In this study, we formulate positive and negative externalities caused by changes in the supply of shared vehicles as ride sharing games. The study aims to understand the price of anarchy (PoA) and its improvement via a coordination technique in ride sharing games. A critical question is whether ride sharing games exhibit a pure Nash equilibrium (pNE) since the PoA bound assumes it. Our result shows a sufficient condition for a ride sharing game to have a finite improvement property and a pNE similar to potential games. This is the first step to analyze PoA bound and its improvement by coordination in ride sharing games. We also show an example of coordinating players in ride sharing games using signaling and evaluate the improvement in the PoA.
Since Rosenthal introduced the congestion game @cite_0 , it has been applied to problems of congestion externalities in several areas such as transportation and communication networks @cite_12 . In this game, players choose a combination of resources and the players' payoffs depend on the congestion level, i.e., the number of players using the same resources. The congestion game is a subclass of potential games, which feature the finite improvement property (FIP). This property guarantees that if each player updates his strategy in response to other players by turns, it will reduce his private cost and improve the common potential function, which eventually reaches a local minimum, the pure Nash equilibrium (pNE), where each player has a deterministic strategy. Because of this property, potential and congestion games have been well studied and have wide applications @cite_2 .
@cite_0 @cite_12 @cite_2
2148042342 2070173191 2076641899 2134373565 2810102121 2056237576 2089015050 2108444537 2072113937 2802735465 1517645949 2759949687 1548210998 1978317291 2074104906 2270201883 2088375330 2133243407 2124034793 1839131291
Congestion games are multi-player games in which players' costs are additive over a set of resources that have anonymous cost functions, with pure strategies corresponding to certain subsets of resources. In a splittable congestion game, each player can choose a convex combination of subsets of resources. We characterize the worst-case price of anarchy — a quantitative measure of the inefficiency of equilibria — in splittable congestion games. Our approximation guarantee is parameterized by the set of allowable resource cost functions, and degrades with the “degree of nonlinearity” of these cost functions. We prove that our guarantee is the best possible for every set of cost functions that satisfies mild technical conditions. We prove our guarantee using a novel “local smoothness” proof framework, and as a consequence the guarantee applies not only to the Nash equilibria of splittable congestion games, but also to all correlated equilibria. We study congestion games where players aim to access a set of resources. Each player has a set of possible strategies and each resource has a function associating the latency it incurs to the players using it. Players are non--cooperative and each wishes to follow a strategy that minimizes her own latency with no regard to the global optimum. Previous work has studied the impact of this selfish behavior on system performance. In this article, we study the question of how much the performance can be improved if players are forced to pay taxes for using resources. Our objective is to extend the original game so that selfish behavior does not deteriorate performance. We consider atomic congestion games with linear latency functions and present both negative and positive results. Our negative results show that optimal system performance cannot be achieved even in very simple games. On the positive side, we show that there are ways to assign taxes that can improve the performance of linear congestion games by forcing players to follow strategies where the total latency suffered is within a factor of 2 of the minimum possible; this result is shown to be tight. Furthermore, even in cases where in the absence of taxes the system behavior may be very poor, we show that the total disutility of players (latency plus taxes) is not much larger than the optimal total latency. Besides existential results, we show how to compute taxes in time polynomial in the size of the game by solving convex quadratic programs. Similar questions have been extensively studied in the model of non-atomic congestion games. To the best of our knowledge, this is the first study of the efficiency of taxes in atomic congestion games. Abstract 6 introduced the class of congestion games and proved that they always possess a Nash equilibrium in pure strategies. Here we obtain conditions for the existence of a strong equilibrium in this class of games, as well as for the equivalence of Nash and strong equilibria. We also give conditions for uniqueness and for Pareto optimality of the Nash equilibrium. Except for a natural monotonicity assumption on the utilities, the conditions are expressed only in terms of the underlying congestion game form. It turns out that avoiding a certain type of bad configuration in the strategy spaces is essential to positive results. Journal of Economic Literature Classification Numbers: C71, C72, D62. A network congestion game is played on a directed, two-terminal network. Every player chooses a route from his origin to his destination. The cost of a route is the sum of the costs of the arcs on it. The arc cost is a function of the number of players who use it. Rosenthal proved that such a game always has a Nash equilibrium in pure strategies. Here we pursue a systematic study of the classes of networks for which a strong equilibrium is guaranteed to exist, under two opposite monotonicity assumptions on the arc cost functions. Our main results are: (a) If costs are increasing, strong equilibrium is guaranteed on extension-parallel networks, regardless of whether the players’ origins and destinations are the same or may differ. (b) If costs are decreasing, and the players have the same origin but possibly different destinations, strong equilibrium is guaranteed on series-parallel networks. (c) If costs are decreasing, and both origins and destinations may differ, strong equilibrium is guaranteed on multiextension-parallel networks. In each case, the network condition is not only sufficient but also necessary in order to guarantee strong equilibrium. These results extend and improve earlier ones by Holzman and Law-Yone in the increasing case, and by in the decreasing case. We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion. In this paper, we introduce and analyze the properties of a class of games, the atomic congestion games on graphs (ACGGs), which is a generalization of the classical congestion games. In particular, an ACGG captures the spatial information that is often ignored in a classical congestion game. This is useful in many networking problems, e.g., wireless networks where interference among the users heavily depends on the spatial information. In an ACGG, a player's payoff for using a resource is a function of the number of players who interact with it and use the same resource. Such spatial information can be captured by a graph. We study fundamental properties of the ACGGs: under what conditions these games possess a pure strategy Nash equilibrium (PNE), or the finite improvement property (FIP), which is sufficient for the existence of a PNE. We show that a PNE may not exist in general, but that it does exist in many important special cases including tree, loop, or regular bipartite networks. The FIP holds for important special cases including systems with two resources or identical payoff functions for each resource. Finally, we present two wireless network applications of ACGGs: power control and channel contention under IEEE 802.11. Congestion games constitute an important class of games in which computing an exact or even approximate pure Nash equilibrium is in general PLS -complete. We present a surprisingly simple polynomial-time algorithm that computes @math -approximate Nash equilibria in these games. In particular, for congestion games with linear latency functions, our algorithm computes @math -approximate pure Nash equilibria in time polynomial in the number of players, the number of resources and @math . It also applies to games with polynomial latency functions with constant maximum degree @math ; there, the approximation guarantee is @math . The algorithm essentially identifies a polynomially long sequence of best-response moves that lead to an approximate equilibrium, the existence of such short sequences is interesting in itself. These are the first positive algorithmic results for approximate equilibria in non-symmetric congestion games. We strengthen them further by proving that, for congestion games that deviate from our mild assumptions, computing @math -approximate equilibria is PLS -complete for any polynomial-time computable @math . We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. We define a new class of games, congestion games with load-dependent failures (CGLFs). In a CGLF each player can choose a subset of a set of available resources in order to try and perform his task. We assume that the resources are identical but that players' benefits from successful completion of their tasks may differ. Each resource is associated with a cost of use and a failure probability which are load-dependent. Although CGLFs in general do not have a pure strategy Nash equilibrium, we prove the existence of a pure strategy Nash equilibrium in every CGLF with nondecreasing cost functions. Moreover, we present a polynomial time algorithm for computing such an equilibrium. We study the outcome of natural learning algorithms in atomic congestion games. Atomic congestion games have a wide variety of equilibria often with vastly differing social costs. We show that in almost all such games, the well- known multiplicative-weights learning algorithm results in convergence to pure equilibria. Our results show that nat- ural learning behavior can avoid bad outcomes predicted by the price of anarchy in atomic congestion games such as the load-balancing game introduced by Koutsoupias and Pa- padimitriou, which has super-constant price of anarchy and has correlated equilibria that are exponentially worse than any mixed Nash equilibrium. Our results identify a set of mixed Nash equilibria that we call weakly stable equilibria. Our notion of weakly stable is defined game-theoretically, but we show that this property holds whenever a stability criterion from the theory of dy- namical systems is satisfied. This allows us to show that in every congestion game, the distribution of play converges to the set of weakly stable equilibria. Pure Nash equilibria are weakly stable, and we show using techniques from algebraic geometry that the converse is true with probability 1 when congestion costs are selected at random independently on each edge (from any monotonically parametrized distribu- tion). We further extend our results to show that players can use algorithms with different (sufficiently small) learn- ing rates, i.e. they can trade off convergence speed and long term average regret differently. Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria—a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly NP-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria. We study bottleneck congestion games where the social cost is determined by the worst congestion on any resource. In the literature, bottleneck games assume player utility costs determined by the worst congested resource in their strategy. However, the Nash equilibria of such games are inefficient since the price of anarchy can be very high and proportional to the number of resources. In order to obtain smaller price of anarchy we introduce exponential bottleneck games, where the utility costs of the players are exponential functions of their congestions. In particular, the delay function for any resource r is MCr, where Cr denotes the number of players that use r, and M is an integer constant. We find that exponential bottleneck games are very efficient and give the following bound on the price of anarchy: O(log |R|), where R is the set of resources. This price of anarchy is tight, since we demonstrate a game with price of anarchy Ω(log |R|). We obtain our tight bounds by using two novel proof techniques: transformation, which we use to convert arbitrary games to simpler games, and expansion, which we use to bound the price of anarchy in a simpler game. We study the impact of combinatorial structure in congestion games on the complexity of computing pure Nash equilibria and the convergence time of best response sequences. In particular, we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time. We show that if the strategy space of each player consists of the bases of a matroid over the set of resources, then the lengths of all best response sequences are polynomially bounded in the number of players and resources. We also prove that this result is tight, that is, the matroid property is a necessary and sufficient condition on the players' strategy spaces for guaranteeing polynomial-time convergence to a Nash equilibrium. In addition, we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games, including first results about the hardness of market sharing games and congestion games for overlay network design. Our approach also yields a short proof for the PLS-completeness of network congestion games. In particular, we show that network congestion games are PLS-complete for directed and undirected networks even in case of linear latency functions. We study the speed of convergence of decentralized dynamics to approximately optimal solutions in potential games. We consider α-Nash dynamics in which a player makes a move if the improvement in his payoff is more than an α factor of his own payoff. Despite the known polynomial convergence of α-Nash dynamics to approximate Nash equilibria in symmetric congestion games [7], it has been shown that the convergence time to approximate Nash equilibria in asymmetric congestion games is exponential [25]. In contrast to this negative result, and as the main result of this paper, we show that for asymmetric congestion games with linear and polynomial delay functions, the convergence time of α-Nash dynamics to an approximate optimal solution is polynomial in the number of players, with approximation ratio that is arbitrarily close to the price of anarchy of the game. In particular, we show this polynomial convergence under the minimal liveness assumption that each player gets at least one chance to move in every T steps. We also prove that the same polynomial convergence result does not hold for (exact) best-response dynamics, showing the α-Nash dynamics is required. We extend these results for congestion games to other potential games including weighted congestion games with linear delay functions, cut games (also called party affiliation games) and market sharing games. We investigate a new class of congestion games, called Totally Unimodular Congestion Games, in which the strategies of each player are expressed as binary vectors lying in a polyhedron defined using a totally unimodular constraint matrix and an integer right-hand side. We study both the symmetric and the asymmetric variants of the game. In the symmetric variant, all players have the same strategy set, i.e. the same constraint matrix and right-hand side. Network congestion games are an example of this class. proved that a pure Nash equilibrium of symmetric network congestion games can be found in strongly polynomial time, while the asymmetric network congestion games are PLS-complete. We give a strongly polynomial-time algorithm to find a pure Nash equilibrium of any symmetric totally unimodular congestion game. We also identify four totally unimodular congestion games, where the players' strategy sets are matchings, vertex covers, edge covers and stable sets of a given bipartite graph. For these games we derive specialized combinatorial algorithms to find a pure Nash equilibrium in the symmetric variant, and show the asymmetric variant is PLS-complete. Imitating successful behavior is a natural and frequently applied approach when facing scenarios for which we have little or no experience upon which we can base our decision. In this paper, we consider such behavior in atomic congestion games. We propose to study concurrent imitation dynamics that emerge when each player samples another player and possibly imitates this agents' strategy if the anticipated latency gain is sufficiently large. Our main focus is on convergence properties. Using a potential function argument, we show that these dynamics converge in a monotonic fashion to stable states. In such a state none of the players can improve their latency by imitating others. As our main result, we show rapid convergence to approximate equilibria. At an approximate equilibrium only a small fraction of agents sustains a latency significantly above or below average. In particular, imitation dynamics behave like fully polynomial time approximation schemes (FPTAS). Fixing all other parameters, the convergence time depends only in a logarithmic fashion on the number of agents. Since imitation processes are not innovative they cannot discover unused strategies. Furthermore, strategies may become extinct with non-zero probability. For the case of singleton games, we show that the probability of this event occurring is negligible. Additionally, we prove that the social cost of a stable state reached by our dynamics is not much worse than an optimal state in singleton congestion games with linear latency functions. While we concentrate on the case of symmetric network congestion games, most of our results do not explicitly use the network structure. They continue to hold accordingly for general symmetric and asymmetric congestion games when each player samples within his commodity. The price of anarchy (POA) is a worst-case measure of the inefficiency of selfish behavior, defined as the ratio of the objective function value of a worst Nash equilibrium of a game and that of an optimal outcome. This measure implicitly assumes that players successfully reach some Nash equilibrium. This drawback motivates the search for inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash and correlated equilibria; or to sequences of outcomes generated by natural experimentation strategies, such as successive best responses or simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly stronger relatives in classes of games with a sum objective. First, we identify a "canonical sufficient condition" for an upper bound of the POA for pure Nash equilibria, which we call a smoothness argument. Second, we show that every bound derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of regret-minimizing players (or "price of total anarchy"). Smoothness arguments also have automatic implications for the inefficiency of approximate and Bayesian-Nash equilibria and, under mild additional assumptions, for bicriteria bounds and for polynomial-length best-response sequences. We also identify classes of games --- most notably, congestion games with cost functions restricted to an arbitrary fixed set --- that are tight, in the sense that smoothness arguments are guaranteed to produce an optimal worst-case upper bound on the POA, even for the smallest set of interest (pure Nash equilibria). Byproducts of our proof of this result include the first tight bounds on the POA in congestion games with non-polynomial cost functions, and the first structural characterization of atomic congestion games that are universal worst-case examples for the POA. We study the equilibria of non-atomic congestion games in which there are two types of players: rational players, who seek to minimize their own delay, and malicious players, who seek to maximize the average delay experienced by the rational players. We study the existence of pure and mixed Nash equilibria for these games, and we seek to quantify the impact of the malicious players on the equilibrium. One counter intuitive phenomenon which we demonstrate is the "windfall of malice": paradoxically, when a myopically malicious player gains control of a fraction of the flow, a fraction of the players change from rational to malicious, the new equilibrium may be more favorable for the remaining rational players than the previous equilibrium. We propose a natural model for agent failures in congestion games. In our model, each of the agents may fail to participate in the game, introducing uncertainty regarding the set of active agents. We examine how such uncertainty may change the Nash equilibria (NE) of the game. We prove that although the perturbed game induced by the failure model is not always a congestion game, it still admits at least one pure Nash equilibrium. Then, we turn to examine the effect of failures on the maximal social cost in any NE of the perturbed game. We show that in the limit case where failure probability is negligible new equilibria never emerge, and that the social cost may decrease but it never increases. For the case of nonnegligible failure probabilities, we provide a full characterization of the maximal impact of failures on the social cost under worst-case equilibrium outcomes.