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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. 
[2] 
M. S. Bazarra, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley and Sons, Inc., New York, 1993. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[10] 
A. Czumaj and B. Vöcking, Tight bounds for worstcase equilibria, in "Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms," (2002), 413420. 
[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. 
[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. 
[13] 
S. C. Dafermos, The traffic assignment problem for multiclassuser transportation network, Transportation Science, 6 (1972), 7387. doi: 10.1287/trsc.6.1.73. 
[14] 
S. Dafermos, Toll pattern for multiclassuser transportation network, Transportation Science, 7 (1973), 211223. doi: 10.1287/trsc.7.3.211. 
[15] 
S. Devarajan, A note on network equilibrium and noncooperative games, Transportaion Research B, 15 (1981), 421426. doi: 10.1016/01912615(81)900266. 
[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. 
[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. 
[18] 
S. D. Flam and Charles Horvath, Network games; adaptations to NashCournot equilibrium, Annals of Operations Research, 64 (1996), 179195. doi: 10.1007/BF02187645. 
[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. 
[20] 
D. R. Han, J. Sun and M. Ang, New bounds for the price of anarchy under nonlinear and asymmetric costs, Optimization, to appear. 
[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. 
[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. 
[23] 
P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 3946. doi: 10.1287/trsc.22.1.39. 
[24] 
A. Haurie and P. Marcotte, On the relationship between NashCournot and Wardrop equlibria, Networks, 15 (1985), 295308. doi: 10.1002/net.3230150303. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[32] 
M. Patriksson, "The Traffic Assignment ProblemModels and Methods," VSP BV, Utrecht, The Netherlands. 
[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. 
[34] 
R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 5359. doi: 10.1002/net.3230030104. 
[35] 
T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236259. doi: 10.1145/506147.506153. 
[36] 
T. Roughgarden, "Selfish Routing and the Price of Anarchy," MIT Press, 2005. 
[37] 
Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," PrenticeHall, 1985. 
[38] 
M. Shubik, "Game Theory in the Social Sciences: Concepts and Solutions," Cambridge, MA and London: The MIT Press, 1985. 
[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. 
[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. 
[41] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325378. 
[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. 
[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. 
[44] 
H. Yang and H. J. Huang, "Mathematical and Economic Theory of Road Pricing," Elsevier, 2005. 
[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. 
[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. 
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. 
[2] 
M. S. Bazarra, H. D. Sherali and C. M. Shetty, "Nonlinear Programming: Theory and Algorithms," John Wiley and Sons, Inc., New York, 1993. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[10] 
A. Czumaj and B. Vöcking, Tight bounds for worstcase equilibria, in "Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms," (2002), 413420. 
[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. 
[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. 
[13] 
S. C. Dafermos, The traffic assignment problem for multiclassuser transportation network, Transportation Science, 6 (1972), 7387. doi: 10.1287/trsc.6.1.73. 
[14] 
S. Dafermos, Toll pattern for multiclassuser transportation network, Transportation Science, 7 (1973), 211223. doi: 10.1287/trsc.7.3.211. 
[15] 
S. Devarajan, A note on network equilibrium and noncooperative games, Transportaion Research B, 15 (1981), 421426. doi: 10.1016/01912615(81)900266. 
[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. 
[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. 
[18] 
S. D. Flam and Charles Horvath, Network games; adaptations to NashCournot equilibrium, Annals of Operations Research, 64 (1996), 179195. doi: 10.1007/BF02187645. 
[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. 
[20] 
D. R. Han, J. Sun and M. Ang, New bounds for the price of anarchy under nonlinear and asymmetric costs, Optimization, to appear. 
[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. 
[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. 
[23] 
P. T. Harker, Multiple equlibrium behaviors on Networks, Transportation Science, 22 (1988), 3946. doi: 10.1287/trsc.22.1.39. 
[24] 
A. Haurie and P. Marcotte, On the relationship between NashCournot and Wardrop equlibria, Networks, 15 (1985), 295308. doi: 10.1002/net.3230150303. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[32] 
M. Patriksson, "The Traffic Assignment ProblemModels and Methods," VSP BV, Utrecht, The Netherlands. 
[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. 
[34] 
R. W. Rosenthal, The network equilibrium problem in integers, Networks, 3 (1973), 5359. doi: 10.1002/net.3230030104. 
[35] 
T. Roughgarden and E. Tardos, How bad is selfish routing, Journal of the ACM, 49 (2002), 236259. doi: 10.1145/506147.506153. 
[36] 
T. Roughgarden, "Selfish Routing and the Price of Anarchy," MIT Press, 2005. 
[37] 
Y. Sheffi, "Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods," PrenticeHall, 1985. 
[38] 
M. Shubik, "Game Theory in the Social Sciences: Concepts and Solutions," Cambridge, MA and London: The MIT Press, 1985. 
[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. 
[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. 
[41] 
J. G. Wardrop, Some theoretical aspects of road traffic research, in "Proceedings of the Institution of Civil Engineers," Part II, (1952), 325378. 
[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. 
[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. 
[44] 
H. Yang and H. J. Huang, "Mathematical and Economic Theory of Road Pricing," Elsevier, 2005. 
[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. 
[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. 
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