Published: August 4, 2008

**Keywords:**game theory, network design, Nash equilibrium, connection game, price of stability

**ACM Classification:**G.2.2, F.2.2, F.2.3, J.4

**AMS Classification:**68W25, 68W40, 05C85, 90B10, 90B18, 91A43, 91A40

**Abstract:**
[Plain Text Version]

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 on the optimal network, and give a polynomial time algorithm to find a $(1+\varepsilon)$-approximate Nash equilibrium on a nearly optimal network. Similarly, for the general connection game we prove that there is a 3-approximate Nash equilibrium on the optimal network, and give an algorithm to find a $(4.65+\varepsilon)$-approximate Nash equilibrium on a network that is close to optimal.