On a Bounded Budget Network Creation Game

We consider a network creation game in which, each player (vertex) has a limited budget to establish links to other players. In our model, each link has a unit cost and each agent tries to minimize its cost which is its local diameter or its total distance to other players in the (undirected) underlying graph of the created network. Two variants of the game are studied: in the MAX version, the cost incurred to a vertex is the maximum distance between that vertex and other vertices, and in the SUM version, the cost incurred to a vertex is the sum of distances between that vertex and other vertices. We prove that in both versions pure Nash equilibria exist, but the problem of finding the best response of a vertex is NP-hard. Next, we study the maximum possible diameter of an equilibrium graph with n vertices in various cases. For infinite numbers of n, we construct an equilibrium tree with diameter circle minus(n) in the MAX version. Also, we prove that the diameter of any equilibrium tree is O(log n) in the SUM version and this bound is tight. When all vertices have unit budgets (i.e. can establish link to just one vertex), the diameter in both versions is O(1). We give an example of equilibrium graph in MAX version, such that all vertices have positive budgets and yet the diameter is as large as Omega(root log n). This interesting result shows that the diameter does not decrease necessarily and may increase as the budgets are increased. For the SUM version, we prove that every equilibrium graph has diameter 2(O)(root log n) when all vertices have positive budgets. Moreover, if the budget of every players is at least k, then every equilibrium graph with diameter more than 3 is k-connected.