We study strong equilibria in symmetric capacitated cost-sharing games. In these games, a graph with designated source @math and sink @math is given, and each edge is associated with some cost. Each agent chooses strategically an @math - @math path, knowing that the cost of each edge is shared equally between all agents using it. Two variants of cost-sharing games have been previously studied: (i) games where coalitions can form, and (ii) games where edges are associated with capacities; both variants are inspired by real-life scenarios. In this work we combine these variants and analyze strong equilibria (profiles where no coalition can deviate) in capacitated games. This combination gives rise to new phenomena that do not occur in the previous variants. Our contribution is two-fold. First, we provide a topological characterization of networks that always admit a strong equilibrium. Second, we establish tight bounds on the efficiency loss that may be incurred due to strategic behavior, as quantified by the strong price of anarchy (and stability) measures. Interestingly, our results are qualitatively different than those obtained in the analysis of each variant alone, and the combination of coalitions and capacities entails the introduction of more refined topology classes than previously studied.
@cite_4: Strong Nash equilibria and Pareto-optimal Nash equilibria are natural and important strengthenings of the Nash equilibrium concept. We study these stronger notions of equilibrium in congestion games, focusing on the relationships between the price of anarchy for these equilibria and that for standard Nash equilibria (which is well understood). For symmetric congestion games with polynomial or exponential latency functions, we show that the price of anarchy for strong and Pareto-optimal equilibria is much smaller than the standard price of anarchy. On the other hand, for asymmetric congestion games with polynomial latencies the strong and Pareto prices of anarchy are essentially as large as the standard price of anarchy; while for asymmetric games with exponential latencies the Pareto and standard prices of anarchy are the same but the strong price of anarchy is substantially smaller. Finally, in the special case of linear latencies, we show that in asymmetric games the strong and Pareto prices of anarchy coincide exactly with the known value @math for standard Nash, but are strictly smaller for symmetric games.
@cite_7: We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent's goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1+e)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65+e)-approximate Nash equilibrium that does not cost much more.
@cite_8: 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.
@cite_10: We study a capacitated symmetric network design game, where each of n agents wishes to construct a path from a network's source to its sink, and the cost of each edge is shared equally among the agents using it. The uncapacitated version of this problem has been introduced by (2003) and has been extensively studied. We find that the consideration of edge capacities entails a significant effect on the quality of the obtained Nash equilibria (NE), under both the utilitarian and the egalitarian objective functions, as well as on the convergence rate to an equilibrium. The following results are established. First, we provide bounds for the price of anarchy (PoA) and the price of stability (PoS) measures with respect to the utilitarian (i.e., sum of costs) and egalitarian (i.e., maximum cost) objective functions. Our main result here is that unlike the uncapacitated version, the network topology is a crucial factor in the quality of NE. Specifically, a network topology has a bounded PoA if and only if it is series-parallel (SP), i.e., a network that is built inductively by series compositions and parallel compositions of SP networks. Second, we show that the convergence rate of best-response dynamics (BRD) may take Ω(n1.5) steps. This is in contrast to the uncapacitated version, where convergence is guaranteed within at most n iterations.
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@cite_1: We study network design games where @math self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as @math . Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of @math for weighted games, where @math is the total weight of all the agents. This almost matches the known upper bound of @math . Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs.
@cite_6: Network design is a fundamental problem for which it is important to understand the effects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions—the Nash equilibria—may look quite different from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context—it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is @math , where @math is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria. We discuss connections to the class of potential games defined by Monderer and Shapley, and extend our results to cases in which users are seeking to balance network design costs with latencies in the constructed network, with stronger results when the network has only delays and no construction costs. We also present bounds on the convergence time of best-response dynamics, and discuss extensions to a weighted game.
@cite_0: Abstract In a route selection game on a network, every player chooses a route from the origin to the destination, which are common to all players. Costs are assigned to road segments in the form of monotone nondecreasing functions of the number of players who use them. Each player incurs a total cost equal to the sum of the costs of the road segments in his route. It is known that such a game always has a Nash equilibrium in pure strategies. Here we obtain a structural characterization of those networks for which a strong equilibrium is guaranteed to exist regardless of the cost assignment. The route selection games based on networks in this class enjoy more stability as well as other desirable properties of equilibrium regarding uniqueness and efficiency.
@cite_2: A class of noncooperative games (of interest in certain applications) is described. Each game in the class is shown to possess at least one Nash equilibrium in pure strategies.
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@cite_13: 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.
@cite_12: A strong equilibrium is a pure Nash equilibrium which is resilient to deviations by coalitions. We define the strong price of anarchy (SPoA) to be the ratio of the worst strong equilibrium to the social optimum. Differently from the Price of Anarchy (defined as the ratio of the worst Nash Equilibrium to the social optimum), it quantifies the loss incurred from the lack of a central designer in settings that allow for coordination. We study the SPoA in two settings, namely job scheduling and network creation. In the job scheduling game we show that for unrelated machines the SPoA can be bounded as a function of the number of machines and the size of the coalition. For the network creation game we show that the SPoA is at most 2. In both cases we show that a strong equilibrium always exists, except for a well defined subset of network creation games.
@cite_11: We study network games in which each player wishes to connect his source and sink, and the cost of each edge is shared among its users either equally (in Fair Connection Games--FCG's) or arbitrarily (in General Connection Games--GCG's). We study the existence and quality of strong equilibria (SE)--strategy profiles from which no coalition can improve the cost of each of its members--in these settings. We show that SE always exist in the following games: (1) Single source and sink FCG's and GCG's. (2) Single source multiple sinks FCG's and GCG's on series parallel graphs. (3) Multi source and sink FCG's on extension parallel graphs. As for the quality of the SE, in any FCG with n players, the cost of any SE is bounded by H(n) (i.e., the harmonic sum), contrasted with the [Theta](n) price of anarchy. For any GCG, any SE is optimal.
Cost-sharing connection games were introduced by @cite_7 , and have been widely studied since. The fair cost-sharing mechanism has been studied in @cite_6 . A key property of fair cost-sharing games is the fact that they admit an exact potential function and thus possess a pure strategy NE @cite_15 @cite_2 . Settings with coalitions in congestion games have been studied by Holzman and Law-Yone @cite_8 @cite_0 , and @cite_11 studied the existence and quality of SE in cost-sharing games. The characterization of network topologies that admit SE in congestion games with monotone cost functions has been later established in @cite_13 . The PoA measure has been introduced by Koutsoupias and Papadimitriou @cite_5 to study the quality of NE in games. The analogue of the PoA with respect to SE (called the strong PoA) was introduced and analyzed by @cite_9 . The SPoA in cost-sharing games was studied in @cite_1 @cite_11 , and was also studied with respect to various additional settings; see, e.g., @cite_4 @cite_12 . The consideration of capacities in cost-sharing games was first suggested by Feldman and Ron @cite_10 .