On Tree Equilibria in Max-Distance Network Creation Games

Qian Wang

Abstract: We 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.