Abstract
We present a network influence game that models players strategically seeding the opinions of nodes embedded in a social network. A social learning dynamic, whereby nodes repeatedly update their opinions to resemble those of their neighbors, spreads the seeded opinions through the network. After a fixed period of time, the dynamic halts and each player’s utility is determined by the relative strength of the opinions held by each node in the network vis-à-vis the other players. We show that the existence of a pure Nash equilibrium cannot be guaranteed in general. However, if the dynamics are allowed to progress for a sufficient amount of time so that a consensus among all of the nodes is obtained, then the existence of a pure Nash equilibrium can be guaranteed. The computational complexity of finding a pure strategy best response is shown to be \(\mathrm {NP}\)-complete, but can be efficiently approximated to within a \((1 - 1/e)\) factor of optimal by a simple greedy algorithm.
This research was conducted while S.D. Johnson was a graduate student at the University of California, Davis.
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Notes
- 1.
- 2.
We use \(\varGamma ^T\) to denote the matrix \(\varGamma \) raised to the Tth power. For matrix transposition, we use the notation \(\varGamma ^\intercal \).
- 3.
A set function \(f : \varOmega \rightarrow \mathbb {R}\) is submodular if, for every \(X \subseteq Y \subset \varOmega \) and element \(x \in \varOmega \setminus Y\), we have \(f(X \cup \{x\}) - f(X) \ge f(Y \cup \{x\}) - f(Y)\).
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Acknowledgements
The authors gratefully acknowledge support from the US Army Research Office MURI Award No. W911NF-13-1-0340 and Cooperative Agreement No. W911NF-09-2-0053.
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Johnson, S.D., George, J., D’Souza, R.M. (2017). Strategic Seeding of Rival Opinions. In: Cheng, J., Hossain, E., Zhang, H., Saad, W., Chatterjee, M. (eds) Game Theory for Networks. GameNets 2016. Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, vol 174. Springer, Cham. https://doi.org/10.1007/978-3-319-47509-7_1
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