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Monotonicity of Equilibria in Nonatomic Congestion Games (2308.00434v2)

Published 1 Aug 2023 in cs.GT and math.OC

Abstract: This paper studies the monotonicity of equilibrium costs and equilibrium loads in nonatomic congestion games, in response to variations of the demands. The main goal is to identify conditions under which a paradoxical non-monotone behavior can be excluded. In contrast to routing games with a single commodity, where the network topology is the sole determinant factor for monotonicity, for general congestion games with multiple commodities the structure of the strategy sets plays a crucial role. We frame our study in the general setting of congestion games, with a special focus on singleton congestion games, for which we establish the monotonicity of equilibrium loads with respect to every demand. We then provide conditions for comonotonicity of the equilibrium loads, i.e., we investigate when they jointly increase or decrease after variations of the demands. We finally extend our study from singleton congestion games to the larger class of constrained series-parallel congestion games, whose structure is reminiscent of the concept of a series-parallel network.

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References (67)
  1. Competition and efficiency in congested markets. Math. Oper. Res., 32(1):1–31.
  2. Infinite Dimensional Analysis. Springer, Berlin, third edition.
  3. Arrow, K. J. (1970). Essays in the Theory of Risk-Bearing. North-Holland Publishing Co., Amsterdam-London.
  4. Attouch, H. (1996). Viscosity solutions of minimization problems. SIAM J. Optim., 6(3):769–806.
  5. Asymptotic analysis for penalty and barrier methods in convex and linear programming. Math. Oper. Res., 22(1):43–62.
  6. Studies in the Economics of Transportation. Yale University Press, New Haven, CT.
  7. On the impact of singleton strategies in congestion games. In 25th European Symposium on Algorithms, volume 87 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 17, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern.
  8. Borch, K. (1962). Equilibrium in a reinsurance market. Econometrica, 30(3):424–444.
  9. Traffic user equilibrium and proportionality. Transportation Res. Part B, 79:149–160.
  10. Braess, D. (1968). Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung, 12:258–268.
  11. On a paradox of traffic planning. Transportation Sci., 39(4):446–450.
  12. Leadership in singleton congestion games: what is hard and what is easy. Artificial Intelligence, 277:103177, 31.
  13. Cominetti, R. (1999). Nonlinear averages and convergence of penalty trajectories in convex programming. In Théra, M. and Tichatschke, R., editors, Ill-posed Variational Problems and Regularization Techniques, pages 65–78, Berlin, Heidelberg. Springer Berlin Heidelberg.
  14. The price of anarchy in routing games as a function of the demand. Math. Program., forthcoming.
  15. Phase transitions of the price-of-anarchy function in multi-commodity routing games. Technical report, arXiv:2305.03459.
  16. On some traffic equilibrium theory paradoxes. Transportation Res. Part B, 18(2):101–110.
  17. Dellacherie, C. (1971). Quelques commentaires sur les prolongements de capacités. In Séminaire de Probabilités, V (Univ. Strasbourg, année universitaire 1969–1970), Lecture Notes in Math., Vol. 191, pages 77–81. Springer, Berlin.
  18. On the connection between Nash equilibria and social optima in electric vehicle charging control games. IFAC-PapersOnLine, 50(1):14320–14325.
  19. The concept of comonotonicity in actuarial science and finance: theory. Insurance Math. Econom., 31(1):3–33. 5th IME Conference (University Park, PA, 2001).
  20. Eppstein, D. (1992). Parallel recognition of series-parallel graphs. Inform. and Comput., 98(1):41–55.
  21. Fisk, C. (1979). More paradoxes in the equilibrium assignment problem. Transportation Res. Part B, 13(4):305–309.
  22. Empirical evidence for equilibrium paradoxes with implications for optimal planning strategies. Transportation Res. Part A, 15(3):245–248.
  23. The structure and complexity of Nash equilibria for a selfish routing game. Theoret. Comput. Sci., 410(36):3305–3326.
  24. Matroids are immune to Braess’ paradox. Math. Oper. Res., 42(3):745–761.
  25. Fukushima, M. (1984). On the dual approach to the traffic assignment problem. Transportation Res. Part B, 18(3):235–245.
  26. Computing Nash equilibria for scheduling on restricted parallel links. Theory Comput. Syst., 47(2):405–432.
  27. Total latency in singleton congestion games. In Deng, X. and Graham, F. C., editors, Internet and Network Economics, pages 381–387, Berlin, Heidelberg. Springer Berlin Heidelberg.
  28. A hydraulic approach to equilibria of resource selection games. In Proceedings of the 2016 ACM Conference on Economics and Computation, EC ’16, page 477, New York, NY, USA. Association for Computing Machinery.
  29. Hall, M. A. (1978). Properties of the equilibrium state in transportation networks. Transportation Sci., 12(3):208–216.
  30. On the existence of pure Nash equilibria in weighted congestion games. Math. Oper. Res., 37(3):419–436.
  31. Toll caps in privatized road networks. European J. Oper. Res., 276(3):947–956.
  32. Equilibrium computation in atomic splittable singleton congestion games. In Integer Programming and Combinatorial Optimization, volume 10328 of Lecture Notes in Comput. Sci., pages 442–454. Springer, Cham.
  33. Parallel recognition and decomposition of two terminal series parallel graphs. Inform. and Comput., 75(1):15–38.
  34. Hoerl, A. E. (1959). Optimum solution of many variables equations. Chemical Engineering Progress, 55:69–78.
  35. Hoerl, A. E. (1962). Application of ridge analysis to regression problems. Chemical Engineering Progress, 58:54–59.
  36. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55–67.
  37. Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design. Transportation Res. Part B, 41(1):4 – 31.
  38. Parametric computation of minimum-cost flows with piecewise quadratic costs. Math. Oper. Res., 47(1):812–846.
  39. Robust multidimensional pricing: separation without regret. Math. Program., 196(1-2):841–874.
  40. Konishi, H. (2004). Uniqueness of user equilibrium in transportation networks with heterogeneous commuters. Transportation Sci., 38(3):315–330.
  41. Co-monotone allocations, Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion. Ann. Oper. Res., 52:97–106.
  42. Decentralized charging control of large populations of plug-in electric vehicles. IEEE Trans. Control Syst. Technol., 21(1):67–78.
  43. How will the presence of autonomous vehicles affect the equilibrium state of traffic networks? IEEE Trans. Control Netw. Syst., 7(1):96–105.
  44. The uniqueness property for networks with several origin-destination pairs. European J. Oper. Res., 237(1):245–256.
  45. Milchtaich, I. (2000). Generic uniqueness of equilibrium in large crowding games. Math. Oper. Res., 25(3):349–364.
  46. Milchtaich, I. (2005). Topological conditions for uniqueness of equilibrium in networks. Math. Oper. Res., 30(1):225–244.
  47. Milchtaich, I. (2006). Network topology and the efficiency of equilibrium. Games Econom. Behav., 57(2):321–346.
  48. Milchtaich, I. (2013). Representation of finite games as network congestion games. Internat. J. Game Theory, 42(4):1085–1096.
  49. A survey of algorithms for distributed charging control of electric vehicles in smart grid. IEEE Trans. Intell. Transportation Syst., 21(11):4497–4515.
  50. Patriksson, M. (2004). Sensitivity analysis of traffic equilibria. Transportation Sci., 38(3):258–281.
  51. Pradeau, T. (2014). Congestion Games with Player-Specific Cost Functions. Phd thesis, Université Paris-Est.
  52. Multivariate comonotonicity. J. Multivariate Anal., 101(1):291–304.
  53. Entropy model for consistent impact-fee assessment. J. Urban Plann. Develop., 115:51–63.
  54. Schmeidler, D. (1982). Subjective probability without additivity. Technical report, The Foerder Institute for Economic Research, Tel-Aviv University.
  55. Schmeidler, D. (1984). Subjective probability and expected utility without additivity. Technical report, Institute of Mathematics and its Applications, University of Minnesota.
  56. Schmeidler, D. (1986). Integral representation without additivity. Proc. Amer. Math. Soc., 97(2):255–261.
  57. Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57(3):571–587.
  58. Tikhonov, A. N. (1943). On the stability of inverse problems. C. R. (Doklady) Acad. Sci. URSS (N.S.), 39:176–179.
  59. Tikhonov, A. N. (1963). On the solution of ill-posed problems and the method of regularization. Dokl. Akad. Nauk SSSR, 151:501–504.
  60. Solutions of Ill-Posed Problems. Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London.
  61. The recognition of series parallel digraphs. SIAM J. Comput., 11(2):298–313.
  62. The interdependence between hospital choice and waiting time – with a case study in urban China. Technical report, arXiv:2306.16256v1.
  63. Wardrop equilibrium and Braess’s paradox for varying demand. Technical report, arXiv preprint arXiv:2310.04256.
  64. Wan, C. (2016). Strategic decentralization in binary choice composite congestion games. European J. Oper. Res., 250(2):531–542.
  65. Wardrop, J. G. (1952). Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers, Part II, volume 1, pages 325–378.
  66. Wilson, R. (1968). The theory of syndicates. Econometrica, 36(1):119–132.
  67. Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55(1):95–115.
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