Abstract
We propose a simple and intuitive cost mechanism which assigns costs for the competitive usage of mresources by n selfish agents. Each agent has an individual demand; demands are drawn according to some probability distribution. The cost paid by an agent for a resource she chooses is the total demand put on the resource divided by the number of agents who chose that same resource. So, resources charge costs in an equitable, fair way, while each resource makes no profit out of the agents.
We call our model the Fair Pricing model. Its fair cost mechanism induces a non-cooperative game among the agents. To evaluate the Nash equilibria of this game, we introduce the Diffuse Price of Anarchy, as an extension of the Price of Anarchy that takes into account the probability distribution on the demands. We prove:
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Pure Nash equilibria may not exist, unless all chosen demands are identical; in contrast, a fully mixed Nash equilibrium exists for all possible choices of the demands. Further on, the fully mixed Nash equilibrium is the unique Nash equilibrium in case there are only two agents.
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In the worst-case choice of demands, the Price of Anarchy is Θ (n); for the special case of two agents, the Price of Anarchy is less than \(2 - \frac{1}{m}\).
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Assume now that demands are drawn from a bounded, independent probability distribution, where all demands are identically distributed and each is at most a (universal for the class) constant times its expectation. Then, the Diffuse Price of Anarchy is at most that same constant, which is just 2 when each demand is distributed symmetrically around its expectation.
Partially supported by the EU within the 6th Framework Programme under contracts 001907 “Dynamically Evolving, Large Scale Information Systems” (DELIS) and 015964 “Algorithmic Principles for Building Efficient Overlay Computers” (AEOLUS), and by the General Secretariat for Research and Technology of the Greek Ministry of Development within the programme PENED 2003.
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References
Billera, L., Heath, D.: Allocation of Shared Costs: A Set of Axioms Yielding a Unique Procedure. Mathematics of Operations Research 7, 32–39 (1982)
Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination Mechanisms. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 345–357. Springer, Heidelberg (2004)
Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The Structure and Complexity of Nash Equilibria for a Selfish Routing Game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 123–134. Springer, Heidelberg (2002)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash Equilibria in Discrete Routing Games with Convex Latency Functions. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 645–657. Springer, Heidelberg (2004)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Extreme Nash Equilibria. In: Blundo, C., Laneve, C. (eds.) ICTCS 2003. LNCS, vol. 2841, pp. 1–20. Springer, Heidelberg (2003)
Hayrapetyan, A., Tardos, É., Wexler, T.: A Network Pricing Game for Selfish Traffic. In: Proceedings of the 24th Annual ACM Symposium on Principles of Distributed Computing, July 2005, pp. 284–291 (2005)
Hoeffding, W.: Probability Inequalities for Sums of Bounded Random Variables. Journal of the American Statistical Association 58, 13–30 (1963)
Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate Equilibria and Ball Fusion. Theory of Computing Systems 36(6), 683–693 (2003)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Koutsoupias, E., Papadimitriou, C.H.: Beyond Competitive Analysis. SIAM Journal on Computing 30(1), 300–317 (2000)
Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: A New Model for Selfish Routing. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 547–558. Springer, Heidelberg (2004)
Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is the Worst-case Nash Equilibrium? In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 551–561. Springer, Heidelberg (2003)
Mavronicolas, M., Spirakis, P.: The Price of Selfish Routing. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 510–519 (2001)
Moulin, H., Shenker, S.: Serial Cost Sharing. Econometrica 60(5), 1009–1037 (1992)
Moulin, H., Shenker, S.: Average Cost Pricing versus Serial Cost Sharing: An Axiomatic Comparison. Journal of Economic Theory 64, 178–201 (1994)
Nash, J.F.: Non-cooperative Games. Annals of Mathematics 54(2), 286–295 (1951)
Papadimitriou, C.H.: Algorithms, Games and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753 (2001)
Shapiro, C., Varian, H.R.: Information Rules – A Strategic Guide to the Network Economy. Harvard Business School Press, Boston (1998)
Spence, M.: Nonlinear Prices and Welfare. Journal of Marketing Research 8, 1–18 (1976)
Varian, H.R.: Prime Discrimination and Social Welfare. American Economic Review 75(4), 870–875 (1985)
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Mavronicolas, M., Panagopoulou, P.N., Spirakis, P.G. (2005). A Cost Mechanism for Fair Pricing of Resource Usage. In: Deng, X., Ye, Y. (eds) Internet and Network Economics. WINE 2005. Lecture Notes in Computer Science, vol 3828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11600930_21
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DOI: https://doi.org/10.1007/11600930_21
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