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Stackelberg games and multiple equilibrium behaviors on networks

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  • Yang, Hai
  • Zhang, Xiaoning
  • Meng, Qiang

Abstract

The classical Wardropian principle assumes that users minimize either individual travel cost or overall system cost. Unlike the pure Wardropian equilibrium, there might be in reality both competition and cooperation among users, typically when there exist oligopoly Cournot-Nash (CN) firms. In this paper, we first formulate a mixed behavior network equilibrium model as variational inequalities (VI) that simultaneously describe the routing behaviors of user equilibrium (UE), system optimum (SO) and CN players, each player is presumed to make routing decision given knowledge of the routing strategies of other players. After examining the existence and uniqueness of solutions, the diagonalization approach is applied to find a mixed behavior equilibrium solution. We then present a Stackelberg routing game on the network in which the SO player is the leader and the UE and CN players are the followers. The UE and CN players route their flows in a mixed equilibrium behavior given the SO player's routing strategy. In contrast, the SO player, realizing how the UE and CN players react to the given strategy, routes its flows to minimize total system travel cost. The Stackelberg game of network flow routing is formulated as a mathematical program with equilibrium constraints (MPEC). Using a marginal function approach, the MPEC is transformed into an equivalent, continuously differentiable single-level optimization problem, where the lower level VI is represented by a differentiable gap function constraint. The augmented Lagrangian method is then used to solve the resulting single-level optimization problem. Some numerical examples are presented to demonstrate the proposed models and algorithms.

Suggested Citation

  • Yang, Hai & Zhang, Xiaoning & Meng, Qiang, 2007. "Stackelberg games and multiple equilibrium behaviors on networks," Transportation Research Part B: Methodological, Elsevier, vol. 41(8), pages 841-861, October.
  • Handle: RePEc:eee:transb:v:41:y:2007:i:8:p:841-861
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    References listed on IDEAS

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    1. Devarajan, Shantayanan, 1981. "A note of network equilibrium and noncooperative games," Transportation Research Part B: Methodological, Elsevier, vol. 15(6), pages 421-426, December.
    2. Meng, Q. & Yang, H. & Bell, M. G. H., 2001. "An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem," Transportation Research Part B: Methodological, Elsevier, vol. 35(1), pages 83-105, January.
    3. Yang, Hai, 1998. "Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration," Transportation Research Part B: Methodological, Elsevier, vol. 32(3), pages 205-218, April.
    4. Yang, Hai & Zhang, Xiaoning & Meng, Qiang, 2004. "Modeling private highways in networks with entry-exit based toll charges," Transportation Research Part B: Methodological, Elsevier, vol. 38(3), pages 191-213, March.
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    1. repec:gam:jsusta:v:10:y:2018:i:4:p:1238-:d:141773 is not listed on IDEAS
    2. (Walker) Wang, Wei & Wang, David Z.W. & Sun, Huijun & Feng, Zengzhe & Wu, Jianjun, 2016. "Braess Paradox of traffic networks with mixed equilibrium behaviors," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 93(C), pages 95-114.
    3. Huang, Yeu-Shiang & Chen, Si-Hen & Ho, Jyh-Wen, 2013. "A study on pricing and delivery strategy for e-retailing systems," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 59(C), pages 71-84.
    4. repec:kap:netspa:v:17:y:2017:i:2:d:10.1007_s11067-016-9335-9 is not listed on IDEAS
    5. repec:spr:joptap:v:153:y:2012:i:2:d:10.1007_s10957-011-9945-9 is not listed on IDEAS
    6. Wan, Cheng, 2016. "Strategic decentralization in binary choice composite congestion games," European Journal of Operational Research, Elsevier, vol. 250(2), pages 531-542.
    7. Wang, Hua & Meng, Qiang & Zhang, Xiaoning, 2014. "Game-theoretical models for competition analysis in a new emerging liner container shipping market," Transportation Research Part B: Methodological, Elsevier, vol. 70(C), pages 201-227.
    8. Andrzej Grzybowski, 2009. "A Note On A Single Vehicle And One Destination Routing Problem And Its Game-Theoretic Models," Advanced Logistic systems, University of Miskolc, Department of Material Handling and Logistics, vol. 3(1), pages 71-76, December.
    9. Farokhi, Farhad & Johansson, Karl H., 2015. "A piecewise-constant congestion taxing policy for repeated routing games," Transportation Research Part B: Methodological, Elsevier, vol. 78(C), pages 123-143.
    10. Van Gorder, Robert A. & Caputo, Michael R., 2010. "Envelope theorems for locally differentiable open-loop Stackelberg equilibria of finite horizon differential games," Journal of Economic Dynamics and Control, Elsevier, vol. 34(6), pages 1123-1139, June.
    11. Zhang, Xiaoning & Yang, Hai & Huang, Hai-Jun, 2008. "Multiclass multicriteria mixed equilibrium on networks and uniform link tolls for system optimum," European Journal of Operational Research, Elsevier, vol. 189(1), pages 146-158, August.
    12. Yang, Hai & Zhang, Xiaoning, 2008. "Existence of anonymous link tolls for system optimum on networks with mixed equilibrium behaviors," Transportation Research Part B: Methodological, Elsevier, vol. 42(2), pages 99-112, February.

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