On Approximate Nash Equilibria in Network Design

In this paper, we study a basic network design game in which n self-interested agents, each having individual connectivity requirements, wish to build a network by purchasing links from a given set of edges. A fundamental cost-sharing mechanism is Shapley cost-sharing, which splits the cost of an edge in a fair manner among the agents using the edge. It is well known that in such games, the price of anarchy is n, while the price of stability is H(n), where H(n) denotes the nth harmonic number. We investigate whether an optimal minimum-cost network represents an attractive, relatively stable state that agents might want to purchase: what extra cost does an agent incur compared to a possible strategy deviation? We employ the concept of α-approximate Nash equilibria, in which no agent can reduce its cost by a factor of more than α. We prove that for single-source games in undirected graphs, every optimal network represents an H(n)-approximate Nash equilibrium. We show that this bound is tight by presenting a matching lower bound. We extend the results to cooperative games in which agents may form coalitions. A combination of strategies forms an α-approximate strong Nash equilibrium if no coalition can deviate such that all members of the coalition reduce their cost by a factor of more than α. We prove that if coalitions of up to c agents are allowed, 1⩽c⩽n, then every optimal network represents a 2c(ln(n/c)+2)-approximate strong Nash equilibrium. We give a corresponding lower bound that is tight up to a small constant factor. Moreover, we extend the results to weighted games and present tight upper and lower bounds on the quality of optimal solutions in noncooperative games. Finally, we show that in general source–sink games and in directed graphs, minimum-cost networks do not represent good states.