# Strategic Network Formation Through an Intermediary

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## Abstract

Settings in which independent self-interested agents form connections with each other are extremely common, and are usually modeled using network formation games. We study a natural extension of network formation games in which the nodes cannot form connections themselves, but instead must do it through an *intermediary*, and must pay the intermediary to form these connections. The price charged by the intermediary is assumed to be determined by its operating costs, which in turn depend on the total amount of connections it facilitates. We investigate the existence and worst-case efficiency (price of anarchy) of stable solutions in these settings, and especially when the intermediary uses common pricing schemes like proportional pricing or marginal cost pricing. For both these pricing schemes we prove existence of stable solutions and completely characterize their structure, as well as generalize these results to a large class of pricing schemes. Our main results are on bounding the price of anarchy in such settings: we show that while marginal cost pricing leads to an upper bound of only 2, i.e., stable solutions are always close to optimal, proportional pricing also performs reasonably well as long as the operating costs of the intermediary are not too convex.

## Keywords

Network formation Price of anarchy## Notes

## References

- 1.Ager, B., Chatzis, N., Feldmann, A., Sarrar, N., Uhlig, S., Willinger, W.: Anatomy of a large european IXP. In: Proceedings of the ACM SIGCOMM 2012 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, pp. 163–174. ACM (2012)Google Scholar
- 2.AMS-IX: Amsterdam internet exchange pricing. https://ams-ix.net/services-pricing/pricing (2015)
- 3.Anshelevich, E., Hoefer, M.: Contribution games in networks. Algorithmica
**63**(1-2), 51–90 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Augustin, B., Krishnamurthy, B., Willinger, W.: IXPs: mapped?. In: Proceedings of the 9th ACM SIGCOMM Conference on Internet Measurement Conference, pp. 336–349. ACM (2009)Google Scholar
- 5.Bala, V., Goyal, S.: A noncooperative model of network formation. Econometrica
**68**(5), 1181–1229 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Calvó-Armengol, A., Ilkilic, R.: Pairwise-stability and nash equilibria in network formation. Int. J. Game Theory
**38**(1), 51–79 (2009)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Cardona Restrepo, J.C., Stanojevic, R.: IXP traffic: a macroscopic view. In: Proceedings of the 7th Latin American Networking Conference, pp. 1–8. ACM (2012)Google Scholar
- 8.Corbo, J., Parkes, D.: The price of selfish behavior in bilateral network formation. In: Proceedings of the Twenty-Fourth Annual ACM Symposium on Principles of Distributed Computing, pp. 99–107. ACM (2005)Google Scholar
- 9.Cormen, T.H.: Introduction to algorithms. MIT Press, Cambridge (2009)zbMATHGoogle Scholar
- 10.DEC-IX: German internet exchange pricing. https://www.de-cix.net/products-services/pricing/ (2015)
- 11.Derks, J., Kuipers, J., Tennekes, M., Thuijsman, F.: Local dynamics in network formation. In: Proceedings of 3rd World Congress of the Game Theory Society (2008)Google Scholar
- 12.Dhamdhere, A., Dovrolis, C.: The internet is flat: modeling the transition from a transit hierarchy to a peering mesh. In: Proceedings of the 6th International Conference, p. 21. ACM (2010)Google Scholar
- 13.Dutta, B., Ghosal, S., Ray, D.: Farsighted network formation. J. Econ. Theory
**122**(2), 143–164 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Epstein, A., Feldman, M., Mansour, Y.: Strong equilibrium in cost sharing connection games. In: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 84–92. ACM (2007)Google Scholar
- 15.EuroIX: European internet exchange association: Starting an IXP – infrastructure and equipment. https://www.euro-ix.net/starting-an-ixp (2015)
- 16.Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.: On a network creation game. In: Proceedings of the 22nd Annual Symposium on Principles of Distributed Computing, pp. 347–351. ACM (2003)Google Scholar
- 17.Jackson, M.O.: A survey of network formation models: stability and efficiency. Group Formation in Economics: Networks, Clubs, and Coalitions pp. 11–49 (2005)Google Scholar
- 18.Jackson, M.O., Wolinsky, A.: A strategic model of social and economic networks. J. Econ. Theory
**71**(1), 44–74 (1996)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Johari, R.: The price of anarchy and the design of scalable resource allocation mechanisms. Algorithmic Game Theory pp. 543–568 (2007)Google Scholar
- 20.Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: STACS 99, pp. 404–413. Springer (1999)Google Scholar
- 21.LINX: London internet exchange service fees. https://www.linx.net/service/servicefees.html (2015)
- 22.Page, F., Resende, J.: Network formation games. http://kenticoqa.uwb.edu/getattachment/business/about/research-series/network-formation-games-uwb.pdf (2013)
- 23.Papadimitriou, C.: Algorithms, games, and the internet. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 749–753. ACM (2001)Google Scholar
- 24.Ryan, P.S., Gerson, J.: A primer on internet exchange points for policymakers and non-engineers http://ssrn.com/abstract=2128103 (2012)
- 25.Shakkottai, S., Fomenkov, M., Koga, R., Krioukov, D., Claffy, K.: Evolution of the internet as-level ecosystem. In: Complex Sciences, pp. 1605–1616. Springer (2009)Google Scholar
- 26.Shakkottai, S., Srikant, R.: Economics of network pricing with multiple isps. IEEE/ACM Trans. Networking
**14**(6), 1233–1245 (2006)CrossRefGoogle Scholar - 27.Tardos, E., Wexler, T.: Network formation games and the potential function method. Algorithmic Game Theory pp. 487–516 (2007)Google Scholar
- 28.Vazirani, V.V.: Combinatorial algorithms for market equilibria. Algorithmic Game Theory pp 103–134 (2007)Google Scholar
- 29.Watts, A.: A dynamic model of network formation. In: Networks and Groups, pp. 337–345. Springer (2003)Google Scholar