On the Windfall and Price of Friendship: Inoculation Strategies on Social Networks

TL;DR

This paper introduces a framework to quantify how caring about friends in social networks influences individual and social outcomes, analyzing virus inoculation games to reveal effects on equilibrium and convergence times.

Contribution

It presents a novel framework for measuring the Windfall of Friendship and analyzes its impact on equilibria and convergence in social network games.

Findings

Windfall of Friendship is never negative.

Social welfare may not increase monotonically with caring levels.

Convergence times are higher in social networks, indicating a price of friendship.

Abstract

This article investigates selfish behavior in games where players are embedded in a social context. A framework is presented which allows us to measure the Windfall of Friendship, i.e., how much players benefit (compared to purely selfish environments) if they care about the welfare of their friends in the social network graph. As a case study, a virus inoculation game is examined. We analyze the corresponding Nash equilibria and show that the Windfall of Friendship can never be negative. However, we find that if the valuation of a friend is independent of the total number of friends, the social welfare may not increase monotonically with the extent to which players care for each other; intriguingly, in the corresponding scenario where the relative importance of a friend declines, the Windfall is monotonic again. This article also studies convergence of best-response sequences. It turns out that in social networks, convergence times are typically higher and hence constitute a price of friendship. While such phenomena may be known on an anecdotal level, our framework allows us to quantify these effects analytically. Our formal insights on the worst case equilibria are complemented by simulations shedding light onto the structure of other equilibria.