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Stochastic effects significantly influence the dynamics of traffic flows. Many dynamic traffic assignment (DTA) models attempt to capture these effects by prescribing a specific ratio that determines how flow splits across different routes based on the routes' costs. In this paper, we propose a new framework for DTA that incorporates the interplay between the routing decisions of each single traffic participant, the stochastic nature of predicting the future state of the network, and the physical flow dynamics. Our framework consists of an edge loading operator modeling the physical flow propagation and a routing operator modeling the routing behavior of traffic participants. The routing operator is assumed to be set-valued and capable to model complex (deterministic) equilibrium conditions as well as stochastic equilibrium conditions assuming that measurements for predicting traffic are noisy. As our main results, we derive several quite general equilibrium existence and uniqueness results which not only subsume known results from the literature but also lead to new results. Specifically, for the new stochastic prediction equilibrium, we show existence and uniqueness under natural assumptions on the probability distribution over the predictions., Comment: 34 pages, 2 figures

The famous edge flow decomposition theorem of Gallai (1958) states that any static edge $s$,$d$-flow in a directed graph can be decomposed into a linear combination of incidence vectors of paths and cycles. In this paper, we study the decomposition problem for the setting of dynamic edge $s$,$d$-flows assuming a quite general dynamic flow propagation model. We prove the following decomposition theorem: For any dynamic edge $s$,$d$-flow with finite support, there exists a decomposition into a linear combination of $s$,$d$-walk inflows and circulations, i.e. edge flows that circulate along cycles with zero transit time. We show that a variant of the classical algorithmic approach of iteratively subtracting walk inflows from the current dynamic edge flow converges to a dynamic circulation. The algorithm terminates in finite time, if there is a lower bound on the minimum edge travel times. We further characterize those dynamic edge flows which can be decomposed purely into linear combinations of $s$,$d$-walk inflows. The proofs rely on the new concept of parameterized network loadings which describe how particles of a different walk flow would hypothetically propagate throughout the network under the fixed travel times induced by the given edge flow. We show several technical properties of this type of network loading and as a byproduct we also derive some general results on dynamic flows which could be of interest outside the context of this paper as well., Comment: 54 pages, 3 figures

Modelling passenger assignments in public transport networks is a fundamental task for city planners, especially when deliberating network infrastructure decisions. A key aspect of a realistic model for passenger assignments is to integrate selfish routing behaviour of passengers on the one hand, and the limited vehicle capacities on the other hand. We formulate a side-constrained user equilibrium model in a schedule-based time-expanded transit network, where passengers are modelled via a continuum of non-atomic agents that want to travel with a fixed start time from a user-specific origin to a destination. An agent's route may comprise several rides along given lines, each using vehicles with hard loading capacities. We give a characterization of (side-constrained) user equilibria via a quasi-variational inequality and prove their existence by generalizing a well-known existence result of Bernstein and Smith (Transp. Sci., 1994). We further derive a polynomial time algorithm for single-commodity instances and an exact finite time algorithm for the multi-commodity case. Based on our quasi-variational characterization, we finally devise a fast heuristic computing user equilibria, which is tested on real-world instances based on data gained from the Hamburg S-Bahn system and the Swiss long-distance train network. It turns out that w.r.t. the total travel time, the computed user-equilibria are quite efficient compared to a system optimum, which neglects equilibrium constraints and only minimizes total travel time., Comment: 32 pages, 9 figures

We study dynamic traffic assignment with side-constraints. We first give a counter-example to a key result from the literature regarding the existence of dynamic equilibria for volume-constrained traffic models in the classical edge-delay model. Our counter-example shows that the feasible flow space need not be convex and it further reveals that classical infinite dimensional variational inequalities are not suited for the definition of side-constrained dynamic equilibria. We propose a new framework for side-constrained dynamic equilibria based on the concept of feasible $\gamma$-deviations of flow particles in space and time. Under natural assumptions, we characterize the resulting equilibria by means of quasi-variational and variational inequalities, respectively. Finally, we establish first existence results for side-constrained dynamic equilibria for the non-convex setting of volume-constraints., Comment: 61 pages, 10 figures

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., Comment: 42 pages

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 may 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 $K \approx 1.1974$ 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 $\alpha$-approximate equilibrium for any $\alpha

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., Comment: 42 pages, 5 figures

This paper studies the existence of pure Nash equilibria in resource graph games, which are a general class of strategic games used to succinctly represent the players' private costs. There is 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 that depends on the load vector of the resources in a specified neighborhood. 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. 1. 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. 2. For games with weighted players, pure Nash equilibria are guaranteed to exist for any choice of the players' strategy space 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. 3. For the special case that the players' strategy sets are matroids, we show that pure Nash equilibria exist under a local monotonicity property, even when cost functions are player-specific. We point out an application of this result to bilevel load balancing games, which are motivated by the study of network infrastructures that are resilient against external attackers and internal congestion effects. 4. Finally, we derive hardness results for deciding whether a given strategy is a pure Nash equilibrium for network routing games and matroid games, respectively.

We consider flows over time within the deterministic queueing model and study the solution concept of instantaneous dynamic equilibrium (IDE) in which flow particles select at every decision point a currently shortest path. The length of such a path is measured by the physical travel time plus the time spent in queues. Although IDE have been studied since the eighties, the efficiency of the solution concept is not well understood. We study the price of anarchy for this model and show an upper bound of order $\mathcal{O}(U\cdot \tau)$ for single-sink instances, where $U$ denotes the total inflow volume and $\tau$ the sum of edge travel times. We complement this upper bound with a family of quite complex instances proving a lower bound of order $\Omega(U\cdot\log\tau)$., Comment: 42 pages, 15 figures

Instantaneous dynamic equilibrium (IDE) is a standard game-theoretic concept in dynamic traffic assignment in which individual flow particles myopically select en route currently shortest paths towards their destination. We analyze IDE within the Vickrey bottleneck model, where current travel times along a path consist of the physical travel times plus the sum of waiting times in all the queues along a path. Although IDE have been studied for decades, several fundamental questions regarding equilibrium computation and complexity are not well understood. In particular, all existence results and computational methods are based on fixed-point theorems and numerical discretization schemes and no exact finite time algorithm for equilibrium computation is known to date. As our main result we show that a natural extension algorithm needs only finitely many phases to converge leading to the first finite time combinatorial algorithm computing an IDE. We complement this result by several hardness results showing that computing IDE with natural properties is NP-hard., Comment: 32 pages, 10 figures

We consider a basic resource allocation game, where the players' strategy spaces are subsets of $R^m$ and cost/utility functions are parameterized by some common vector $u\in R^m$ and, otherwise, only depend on the own strategy choice. A strategy of a player can be interpreted as a vector of resource consumption and a joint strategy profile naturally leads to an aggregate consumption vector. Resources can be priced, that is, the game is augmented by a price vector $\lambda\in R^m_+$ and players have quasi-linear overall costs/utilities meaning that in addition to the original costs/utilities, a player needs to pay the corresponding price per consumed unit. We investigate the following question: for which aggregated consumption vectors $u$ can we find prices $\lambda$ that induce an equilibrium realizing the targeted consumption profile? For answering this question, we revisit a well-known duality-based framework and derive several characterizations of the existence of such $u$ and $\lambda$ using convexification techniques. We show that for finite strategy spaces or certain concave games, the equilibrium existence problem reduces to solving a well-structured LP. We then consider a class of monotone aggregative games having the property that the cost/utility functions of players may depend on the induced load of a strategy profile. For this class, we show a sufficient condition of enforceability based on the previous characterizations. We demonstrate that this framework can help to unify parts of four largely independent streams in the literature: tolls in transportation systems, Walrasian market equilibria, trading networks and congestion control in communication networks., Comment: 40 pages, updated and extended version

We study oligopolistic competition in service markets where firms offer a service to customers. The service quality of a firm - from the perspective of a customer - depends on the congestion and the charged price. A firm can set a price for the service offered and additionally decides on the service capacity in order to mitigate congestion. The total profit of a firm is derived from the gained revenue minus the capacity investment cost. Firms simultaneously set capacities and prices in order to maximize their profit and customers subsequently choose the services with lowest combined cost (congestion and price). For this basic model, Johari et al. (2010) derived the first existence and uniqueness results of pure Nash equilibria (PNE) assuming mild conditions on congestion functions. Their existence proof relies on Kakutani's fixed-point theorem and a key assumption for the theorem to work is that demand for service is elastic (modeled by a smooth and strictly decreasing inverse demand function). In this paper, we consider the case of perfectly inelastic demand, i.e. there is a fixed volume of customers requesting service. This scenario applies to realistic cases where customers are not willing to drop out of the market, e.g. if prices are regulated by reasonable price caps. We investigate existence, uniqueness and quality of PNE for models with inelastic demand and price caps. We show that for linear congestion cost functions, there exists a PNE. This result requires a completely new proof approach compared to previous approaches, since the best response correspondences of firms may be empty, thus standard fixed-point arguments are not directly applicable. We show that the game is C-secure (see McLennan et al. (2011)), which leads to the existence of PNE. We furthermore show that the PNE is unique, and that the efficiency compared to a social optimum is unbounded in general., Comment: A one-page abstract of this paper appeared in the proceedings of the 15th International Conference on Web and Internet Economics (WINE 2019)

We study dynamic network flows and introduce a notion of instantaneous dynamic equilibrium (IDE) requiring that for any positive inflow into an edge, this edge must lie on a currently shortest path towards the respective sink. We measure current shortest path length by current waiting times in queues plus physical travel times. As our main results, we show: 1. existence and constructive computation of IDE flows for single-source single-sink networks assuming constant network inflow rates, 2. finite termination of IDE flows for multi-source single-sink networks assuming bounded and finitely lasting inflow rates, 3. the existence of IDE flows for multi-source multi-sink instances assuming general measurable network inflow rates, 4. the existence of a complex single-source multi-sink instance in which any IDE flow is caught in cycles and flow remains forever in the network., Comment: 40 pages, shorter version published in the "Proceedings of the 20th Conference on Integer Programming and Combinatorial Optimization, 2019"

In this paper, we compute $\epsilon$-approximate Nash equilibria in atomic splittable polymatroid congestion games with convex Lipschitz continuous cost functions. The main approach relies on computing a pure Nash equilibrium for an associated integrally-splittable congestion game, where players can only split their demand in integral multiples of a common packet size. It is known that one can compute pure Nash equilibria for integrally-splittable congestion games within a running time that is pseudo-polynomial in the aggregated demand of the players. As the main contribution of this paper, we decide for every $\epsilon>0$, a packet size $k_{\epsilon}$ and prove that the associated $k_{\epsilon}$-splittable Nash equilibrium is an $\epsilon$-approximate Nash equilibrium for the original game. We further show that our result applies to multimarket oligopolies with decreasing, concave Lipschitz continuous price functions and quadratic production costs: there is a polynomial time transformation to atomic splittable polymatroid congestion games implying that we can compute $\epsilon$-approximate Cournot-Nash equilibria within pseudo-polynomial time., Comment: 10 pages. arXiv admin note: text overlap with arXiv:1612.00190

In cost sharing games with delays, a set of agents jointly allocates a finite subset of resources. Each resource has a fixed cost that has to be shared by the players, and each agent has a nonshareable player-specific delay for each resource. A prominent example is uncapacitated facility location (UFL), where facilities need to be opened (at a shareable cost) and clients want to connect to opened facilities. Each client pays a cost share and his non-shareable physical connection cost. Given any profile of subsets allocated by the agents, a separable cost sharing protocol determines cost shares that satisfy budget balance on every resource and separability over the resources. Moreover, a separable protocol guarantees existence of pure Nash equilibria in the induced strategic game for the agents. In this paper, we study separable cost sharing protocols in several general combinatorial domains. We provide black-box reductions to reduce the design of a separable cost-sharing protocol to the design of an approximation algorithm for the underlying cost minimization problem. In this way, we obtain new separable cost-sharing protocols in games based on arbitrary player-specific matroids, single-source connection games without delays, and connection games on $n$-series-parallel graphs with delays. All these reductions are efficiently computable - given an initial allocation profile, we obtain a cheaper profile and separable cost shares turning the profile into a pure Nash equilibrium. Hence, in these domains any approximation algorithm can be used to obtain a separable cost sharing protocol with a price of stability bounded by the approximation factor.

We consider a network pricing game on a parallel network with congestion effects in which link owners set tolls for travel so as to maximize profit. A central authority is able to regulate this competition by means of a (uniform) price cap. The first question we want to answer is how such a cap should be designed in order to minimize the total congestion. We provide an algorithm that finds an optimal price cap for networks with affine latency functions and a full support Wardrop equilibrium. Second, we consider the induced network performance at an optimal price cap. We show that for two link networks with affine latency functions, the congestion costs at the optimal price cap are at most $8/7$ times the optimal congestion costs. For more general latency functions, this bound goes up to 2 under the assumption that an uncapped Nash equilibrium exists. However, in general such an equilibrium need not exist and this can be used to show that optimal price caps can induce arbitrarily inefficient flows., Comment: 25 pages, 8 figures

In a seminal paper, Chen, Roughgarden and Valiant 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 budget-balanced and 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., Comment: 60 pages, 69 figures, OR 2017 Berlin, WINE 2017 Bangalore

We study the equilibrium computation problem for two classical resource allocation games: atomic splittable congestion games and multimarket Cournot oligopolies. For atomic splittable congestion games with singleton strategies and player-specific affine cost functions, we devise the first polynomial time algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and computes the exact equilibrium assuming rational input. The idea is to compute an equilibrium for an associated integrally-splittable singleton congestion game in which the players can only split their demands in integral multiples of a common packet size. While integral games have been considered in the literature before, no polynomial time algorithm computing an equilibrium was known. Also for this class, we devise the first polynomial time algorithm and use it as a building block for our main algorithm. We then develop a polynomial time computable transformation mapping a multimarket Cournot competition game with firm-specific affine price functions and quadratic costs to an associated atomic splittable congestion game as described above. The transformation preserves equilibria in either games and, thus, leads -- via our first algorithm -- to a polynomial time algorithm computing Cournot equilibria. Finally, our analysis for integrally-splittable games implies new bounds on the difference between real and integral Cournot equilibria. The bounds can be seen as a generalization of the recent bounds for single market oligopolies obtained by Todd [2016]., Comment: This version contains some typo corrections only

We study the sensitivity of optimal solutions of convex separable optimization problems over an integral polymatroid base polytope with respect to parameters determining both the cost of each element and the polytope. Under convexity and a regularity assumption on the functional dependency of the cost function with respect to the parameters, we show that reoptimization after a change in parameters can be done by elementary local operations. Applying this result, we derive that starting from any optimal solution there is a new optimal solution to new parameters such that the L1-norm of the difference of the two solutions is at most two times the L1 norm of the difference of the parameters. We apply these sensitivity results to a class of non-cooperative polymatroid games and derive the existence of pure Nash equilibria. We complement our results by showing that polymatroids are the maximal combinatorial structure enabling these results. For any non-polymatroid region, there is a corresponding optimization problem for which the sensitivity results do not hold. In addition, there is a game where the players strategies are isomorphic to the non-polymatroid region and that does not admit a pure Nash equilibrium., Comment: Succeeds arXiv paper 1407.7650. arXiv admin note: text overlap with arXiv:1407.7650

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 nonmatroidal 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., Comment: 17 pages

The famous Braess paradox describes the following phenomenon: It might happen that the improvement of resources, like building a new street within a congested network, may in fact lead to larger costs for the players in an equilibrium. In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur. In a nutshell, bases of matroids are exactly this maximal structure. We prove our characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest., Comment: 21 pages

We study competitive resource allocation problems in which players distribute their demands integrally on a set of resources subject to player-specific submodular capacity constraints. Each player has to pay for each unit of demand a cost that is a nondecreasing and convex function of the total allocation of that resource. This general model of resource allocation generalizes both singleton congestion games with integer-splittable demands and matroid congestion games with player-specific costs. As our main result, we show that in such general resource allocation problems a pure Nash equilibrium is guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure Nash equilibrium., Comment: 17 pages

Public transit systems in urban areas usually require large state subsidies, primarily due to high fare evasion rates. In this paper, we study new models for optimizing fare inspection strategies in transit networks based on bilevel programming. In the first level, the leader (the network operator) determines probabilities for inspecting passengers at different locations, while in the second level, the followers (the fare-evading passengers) respond by optimizing their routes given the inspection probabilities and travel times. To model the followers' behavior we study both a non-adaptive variant, in which passengers select a path a priori and continue along it throughout their journey, and an adaptive variant, in which they gain information along the way and use it to update their route. For these problems, which are interesting in their own right, we design exact and approximation algorithms and we prove a tight bound of 3/4 on the ratio of the optimal cost between adaptive and non-adaptive strategies. For the leader's optimization problem, we study a fixed-fare and a flexible-fare variant, where ticket prices may or may not be set at the operator's will. For the latter variant, we design an LP based approximation algorithm. Finally, using a local search procedure that shifts inspection probabilities within an initially determined support set, we perform an extensive computational study for all variants of the problem on instances of the Dutch railway and the Amsterdam subway network. This study reveals that our solutions are within 95% of theoretical upper bounds drawn from the LP relaxation.

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 introduce a quantitative approach to comparative statics that allows to bound the maximum effect of an exogenous parameter change on a system's equilibrium. The motivation for this approach is a well known paradox in multimarket Cournot competition, where a positive price shock on a monopoly market may actually reduce the monopolist's profit. We use our approach to quantify for the first time the worst case profit reduction for multimarket oligopolies exposed to arbitrary positive price shocks. For markets with affine price functions and firms with convex cost technologies, we show that the relative profit loss of any firm is at most 25% no matter how many firms compete in the oligopoly. We further investigate the impact of positive price shocks on total profit of all firms as well as on social welfare. We find tight bounds also for these measures showing that total profit and social welfare decreases by at most 25% and 16.6%, respectively. Finally, we show that in our model, mixed, correlated and coarse correlated equilibria are essentially unique, thus, all our bounds apply to these game solutions as well., Comment: 23 pages, 1 figure

We revisit a classical problem in transportation, known as the continuous (bilevel) network design problem, CNDP for short. We are given a graph for which the latency of each edge depends on the ratio of the edge flow and the capacity installed. The goal is to find an optimal investment in edge capacities so as to minimize the sum of the routing cost of the induced Wardrop equilibrium and the investment cost. While this problem is considered as challenging in the literature, its complexity status was still unknown. We close this gap showing that CNDP is strongly NP-complete and APX-hard, both on directed and undirected networks and even for instances with affine latencies. As for the approximation of the problem, we first provide a detailed analysis for a heuristic studied by Marcotte for the special case of monomial latency functions (Mathematical Programming, Vol.~34, 1986). Specifically, we derive a closed form expression of its approximation guarantee for arbitrary sets S of allowed latency functions. Second, we propose a different approximation algorithm and show that it has the same approximation guarantee. As our final -- and arguably most interesting -- result regarding approximation, we show that using the better of the two approximation algorithms results in a strictly improved approximation guarantee for which we give a closed form expression. For affine latencies, e.g., this algorithm achieves a 1.195-approximation which improves on the 5/4 that has been shown before by Marcotte. We finally discuss the case of hard budget constraints on the capacity investment., Comment: 27 pages

In resource buying games a set of players jointly buys a subset of a finite resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a resource e depends on the number (or load) of players using e, and has to be paid completely by the players before it becomes available. Each player i needs at least one set of a predefined family S_i in 2^E to be available. Thus, resource buying games can be seen as a variant of congestion games in which the load-dependent costs of the resources can be shared arbitrarily among the players. A strategy of player i in resource buying games is a tuple consisting of one of i's desired configurations S_i together with a payment vector p_i in R^E_+ indicating how much i is willing to contribute towards the purchase of the chosen resources. In this paper, we study the existence and computational complexity of pure Nash equilibria (PNE, for short) of resource buying games. In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the S_i's and the behavior of the cost functions. We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit a PNE.

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. We study the fundamental problem of the existence of pure Nash equilibria (PNE for short) in congestion games with variable demands. We call a set of cost functions C consistent if every congestion game with variable demands and cost functions in C possesses a PNE. We say that C is FIP consistent if every such game possesses the alpha-Finite Improvement Property for every alpha>0. Our main results are structural characterizations of consistency and FIP consistency for twice continuously differentiable cost functions. Specifically, we show 1. C is consistent if and only if C contains either only affine functions or only homogeneously exponential functions (c(x) = a exp(p x)). 2. C is FIP consistent if and only if C contains only affine functions. Our results provide a complete characterization of consistency of cost functions revealing structural differences to congestion games with fixed demands (weighted congestion games), where in the latter even inhomogeneously exponential functions are FIP consistent. Finally, we study consistency and FIP consistency of cost functions in a slightly different class of games, where every player experiences the same cost on a resource (uniform cost model). We give a characterization of consistency and FIP consistency showing that only homogeneously exponential functions are consistent.

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium.