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## Traffic congestion pricing via network congestion game approach

 1 School of Mathematics, Southeast University, Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, Nanjing 210096, China 2 Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, School of Mathematics, Southeast University, Nanjing 210096, China 3 Intelligent Transportation System Research Center, Southeast University, Nanjing 210096, China 4 Beijing Urban Construction Design and Development Group Co., Ltd, Beijing 100000, China

* Corresponding author: Jianquan Lu

Received  January 2020 Revised  March 2020 Published  May 2020

Fund Project: This work was supported by the National Natural Science Foundation of China under Grant No. 61973078, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK20170019, "333 Engineering" Foundation of Jiangsu Province of China under Grant BRA2019260, and Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under Grant No. BM2017002

This paper investigates the optimization of traffic congestion systems via network congestion game approach. Firstly, using the semi-tensor product(STP) of matrices, the matrix expression of network congestion game is obtained. Secondly, a necessary and sufficient condition is proposed to guarantee that the traffic systems can be transformed into network congestion game with given performance criterion as its weighted potential function. Then an algorithm is provided to design the traffic congestion price in the case that conversion can be established. Thirdly, by designing proper learning rule, the optimization of traffic systems can be achieved when individuals optimize their own utility function. Moreover, two special cases which make our results more accord with reality and rich. Finally, an example is exploited to demonstrate the effectiveness of our obtained results.

Citation: Jing Zhang, Jianquan Lu, Jinde Cao, Wei Huang, Jianhua Guo, Yun Wei. Traffic congestion pricing via network congestion game approach. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020378
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##### References:
The approach of network congestion game
The traffic network graph
system objective function $P(a)$
 a 111 112 121 122 211 212 221 222 $P(a)$ 63 70 72 71 72 71 72 62
 a 111 112 121 122 211 212 221 222 $P(a)$ 63 70 72 71 72 71 72 62
Utility matrix of network congestion game
 u\s 111 112 121 122 211 212 221 222 $u_1$ 8 18 20 26 26 20 20 8 $u_2$ 8 18 26 20 20 26 20 8 $u_3$ 41 48 46 45 46 45 49 40
 u\s 111 112 121 122 211 212 221 222 $u_1$ 8 18 20 26 26 20 20 8 $u_2$ 8 18 26 20 20 26 20 8 $u_3$ 41 48 46 45 46 45 49 40
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