Congestion games are a classical type of games studied in game theory, in which n players choose a resource, and their individual cost increases with the number of other players choosing the same resource. In network congestion games (NCGs), the resources correspond to simple paths in a graph, e.g. representing routing options from a source to a target. In this paper, we introduce a variant of NCGs, referred to as dynamic NCGs: in this setting, players take transitions synchronously, they select their next transitions dynamically, and they are charged a cost that depends on the number of players simultaneously using the same transition.
We study, from a complexity perspective, standard concepts of game theory in dynamic NCGs: social optima, Nash equilibria, and subgame perfect equilibria. Our contributions are the following: the existence of a strategy profile with social cost bounded by a constant is in PSPACE and NP-hard. (Pure) Nash equilibria always exist in dynamic NCGs; the existence of a Nash equilibrium with bounded cost can be decided in EXPSPACE, and computing a witnessing strategy profile can be done in doubly-exponential time. The existence of a subgame perfect equilibrium with bounded cost can be decided in 2EXPSPACE, and a witnessing strategy profile can be computed in triply-exponential time.
This paper shows that the PoA in non-atomic congestion games is H{ö}lder continuous w.r.t. combined disturbance on cost functions and demands. We then apply this result to the convergence analysis of the PoA.
We study a new class of games which generalizes congestion games and its bottleneck variant. We introduce congestion games with mixed objectives to model network scenarios in which players seek to optimize for latency and bandwidths alike. We characterize the existence of pure Nash equilibria (PNE) and the convergence of improvement dynamics. For games that do not possess PNE we give bounds on the approximation ratio of approximate pure Nash equilibria.
We put forward a new model of congestion games where agents have uncertainty over the routes used by other agents. We take a non-probabilistic approach, assuming that each agent knows that the number of agents using an edge is within a certain range. Given this uncertainty, we model agents who either minimize their worst-case cost (WCC) or their worst-case regret (WCR), and study implications on equilibrium existence, convergence through adaptive play, and efficiency. Under the WCC behavior the game reduces to a modified congestion game, and welfare improves when agents have moderate uncertain…
Computing an equilibrium in congestion games can be challenging when the number of players is large. Yet, it is a problem to be addressed in practice, for instance to forecast the state of the system and be able to control it. In this work, we analyze the case of generalized atomic congestion games, with coupling constraints, and with players that are heterogeneous through their action sets and their utility functions. We obtain an approximation of the variational Nash equilibria---a notion generalizing Nash equilibria in the presence of coupling constraints---of a large atomic congestion game…
We study the impact of player capability on social welfare in congestion games. We introduce a new game, the Distance-bounded Network Congestion game (DNC), as the basis of our study. DNC is a symmetric network congestion game with a bound on the number of edges each player can use. We show that DNC is PLS-complete in contrast to standard symmetric network congestion games which are in P. To model different player capabilities, we propose using programs in a Domain-Specific Language (DSL) to compactly represent player strategies. We define a player's capability as the maximum size of the progr…
The travel costs of the players (travelers) in anonymous congestion games depend on their choices of routes and also on the states of the transportation network such as incidents, weather, and road work. In this extended abstract, we consider an incomplete-information environment in which the realizations of the states are unobserved by the travelers. We study how a planner can incentivize the travelers to behave in her favor by strategically designing what and how the travelers get informed about the realizations of the states.
We provide a unifying, black-box tool for establishing existence of approximate equilibria in weighted congestion games and, at the same time, bounding their Price of Stability. Our framework can handle resources with general costs--including, in particular, decreasing ones--and is formulated in terms of a set of parameters which are determined via elementary analytic properties of the cost functions. We demonstrate the power of our tool by applying it to recover the recent result of Caragiannis and Fanelli [ICALP'19] for polynomial congestion games; improve upon the bounds for fair cost shari…
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 th…
We study the computation of approximate pure Nash equilibria in Shapley value (SV) weighted congestion games, introduced in [19]. This class of games considers weighted congestion games in which Shapley values are used as an alternative (to proportional shares) for distributing the total cost of each resource among its users. We focus on the interesting subclass of such games with polynomial resource cost functions and present an algorithm that computes approximate pure Nash equilibria with a polynomial number of strategy updates. Since computing a single strategy update is hard, we apply samp…
Congestion games constitute an important class of non-cooperative games which was introduced by Rosenthal in 1973. In recent years, several extensions of these games were proposed to incorporate aspects that are not captured by the standard model. Examples of such extensions include the incorporation of risk sensitive players, the modeling of altruistic player behavior and the imposition of taxes on the resources. These extensions were studied intensively with the goal to obtain a precise understanding of the inefficiency of equilibria of these games. In this paper, we introduce a new model of…
Rosenthal（1973）引入了拥塞博弈类，并证明了它们在纯策略下总是存在一个纳什均衡。Fotakis et al.（2005）引入了贪婪策略元组的概念，其中参与者顺序地并且可重复地选择一个策略，该策略是对前参与者选择的策略的最佳反应。前一种解决方案的概念是由对参与者的理性及其常识的强烈假设驱动的，而后者对参与者的行为假设很少。从Fotakis \cite{fotakis10}可以得出，对于树可表示拥塞博弈，贪婪行为导致NE。本文给出了这两个解概念等价的充要条件。这种等同性增强了这些概念作为环境的现实结果的可行性。单调对称博弈出现这种等价的条件是策略集具有树形，或者等价地是一个“扩张并行图”。
We study a model of selfish resource allocation that seeks to incorporate dependencies among resources as they exist in modern networked environments. Our model is inspired by utility functions with constant elasticity of substitution (CES) which is a well-studied model in economics. We consider congestion games with different aggregation functions. In particular, we study $L_p$ norms and analyze the existence and complexity of (approximate) pure Nash equilibria. Additionally, we give an almost tight characterization based on monotonicity properties to describe the set of aggregation functions…
This work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it as not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness wh…
We consider the problem of coordination via signaling in network congestion games to improve social welfare deteriorated by incomplete information about traffic flow. Traditional studies on signaling, which focus on exogenous factors of congestion and ignore congestion externalities, fail to discuss the oscillations of traffic flow. To address this gap, we formulate a problem of designing a coordination signal on endogenous information about traffic flow and introduce a it self-fulfilling characteristic of a signal that guarantees an outcome flow consistent with the signal itself without causi…
We study natural improvement dynamics in weighted congestion games with polynomial latencies of maximum degree $d\geq 1$. We focus on two problems regarding the existence and efficiency of approximate pure Nash equilibria, with a reasonable small approximation factor, in these games. By exploiting a simple technique, we firstly show that such a game always admits a $d$-approximate potential function. This implies that every sequence of $d$-approximate improvement moves by the players leads to a $d$-approximate pure Nash equilibrium. As a corollary, we also obtain that, under mild assumptions o…
This article studies the user behavior in non-atomic congestion games. We consider non-atomic congestion games with continuous and non-decreasing functions and investigate the limit of the price of anarchy when the total user volume approaches infinity. We deepen the knowledge on {\em asymptotically well designed games} \cite{Wu2017Selfishness}, {\em limit games} \cite{Wu2017Selfishness}, {\em scalability} \cite{Wu2017Selfishness} and {\em gaugeability} \cite{Colini2017b} that were recently used in the limit analyses of the price of anarchy for non-atomic congestion games. We develop a unified…
We present a deterministic polynomial-time algorithm for computing $d^{d+o(d)}$-approximate (pure) Nash equilibria in (proportional sharing) weighted congestion games with polynomial cost functions of degree at most $d$. This is an exponential improvement of the approximation factor with respect to the previously best deterministic algorithm. An appealing additional feature of the algorithm is that it only uses best-improvement steps in the actual game, as opposed to the previously best algorithms, that first had to transform the game itself. Our algorithm is an adaptation of the seminal algor…
Congestion games constitute an important class of games to model resource allocation by different users. As computing an exact or even an approximate pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011) present a polynomial-time algorithm that computes a ($2 + ε$)-approximate pure Nash equilibria for games with linear cost functions and further results for polynomial cost functions. We show that this factor can be improved to $(1.61+ε)$ and further improved results for polynomial cost functions, by a seemingly simple modification to their algorithm by allowing for the co…
We consider non-atomic network congestion games with heterogeneous players where the latencies of the paths are subject to some bounded deviations. This model encompasses several well-studied extensions of the classical Wardrop model which incorporate, for example, risk-aversion, altruism or travel time delays. Our main goal is to analyze the worst-case deterioration in social cost of a perturbed Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow. We show that for homogeneous players perturbed Nash flows coincide with approximate Nash flows and derive tight bou…
Are rewards or penalties more effective in influencing user behavior? This work compares the effectiveness of subsidies and tolls in incentivizing users in congestion games. The predominantly studied method of influencing user behavior in network routing problems is to institute taxes which alter users' observed costs in a manner that causes their self-interested choices to more closely align with a system-level objective. Another feasible method to accomplish the same goal is to subsidize the users' actions that are preferable from a system-level perspective. We show that, when users behave s…
We introduce a novel framework to model limited lookahead in congestion games. Intuitively, the players enter the game sequentially and choose an optimal action under the assumption that the $k-1$ subsequent players play subgame-perfectly. Our model naturally interpolates between outcomes of greedy best-response ($k=1$) and subgame-perfect outcomes ($k=n$, the number of players). We study the impact of limited lookahead (parameterized by $k$) on the stability and inefficiency of the resulting outcomes. As our results reveal, increased lookahead does not necessarily lead to better outcomes; in …
This paper provides a comprehensive convergence analysis of the PoA of both pure and mixed Nash equilibria in atomic congestion games with unsplittable demands.
Reinforcement Learning Algorithms (RLA) are useful machine learning tools to understand how decision makers react to signals. It is known that RLA converge towards the pure Nash Equilibria (NE) of finite congestion games and more generally, finite potential games. For finite congestion games, only separable cost functions are considered. However, non-separable costs, which depend on the choices of all players instead of only those choosing the same resource, may be relevant in some circumstances, like in smart charging games. In this paper, finite congestion games with non-separable costs are …
Markov decision process (MDP) congestion game is an extension of classic congestion games, where a continuous population of selfish agents solves Markov decision processes with congestion: the payoff of a strategy decreases as more population uses it. We draw parallels between key concepts from capacitated congestion games and MDP. In particular, we show that population mass constraints in MDP congestion games are equivalent to imposing tolls/incentives on the reward function, which can be utilized by social planners to achieve auxiliary objectives. We demonstrate such methods in a simulated S…
We study the problem of computing Stackelberg equilibria Stackelberg games whose underlying structure is in congestion games, focusing on the case where each player can choose a single resource (a.k.a. singleton congestion games) and one of them acts as leader. In particular, we address the cases where the players either have the same action spaces (i.e., the set of resources they can choose is the same for all of them) or different ones, and where their costs are either monotonic functions of the resource congestion or not. We show that, in the case where the players have different action spa…
We give exponential lower bounds on the Price of Stability (PoS) of weighted congestion games with polynomial cost functions. In particular, for any positive integer $d$ we construct rather simple games with cost functions of degree at most $d$ which have a PoS of at least $\varOmega(Φ_d)^{d+1}$, where $Φ_d\sim d/\ln d$ is the unique positive root of equation $x^{d+1}=(x+1)^d$. This almost closes the huge gap between $\varTheta(d)$ and $Φ_d^{d+1}$. Our bound extends also to network congestion games. We further show that the PoS remains exponential even for singleton games. More generally, we p…
Network congestion games are a convenient model for reasoning about routing problems in a network: agents have to move from a source to a target vertex while avoiding congestion, measured as a cost depending on the number of players using the same link. Network congestion games have been extensively studied over the last 40 years, while their extension with timing constraints were considered more recently. Most of the results on network congestion games consider blind strategies: they are static, and do not adapt to the strategies selected by the other players. We extend the recent results of …
Game-theoretical approach to the analysis of parallel algorithms is proposed. The approach is based on presentation of the parallel computing as a congestion game. In the game processes compete for resources such as core of a central processing unit and a communication subsystem. There are players, resources and payoffs (time delays) of players which depend on resources usage. Comparative analysis of various optimality principles in the proposed model may be performed.
We consider a non-atomic congestion game where each decision maker performs selfish optimization over states of a common MDP. The decision makers optimize for their own expected costs, and influence each other through congestion effects on the state-action costs. We analyze on the sensitivity of MDP congestion game equilibria to uncertainty and perturbations in the state-action costs by applying an implicit function type analysis. The occurrence of a stochastic Braess paradox is defined, analyzed based on sensitivity of game equilibria and demonstrated in simulation. We further analyze how the…
We investigate a new class of congestion games, called Totally Unimodular (TU) Congestion Games, where the players' strategies are binary vectors inside polyhedra defined by totally unimodular constraint matrices. Network congestion games belong to this class. In the symmetric case, when all players have the same strategy set, we design an algorithm that finds an optimal aggregated strategy and then decomposes it into the single players' strategies. This approach yields strongly polynomial-time algorithms to (i) find a pure Nash equilibrium, and (ii) compute a socially optimal state, if the de…
We study the problem of finding Stackelberg equilibria in games with a massive number of players. So far, the only known game instances in which the problem is solved in polynomial time are some particular congestion games. However, a complete characterization of hard and easy instances is still lacking. In this paper, we extend the state of the art along two main directions. First, we focus on games where players' actions are made of multiple resources, and we prove that the problem is NP-hard and not in Poly-APX unless P = NP, even in the basic case in which players are symmetric, their acti…
The Multiplicative Weights Update (MWU) method is a ubiquitous meta-algorithm that works as follows: A distribution is maintained on a certain set, and at each step the probability assigned to element $γ$ is multiplied by $(1 -εC(γ))>0$ where $C(γ)$ is the "cost" of element $γ$ and then rescaled to ensure that the new values form a distribution. We analyze MWU in congestion games where agents use \textit{arbitrary admissible constants} as learning rates $ε$ and prove convergence to \textit{exact Nash equilibria}. Our proof leverages a novel connection between MWU and the Baum-Welch algorithm, …
In this paper, we compute $ε$-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…
In congestion games, users make myopic routing decisions to jam each other, and the social planner with the full information designs mechanisms on information or payment side to regulate. However, it is difficult to obtain time-varying traffic conditions, and emerging crowdsourcing platforms (e.g., Waze and Google Maps) provide a convenient way for mobile users travelling on the paths to learn and share the traffic conditions over time. When congestion games meet mobile crowdsourcing, it is critical to incentive selfish users to change their myopic routing policy and reach the best exploitatio…
The fast-growing market of autonomous vehicles, unmanned aerial vehicles, and fleets in general necessitates the design of smart and automatic navigation systems considering the stochastic latency along different paths in the traffic network. The longstanding shortest path problem in a deterministic network, whose counterpart in a congestion game setting is Wardrop equilibrium, has been studied extensively, but it is well known that finding the notion of an optimal path is challenging in a traffic network with stochastic arc delays. In this work, we propose three classes of risk-averse equilib…
Under the Markov decision process (MDP) congestion game framework, we study the problem of enforcing population distribution constraints on a population of players with stochastic dynamics and coupled congestion costs. Existing research demonstrates that the constraints on the players' population distribution can be satisfied by enforcing tolls. However, computing the minimum toll value for constraint satisfaction requires accurate modeling of the player's congestion costs. Motivated by settings where an accurate congestion cost model is unavailable (e.g. transportation networks), we consider …
