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Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks
Existence of anonymous link tolls for decentralizing an oligopolistic game and the efficiency analysis
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 
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), 183197. 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), 5171. Google Scholar 
[4] 
C. K. Chau and K. M. Sim, The price of anarchy for nonatomic congestion games with symmetric cost maps and elastic demands, Operations Research Letters, 31 (2003), 327334. doi: 10.1016/S01676377(03)000300. 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), 521530. 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), 98107. doi: 10.1145/779928.779941. Google Scholar 
[7] 
R. Cominetti, J. R. Correa and N. E. StierMoses, The impact of oligopolistic competition in networks, Operation Research, 57 (2009), 14211437. 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), 961976. 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), 5973. doi: 10.1007/9783540259602_5. Google Scholar 
[10] 
A. Czumaj and B. Vöcking, Tight bounds for worstcase equilibria, in "Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms," (2002), 413420. Google Scholar 
[11] 
S. C. Dafermos and F. T. Sparrow, Optimal resource allocation and toll patterns in useroptimized transport network, Journal of Transport Economics and Policy, 5 (1971), 198200. Google Scholar 
[12] 
S. C. Dafermos, An extended traffic assignment modal with applications to twoway traffic, Transportation Science, 5 (1971), 366389. doi: 10.1287/trsc.5.4.366. Google Scholar 
[13] 
S. C. Dafermos, The traffic assignment problem for multiclassuser transportation network, Transportation Science, 6 (1972), 7387. doi: 10.1287/trsc.6.1.73. Google Scholar 
[14] 
S. Dafermos, Toll pattern for multiclassuser transportation network, Transportation Science, 7 (1973), 211223. doi: 10.1287/trsc.7.3.211. Google Scholar 
[15] 
S. Devarajan, A note on network equilibrium and noncooperative games, Transportaion Research B, 15 (1981), 421426. doi: 10.1016/01912615(81)900266. Google Scholar 
[16] 
R. B. Dial, Minimalrevenue congestion pricing part I: A fast algorithm for the singleorigin case, Transportation Research B, 33 (1999), 189202. doi: 10.1016/S01912615(98)000265. Google Scholar 
[17] 
R. B. Dial, Minimalrevenue congestion pricing Part II: An efficient algorithm for the general case, Transportation Research B, 34 (2000), 645665. doi: 10.1016/S01912615(99)000466. Google Scholar 
[18] 
S. D. Flam and Charles Horvath, Network games; adaptations to NashCournot equilibrium, Annals of Operations Research, 64 (1996), 179195. 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), 300316. 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), 237257. doi: 10.1504/IJMMNO.2010.031751. Google Scholar 
[23] 
P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 3946. doi: 10.1287/trsc.22.1.39. Google Scholar 
[24] 
A. Haurie and P. Marcotte, On the relationship between NashCournot and Wardrop equlibria, Networks, 15 (1985), 295308. 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 AnalysisPapers in Honor of Michael Florian" (eds. M. Gendreau and P. Marcotte), Kluwer Academic Publishers, Norwell (2002), 135145. 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), 312. 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), 268276. doi: 10.1109/FOCS.2004.26. Google Scholar 
[28] 
E. Koutsoupias and C. H. Papadimitriou, Worstcase equilibria, in "Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science," 1563 (1999), 404413. doi: 10.1007/3540491163_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), 305314. 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), 265284. Google Scholar 
[31] 
M. Netter, Equilibrium and marginalcost pricing on a road network with several traffic flow types, in "Proceedings of 5th International Symposium on Transportation and Traffic Theory," (1971), 155163. Google Scholar 
[32] 
M. Patriksson, "The Traffic Assignment ProblemModels and Methods,", VSP BV, (). Google Scholar 
[33] 
G. Perakis, The "price of anarchy" under nonlinear and asymmetric costs, Mathematics of Operations Research, 32 (2007), 614628. doi: 10.1287/moor.1070.0258. Google Scholar 
[34] 
R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 5359. doi: 10.1002/net.3230030104. Google Scholar 
[35] 
T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236259. 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," PrenticeHall, 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), 357374. 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 ACMSIAM Symposium on Discrete Algorithms," (2007), 11331142. Google Scholar 
[41] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325378. Google Scholar 
[42] 
H. Yang and H. J. Huang, Principle of marginalcost pricing: How does it work in a general network?, Transportation Research A, 32 (1998), 4554. doi: 10.1016/S09658564(97)000189. Google Scholar 
[43] 
H. Yang and H. J. Huang, The multiclass, multicriteria traffic network equilibrium and system optimum problem, Transportation Research B, 38 (2004), 115. doi: 10.1016/S01912615(02)000747. 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), 99112. 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), 146158. 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), 183197. 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), 5171. Google Scholar 
[4] 
C. K. Chau and K. M. Sim, The price of anarchy for nonatomic congestion games with symmetric cost maps and elastic demands, Operations Research Letters, 31 (2003), 327334. doi: 10.1016/S01676377(03)000300. 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), 521530. 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), 98107. doi: 10.1145/779928.779941. Google Scholar 
[7] 
R. Cominetti, J. R. Correa and N. E. StierMoses, The impact of oligopolistic competition in networks, Operation Research, 57 (2009), 14211437. 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), 961976. 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), 5973. doi: 10.1007/9783540259602_5. Google Scholar 
[10] 
A. Czumaj and B. Vöcking, Tight bounds for worstcase equilibria, in "Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms," (2002), 413420. Google Scholar 
[11] 
S. C. Dafermos and F. T. Sparrow, Optimal resource allocation and toll patterns in useroptimized transport network, Journal of Transport Economics and Policy, 5 (1971), 198200. Google Scholar 
[12] 
S. C. Dafermos, An extended traffic assignment modal with applications to twoway traffic, Transportation Science, 5 (1971), 366389. doi: 10.1287/trsc.5.4.366. Google Scholar 
[13] 
S. C. Dafermos, The traffic assignment problem for multiclassuser transportation network, Transportation Science, 6 (1972), 7387. doi: 10.1287/trsc.6.1.73. Google Scholar 
[14] 
S. Dafermos, Toll pattern for multiclassuser transportation network, Transportation Science, 7 (1973), 211223. doi: 10.1287/trsc.7.3.211. Google Scholar 
[15] 
S. Devarajan, A note on network equilibrium and noncooperative games, Transportaion Research B, 15 (1981), 421426. doi: 10.1016/01912615(81)900266. Google Scholar 
[16] 
R. B. Dial, Minimalrevenue congestion pricing part I: A fast algorithm for the singleorigin case, Transportation Research B, 33 (1999), 189202. doi: 10.1016/S01912615(98)000265. Google Scholar 
[17] 
R. B. Dial, Minimalrevenue congestion pricing Part II: An efficient algorithm for the general case, Transportation Research B, 34 (2000), 645665. doi: 10.1016/S01912615(99)000466. Google Scholar 
[18] 
S. D. Flam and Charles Horvath, Network games; adaptations to NashCournot equilibrium, Annals of Operations Research, 64 (1996), 179195. 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), 300316. 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), 237257. doi: 10.1504/IJMMNO.2010.031751. Google Scholar 
[23] 
P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 3946. doi: 10.1287/trsc.22.1.39. Google Scholar 
[24] 
A. Haurie and P. Marcotte, On the relationship between NashCournot and Wardrop equlibria, Networks, 15 (1985), 295308. 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 AnalysisPapers in Honor of Michael Florian" (eds. M. Gendreau and P. Marcotte), Kluwer Academic Publishers, Norwell (2002), 135145. 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), 312. 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), 268276. doi: 10.1109/FOCS.2004.26. Google Scholar 
[28] 
E. Koutsoupias and C. H. Papadimitriou, Worstcase equilibria, in "Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science," 1563 (1999), 404413. doi: 10.1007/3540491163_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), 305314. 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), 265284. Google Scholar 
[31] 
M. Netter, Equilibrium and marginalcost pricing on a road network with several traffic flow types, in "Proceedings of 5th International Symposium on Transportation and Traffic Theory," (1971), 155163. Google Scholar 
[32] 
M. Patriksson, "The Traffic Assignment ProblemModels and Methods,", VSP BV, (). Google Scholar 
[33] 
G. Perakis, The "price of anarchy" under nonlinear and asymmetric costs, Mathematics of Operations Research, 32 (2007), 614628. doi: 10.1287/moor.1070.0258. Google Scholar 
[34] 
R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 5359. doi: 10.1002/net.3230030104. Google Scholar 
[35] 
T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236259. 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," PrenticeHall, 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), 357374. 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 ACMSIAM Symposium on Discrete Algorithms," (2007), 11331142. Google Scholar 
[41] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325378. Google Scholar 
[42] 
H. Yang and H. J. Huang, Principle of marginalcost pricing: How does it work in a general network?, Transportation Research A, 32 (1998), 4554. doi: 10.1016/S09658564(97)000189. Google Scholar 
[43] 
H. Yang and H. J. Huang, The multiclass, multicriteria traffic network equilibrium and system optimum problem, Transportation Research B, 38 (2004), 115. doi: 10.1016/S01912615(02)000747. 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), 99112. 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), 146158. doi: 10.1016/j.ejor.2007.05.004. Google Scholar 
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