Bounded Budget Connection (BBC) Games or How to make friends and influence people, on a budget

TL;DR

This paper introduces and analyzes the Bounded Budget Connection (BBC) game, exploring the existence, structure, and properties of Nash equilibria in network formation under budget constraints, with implications for social and overlay networks.

Contribution

It establishes complexity results, existence conditions, and structural properties of Nash equilibria in BBC games, including uniform and fractional variants, and analyzes stability and fairness.

Findings

NP-hardness of equilibrium existence in general BBC games

Existence of pure Nash equilibrium in fractional BBC games

Stable, fair graphs exist in (n,k)-uniform BBC games

Abstract

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. However, in a natural variant, fractional BBC games - where it is permitted to buy fractions of links - a pure Nash equilibrium always exists. 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. We also study a special family of regular graphs in which all nodes imitate the "same" buying pattern, and show that if n is sufficiently large no such regular graph can be a pure Nash equilibrium. We analyze best-response walks on the configuration defined by the uniform game. Lastly, we extend our results to the case where each node seeks to minimize its maximum distance to the other nodes.