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Linear Tolls Suffice: New Bounds and Algorithms for Tolls in Single Source Networks

  • Lisa Fleischer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3142)

Abstract

We show that tolls that are linear in the latency of the maximum latency path are necessary and sufficient to induce heterogeneous network users to independently choose routes that lead to traffic with minimum average latency. This improves upon the earlier bound of \(O(n^3l_{\max})\) given by Cole, Dodis, and Roughgarden in STOC 03. (Here, n is the number of vertices in the network; and l max is the maximum latency of any edge.) Our proof is also simpler, relating the Nash flow to the optimal flow as flows rather than cuts.

We model the set of users as the set [0,1] ordered by their increasing willingness to pay tolls to reduce latency — their valuation of time. Cole, et al. give an algorithm that computes optimal tolls for a bounded number of agent valuations, under the very strong assumption that they know which path each user type takes in the Nash flow imposed by these (unknown) tolls. We show that in series parallel graphs, the set of paths travelled by users in any Nash flow with optimal tolls is independent of the distribution of valuations of time of the users. In particular, for any continuum of users (not restricted to a finite number of valuation classes) in series parallel graphs, we show how to compute these paths without knowing α.

We give a simple example to demonstrate that if the graph is not series parallel, then the set of paths travelled by users in the Nash flow depends critically on the distribution of users’ valuations of time.

Keywords

Longe Path Valuation Function Latency Path Congestion Game Optimal Toll 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)Google Scholar
  2. 2.
    Beckman, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)Google Scholar
  3. 3.
    Border, K.C.: Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge (1985)Google Scholar
  4. 4.
    Carstensen, P.J.: Parametric cost shortest path problems. Unpoblished Bellcore memo (1984)Google Scholar
  5. 5.
    Cole, R., Dodis, Y., Roughgarden, T.: Pricing network edges for heterogeneous selfish users. In: Proc. 35th Annual ACM Symposium on the Theory of Computing (2003)Google Scholar
  6. 6.
    Dafermos, S.C.: Toll patterns for multiclass-user transportation networks. Transportation Sci. 7, 211–223 (1973)CrossRefGoogle Scholar
  7. 7.
    Fleischer, L., Jain, K., Mahdian, M.: Taxes for heterogeneous selfish users in a multicommodity network (April 2004) (submitted)Google Scholar
  8. 8.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Pigou, A.C.: The Economics of Welfare. Macmillan, Basingstoke (1920)Google Scholar
  10. 10.
    Roughgarden, T., Tardos, E.: How bad is selfish routing? In: IEEE Symposium on Foundations of Computer Science, pp. 93–102 (2000)Google Scholar
  11. 11.
    Smith, M.J.: The marginal cost taxation of a transportation network. Trans. Res. Ser. B 13, 237–242 (1979)CrossRefGoogle Scholar
  12. 12.
    Wardrop, J.G.: Some theoretical aspects of rad traffic research. Proc. Institute of Civil Engineers, Pt. II 1, 325–378 (1952)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Lisa Fleischer
    • 1
    • 2
  1. 1.T. J. Watson ResearchIBMYorktown Heights
  2. 2.Carnegie Mellon UniversityPittsburghUSA

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