This work focuses on the design of taxes in atomic congestion games, a commonly studied model for competitive resource sharing. While most related studies focus on optimizing either the worst- or best-case performance (i.e., Price of Anarchy (PoA) or Price of Stability (PoS)), we investigate whether optimizing for the PoA has consequences on the PoS. Perhaps surprisingly, our results reveal a fundamental trade-off between the two performance metrics. Our main result demonstrates that the taxation rule that optimizes the PoA inherits a matching PoS, implying that the best outcome is no better than the worst outcome under such a design choice. We then study this trade-off in terms of the Pareto frontier between the PoA and PoS. Our results also establish that any taxes with PoS equal to 1 incur a much higher PoA, and that, in several well-studied cases, the untaxed setting lies strictly above the Pareto frontier.
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.
This paper provides a comprehensive convergence analysis of the PoA of both pure and mixed Nash equilibria in atomic congestion games with unsplittable demands.
We generalize the notions of user equilibrium and system optimum to non-atomic congestion games with stochastic demands. We establish upper bounds on the price of anarchy for three different settings of link cost functions and demand distributions, namely, (a) affine cost functions and general distributions, (b) polynomial cost functions and general positive-valued distributions, and (c) polynomial cost functions and the normal distributions. All the upper bounds are tight in some special cases, including the case of deterministic demands.
We study {\em bottleneck congestion games} where the social cost is determined by the worst congestion of any resource. These games directly relate to network routing problems and also job-shop scheduling problems. In typical bottleneck congestion games, the utility costs of the players are determined by the worst congested resources that they use. However, the resulting Nash equilibria are inefficient, since the price of anarchy is proportional on the number of resources which can be high. Here we show that we can get smaller price of anarchy with the bottleneck social cost metric. We introdu…
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 …
One of the main results shown through Roughgarden's notions of smooth games and robust price of anarchy is that, for any sum-bounded utilitarian social function, the worst-case price of anarchy of coarse correlated equilibria coincides with that of pure Nash equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions and fo…
When users lack specific knowledge of various system parameters, their uncertainty may lead them to make undesirable deviations in their decision making. To alleviate this, an informed system operator may elect to signal information to uninformed users with the hope of persuading them to take more preferable actions. In this work, we study public and truthful signalling mechanisms in the context of Bayesian congestion games on parallel networks. We provide bounds on the possible benefit a signalling policy can provide with and without the concurrent use of monetary incentives. We find that tho…
We consider non-cooperative unsplittable congestion games where players share resources, and each player's strategy is pure and consists of a subset of the resources on which it applies a fixed weight. Such games represent unsplittable routing flow games and also job allocation games. The congestion of a resource is the sum of the weights of the players that use it and the player's cost function is the sum of the utilities of the resources on its strategy. The social cost is the total weighted sum of the player's costs. The quality of Nash equilibria is determined by the price of anarchy ($PoA…
Max-Planck-Institut für Informatik（马克斯·普朗克信息学研究所）；University of Athens（雅典大学）；Patras University（帕特雷大学）
We study the performance of approximate Nash equilibria for linear congestion games. We consider how much the price of anarchy worsens and how much the price of stability improves as a function of the approximation factor $ε$. We give (almost) tight upper and lower bounds for both the price of anarchy and the price of stability for atomic and non-atomic congestion games. Our results not only encompass and generalize the existing results of exact equilibria to $ε$-Nash equilibria, but they also provide a unified approach which reveals the common threads of the atomic and non-atomic price of ana…
We study the inefficiency of equilibria for various classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i's perceived cost is a convex combination of 1-α_i times his direct cost and α_i times the social cost. Tuning the parameters α_i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of linear congestion games, fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concept…
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…
The price of anarchy (PoA) is a popular metric for analyzing the inefficiency of self-interested decision making. Although its study is widespread, characterizing the PoA can be challenging. A commonly employed approach is based on the smoothness framework, which provides tight PoA values under the assumption that the system objective consists in the sum of the agents' individual welfares. Unfortunately, several important classes of problems do not satisfy this requirement (e.g., taxation in congestion games), and our first result demonstrates that the smoothness framework does *not* tightly c…
We study {\em bottleneck routing games} where the social cost is determined by the worst congestion on any edge in the network. In the literature, bottleneck games assume player utility costs determined by the worst congested edge in their paths. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high and proportional to the size of the network. In order to obtain smaller price of anarchy we introduce {\em exponential bottleneck games} where the utility costs of the players are exponential functions of their congestions. We find that exponential b…
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.…
We analyze the network congestion game with atomic players, asymmetric strategies, and the maximum latency among all players as social cost. This important social cost function is much less understood than the average latency. We show that the price of anarchy is at most two, when the network is a ring and the link latencies are linear. Our bound is tight. This is the first sharp bound for the maximum latency objective.
We study bottleneck routing games where the social cost is determined by the worst congestion on any edge in the network. Bottleneck games have been studied in the literature by having the player's utility costs to be determined by the worst congested edge in their paths. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high with respect to the parameters of the game. In order to obtain smaller price of anarchy we explore {\em exponential bottleneck games} where the utility costs of the players are exponential functions on the congestion of the …
Two important metrics for measuring the quality of routing paths are the maximum edge congestion $C$ and maximum path length $D$. Here, we study bicriteria in routing games where each player $i$ selfishly selects a path that simultaneously minimizes its maximum edge congestion $C_i$ and path length $D_i$. We study the stability and price of anarchy of two bicriteria games: - {\em Max games}, where the social cost is $\max(C,D)$ and the player cost is $\max(C_i, D_i)$. We prove that max games are stable and convergent under best-response dynamics, and that the price of anarchy is bounded above …
This paper examines the behavior of the price of anarchy as a function of the traffic inflow in nonatomic congestion games with multiple origin-destination (O/D) pairs. Empirical studies in real-world networks show that the price of anarchy is close to 1 in both light and heavy traffic, thus raising the question: can these observations be justified theoretically? We first show that this is not always the case: the price of anarchy may remain a positive distance away from 1 for all values of the traffic inflow, even in simple three-link networks with a single O/D pair and smooth, convex costs. …
Price of anarchy quantifies the degradation of social welfare in games due to the lack of a centralized authority that can enforce the optimal outcome. At its antipodes, mechanism design studies how to ameliorate these effects by incentivizing socially desirable behavior and implementing the optimal state as equilibrium. In practice, the responsiveness to such measures depends on the wealth of each individual. This leads to a natural, but largely unexplored, question. Does optimal mechanism design entrench, or maybe even exacerbate, social inequality? We study this question in nonatomic conges…
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 …
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…
The congestion pricing is an efficient allocation approach to mediate demand and supply of network resources. Different from the previous pricing using Affine Marginal Cost (AMC), we focus on studying the game between network coding and routing flows sharing a single link when users are price anticipating based on an Average Cost Sharing (ACS) pricing mechanism. We characterize the worst-case efficiency bounds of the game compared with the optimal, i.e., the price-of anarchy (POA), which can be low bound 50% with routing only. When both network coding and routing are applied, the POA can be as…
We seek to understand the fundamental mathematics governing infrastructure-scale interactions between humans and machines, particularly when the machines' intended purpose is to influence and optimize the behavior of the humans. To that end, this paper investigates the worst-case congestion that can arise in nonatomic network congestion games when a fraction of the traffic is completely altruistic (e.g., benevolent self-driving cars) and the remainder is completely selfish (e.g., human commuters). We study the worst-case harm of altruism in such scenarios in terms of the perversity index, or t…
We study the issues of existence and inefficiency of pure Nash equilibria in linear congestion games with altruistic social context, in the spirit of the model recently proposed by de Keijzer {\em et al.} \cite{DSAB13}. In such a framework, given a real matrix $Γ=(γ_{ij})$ specifying a particular social context, each player $i$ aims at optimizing a linear combination of the payoffs of all the players in the game, where, for each player $j$, the multiplicative coefficient is given by the value $γ_{ij}$. We give a broad characterization of the social contexts for which pure Nash equilibria are a…
The price of anarchy has become a standard measure of the efficiency of equilibria in games. Most of the literature in this area has focused on establishing worst-case bounds for specific classes of games, such as routing games or more general congestion games. Recently, the price of anarchy in routing games has been studied as a function of the traffic demand, providing asymptotic results in light and heavy traffic. The aim of this paper is to study the price of anarchy in nonatomic routing games in the intermediate region of the demand. To achieve this goal, we begin by establishing some smo…
We study a static game played by a finite number of agents, in which agents are assigned independent and identically distributed random types and each agent minimizes its objective function by choosing from a set of admissible actions that depends on its type. The game is anonymous in the sense that the objective function of each agent depends on the actions of other agents only through the empirical distribution of their type-action pairs. We study the asymptotic behavior of Nash equilibria, as the number of agents tends to infinity, first by deriving laws of large numbers characterizes almos…
Route controlled autonomous vehicles could have a significant impact in reducing congestion in the future. Before applying multi-agent reinforcement learning algorithms to route control, we can model the system using a congestion game to predict and mitigate potential issues. We consider the problem of distributed operating systems in a transportation network that control the routing choices of their assigned vehicles. We formulate an associated network control game, consisting of multiple actors seeking to optimise the social welfare of their assigned subpopulations in an underlying nonatomic…
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…
Recent years have witnessed a growing number of attack vectors against increasingly interconnected traffic networks. Informational attacks have emerged as the prominent ones that aim to poison traffic data, misguide users, and manipulate traffic patterns. To study the impact of this class of attacks, we propose a game-theoretic framework where the attacker, as a Stackelberg leader, falsifies the traffic conditions to change the traffic pattern predicted by the Wardrop traffic equilibrium, achieved by the users, or the followers. The intended shift of the Wardrop equilibrium is a consequence of…
We study a general scenario of simultaneous contests that allocate prizes based on equal sharing: each contest awards its prize to all players who satisfy some contest-specific criterion, and the value of this prize to a winner decreases as the number of winners increases. The players produce outputs for a set of activities, and the winning criteria of the contests are based on these outputs. We consider two variations of the model: (i) players have costs for producing outputs; (ii) players do not have costs but have generalized budget constraints. We observe that these games are exact potenti…
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…
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…
We introduce a new class of games, called social contribution games (SCGs), where each player's individual cost is equal to the cost he induces on society because of his presence. Our results reveal that SCGs constitute useful abstractions of altruistic games when it comes to the analysis of the robust price of anarchy. We first show that SCGs are altruism-independently smooth, i.e., the robust price of anarchy of these games remains the same under arbitrary altruistic extensions. We then devise a general reduction technique that enables us to reduce the problem of establishing smoothness for …
In this work, we consider the problem of minimising the social cost in atomic congestion games. For this problem, we provide tight computational lower bounds along with taxation mechanisms yielding polynomial time algorithms with optimal approximation. Perhaps surprisingly, our results show that indirect interventions, in the form of efficiently computed taxation mechanisms, yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the agents' actions. It follows that no other tractable approach geared at incentivizing desirable sys…
In this paper, we study a routing and travel-mode choice problem for mobility systems with a multimodal transportation network as a ``mobility game" with coupled action sets. We develop a game-theoretic framework to study the impact on efficiency of the travelers' behavioral decision-making. In our framework, we introduce a mobility ``pricing mechanism," in which we model traffic congestion using linear cost functions while also considering the waiting times at different transport hubs. We show that the travelers' selfish actions lead to a pure-strategy Nash equilibrium. We then perform a Pric…
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…
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…
Routing games are amongst the most studied classes of games. Their two most well-known properties are that learning dynamics converge to equilibria and that all equilibria are approximately optimal. In this work, we perform a stress test for these classic results by studying the ubiquitous dynamics, Multiplicative Weights Update, in different classes of congestion games, uncovering intricate non-equilibrium phenomena. As the system demand increases, the learning dynamics go through period-doubling bifurcations, leading to instabilities, chaos and large inefficiencies even in the simplest case …
We investigate traffic routing both from the perspective of theory as well as real world data. First, we introduce a new type of games: $θ$-free flow games. Here, commuters only consider, in their strategy sets, paths whose free-flow costs (informally their lengths) are within a small multiplicative $(1+θ)$ constant of the optimal free-flow cost path connecting their source and destination, where $θ\geq0$. We provide an exhaustive analysis of tight bounds on PoA($θ$) for arbitrary classes of cost functions, both in the case of general congestion/routing games as well as in the special case of …
We consider a general class of Bayesian Games where each players utility depends on his type (possibly multidimensional) and on the strategy profile and where players' types are distributed independently. We show that if their full information version for any fixed instance of the type profile is a smooth game then the Price of Anarchy bound implied by the smoothness property, carries over to the Bayes-Nash Price of Anarchy. We show how some proofs from the literature (item bidding auctions, greedy auctions) can be cast as smoothness proofs or be simplified using smoothness. For first price it…
We propose a model of discrete time dynamic congestion games with atomic players and a single source-destination pair. The latencies of edges are composed by free-flow transit times and possible queuing time due to capacity constraints. We give a precise description of the dynamics induced by the individual strategies of players and of the corresponding costs, either when the traffic is controlled by a planner, or when players act selfishly. In parallel networks, optimal and equilibrium behavior eventually coincides, but the selfish behavior of the first players has consequences that cannot be…
We present a general technique, based on a primal-dual formulation, for analyzing the quality of self-emerging solutions in weighted congestion games. With respect to traditional combinatorial approaches, the primal-dual schema has at least three advantages: first, it provides an analytic tool which can always be used to prove tight upper bounds for all the cases in which we are able to characterize exactly the polyhedron of the solutions under analysis; secondly, in each such a case the complementary slackness conditions give us an hint on how to construct matching lower bounding instances; t…
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…
The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the objective function value of an equilibrium --- solution of the game --- and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games. In the paper, we present an approach based on linear programming duality to stu…
Best response (BR) dynamics is a natural method by which players proceed toward a pure Nash equilibrium via a local search method. The quality of the equilibrium reached may depend heavily on the order by which players are chosen to perform their best response moves. A {\em deviator rule} $S$ is a method for selecting the next deviating player. We provide a measure for quantifying the performance of different deviator rules. The {\em inefficiency} of a deviator rule $S$ is the maximum ratio, over all initial profiles $p$, between the social cost of the worst equilibrium reachable by $S$ from $…
In this work we completely characterize how the frequency with which each player participates in the game dynamics affects the possibility of reaching efficient states, i.e., states with an approximation ratio within a constant factor from the price of anarchy, within a polynomially bounded number of best responses. We focus on the well known class of congestion games and we show that, if each player is allowed to play at least once and at most $β$ times any $T$ best responses, states with approximation ratio $O(β)$ times the price of anarchy are reached after $T \lceil \log \log n \rceil$ bes…
