Boundedly Rational User Equilibria (BRUE) capture situations where all agents on a transportation network are electing the fastest option up to some time indifference, and serve as a relaxation of User Equilibria (UE), where each agent exactly minimizes their travel time. We study how the social cost under BRUE departs from that of UE in the context of static demand and stochastic costs, along with the implications of BRUE on the optimal signaling scheme of a benevolent central planner. We show that the average excess time is sublinear in the maximum time indifference of the agents, though such aggregate may hide disparity between populations and the sublinearity constant depends on the topology of the network. Regarding the design of public signals, even though in the limit where agents are totally indifferent, it is optimal to not reveal any information, there is in general no trend in how much information is optimally disclosed to agents. What is more, an increase in information disclosed may either harm or benefit agents as a whole.
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 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…
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…
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 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 e…
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.
This paper concerns computing approximate pure Nash equilibria in weighted congestion games, which has been shown to be PLS-complete. With the help of $\hatΨ$-game and approximate potential functions, we propose two algorithms based on best response dynamics, and prove that they efficiently compute $\fracρ{1-ε}$-approximate pure Nash equilibria for $ρ= d!$ and $ρ=\frac{2\cdot W\cdot(d+1)}{2\cdot W+d+1}\le {d + 1}$, respectively, when the weighted congestion game has polynomial latency functions of degree at most $d \ge 1$ and players' weights are bounded from above by a constant $W \ge 1$. Thi…
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…
On the Fragility of Worst-Case Nash Equilibria in Atomic Congestion Games
University of Colorado at Colorado Springs（科罗拉多大学科罗拉多斯普林斯分校）；Politecnico di Torino（都灵理工大学）；Thayer School of Engineering, Dartmouth College（达特茅斯学院塔克工程学院）
We show that computing an equilibrium in atomic splittable congestion games with player-specific affine cost functions $l_{e,i}(x) = a_{e,i} x + b_{e,i}$ is $\mathsf{PPAD}$-complete. To prove that the problem is contained in $\mathsf{PPAD}$, we develop a homotopy method that traces an equilibrium for varying flow demands of the players. A key technique for this method is to describe the evolution of the equilibrium locally by a novel block Laplacian matrix. Using the properties of this matrix give rise to a path following formulation for computing an equilibrium where states correspond to supp…
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…
We study the existence of approximate pure Nash equilibria ($α$-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree $d$. Previously it was known that $d$-approximate equilibria always exist, while nonexistence was established only for small constants, namely for $1.153$-PNE. We improve significantly upon this gap, proving that such games in general do not have $\tildeΘ(\sqrt{d})$-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a spec…
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…
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 playe…
With increasing renewable penetration in power systems, a prominent challenge in efficient and reliable power system operation is handling the uncertainties inherent in the renewable generation. In this paper, we propose a simple two-settlement market mechanism in which renewable power producers (RPPs) participate, so that a) the independent system operator (ISO) does not need to consider the uncertainties of the renewables in its economic dispatch, and yet b) the market equilibrium is shown to approach social efficiency as if the ISO solves a stochastic optimization taking into account all th…
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…
In this work, we seek a more refined understanding of the complexity of local optimum computation for Max-Cut and pure Nash equilibrium (PNE) computation for congestion games with weighted players and linear latency functions. We show that computing a PNE of linear weighted congestion games is PLS-complete either for very restricted strategy spaces, namely when player strategies are paths on a series-parallel network with a single origin and destination, or for very restricted latency functions, namely when the latency on each resource is equal to the congestion. Our results reveal a remarkabl…
We consider the question of whether, and in what sense, Wardrop equilibria provide a good approximation for Nash equilibria in atomic unsplittable congestion games with a large number of small players. We examine two different definitions of small players. In the first setting, we consider games where each player's weight is small. We prove that when the number of players goes to infinity and their weights to zero, the random flows in all (mixed) Nash equilibria for the finite games converge in distribution to the set of Wardrop equilibria of the corresponding nonatomic limit game. In the seco…
This paper provides a comprehensive convergence analysis of the PoA of both pure and mixed Nash equilibria in atomic congestion games with unsplittable demands.
A key challenge in responding to public health crises such as COVID-19 is the difficulty of predicting the results of feedback interconnections between the disease and society. As a step towards understanding these interconnections, we pose a simple game-theoretic model of a global pandemic in which individuals can choose where to live, and we investigate the global behavior that may emerge as a result of individuals reacting locally to the competing costs of isolation and infection. We study the game-theoretic equilibria that emerge from this setup when the population is composed of either se…
A central question in routing games has been to establish conditions for the uniqueness of the equilibrium, either in terms of network topology or in terms of costs. This question is well understood in two classes of routing games. The first is the non-atomic routing introduced by Wardrop on 1952 in the context of road traffic in which each player (car) is infinitesimally small; a single car has a negligible impact on the congestion. Each car wishes to minimize its expected delay. Under arbitrary topology, such games are known to have a convex potential and thus a unique equilibrium. The secon…
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 …
Institute of Information Management, National Chiao Tung University（台湾交通大学信息管理研究所）；Institute of Information Science, Academia Sinica（中央研究院信息科学研究所）
Different types of dynamics have been studied in repeated game play, and one of them which has received much attention recently consists of those based on "no-regret" algorithms from the area of machine learning. It is known that dynamics based on generic no-regret algorithms may not converge to Nash equilibria in general, but to a larger set of outcomes, namely coarse correlated equilibria. Moreover, convergence results based on generic no-regret algorithms typically use a weaker notion of convergence: the convergence of the average plays instead of the actual plays. Some work has been done s…
This paper gives a complete analysis of worst-case equilibria for various versions of weighted congestion games with two players and affine cost functions. The results are exact price of anarchy bounds which are parametric in the weights of the two players, and establish exactly how the primitives of the game enter into the quality of equilibria. Interestingly, some of the worst-cases are attained when the players' weights only differ slightly. Our findings also show that sequential play improves the price of anarchy in all cases, however, this effect vanishes with an increasing difference in …
University of Augsburg（奥格斯堡大学）；University of Applied Sciences Augsburg（奥格斯堡应用技术大学）；TU Berlin（柏林工业大学）；KU Leuven（鲁汶大学）
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 uniq…
We study the equilibrium behavior in a multi-commodity selfish routing game with many types of uncertain users where each user over- or under-estimates their congestion costs by a multiplicative factor. Surprisingly, we find that uncertainties in different directions have qualitatively distinct impacts on equilibria. Namely, contrary to the usual notion that uncertainty increases inefficiencies, network congestion actually decreases when users over-estimate their costs. On the other hand, under-estimation of costs leads to increased congestion. We apply these results to urban transportation ne…
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, …
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 i…
Princeton University（普林斯顿大学）；University of Oregon（俄勒冈大学）；University of Chicago（芝加哥大学）
Most algorithmic studies on multi-agent information design so far have focused on the restricted situation with no inter-agent externalities; a few exceptions investigated truly strategic games such as zero-sum games and second-price auctions but have all focused only on optimal public signaling. This paper initiates the algorithmic information design of both \emph{public} and \emph{private} signaling in a fundamental class of games with negative externalities, i.e., singleton congestion games, with wide application in today's digital economy, machine scheduling, routing, etc. For both public …
How can we design mechanisms to promote efficient use of shared resources? Here, we answer this question in relation to the well-studied class of atomic congestion games, used to model a variety of problems, including traffic routing. Within this context, a methodology for designing tolling mechanisms that minimize the system inefficiency (price of anarchy) exploiting solely local information is so far missing in spite of the scientific interest. In this manuscript we resolve this problem through a tractable linear programming formulation that applies to and beyond polynomial congestion games.…
Credit-based congestion pricing (CBCP) has emerged as a mechanism to alleviate the social inequity concerns of road congestion pricing - a promising strategy for traffic congestion mitigation - by providing low-income users with travel credits to offset some of their toll payments. While CBCP offers immense potential for addressing inequity issues that hamper the practical viability of congestion pricing, the deployment of CBCP in practice is nascent, and the potential efficacy and optimal design of CBCP schemes have yet to be formalized. In this work, we study the design of CBCP schemes to ac…
The University of Tokyo（东京大学）；NTT Communication Science Laboratories（NTT通信科学实验室）
We address Stackelberg models of combinatorial congestion games (CCGs); we aim to optimize the parameters of CCGs so that the selfish behavior of non-atomic players attains desirable equilibria. This model is essential for designing such social infrastructures as traffic and communication networks. Nevertheless, computational approaches to the model have not been thoroughly studied due to two difficulties: (I) bilevel-programming structures and (II) the combinatorial nature of CCGs. We tackle them by carefully combining (I) the idea of \textit{differentiable} optimization and (II) data structu…
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…
