Bounded budget connection (BBC) games or how to make friends and influence people, on a budget

Motivated by applications in social networks, peer-to-peer and overlay networks, we define and study the Bounded Budget Connection (BBC) game - we have a collection of n players or nodes each of whom has a budget for purchasing links; each link has a cost as well as a length and each node has a set of preference weights for each of the remaining nodes; the objective of each node is to use its budget to buy a set of outgoing links so as to minimize its sum of preference-weighted distances to the remaining nodes. We study the structural and complexity-theoretic properties of pure Nash equilibria in BBC games. We show that determining the existence of a pure Nash equilibrium in general BBC games is NP-hard. A major focus is the study of (n,k)-uniform BBC games - those in which all link costs, link lengths and preference weights are equal (to 1) and all budgets are equal (to k). We show that a pure Nash equilibrium or stable graph exists for all (n,k)-uniform BBC games and that all stable graphs are essentially fair (i.e. all nodes have similar costs). We provide an explicit construction of a family of stable graphs that spans the spectrum from minimum total social cost to maximum total social cost. To be precise we show that that the price of stability is Θ(1) and the price of anarchy is Ω(√n/k / log_k n) and O(√n/log_k n).

We analyze best-response walks on the configuration space defined by the uniform game, and show that starting from any initial configuration, strong connectivity is reached within n² rounds. We demonstrate that convergence to a pure Nash equilibrium is not guaranteed by demonstrating the existence of an explicit loop which also proves that even uniform BBC games are not potential games. Lastly, we extend our results to the case where each node seeks to minimize its maximum distance to the other nodes.