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April  2011, 7(2): 347-364. doi: 10.3934/jimo.2011.7.347

Existence of anonymous link tolls for decentralizing an oligopolistic game and the efficiency analysis

Deren Han 1, and  Xiaoming Yuan 2,

1. 

School of Mathematical Sciences and Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210046, China
2. 

Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received  January 2010 Revised  January 2011 Published  April 2011

To apply the traditional marginal-cost pricing to drive a user equilibrium of the oligopolistic game to the system optimum, it requires to classify the users into different classes and then charge discriminatory tolls across user classes. By realizing the difficulty of discriminating users when they differ in some unobservable ways, Yang and Zhang investigated existence of anonymous link tolls for transportation networks recently. In this paper, we consider the anonymous link tolls for the oligopolistic game with nonseparable, nonlinear and asymmetric cost functions with fixed demands. With similar techniques developed by Yang and Zhang, we first prove the existence of anonymous link tolls to decentralize the system optimum to a user equilibrium. Then, by deriving some bounds on the so-called price of anarchy, we analyze the efficiency of such a toll strategy when the tolls are considered as part of the system cost.
Keywords: Asymmetric cost functions., System optimum, Anonymous link tolls, Oligopolistic games, Price of anarchy, User equilibrium.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Deren Han, Xiaoming Yuan. Existence of anonymous link tolls for decentralizing an oligopolistic game and the efficiency analysis. Journal of Industrial & Management Optimization, 2011, 7 (2) : 347-364. doi: 10.3934/jimo.2011.7.347
References:
[1]

R. P. Agdeppa, N. Yamashita and M. Fukushima, An implicit programming approach for the road pricing problem with nonadditive route costs, Journal of Industrial and Management Optimization, 4 (2008), 183-197.  Google Scholar

[2]

M. S. Bazarra, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley and Sons, Inc., New York, 1993. Google Scholar

[3]

P. Bergendorff, D. W. Hearn and M. V. Ramana, Congestion toll pricing of traffic networks, in "Network Optimization" (eds. P. M. Pardalos, D. W. Hearn and W. W. Hager), Springer, (1997), 51-71.  Google Scholar

[4]

C. K. Chau and K. M. Sim, The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands, Operations Research Letters, 31 (2003), 327-334. doi: 10.1016/S0167-6377(03)00030-0.  Google Scholar

[5]

R. Cole, Y. Dodis and T. Roughgarden, Pricing network edges for heterogeneous selfish users, in "Proceedings of the 4th ACM Conference on Electronic Commerce," (2003), 521-530.  Google Scholar

[6]

R. Cole, Y. Dodis and T. Roughgarden, How much can tolls help selfish routing, in "Proceedings of the 4th ACM Conference on Electronic Commerce," (2003), 98-107. doi: 10.1145/779928.779941.  Google Scholar

[7]

R. Cominetti, J. R. Correa and N. E. Stier-Moses, The impact of oligopolistic competition in networks, Operation Research, 57 (2009), 1421-1437. doi: 10.1287/opre.1080.0653.  Google Scholar

[8]

J. Correa, A. Schulz and N. Stier Moses, Selfish routing in capacitated networks, Mathematics of Operations Research, 29 (2004), 961-976. doi: 10.1287/moor.1040.0098.  Google Scholar

[9]

J. Correa, A. Schulz and N. Stier Moses, Computational complexity, fairness, and the price of anarchy of the maximum latency problem, in "Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science" (eds. D. Bienstock and G. Nemhauser), Springer Berlin, (2004), 59-73. doi: 10.1007/978-3-540-25960-2_5.  Google Scholar

[10]

A. Czumaj and B. Vöcking, Tight bounds for worst-case equilibria, in "Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms," (2002), 413-420.  Google Scholar

[11]

S. C. Dafermos and F. T. Sparrow, Optimal resource allocation and toll patterns in user-optimized transport network, Journal of Transport Economics and Policy, 5 (1971), 198-200. Google Scholar

[12]

S. C. Dafermos, An extended traffic assignment modal with applications to two-way traffic, Transportation Science, 5 (1971), 366-389. doi: 10.1287/trsc.5.4.366.  Google Scholar

[13]

S. C. Dafermos, The traffic assignment problem for multiclass-user transportation network, Transportation Science, 6 (1972), 73-87. doi: 10.1287/trsc.6.1.73.  Google Scholar

[14]

S. Dafermos, Toll pattern for multiclass-user transportation network, Transportation Science, 7 (1973), 211-223. doi: 10.1287/trsc.7.3.211.  Google Scholar

[15]

S. Devarajan, A note on network equilibrium and non-cooperative games, Transportaion Research B, 15 (1981), 421-426. doi: 10.1016/0191-2615(81)90026-6.  Google Scholar

[16]

R. B. Dial, Minimal-revenue congestion pricing part I: A fast algorithm for the single-origin case, Transportation Research B, 33 (1999), 189-202. doi: 10.1016/S0191-2615(98)00026-5.  Google Scholar

[17]

R. B. Dial, Minimal-revenue congestion pricing Part II: An efficient algorithm for the general case, Transportation Research B, 34 (2000), 645-665. doi: 10.1016/S0191-2615(99)00046-6.  Google Scholar

[18]

S. D. Flam and Charles Horvath, Network games; adaptations to Nash-Cournot equilibrium, Annals of Operations Research, 64 (1996), 179-195. doi: 10.1007/BF02187645.  Google Scholar

[19]

L. Fleischer, K. Jain and M. Mahdain, Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games, in "Proceedings of the 45th Annual IEEE Symposium on Foundations of Compuer Science," 2004. doi: 10.1109/FOCS.2004.69.  Google Scholar

[20]

D. R. Han, J. Sun and M. Ang, New bounds for the price of anarchy under nonlinear and asymmetric costs,, Optimization, ().   Google Scholar

[21]

D. R. Han, H. K. Lo, J. Sun and H. Yang, The toll effect on price of anarchy when costs are nonlinear and asymmetric, European Journal of Operational Research, 186 (2008), 300-316. doi: 10.1016/j.ejor.2007.01.027.  Google Scholar

[22]

D. R. Han, H. Yang and X. M. Yuan, The efficiency analysis for oligopolistic games when cost functions are nonseparable, International Journal of Mathematical Modelling and Numerical Optimisation, 1 (2010), 237-257. doi: 10.1504/IJMMNO.2010.031751.  Google Scholar

[23]

P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 39-46. doi: 10.1287/trsc.22.1.39.  Google Scholar

[24]

A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equlibria, Networks, 15 (1985), 295-308. doi: 10.1002/net.3230150303.  Google Scholar

[25]

D. W. Hearn and M. B. Yildirim, A toll pricing framework for traffic assignment problem with elastic demand, in "Current Trends in Transportation and Network Analysis-Papers in Honor of Michael Florian" (eds. M. Gendreau and P. Marcotte), Kluwer Academic Publishers, Norwell (2002), 135-145.  Google Scholar

[26]

G. Karakostas and S. G. Kolliopoulos, The efficiency of optimal taxes, in "Proceedings of 1st Workshop on Combinatorial and Algorithmic Aspects of Networking," (2004), 3-12.  Google Scholar

[27]

G. Karakostas and S. Kolliopoulos, Edge pricing of multicommodity networks for heterogeneous users, in "Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science," (2004), 268-276. doi: 10.1109/FOCS.2004.26.  Google Scholar

[28]

E. Koutsoupias and C. H. Papadimitriou, Worst-case equilibria, in "Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science," 1563 (1999), 404-413. doi: 10.1007/3-540-49116-3_38.  Google Scholar

[29]

G. S. Liu and J. Z. Zhang, Decision making of transportation plan, a bilevel transportation problem approach, Journal of Industrial and Management Optimization, 1 (2005), 305-314.  Google Scholar

[30]

S. Mehrotra and J. Sun, An interior point algorithm for solving smooth convex programs based on Newton's method, in "Contemporary Mathematics" (eds. B. Lagarias and M. Todd), 114 (1991), 265-284.  Google Scholar

[31]

M. Netter, Equilibrium and marginal-cost pricing on a road network with several traffic flow types, in "Proceedings of 5th International Symposium on Transportation and Traffic Theory," (1971), 155-163. Google Scholar

[32]

M. Patriksson, "The Traffic Assignment Problem--Models and Methods,", VSP BV, ().   Google Scholar

[33]

G. Perakis, The "price of anarchy" under nonlinear and asymmetric costs, Mathematics of Operations Research, 32 (2007), 614-628. doi: 10.1287/moor.1070.0258.  Google Scholar

[34]

R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 53-59. doi: 10.1002/net.3230030104.  Google Scholar

[35]

T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236-259. doi: 10.1145/506147.506153.  Google Scholar

[36]

T. Roughgarden, "Selfish Routing and the Price of Anarchy," MIT Press, 2005. Google Scholar

[37]

Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," Prentice-Hall, 1985. Google Scholar

[38]

M. Shubik, "Game Theory in the Social Sciences: Concepts and Solutions," Cambridge, MA and London: The MIT Press, 1985. Google Scholar

[39]

J. Sun, A convergence analysis for a convex version of Dikin's algorithm, Annals of Operations Research, 62 (1996), 357-374. doi: 10.1007/BF02206823.  Google Scholar

[40]

C. Swamy, The effectiveness of Stackelberg strategies and tolls for network congestion games, in "Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms," (2007), 1133-1142.  Google Scholar

[41]

J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325-378. Google Scholar

[42]

H. Yang and H. J. Huang, Principle of marginal-cost pricing: How does it work in a general network?, Transportation Research A, 32 (1998), 45-54. doi: 10.1016/S0965-8564(97)00018-9.  Google Scholar

[43]

H. Yang and H. J. Huang, The multi-class, multi-criteria traffic network equilibrium and system optimum problem, Transportation Research B, 38 (2004), 1-15. doi: 10.1016/S0191-2615(02)00074-7.  Google Scholar

[44]

H. Yang and H. J. Huang, "Mathematical and Economic Theory of Road Pricing," Elsevier, 2005. Google Scholar

[45]

H. Yang and X. N. Zhang, Existence of anonymous link tolls for system optimum on networks with mixed equilibrium behaviors, Transportation Research B, 42 (2008), 99-112. doi: 10.1016/j.trb.2007.07.001.  Google Scholar

[46]

X. N. Zhang, H. Yang and H. J. Huang, Multiclass multicriteria mixed equilibrium on networks and uniform link tolls for system optimum, European Journal of Operational Research, 189 (2008), 146-158. doi: 10.1016/j.ejor.2007.05.004.  Google Scholar

show all references

References:
[1]

R. P. Agdeppa, N. Yamashita and M. Fukushima, An implicit programming approach for the road pricing problem with nonadditive route costs, Journal of Industrial and Management Optimization, 4 (2008), 183-197.  Google Scholar

[2]

M. S. Bazarra, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley and Sons, Inc., New York, 1993. Google Scholar

[3]

P. Bergendorff, D. W. Hearn and M. V. Ramana, Congestion toll pricing of traffic networks, in "Network Optimization" (eds. P. M. Pardalos, D. W. Hearn and W. W. Hager), Springer, (1997), 51-71.  Google Scholar

[4]

C. K. Chau and K. M. Sim, The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands, Operations Research Letters, 31 (2003), 327-334. doi: 10.1016/S0167-6377(03)00030-0.  Google Scholar

[5]

R. Cole, Y. Dodis and T. Roughgarden, Pricing network edges for heterogeneous selfish users, in "Proceedings of the 4th ACM Conference on Electronic Commerce," (2003), 521-530.  Google Scholar

[6]

R. Cole, Y. Dodis and T. Roughgarden, How much can tolls help selfish routing, in "Proceedings of the 4th ACM Conference on Electronic Commerce," (2003), 98-107. doi: 10.1145/779928.779941.  Google Scholar

[7]

R. Cominetti, J. R. Correa and N. E. Stier-Moses, The impact of oligopolistic competition in networks, Operation Research, 57 (2009), 1421-1437. doi: 10.1287/opre.1080.0653.  Google Scholar

[8]

J. Correa, A. Schulz and N. Stier Moses, Selfish routing in capacitated networks, Mathematics of Operations Research, 29 (2004), 961-976. doi: 10.1287/moor.1040.0098.  Google Scholar

[9]

J. Correa, A. Schulz and N. Stier Moses, Computational complexity, fairness, and the price of anarchy of the maximum latency problem, in "Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science" (eds. D. Bienstock and G. Nemhauser), Springer Berlin, (2004), 59-73. doi: 10.1007/978-3-540-25960-2_5.  Google Scholar

[10]

A. Czumaj and B. Vöcking, Tight bounds for worst-case equilibria, in "Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms," (2002), 413-420.  Google Scholar

[11]

S. C. Dafermos and F. T. Sparrow, Optimal resource allocation and toll patterns in user-optimized transport network, Journal of Transport Economics and Policy, 5 (1971), 198-200. Google Scholar

[12]

S. C. Dafermos, An extended traffic assignment modal with applications to two-way traffic, Transportation Science, 5 (1971), 366-389. doi: 10.1287/trsc.5.4.366.  Google Scholar

[13]

S. C. Dafermos, The traffic assignment problem for multiclass-user transportation network, Transportation Science, 6 (1972), 73-87. doi: 10.1287/trsc.6.1.73.  Google Scholar

[14]

S. Dafermos, Toll pattern for multiclass-user transportation network, Transportation Science, 7 (1973), 211-223. doi: 10.1287/trsc.7.3.211.  Google Scholar

[15]

S. Devarajan, A note on network equilibrium and non-cooperative games, Transportaion Research B, 15 (1981), 421-426. doi: 10.1016/0191-2615(81)90026-6.  Google Scholar

[16]

R. B. Dial, Minimal-revenue congestion pricing part I: A fast algorithm for the single-origin case, Transportation Research B, 33 (1999), 189-202. doi: 10.1016/S0191-2615(98)00026-5.  Google Scholar

[17]

R. B. Dial, Minimal-revenue congestion pricing Part II: An efficient algorithm for the general case, Transportation Research B, 34 (2000), 645-665. doi: 10.1016/S0191-2615(99)00046-6.  Google Scholar

[18]

S. D. Flam and Charles Horvath, Network games; adaptations to Nash-Cournot equilibrium, Annals of Operations Research, 64 (1996), 179-195. doi: 10.1007/BF02187645.  Google Scholar

[19]

L. Fleischer, K. Jain and M. Mahdain, Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games, in "Proceedings of the 45th Annual IEEE Symposium on Foundations of Compuer Science," 2004. doi: 10.1109/FOCS.2004.69.  Google Scholar

[20]

D. R. Han, J. Sun and M. Ang, New bounds for the price of anarchy under nonlinear and asymmetric costs,, Optimization, ().   Google Scholar

[21]

D. R. Han, H. K. Lo, J. Sun and H. Yang, The toll effect on price of anarchy when costs are nonlinear and asymmetric, European Journal of Operational Research, 186 (2008), 300-316. doi: 10.1016/j.ejor.2007.01.027.  Google Scholar

[22]

D. R. Han, H. Yang and X. M. Yuan, The efficiency analysis for oligopolistic games when cost functions are nonseparable, International Journal of Mathematical Modelling and Numerical Optimisation, 1 (2010), 237-257. doi: 10.1504/IJMMNO.2010.031751.  Google Scholar

[23]

P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 39-46. doi: 10.1287/trsc.22.1.39.  Google Scholar

[24]

A. Haurie and P. Marcotte, On the relationship between Nash-Cournot and Wardrop equlibria, Networks, 15 (1985), 295-308. doi: 10.1002/net.3230150303.  Google Scholar

[25]

D. W. Hearn and M. B. Yildirim, A toll pricing framework for traffic assignment problem with elastic demand, in "Current Trends in Transportation and Network Analysis-Papers in Honor of Michael Florian" (eds. M. Gendreau and P. Marcotte), Kluwer Academic Publishers, Norwell (2002), 135-145.  Google Scholar

[26]

G. Karakostas and S. G. Kolliopoulos, The efficiency of optimal taxes, in "Proceedings of 1st Workshop on Combinatorial and Algorithmic Aspects of Networking," (2004), 3-12.  Google Scholar

[27]

G. Karakostas and S. Kolliopoulos, Edge pricing of multicommodity networks for heterogeneous users, in "Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science," (2004), 268-276. doi: 10.1109/FOCS.2004.26.  Google Scholar

[28]

E. Koutsoupias and C. H. Papadimitriou, Worst-case equilibria, in "Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science," 1563 (1999), 404-413. doi: 10.1007/3-540-49116-3_38.  Google Scholar

[29]

G. S. Liu and J. Z. Zhang, Decision making of transportation plan, a bilevel transportation problem approach, Journal of Industrial and Management Optimization, 1 (2005), 305-314.  Google Scholar

[30]

S. Mehrotra and J. Sun, An interior point algorithm for solving smooth convex programs based on Newton's method, in "Contemporary Mathematics" (eds. B. Lagarias and M. Todd), 114 (1991), 265-284.  Google Scholar

[31]

M. Netter, Equilibrium and marginal-cost pricing on a road network with several traffic flow types, in "Proceedings of 5th International Symposium on Transportation and Traffic Theory," (1971), 155-163. Google Scholar

[32]

M. Patriksson, "The Traffic Assignment Problem--Models and Methods,", VSP BV, ().   Google Scholar

[33]

G. Perakis, The "price of anarchy" under nonlinear and asymmetric costs, Mathematics of Operations Research, 32 (2007), 614-628. doi: 10.1287/moor.1070.0258.  Google Scholar

[34]

R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 53-59. doi: 10.1002/net.3230030104.  Google Scholar

[35]

T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236-259. doi: 10.1145/506147.506153.  Google Scholar

[36]

T. Roughgarden, "Selfish Routing and the Price of Anarchy," MIT Press, 2005. Google Scholar

[37]

Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," Prentice-Hall, 1985. Google Scholar

[38]

M. Shubik, "Game Theory in the Social Sciences: Concepts and Solutions," Cambridge, MA and London: The MIT Press, 1985. Google Scholar

[39]

J. Sun, A convergence analysis for a convex version of Dikin's algorithm, Annals of Operations Research, 62 (1996), 357-374. doi: 10.1007/BF02206823.  Google Scholar

[40]

C. Swamy, The effectiveness of Stackelberg strategies and tolls for network congestion games, in "Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms," (2007), 1133-1142.  Google Scholar

[41]

J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325-378. Google Scholar

[42]

H. Yang and H. J. Huang, Principle of marginal-cost pricing: How does it work in a general network?, Transportation Research A, 32 (1998), 45-54. doi: 10.1016/S0965-8564(97)00018-9.  Google Scholar

[43]

H. Yang and H. J. Huang, The multi-class, multi-criteria traffic network equilibrium and system optimum problem, Transportation Research B, 38 (2004), 1-15. doi: 10.1016/S0191-2615(02)00074-7.  Google Scholar

[44]

H. Yang and H. J. Huang, "Mathematical and Economic Theory of Road Pricing," Elsevier, 2005. Google Scholar

[45]

H. Yang and X. N. Zhang, Existence of anonymous link tolls for system optimum on networks with mixed equilibrium behaviors, Transportation Research B, 42 (2008), 99-112. doi: 10.1016/j.trb.2007.07.001.  Google Scholar

[46]

X. N. Zhang, H. Yang and H. J. Huang, Multiclass multicriteria mixed equilibrium on networks and uniform link tolls for system optimum, European Journal of Operational Research, 189 (2008), 146-158. doi: 10.1016/j.ejor.2007.05.004.  Google Scholar

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