On Tree Equilibria in Max-Distance Network Creation Games

Abstract

AbstractWe study the Nash equilibrium and the price of anarchy in the max-distance network creation game. The network creation game, first introduced and studied by Fabrikant et al. [18], is a classic model for real-world networks from a game-theoretic point of view. In a network creation game with n selfish vertex agents, each vertex can build undirected edges incident to a subset of the other vertices. The goal of every agent is to minimize its creation cost plus its usage cost, where the creation cost is the unit edge cost \(\alpha \) times the number of edges it builds, and the usage cost is the sum of distances to all other agents in the resulting network. The max-distance network creation game, introduced and studied by Demaine et al. [15], is a key variant of the original game, where the usage cost takes into account the maximum distance instead. The main result of this paper shows that for \(\alpha > 19\) all equilibrium graphs in the max-distance network creation game must be trees, while the best bound in previous work is \(\alpha > 129\) [25]. We also improve the constant upper bound on the price of anarchy to 3 for tree equilibria. Our work brings new insights into the structure of Nash equilibria and takes one step forward in settling the tree conjecture in the max-distance network creation game.KeywordsNetwork creation gameTree equilibriumPrice of anarchy