Social Network Games with Obligatory Product Selection

TL;DR

This paper introduces social network games with obligatory product adoption, providing algorithms for cycle graphs, proving NP-completeness for general graphs, and analyzing paradoxes like Braess in social networks.

Contribution

It formalizes social network games with obligatory choices, offers polynomial algorithms for cycle graphs, proves complexity results for general graphs, and explores paradoxical effects in product adoption.

Findings

Polynomial-time algorithm for Nash equilibrium on cycle graphs

NP-completeness of equilibrium existence in general graphs

Existence of paradoxes where adding products worsens outcomes

Abstract

Recently, Apt and Markakis introduced a model for product adoption in social networks with multiple products, where the agents, influenced by their neighbours, can adopt one out of several alternatives (products). To analyze these networks we introduce social network games in which product adoption is obligatory. We show that when the underlying graph is a simple cycle, there is a polynomial time algorithm allowing us to determine whether the game has a Nash equilibrium. In contrast, in the arbitrary case this problem is NP-complete. We also show that the problem of determining whether the game is weakly acyclic is co-NP hard. Using these games we analyze various types of paradoxes that can arise in the considered networks. One of them corresponds to the well-known Braess paradox in congestion games. In particular, we show that social networks exist with the property that by adding an additional product to a specific node, the choices of the nodes will unavoidably evolve in such a way that everybody is strictly worse off.