from the lower bound, while the current method deviates a maximum of 300%.

In the context of a network implementing congestion pricing, we focus on user strategies to determine their willingness to pay parameter. We argue that users downloading a file in fixed time, or users running a multimedia application will have other strategies to decide on their payment than simply to maximize their surplus measured by a concave utility function minus cost. Instead, we formulate the download task as an optimal control problem and account for dynamic changes of the state of congestion by using (online) model predictive control techniques. Finally, we develop online control strategies in order to automatically adapt the willingness to pay parameter.

The inter-domain resource exchange (iREX) architecture uses economic market mechanisms to automate the deployment of end to end (E2E) inter-domain (ID) quality of service (QoS) policy among resource consumer and resource provider Internet Service Providers (ISPs). Previous simulation results have shown that iREX allows more coexisting ID policy deployments with less network congestion when compared to the existing method. In this paper we explore iREX's network load distribution efficiency limits by comparing iREX's performance to a lower bound for network congestion. We present an analytical model of iREX in terms of a min-cost flow problem, and numerical results of efficiency loss between iREX simulations and derived optimal solutions based on multi-commodity flow optimization models. Our results show that for nominal to high traffic loads of 50% or more, iREX deviates a maximum of approximately 30% from the derived lower bound, while the current method deviates a maximum of 350%.

This paper deals with a new pricing approach in utility fair networks, where the user's application is associated with a utility function. We allow users to have concave as well as non-concave utility functions. Bandwidth is allocated such that utility values of applications are shared fairly. In this work, we derive a fairness measure for utility functions that takes their specific shape into account. Based on this fairness measure, we present a simple pricing mechanism: the user announces his utility function and the network charges in accordance with the fairness measure. Then, we apply our pricing mechanism to a content provider's network. In our model customers want to scale their utilities to achieve their goals (e.g. file download, multimedia streaming) in a cost optimal way. In this regard, we formulate a download problem with predefined, deadline as an optimal control problem and account for dynamic changes of the state of congestion by using (online) model predictive control techniques. Finally, we develop online control strategies and implement them in a user agent (UA) that automatically scales the utilities.

This paper deals with a new approach to integrate congestion control for real-time applications and elastic traffic into a unified framework. In our previous work, we proposed a new fairness criterion, utility proportional fairness, that takes characteristics of real-time applications into account. We complement this framework by deriving a general method to generate utility functions for layered multimedia applications. Finally, we demonstrate our approach through ns-simulations.