On Tree Equilibria in Max-Distance Network Creation Games

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., 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 Demaineet al., 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 $\alpha \geq 23$ all equilibrium graphs in the max-distance network creation game must be trees, while the best bound in previous work is $\alpha > 129$. 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 so-called tree conjecture in the max-distance network creation game.