On the Windfall and price of friendship: Inoculation strategies on social networks

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.

Altruism and Its Impact on the Price of Anarchy

“…Pure Nash equilibria may not exist in this game, but Chen et al [2010] show that under a natural opt-out dynamic, the price of anarchy is bounded by 1/α. Meier et al [2008] study a similar model in which altruism is replaced by "friendship," meaning that individuals only care about the utility of their neighbors in the network, not all nodes.…”

“…The idea of considering "friendship networks" (and thus nonuniform altruism across pairs of individuals), which is pursued by Meier et al [2008] is also present in several other articles on "social games." For instance, such analyses are carried out in analyzing the price of anarchy and stability in Anshelevich et al [2012] and Buehler et al [2011].…”

Competing Memes Propagation on Networks: A Network Science Perspective

“…Meier et al [32] studied inoculation games in social networks, where each node selfishly decides whether or not to protect itself. The game between a virus and an alert over a network was investigated by Aspnes et al [33].…”

On Non-cooperative Genomic Privacy

“…Following an idea introduced in [21] for social networks, we define a familial factor α ∈ [0, 1] that conveys the fact that relatives tend to be altruistic among themselves. We raise this factor to the power k(i, j) ∈ N * that represents the degree of kinship between relatives i and j.…”

“…The price of stability (PoS) also measures this inefficiency but, assuming that players coordinate amongst themselves, considers the best NE instead of the worst one, i.e., is defined as U * (s)/ max NE U (s) [4]. Following the notion of windfall of friendship (WoF) proposed in [21], we define the windfall of kinship (WoK) as the ratio between the social welfare of the worst FNE and the social welfare of the worst NE:…”