Abstract
In this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the \(\mathsf{FMNE}\) Conjecture, in selfish routing for the special case of n identical users over two (identical) parallel links. We introduce a new measure of Social Cost, defined as the expectation of the square of the maximum congestion on a link; we call it Quadratic Maximum Social Cost. A Nash equilibrium is a stable state where no user can improve her (expected) latency by switching her mixed strategy; a worst-case Nash equilibrium is one that maximizes Quadratic Maximum Social Cost. In the fully mixed Nash equilibrium, all mixed strategies achieve full support.
Formulated within this framework is yet another facet of the \(\mathsf{FMNE}\) Conjecture, which states that the fully mixed Nash equilibrium is the worst-case Nash equilibrium. We present an extensive proof of the \(\mathsf{FMNE}\) Conjecture; the proof employs a combination of combinatorial arguments and analytical estimations.
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Bernstein, S.: Démonstration du théoreme de Weierstrass fondée sur le calcul des probabilities. Commun. Soc. Math. Kharkow 2(13), 1–2 (1912)
Elsässer, R., Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: A simple graph-theoretic model for selfish restricted scheduling. In: Proceedings of the 1st International Workshop on Internet and Network Economics. Lecture Notes in Computer Science, vol. 3828, pp. 195–209. Springer, Berlin (2005)
Feller, W.: An Introduction to Probability Theory and its Applications. 3rd edn. Wiley, New York (1968)
Ferrante, A., Parente, M.: Existence of Nash equilibria in selfish routing problems. In: Proceedings of the 11th International Colloquium on Structural Information and Communication Complexity. Lecture Notes in Computer Science, vol. 3104, pp. 149–160. Springer, New York (2004)
Fischer, S., Vöcking, B.: On the structure and complexity of worst-case equilibria. Theor. Comput. Sci. 378(2), 165–174 (2007)
Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of Nash equilibria for a selfish routing game. In: Proceedings of the 29th International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 2380, pp. 123–134. Springer, Berlin (2002). Extended version accepted to Theor. Comput. Sci., December 2007
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theor. Comput. Sci. 343(1–2), 133–157 (2005)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash equilibria in discrete routing games with convex latency functions. J. Comput. Syst. Sci. 74(7), 1199–1225 (2008)
Gairing, M., Monien, B., Tiemann, K.: Selfish routing with incomplete information. Theory Comput. Syst. 42(1), 91–130 (2008)
Georgiou, Ch., Pavlides, Th., Philippou, A.: Uncertainty in selfish routing. In: Proceedings of the 20th IEEE International Parallel and Distributed Processing Symposium, p. 105 (2006). CD-ROM
Goussevskala, O., Oswald, Y.A., Wattenhofer, R.: Complexity in geometric SINR. In: Proceedings of the 8th ACM International Symposium on Mobile Ad Hoc Networking and Computing, pp. 100–109 (2007)
Kaplansky, I.: A contribution to von-Neumann’s theory of games. Ann. Math. 46(3), 474–479 (1945)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Proceedings of the 16th International Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1563, pp. 404–413. Springer, Berlin (1999)
Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: A new model for selfish routing. Theor. Comput. Sci. 406(3), 187–206 (2008)
Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is the worst-case Nash equilibrium? In: Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 2747, pp. 551–561. Springer, Berlin (2003)
Mavronicolas, M., Panagopoulou, P., Spirakis, P.: Cost sharing mechanisms for fair pricing of resource usage. Algorithmica 52(1), 19–43 (2008)
Mavronicolas, M., Spirakis, P.: The price of selfish routing. Algorithmica 48(1), 91–126 (2007)
Milchtaich, I.: Congestion games with player-specific payoff function. Games Econ. Behav. 13(1), 111–124 (1996)
Nash, J.F.: Equilibrium points in N-person games. Proc. Natt. Acad. Sci. USA 36, 48–49 (1950)
Nash, J.F.: Non-cooperative games. Ann. Math. 54(2), 286–295 (1951)
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A preliminary version of this work appeared in the Proceedings of the 1st International Symposium on Algorithmic Game Theory, pp. 145–157, Vol. 4997, Lecture Notes in Computer Science, Springer-Verlag, April/May 2008. This work has been partially supported by the \(\mathsf{IST}\) Program of the European Union under contract number 15964 ( \(\mathsf{AEOLUS}\) ).
Part of the work of M. Mavronicolas was performed while visiting the Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Germany.
Part of the work of A. Pieris was performed while at the Department of Computer Science, University of Cyprus, Cyprus, and while visiting the Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Germany.
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Feldmann, R., Mavronicolas, M. & Pieris, A. Facets of the Fully Mixed Nash Equilibrium Conjecture. Theory Comput Syst 47, 60–112 (2010). https://doi.org/10.1007/s00224-009-9199-1
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DOI: https://doi.org/10.1007/s00224-009-9199-1