The Price of Anarchy of a Network Creation Game with Exponential Payoff
We analyze a graph process (or network creation game) where the vertices as players can establish mutual relations between each other at a fixed price. Each vertex receives income from every other vertex, exponentially decreasing with their distance. To establish an edge, both players have to make a consent acting selfishly. This process has originially been proposed in economics to analyse social networks of cooperation. Though the exponential payoff is a desirable principle to model the benefit of distributed systems, it has so far been an obstacle for analysis.
We show that the process has a positive probability to cycle. We reduce the creation rule with payoff functions to graph theoretic criteria. Moreover, these criteria can be evaluated locally. This allows us to thoroughly reveal the structure of all stable states. In addition, the question for the price of anarchy can be reduced to counting the maximum number of edges of a stable graph. This together with a probabilistic argument allows to determine the price of anarchy exactly.
Unable to display preview. Download preview PDF.
- 1.Albers, S., Eilts, S., Even-Dar, E., Mansour, Y., Roditty, L.: On Nash equilibria for a network creation game. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms (SODA 2006), pp. 89–98 (2006)Google Scholar
- 3.Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost alloccation. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 295–304 (2004)Google Scholar
- 4.Anshelevich, E., Dasgupta, A., Tardos, E., Wexler, T.: Near-optimal network design with selfish agents. In: Proceedings of 35th Annual ACM Symposium on Theory of Computing (STOC 2003), pp. 511–520 (2003)Google Scholar
- 7.Corbo, J., Parkes, D.C.: The price of selfish behavior in bilateral network formation. In: Proceedings of the 24th ACM Symposium on Principles of Distributed Computing (PODC 2005) (2005)Google Scholar
- 8.Demaine, E.D., Hajiaghayi, M.T., Mahini, H., Zadimoghaddam, M.: The price of anarchy in network creation games. In: Proceedings of the 26th Annual ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing (PODC 2007) (2007)Google Scholar
- 9.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 (PODC 2003), pp. 347–351 (2003)Google Scholar
- 10.Jackson, M.O.: A survey of models of network formation: Stability and efficiency. In: Demange, G., Wooders, M. (eds.) Group Formation in Economics: Networks, Clubs and Coalitions, ch. 1, pp. 11–57. Cambridge University Press, Cambridge (2004)Google Scholar
- 11.Jackson, M.O., Wolinsky, A.: A strategic model of social and economic networks. Journal of Economic Theory 71, 44–74 (1996); reprinted in Networks and Groups: Models of Strategic Formation, edited by Dutta and Jackson, Springer, Heidelberg (2003)Google Scholar
- 13.Melendez-Jiminez, M.A.: Network formation and coordination: Bargaining the division of link costs. In: Presented at 57th European Meeting of the Econometric Society (August 2002), http://www.webdeptos.uma.es/THEconomica.net1.pdf
- 14.Nash, J.F.: Non-cooperative games. Annals of Mathematics 54(286–295), 2951Google Scholar