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 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 + ε)-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 + ε)-approximate Nash equilibrium that does not cost much more.

In a network with selfish users, designing and deploying a protocol determines the rules of the game by which end users interact with each other and with the network. We study the problem of designing a protocol to optimize the equilibrium behavior of the induced network game. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal costsharing protocols for undirected and directed graphs, single-sink and multicommodity networks, different classes of cost-sharing methods, and different measures of the inefficiency of equilibria. One of our main technical tools is a complete characterization of the uniform cost-sharing protocols—protocols that are designed without foreknowledge of or assumptions on the network in which they will be deployed. We use this characterization result to identify the optimal uniform protocol in several scenarios: for example, the Shapley protocol is optimal in directed graphs, while the optimal protocol in undirected graphs, a simple priority scheme, has exponentially smaller worst-case price of anarchy than the Shapley protocol. We also provide several matching upper and lower bounds on the bestpossible performance of non-uniform cost-sharing protocols.

We study network design games where n 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 n. Therefore, recent 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 using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first super-constant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of Ω(log W / log log W) for weighted games, where W is the total weight of all the agents. This almost matches the known upper bound of O(log W). 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.

We consider a multicast game played by a set of selfish noncooperative players (i.e., nodes) on a rooted undirected graph. Players arrive one by one and each connects to the root by greedily choosing a path minimizing its cost; the cost of using an edge is split equally among all users using the edge. How large can the sum of the players ’ costs be, compared to the cost of a “socially optimal” solution, defined to be a minimum Steiner tree connecting the players to the root? We show that the ratio is O(log 2 n) and Ω(log n), when there are n players. One can view this multicast game as a variant of ONLINE STEINER TREE with a different cost sharing mechanism. Furthermore, we consider what happens if the players, in a second phase, are allowed to change their paths in order to decrease their costs. Thus, in the second phase players play best response dynamics until eventually a Nash equilibrium is reached. We show that the price of anarchy is O(log 3 n) and Ω(log n). We also make progress towards understanding the challenging case where arrivals and path changes by existing terminals are interleaved. In particular, we analyze the interesting special case where the terminals fire in random order and prove that the cost of the solution produced (with arbitrary interleaving of actions) is at most O(polylog(n) √ n) times the optimum.

Abstract. We study the survivable version of the game theoretic network formation model known as the Connection Game, originally introduced in [5]. In this model, players attempt to connect to a common source node in a network by purchasing edges, and sharing their costs with other players. We introduce the survivable version of this game, where each player desires 2 edge-disjoint connections between her pair of nodes instead of just a single connecting path, and analyze the quality of exact and approximate Nash equilibria. This version is significantly different from the original Connection Game and have more complications than the existing literature on arbitrary cost-sharing games since we consider the formation of networks that involve many cycles. For the special case where each node represents a player, we show that Nash equilibria are guaranteed to exist and price of stability is 1, i.e., there always exists a stable solution that is as good as the centralized optimum. For the general version of the Survivable Connection Game, we show that there always exists a 2-approximate Nash equilibrium that is as good as the centralized optimum. To obtain the result, we use an approximation algorithm technique that compares the strategy of each player with only a carefully selected subset of her strategy space as well as proving new results about the laminar structure of survivable networks, which may be of independent interest in classical settings. Furthermore, if a player is only allowed to deviate by changing the payments on one of her connection paths at a time, instead of both of them at once, we prove that the price of stability is 1. We also discuss the time complexity issues. 1

We give a brief and biased survey of the past, present, and future of research on the interface of theoretical computer science and game theory. 1

Abstract. We consider a cost sharing system where users are selfish and act according to their own interest. There is a set of facilities and each facility provides services to a subset of the users. Each user is interested in purchasing a service, and will buy it from the facility offering it at the lowest cost. The notion of social welfare is defined to be the total cost of the facilities chosen by the users. A central authority can encourage the purchase of services by offering subsidies that reduce their price, in order to improve the social welfare. The subsidies are financed by taxes collected from the users. Specifically, we investigate a noncooperative game, where users join the system, and act according to their best response. We model the system as an instance of a set cover game, where each element is interested in selecting a cover minimizing its payment. The subsidies are updated dynamically, following the selfish moves of the elements and the taxes collected due to their payments. Our objective is to design a dynamic subsidy mechanism that improves on the social welfare while collecting as taxes only a small fraction of the sum of the payments of the users. The performance of such a subsidy mechanism is thus defined by two different quality parameters: (i) the price of anarchy, defined as the ratio between the social welfare cost of the Nash equilibrium obtained and the cost of an optimal solution; and (ii) the taxation ratio, defined as the fraction of payments collected as taxes from the users. 1

Abstract. We consider a process called Group Network Formation Game, which represents the scenario when strategic agents are building a network together. In our game, agents can have extremely varied connectivity requirements, and attempt to satisfy those requirements by purchasing links in the network. We show a variety of results about equilibrium properties in such games, including the fact that the price of stability is 1 when all nodes in the network are owned by players, and that doubling the number of players creates an equilibrium as good as the optimum centralized solution. For the general case, we show the existence of a 2-approximate Nash equilibrium that is as good as the centralized optimum solution, as well as how to compute good approximate equilibria in polynomial time. Our results essentially imply that for a variety of connectivity requirements, giving agents more freedom can paradoxically result in more efficient outcomes. 1

Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network costsharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that only induce network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs, and that simple priority protocols are essentially optimal in undirected graphs.