A dynamic parking charge optimal control model under perspective of commuters’ evolutionary game behavior

Accepted Manuscript A dynamic parking charge optimal control model under perspective of commuters’ evolutionary game behavior XuXun Lin, PengCheng Yuan

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S0378-4371(17)30783-5 http://dx.doi.org/10.1016/j.physa.2017.08.063 PHYSA 18509

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Physica A

Received date : 30 March 2017 Revised date : 22 June 2017 Please cite this article as: X. Lin, P. Yuan, A dynamic parking charge optimal control model under perspective of commuters’ evolutionary game behavior, Physica A (2017), http://dx.doi.org/10.1016/j.physa.2017.08.063 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A dynamic parking charge optimal control model under perspective of commuters’ evolutionary game behavior* XuXun Lina, PengCheng Yuan b, ∗ a. Business School of Changzhou University, Jiangsu Changzhou, 213164, P R China b. Business School, University of Shanghai for Science and Technology, Shanghai 200093, P R China

Abstract: In this research we consider commuters’ dynamic learning effect by modeling the trip mode choice behavior from a new perspective of dynamic evolutionary game theory. We explore the behavior pattern of different types of commuters and study the evolution path and equilibrium properties under different traffic conditions. We further establish a dynamic parking charge optimal control (referred to as DPCOC) model to alter commuters’ trip mode choice while minimizing the total social cost. Numerical tests show. (1) Under fixed parking fee policy, the evolutionary results are completely decided by the travel time and the only method for public transit induction is to increase the parking charge price. (2) Compared with fixed parking fee policy, DPCOC policy proposed in this research has several advantages. Firstly, it can effectively turn the evolutionary path and evolutionary stable strategy to a better situation while minimizing the total social cost. Secondly, it can reduce the sensitivity of trip mode choice behavior to traffic congestion and improve the ability to resist interferences and emergencies. Thirdly, it is able to control the private car proportion to a stable state and make the trip behavior more predictable for the transportation management department. The research results can provide theoretical basis and decision-making references for commuters’ mode choice prediction, dynamic setting of urban parking charge prices and public transit induction.

Key words: Trip mode; Asymmetric evolutionary game; Parking charge; Dynamic optimal control

1. Introduction In recent years, with the growth of world economic and the residents’ income level, the vehicle number of major cities in the world has increased dramatically. However, on the contrary, the growth of traffic supply facilities is lagging behind because of the investment/resource constraints and other reasons. The traffic supply and demand contradiction is obvious[1]. Nowadays, many cities in the world are encountering the parking difficult problems. Road driving speed is declining, congestion area is expanding, urban development is facing severe challenges. It has been proved infeasible that relying solely on the transport facilities construction to alleviate traffic needs[2].So, in recent years, many new methods are used to solve these social problems. For example, many researchers tend to use dynamic and nonlinear methods (such as dynamic control, chaos & fractal theory etc.) to study the economic phenomena (oil price, stock market, gas market and traffic congestion etc) in social Sciences [3-6]. Parking charge is one of the most commonly used strategies for congestion management. It usually imposes a certain charge on commuters who drive into congested traffic area such as urban downtown. It helps to induce bus trip and reduce urban traffic flow by increasing the trip cost of private car mode and has been proved to be an effective approach for congestion prevention [7, 8]. Previous research about parking charge strategies mainly focuses on the policy implementation in view of trip behavior [9-11], congestion tolls [12] and dynamic traffic flows [13-18]. These works point out that parking charge may be more effective than road congestion tolls in influencing commuters’ trip behavior [8, 19, 20] and reducing traffic congestion [21-23]. Van and Russo point out that charging a fixed price per day can induce a welfare loss of at least 4% of the organization’s parking cost [24]. Pierce examines the most promising parking charge policies for public garages [25]. Chang and Chun etc propose a pricing strategy for privately-owned parking services [26]. Although previous research has achieved many results, most of the previous research only assumes that commuters have full information of the traffic situation and always make the optimal mode and route choices, thus seldom consider the ‘dynamic learning effect’. In reality, commuters need a period of time to adapt to the transportation and they keep making adjustments to their trip behavior according to their past experiences. This process is defined as ‘dynamic learning effect’ [27-29]. In this research, we consider commuters’ dynamic learning effect in formulation of the optimal parking charge policy. ‘Dynamic learning effect’ can be explained and modeled by fractal geometry [30, 31] and evolution game theory [27, 28, 32]. Game theory is first applied in transportation reliability to model travel time randomness and commuters’ risk

*

∗ Corresponding author. E-mail addresses: [email protected] (P. Yuan), [email protected] (X. Lin).

avoidance [33, 34]. Furthermore, game theory is used to analyze commuters’ trip mode choice behavior [35], relationship between commuters and public transportation management department [36]. It is also used in the research of optimal pricing strategy for transportation facilities [37] and transportation network design [38]. On the basis of static analysis, dynamic game model is applied in trip behavior. Su and Chang etc study urban public traffic networks from the viewpoint of complex networks and game theory. They investigate an urban public traffic network of Beijing, and propose a simplified game theory model to simulate the evolution of the traffic network [39]. Farokhi and Johansson use repeated game to establish a piecewise-constant congestion taxing model under situations of time-varying demand [40]. Nakata and Yamauchi etc find that several social dilemma structures can be represented by n-person Prisoner’s Dilemma games in certain traffic flow phases at a bottleneck caused by a lane-closing section. So they establish the stochastic Nishinari–Fukui–Schadschneider model to simulate real traffic flow [41]. Although great knowledge on the theory of parking strategy has been accumulated, several issues remain open for further investigation. Firstly, Most of the research assumes that the commuters have complete rationality, thus neglect commuters’ dynamic learning and adaptation process, which is inconsistent with the real situation. Secondly, Most of the models assume that different game players have symmetric influence on each other. Thirdly, Most of the research seldom mentions the optimal control of dynamic parking charge policy. The innovation of this research is twofold. Firstly, by using dynamic evolutionary game theory, we consider commuters’ learning effect and the dynamic evolution process in their trip mode choice behavior which is rarely considered in previous research. Secondly, we apply dynamic control theory and propose a dynamic parking charge optimal control (referred to as DPCOC) model which is able to alter commuters’ trip mode choice behavior while minimizing the total social cost. To the best of our knowledge, it is the first time this theory is used in optimal dynamic parking charge price problem. Also, we demonstrate that the DPCOC policy has significant advantage over fixed parking fee policy. The results can be applied for the dynamic optimal formulation and implementation of parking charge price setting to induce bus trip, ease traffic congestion and reduce social costs. The rest of the research is structured as follows. After the introduction, Section 2 describes the evolution game process of commuters’ trip mode choice behavior. Section 3 establishes the dynamic optimal parking charge control model in view of minimizing the total social cost. Section 4 provides a numerical example to conduct comparison analysis between the fixed parking fee policy and dynamic optimal control policy. Finally, we finish the research with conclusion remarks and a discussion of future works in Section 5.

2. The second order asymmetric evolution game of commute choice behavior 2.1 Travel mode choice and model assumptions Commuters choose their trip mode and keep adjusting their daily decisions according to the traffic situation, and finally a stable state is reached. This whole process can be seen as a game behavior facing limited traffic travel time and space resources. For simplicity, we consider the following two different types of commuters as game players. City downtown commuters (referred to as ‘downtown commuters’): This type of commuters live in downtown area, they have two alternatives: drive to work or travel by public transport. We set the mode selection strategy set for downtown commuters as I1= {1: private car; 2: public transit}. City suburban commuters (referred to as ‘suburban commuters’): This type of commuters live in suburban area but work in downtown centers, they also have two alternatives: drive directly to their working place, or drive to the Park & Ride facility nearest to the urban center and transfer to the subway. We have the mode selection strategy set for suburban commuters as I2= {1: private car; 2: Park & Ride}. We assume that the trip cost in suburban area is trivial compared to urban area, thus we ignore the suburban trip cost and mainly focus on the scenario within the downtown area. Due to limited road travel capacity, if both types of commuters choose to travel by private car, there will be serious traffic congestion, and part of the commuters will be forced to choose public transit. Based on the theory of dynamic evolutionary game we make the following assumptions to simplify the situation. First, private car trip cost is composed of travel time and parking charge. Second, public transit travel cost is composed of travel time alone. Third, to encourage transit travel mode, the Park & Ride facility offer free-parking service.

Fourth, compared with the congestion in downtown area, traffic condition is much better in suburban area, the driving time and cost is neglected in Park & Ride trip mode. Last, to limit private car trip in downtown area, certain parking charges are imposed on private cars entering the downtown area. 2.2 Model establishment and equilibrium point calculation All acronyms for parameters and variables in the model are listed in table 1. Table 1 Acronyms Stratification

symbols

definition

𝑡0

Game starting time

𝑡𝑓

Game finishing time

𝑇𝐴

Private car trip time when both types of commuters drive their private cars to work

𝑇𝐵

Private car trip time for suburban commuters when urban commuters take public transit to work

Parameters

Variables

𝑇𝐶

Private car trip time for urban commuters when suburban commuters choose Park & Ride to work

𝐵

Travel time by public transit

𝑞

Parking charge base price for private car

𝜃

Cost-time conversion coefficient

ℎ0

Cost conversion coefficient of private car commuters proportion

ℎ1

Cost conversion coefficient of the dynamic control function

𝑡

System time

𝑥(𝑡)

Proportion of suburban commuters taking private car

𝑦(𝑡)

Proportion of urban commuters taking private car

𝛩(𝑡)

Dynamic control function on the parking charge

According to 2.1, we can calculate the utility under different game strategies. In all, there are four scenarios according to commuters’ trip mode choice: Scenario 1: when both types of commuters choose private car (refer to the left-upper of the payoff matrix in table 2). The travel time is TA, θq is the equivalent travel time caused by parking charge, then for both types of commuters, the utility is −𝜃𝑞 − 𝑇𝐴 . Scenario 2: When suburban commuters choose private car, and downtown commuters choose public transit (refer to the left-upper of the payoff matrix in table 2). For suburban commuters, the travel time is TB, so the total utility is −𝜃𝑞 − 𝑇𝐵 , for downtown commuters, the utility is –B. Scenario 3: When downtown commuters choose private car, and suburban commuters choose Park & Ride (refer to the left-upper of the payoff matrix in table 2). For downtown commuters, the travel time is TC, so the total utility is −𝜃𝑞 − 𝑇𝐶 , for downtown commuters, the utility is –B. Scenario 4: When both types of commuters choose bus or Park & Ride (refer to the left-upper of the payoff matrix in table 2). In this case parking charge is imposed on the commuters, both types of commuters have the utility as –B. Finally, we get the single stage game payoff matrix of trip mode choice as shown in table 2. Table 2 Payoff matrix of single stage game Downtown commuters Suburban commuters 1：private car

2：public transit

1: private car

−𝜃𝑞 − 𝑇𝐴 , −𝜃𝑞 − 𝑇𝐴

−𝜃𝑞 − 𝑇𝐵 , −𝐵

2: Park & Ride

−𝐵, −𝜃𝑞 − 𝑇𝐶

 B,  B

When both type of commuters drive their private cars to work, all private car travel into the transportation system, the roads should become the most congested. The private car travel time TA should be the longest. When one type of commuters takes bus or Park & Ride to work, there will be fewer private cars in the transportation system, making the roads less congested. The private car travel time TB or TC should be shorter than TA. So we give the following two assumptions: (1) TA>TB and (2) TA>TC. The bus trip and Park & Ride are two different types of commute and have different influence on the transportation system as well as private car travel time. To distinguish the different influence, we make the third assumption: (3) TB≠TC. For suburban commuters, we define the expected utility of private car and Park & Ride as 𝑈1𝐶 and 𝑈1𝑃 respectively, the group average utility as 𝑈1 . We then have the following relations. 𝑈1𝐶 = 𝑦(−𝜃𝑞 − 𝑇𝐴 ) + (1 − 𝑦)(−𝜃𝑞 − 𝑇𝐵 )

(1)

𝑈1𝑃 = 𝑦(−𝐵) + (1 − 𝑦)(−𝐵)

(2)

𝑈1 = 𝑥(𝑈1𝐶 ) + (1 − 𝑥)(𝑈1𝑝 )

(3)

Similarly, for downtown commuters we define the expected utility of private car and public transit as 𝑈2𝐶 and𝑈2𝑝 , the group average utility as 𝑈2 , and we have the following relations. 𝑈2𝐶 = 𝑥(−𝜃𝑞 − 𝑇𝐴 ) + (1 − 𝑥)(−𝜃𝑞 − 𝑇𝐶 )

(4)

𝑈2𝑃 = 𝑥(−𝐵) + (1 − 𝑥)(−𝐵)

(5)

𝑈2 = 𝑦(𝑈2𝐶 ) + (1 − 𝑦)(𝑈2𝑝 )

(6)

Based on evolutionary game theory [42], the replicated dynamic equations for both types of commuters can be obtained as follows. 𝑑𝑥 𝑑𝑡 {𝑑𝑦 𝑑𝑡

= 𝑥(𝑈1𝐶 − 𝑈1 ) = 𝑦(𝑈2𝐶 − 𝑈2 )

(7)

Substitute (1) ~ (6) into (7), we have the evolutionary equation as follows. 𝑑𝑥 𝑑𝑡 {𝑑𝑦 𝑑𝑡

= 𝑥(1 − 𝑥)(𝐶1 − 𝐶2 𝑦) = 𝑦(1 − 𝑦)(𝐷1 − 𝐷2 𝑥)

(8)

The parameters in (8) are calculated as follows. 𝐶1 = −𝜃𝑞 − 𝑇𝐶 + 𝐵 𝐶2 = 𝑇𝐴 − 𝑇𝐶 { (9) 𝐷1 = −𝜃𝑞 − 𝑇𝐵 + 𝐵 𝐷2 = 𝑇𝐴 − 𝑇𝐵 According to the assumption on TA, TB and TC, it is obvious that 𝐶2 > 0 and 𝐷2 > 0. Let𝑑𝑥/𝑑𝑡 = 0, 𝑑𝑦/𝑑𝑡 = 0 in (8), we get five different kinds of equilibrium points: (0, 0), (0, 1), (1, 0), (1, 1), (𝐷1 /𝐷2 , 𝐶1 /𝐶2 ). However, 0 ≤ x, y ≤ 1, so the equilibrium points should lie within [0, 1], thus (𝐷1 /𝐷2 , 𝐶1 /𝐶2 ) is the qualified equilibrium point only if 0 ≤ 𝐷1 /𝐷2 ≤ 1, and 0 ≤ 𝐶1 /𝐶2 ≤ 1. We further discuss the stability properties of the equilibrium points in Section 2.3. 2.3 Replicated dynamic equation and model stability analysis In this section we analyze the stability properties of the five equilibrium points. The system shown in (8) is denoted as System 1, shown as below in (10). 𝑑𝑥

𝐶 𝑑𝑡 (𝑑𝑦 )=( 1 0 𝑑𝑡

0 𝑥 −𝐶 𝑥2 −𝐶 𝑥𝑦+𝐶 𝑥2 𝑦 ) ( ) + (−𝐷 1𝑦2−𝐷2 𝑥𝑦+𝐷2 𝑦2𝑥) 𝐷1 𝑦 1 2 2

The linear approximation of System 1 is denoted as System 2, which can be easily deduced as (11).

(10)

𝑑𝑥

𝐶 𝑑𝑡 (𝑑𝑦 )=( 1 0

0 𝑥 )( ) 𝐷1 𝑦

𝑑𝑡

(11)

We then have the relation between System 1 and System 2 as Theorem 1. Theorem 1: The analysis of stability properties for the Node, focus and saddle points of System 1 can be turned to the analysis of System 2 Proof: Let 𝑍 = (𝑦𝑥 ) , denote the nonlinear part of System 1 as 𝐺(𝑍) , let 𝑟 = √𝑥 2 + 𝑦 2 and 𝛼 = 1/2 , we 2

2

−𝐶1 𝑥 −𝐶2 𝑥𝑦+𝐶2 𝑥 𝑦 have (−𝐷 ) = 𝐺(𝑍). From the following inequalities (12) 𝑦 2 −𝐷 𝑥𝑦+𝐷 𝑦 2 𝑥 1

2

2

𝑥2 𝑟 1+𝛼

≤

𝑥 2 +𝑦 2 𝑟 1+1⁄2

|𝑥𝑦|

≤

𝑟 1+𝛼 |𝑥 2 𝑦|

{

𝑟 1+𝛼 |𝑦 2 𝑥| 𝑟 1+𝛼

𝑟2

≤

≤ 𝑟 1⁄2

𝑟 3⁄2 1 𝑟2 1 2 𝑟 3⁄2 𝑟 2𝑟

≤ ≤

𝑟 3⁄2 𝑟2𝑟 𝑟 3⁄2

≤ √𝑟 2

(12)

≤ 𝑟 3⁄2 ≤ 𝑟 3⁄2

We can prove that there exists a α >0 which satisfies the following properties. 𝑥2

|𝑥2 𝑦|

𝑦2

|𝑦2 𝑥|

|𝑥𝑦|

lim 𝑟(1+𝛼) = 0, lim 𝑟(1+𝛼) = 0, lim 𝑟(1+𝛼) = 0, lim 𝑟(1+𝛼) = 0 and lim 𝑟(1+𝛼) = 0.

𝑟−>0

𝑟−>0

𝑟−>0

𝑟−>0

𝑟−>0

(13)

According to the expression of 𝐺(𝑍) we get the final result as in (14).

lim

‖G(Z)‖

𝑟−>0 𝑟 1+𝛼

=0

(14)

According to the dynamic system theory [43], Theorem 1 holds as long as condition shown by (14) is satisfied. According to Theorem 1, we can solve the problem simply by analyzing the analytical properties of equilibrium point of System 2, let {

𝐹1 = 𝑥(1 − 𝑥)(𝐶1 − 𝐶2 𝑦) 𝐹2 = 𝑦(1 − 𝑦)(𝐷1 − 𝐷2 𝑥)

(15)

The Jacobian matrix of System 2 can be calculated as follows. 𝜕(𝐹1 𝐹1 ) 𝜕(𝑥𝑦)

(𝐶 𝑦 − 𝐶1 )(2𝑥 − 1) =( 2 𝐷2 𝑦(𝑦 − 1)

𝐶2 𝑥(𝑥 − 1) ) (2𝑦 − 1)(𝐷2 𝑥 − 𝐷1 )

(16)

Denote the Jacobian matrix at equilibrium point (𝑥 ∗ , 𝑦 ∗ ) as 𝐽(𝑥∗,𝑦∗ ) . Let 𝑃(𝑥∗ ,𝑦∗) = −𝑇𝑟[𝐽(𝑥 ∗,𝑦∗) ] , 𝐷(𝑥 ∗,𝑦∗) = 2 𝑑𝑒𝑡[ 𝐽(𝑥∗,𝑦∗) ] and ∆(𝑥∗,𝑦∗ ) = 𝑝(𝑥 ∗ ,𝑦 ∗ ) − 4𝐷(𝑥 ∗ ,𝑦 ∗ ) , where Tr is the matrix trace and det is the matrix determinant. In order to

explain the role of the commuters’ trip mode choice behavior and the role of traffic management measures, we analyze the convergence and stability of the evolutionary game in trip mode choice behavior under the following three different traffic situations respectively. The results are shown in table 3~5. Situation 1：𝐵 > 𝑇𝐴 + 𝜃𝑞. Situation 1 indicates that the bus travel time is longer than private car trip time for both types of commuters. According to (9) we have 𝐷1 /𝐷2 > 1 and 𝐶1 /𝐶2 > 1. We get all four equilibrium points for System 2: (0, 0), (0, 1), (1, 0) and (1, 1). The analytical properties of each equilibrium point are shown in Table 3. Table 3 Analytical properties of commuters’ trip mode choice evolution in situation 1 Equilibrium point

𝑃(𝑥∗,𝑦∗)

𝐷(𝑥 ∗,𝑦∗ )

∆(𝑥∗ ,𝑦∗)

(0, 0)

<0

>0

>= 0

(0, 1)

>0

<0

>0

Analytical properties Unstable Node Saddle

(1, 0)

>0

<0

>0

Saddle

(1, 1)

>0

>0

=0

Stable degenerate Node

Table 3 shows that for all the four equilibrium points there exists only one stable point (1, 1). The dynamic game has one unique evolutionary stable strategy. This result indicates that when bus travel time is long, both types of commuters will finally choose to travel by private car and the whole transportation system has a higher possibility of traffic congestion. Situation 2：𝑚𝑎𝑥(𝑇𝐵 , 𝑇𝐶 ) + 𝜃𝑞 < 𝐵 < 𝑇𝐴 + 𝜃𝑞. Situation 2 indicates a moderate bus travel time. In this situation, five equilibrium points can be obtained as (0, 0), (0, 1), (1, 0), (1, 1) and (𝐷1 /𝐷2 , 𝐶1 /𝐶2 ). The analytical properties of each equilibrium point are shown in Table 4. Table 4 Analytical properties of commuters’ trip mode choice evolution in situation 2 Equilibrium point

𝑃(𝑥∗,𝑦∗)

𝐷(𝑥 ∗,𝑦∗)

∆(𝑥∗ ,𝑦∗)

Analytical properties

(0, 0)

<0

>0

>= 0

Unstable Node

(0, 1)

>0

>0

>0

Stable Node

(1, 0)

>0

>0

>0

Stable Node

(1, 1)

<0

>0

=0

Unstable degenerate Node

(𝐷1 /𝐷2 , 𝐶1 /𝐶2 )

=0

<0

>0

Saddle

Table 4 shows there exist two stable equilibrium points: (0, 1) and (1, 0) which means the game has two types of evolutionary stable strategy. This indicates that when bus travel time increases to a certain level, part of the commuters begin to change their choice behavior and take public transit to work. Finally, all the urban and suburban commuters will choose private car or public transit. Situation 3：𝐵 < 𝑚𝑖𝑛(𝑇𝐵 , 𝑇𝐶 ) + 𝜃𝑞. Situation 3 indicates bus travel time shorter than both types of private car travel time. According to (9) we have 𝐷1 /𝐷2 < 0 and 𝐶1 /𝐶2 < 0. We get all four equilibrium points for System 2: (0, 0), (0, 1), (1, 0) and (1, 1). The analytical properties of each equilibrium point are shown in Table 5. Table 5 Analytical properties of commuters’ trip mode choice evolution in situation 3 Equilibrium point

𝑃(𝑥∗ ,𝑦∗)

𝐷(𝑥∗ ,𝑦∗)

∆(𝑥 ∗,𝑦∗)

Analytical properties

（0，0）

>0

>0

>0

Stable Node

（0，1）

>0

<0

>0

Saddle

（1，0）

>0

<0

>0

Saddle

（1，1）

<0

>0

=0

Unstable degenerate Node

Table 5 shows that there exists a unique stable equilibrium point: (0, 0), which is also the game evolutionary stable strategy. This indicates that when bus travel time decreases to less than private car trip time, all the commuters will choose public transit to work. According to Theorem 1, System 1 has the same equilibrium points and the corresponding analytical properties as System 2. Section 2 shows that under fixed parking fee policy, bus travel time plays a decisive role on the evolutionary path and final results. This helps to explain why fixed parking fee policy can hardly change commuters’ behavior under situation of long bus travel time. Under fixed parking fee policy, the price may be beyond commuters’ acceptable range and cause huge social cost and other problems. Next, we propose a dynamic parking charge optimal control (referred to as DPCOC) model. The model can set the proper price according to different traffic situations. By doing so, it can change commuters’ mode choice behavior while in the mean time minimize the total social cost for a certain period of time. Also, numerical test

shows the advantage of DPCOC over fixed parking fee policy.

3. Dynamic parking charge optimal control policy 3.1 Model establishment In this section we propose a DPCOC policy based on the asymmetric evolutionary game model in Section 2. To consider the dynamic parking charge optimal control policy, we add a dynamic control function 𝛩(𝑡) on the parking charge price q, and expand System 1 to the following form. 𝑑𝑥 𝑑𝑡 {𝑑𝑦 𝑑𝑡

= 𝑥(1 − 𝑥)[𝐶1 (𝛩) − 𝐶2 𝑦] = 𝑦(1 − 𝑦)[𝐷1 (𝛩) − 𝐷2 𝑥]

(17)

All other parameters are the same as System 1, and the specific expressions of parameters in (17) are as follows. 𝐶1 (𝛩) = −𝜃𝑞(1 − 𝛩) − 𝑇𝐶 + 𝐵 𝐶2 = 𝑇𝐴 − 𝑇𝐶 { 𝐷1 (𝛩) = −𝜃𝑞(1 − 𝛩) − 𝑇𝐵 + 𝐵 𝐷2 = 𝑇𝐴 − 𝑇𝐵

(18)

For the sake of simplicity, System shown in (17) is expressed in form of vectors. 𝑑𝑋 𝑑𝑡

= 𝐹(𝑡, 𝑋, 𝛩)

(19)

Where X is the vector (x, y), 𝐹(𝑡, 𝑋, 𝛩) is the vector function in the right side of (17). 𝐹: 𝑅1 × 𝑅 2 × 𝑅1 → 𝑅 2

(20)

The purpose of the optimal control is to minimize the total social cost. In this research we consider the following two steps. Step1: In order to induce bus trip and reduce private car trip, we intend to minimize the proportion of private car trip as much as possible. Step 2: In order to maintain traffic order and reduce the cost of transportation policy implementation, we intend to minimize the total control cost. Let 𝑈𝛩 denote the space formed by all the optional control function 𝛩(𝑡). Let U denote the value range of all the functions in 𝑈𝛩 . As the parking charge price cannot be infinitely high, for example, it should be within the commuters’ acceptable range. So it is reasonable to make the assumption that U is a closed and bounded set. Specifically, let 𝑈 = [−𝑀, 𝑀], where M is a positive real number large enough. ℎ0 and ℎ1 are defined in table 1. These two parameters also serve as the balance coefficient between different variables. The total cost of road congestion caused by private car commuters at time t can be calculated as ℎ0 [𝑥(𝑡) + 𝑦(𝑡)]. Referring to the research of [44] we assume the control cost caused by the implementation of DPCOC at time t can be calculated as ℎ1 𝛩(𝑡)2 . We can calculate the social cost at time t as follows. 𝐿(𝑡, 𝑥, 𝑦, 𝛩) = ℎ0 [𝑥(𝑡) + 𝑦(𝑡)] + ℎ1 𝛩(𝑡)2

(21)

In view of total social cost minimization, we have the objective function as follows. 𝑡

𝑚𝑖𝑛 𝑅(𝛩) = ∫𝑡0𝑓 𝐿(𝑡, 𝑥, 𝑦, 𝛩)𝑑𝑡 𝛩∈𝑈𝛩

(22)

The meaning of 𝑡0 and 𝑡𝑓 is shown in table 1. 3.2 Model analysis of DPCOC We prove the existence of solution of DPCOC in Theorem 2. Theorem 2: There exists an optimal control solution 𝛩(𝑡)∗ for DPCOC shown in (17) ~ (22). Proof: According to the optimal control theory of [45] (Theorem 4.1, p68), we have the sufficient condition for the

establishment of Theorem 2 as follows. First, {(𝑋0 , 𝛩)：𝑋0 is the initial value for X in (19), i.e. 𝑋0 = 𝑋(𝑡0 ), 𝛩 ∈ 𝑈𝛩 } is a non-empty set. Second, 𝑈 is a compact set. Third, 𝐿(𝑡, 𝑥, 𝑦, 𝛩) in (23) is a continuous function, the initial and boundary conditions of system (17) are all continuous functions. Fourth, let 𝑊(𝑡, 𝑋) = {𝐹(𝑡, 𝑋, 𝛩): 𝛩 ∈ 𝑈𝛩 }, then ∀(𝑡, 𝑋) ∈ 𝑅1 × 𝑅 2 , 𝑊(𝑡, 𝑋) is a convex set. Last, there exists a continuous function g, which satisfies that 𝐿(𝑡, 𝑥, 𝑦, 𝛩) ≥ 𝑔(𝛩) and |𝛩|−1 𝑔(𝛩) → +∞ as |𝛩| → +∞. Obviously, the first condition holds. According to the expression of U, U is a closed and bounded set in 𝑅1 space, i.e. U is a compact set, so the second condition holds. According to the expression of F in (17) ~ (19) its initial function and boundary function are all continuous, so the third condition holds. For ∀(𝑡, 𝑋) ∈ 𝑅1 × 𝑅 2 , let 𝑓1 , 𝑓2 ∈ 𝑊(𝑡, 𝑋), according to the definition of 𝑊(𝑡, 𝑋) we know that there exists 𝛩1 and 𝛩2 which holds 𝑓𝑘 = 𝐹(𝑡, 𝑥, 𝑦, 𝛩𝑘 (𝑡)), where 𝛩𝑘 (𝑡) ∈ 𝑈, 𝑘 = 1, 2. According to (17) ~ (19) we can see that F is linear with respect to 𝛩𝑘 (𝑡) when t and X are fixed, then 𝑓1 and 𝑓2 can be written as follows. {

𝑓1 = (𝑆1 𝛩1 (𝑡) + 𝐾1 , 𝑆2 𝛩1 (𝑡) + 𝐾2 ) 𝑓2 = (𝑆1 𝛩2 (𝑡) + 𝐾1 , 𝑆2 𝛩2 (𝑡) + 𝐾2 )

(23)

Where S1, S2, K1 and K2 are all constants. Let 𝑒 be a arbitrary real number in interval [0, 1], then we have 𝑒𝑓1 + (1 − 𝑒)𝑓2 = (𝑆1 [𝑒𝛩1 (𝑡) + (1 − 𝑒)𝛩2 (𝑡)] + 𝐾1 , 𝑆2 [𝑒𝛩1 (𝑡) + (1 − 𝑒)𝛩2 (𝑡)] + 𝐾2 )

(24)

Because 𝑈 is a bounded interval in 𝑅1 space, 𝑈 is also a convex set; as 𝛩𝑘 (𝑡) ∈ 𝑈, 𝑘 = 1,2, so 𝑒𝛩1 (𝑡) + (1 − 𝑒)𝛩1 (𝑡) ∈ 𝑈, i.e. there exists a 𝛩3 (𝑡) ∈ 𝑈, which can satisfy the following two conditions. (1)𝛩3 (𝑡) = 𝑒𝛩1 (𝑡) + (1 − 𝑒)𝛩2 (𝑡). (2) 𝑒𝑓1 + (1 − 𝑒)𝑓2 = (𝑆1 𝛩3 (𝑡) + 𝐾1 , 𝑆2 𝛩3 (𝑡) + 𝐾2 ) = 𝐹(𝑡, 𝑋, 𝛩3 (𝑡)).

(25) (26)

Therefore, 𝑒𝑓1 + (1 − 𝑒)𝑓2 ∈ 𝑊(𝑡, 𝑋) and the fourth condition holds. Finally, let 𝑔(𝛩) = 𝛩2 , it is obvious that |𝛩|−1 𝑔(𝛩) → +∞ as |𝛩| → +∞. As ℎ0 [𝑥(𝑡) + 𝑦(𝑡)] ≥ 0, so it is obvious that 𝐿(𝑡, 𝑥, 𝑦, 𝛩) ≥ 𝑔(𝛩). Therefore, the last condition holds. In summary, all of the five sufficient conditions for the establishment of Theorem 2 hold, so the existence of the optimal control solution 𝛩(𝑡)∗ is proved. Theorem 2 guarantees the existence of the optimal control system shown in (17) ~ (22). According to Pontryagin minimum theory, we obtain the corresponding necessary conditions. Using the differential equations (17) as the state equations, together with (18) and (21) we establish the Hamilton function as follows. H(𝑥, 𝑦, 𝜆1 , 𝜆2 , 𝛩, 𝑡) = ℎ0 [𝑥(𝑡) + 𝑦(𝑡)] + ℎ1 𝛩(𝑡)2 + 𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)][𝐶1 (𝛩) − 𝐶2 𝑦(𝑡)] + 𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)][𝐷1 (𝛩) − 𝐷2 𝑥(𝑡)]

(27)

Adjoint equations: 𝑑𝜆1 (𝑡) 𝑑𝑡 𝑑𝜆2 (𝑡) 𝑑𝑡

=− =−

𝜕𝐻 𝜕𝑥 𝜕𝐻 𝜕𝑦

(28) (29)

Boundary conditions: 𝜆1 (𝑇) = 𝜆2 (𝑇) = 0 By using the Hamilton function in (27), we have the solution algorithm in Theorem 3. Theorem 3: The optimal control solution 𝛩(𝑡)∗ can be calculated as follows.

(30)

−

1 2ℎ1

{𝜃𝑞𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)]} < −𝑀 𝛩(𝑡) = 𝑀

−𝑀 ≤ −

1 2ℎ1

𝛩(𝑡) = −

{𝜃𝑞𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)]} ≤ 𝑀 1 2ℎ1

(31)

{𝜃𝑞𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)]}

1

{ − 2ℎ1 {𝜃𝑞𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)]} > 𝑀 𝛩(𝑡) = 𝑀 Proof: According to the necessary conditions of Pontryagin minimum theory, the optimal control solution 𝛩(𝑡)∗ must

satisfy the following condition. 𝜕𝐻

∗ | 𝜕𝛩 𝛩=𝛩

= 2ℎ1 𝛩(𝑡)∗ + 𝜃𝑞 𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞 𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)] = 0

(32)

Then we have 𝛩(𝑡)∗ = −

1 2ℎ1

{𝜃𝑞 𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)] + 𝜃𝑞 𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)]}

(33)

However, as stated above, the control function 𝛩(𝑡) is a bounded function and lies within period of [−𝑀, 𝑀], 𝑀 > 0. By comparing (33) with M and -M, it is easy to get (31) in Theorem 3. Therefore, the solution of the optimal control system can be finally turned to the solution of the following differential equation groups as follows. 𝑑𝑥(𝑡) 𝑑𝑡 𝑑𝑦(𝑡) 𝑑𝑡 𝑑𝜆1 (𝑡) 𝑑𝑡 𝑑 𝜆2 (𝑡) 𝑑𝑡

{

= 𝑥(𝑡)[1 − 𝑥(𝑡)][−𝜃𝑞(1 − 𝛩(𝑡)) − 𝑇𝐶 + 𝐵 − (𝑇𝐴 − 𝑇𝐶 )𝑦(𝑡)] = 𝑦(𝑡)[1 − 𝑦(𝑡)][−𝜃𝑞(1 − 𝛩(𝑡)) − 𝑇𝐵 + 𝐵 − (𝑇𝐴 − 𝑇𝐵 )𝑥(𝑡)]

= −ℎ0 − 𝜆1 (𝑡)[1 − 2𝑥(𝑡)][−𝜃𝑞(1 − 𝛩(𝑡)) − 𝑇𝐶 + 𝐵 − (𝑇𝐴 − 𝑇𝐶 )𝑦(𝑡)]

(34)

+𝜆2 (𝑡)𝑦(𝑡)[1 − 𝑦(𝑡)](𝑇𝐴 − 𝑇𝐵 ) = −ℎ0 − 𝜆2 (𝑡)[1 − 2𝑦(𝑡)][−𝜃𝑞(1 − 𝛩(𝑡)) − 𝑇𝐵 + 𝐵 − (𝑇𝐴 − 𝑇𝐵 )𝑥(𝑡)] +𝜆1 (𝑡)𝑥(𝑡)[1 − 𝑥(𝑡)](𝑇𝐴 − 𝑇𝐶 )

Initial conditions: 𝑥(𝑡0 ) = 𝑥0 , 𝑦(𝑡0 ) = 𝑦0

(35)

𝜆1 (𝑇) = 0, 𝜆2 (𝑇) = 0

(36)

Boundary conditions: We firstly calculate 𝑥(𝑡), 𝑦(𝑡), 𝜆1 (𝑡) and 𝜆2 (𝑡) by solving the differential equations shown in (34) ~ (36), then the optimal control solution 𝛩(𝑡)∗ of DPCOC can be obtained according to (31).

4. Numerical test and policy comparison analysis On the basis of theoretical analysis, we further verify the model validity by numerical test and conduct comparison analysis between fixed parking fee policy and DPCOC. The parameter values for the numerical test are listed in table 6. Table 6 Parameter values Parameter

𝑡0

𝑡𝑓

𝑇𝐴

𝑇𝐵

𝑇𝐶

𝜃

q

ℎ0

ℎ1

Value

0

2

10

9

8

1

15

2

2

Firstly, the evolution process of commuting behavior under fixed parking fee policy is calculated. The evolution phase diagrams as well as the analytic properties of the equilibrium points are investigated respectively under three different conditions in Section 2.3. Secondly, the evolution process under DPCOC is calculated and comparison analysis is carried out. Finally, we calculate the total social cost and demonstrate the advantage of DPCOC over fixed parking fee policy. For reading convenience, we refer to ‘parking charge base price’ as ‘base price’ for the rest part of this research.

4.1 Numerical test of fixed parking fee policy Situation 1: Let 𝐵 = 29 to satisfy 𝐵 > 𝑇𝐴 + 𝜃𝑞, the evolution phase diagram for both types of commuters is as follows.

Fig.1 Evolution phase diagram of private cars in situation 1

Fig.2 Evolution phase diagram under different base prices in situation 1 (x0=y0=0.4)

Situation 2: Let B=24 to satisfy 𝑚𝑖𝑛(𝑇𝐵 , 𝑇𝐶 ) + 𝜃𝑞 < 𝐵 < +𝜃𝑞, the evolution phase diagram for both types of commuters are as follows.

Fig.3 Evolution phase diagram of private cars in situation 2

Fig.4 Evolution phase diagram under different base prices in situation 2 (x0=y0=0.4)

Situation 3: Let B=18 to satisfy 𝐵 < 𝑚𝑖𝑛(𝑇𝐵 , 𝑇𝐶 ) + 𝜃𝑞, the evolution phase diagram for both types of commuters are as follows.

Fig.5 Evolution phase diagram of private cars in situation 3

Fig.6 Evolution phase diagram under different base prices in situation 3 (x0=y0=0.4)

Figure 1, 3 and 5 demonstrate the dynamic evolution phase of the commute choice behavior under fixed parking fee policy. Obviously, with the gradual decrease of the bus travel time, the evolutionary stable strategy shows that both types of commuters are inclined to take public transit. The result is consistent with the theoretical deduction in Section 2. The model correctness is thus verified. Figure 2, 4 and 6 demonstrate the influence of base price on the commute choice behavior under three situations respectively. With the increase of base price, the private car commute proportion under all three situations decreases and the evolutionary stable strategy transfers to a new state. Under conditions of fixed parking fee policy, the evolutionary path,

equilibrium properties and evolutionary stable strategy are completely decided by travel cost of trip mode choice. Therefore, under fixed parking fee policy, we mainly use the following methods to effectively conduct bus trip induction. First, reduce bus travel time, however, this is not easy to realize and may need huge investment. Second, alter the evolutionary stable strategy by increasing the parking charge price to reduce private car commute and induce public transit. As we have stated in Section 2, under situations of fixed parking fee policy, when the parking charge price increases it may cause high social cost, thus has limitations in reality. Next, we demonstrate the significant advantage of DPCOC over fixed parking fee policy. 4.2 The numerical tests of DPCOC policy 4.2.1 Disturbance effect of bus travel time B on commute choice evolution For calculation simplicity we define U as [-0.8, 0.8], let x0  y0  0.1 . Let B vary from 27 to 18. All other parameters are the same as table 6. We use fourth-order Runge-Kutta combined with boundary value-Residuals convergence algorithm (through Bvp4C function in Mat-lab) to solve the equations in (30) ~ (31). The disturbance effect is shown as follows.

Fig.7 Disturbance effect by bus travel time on the trajectory of optimal control

Fig.8 Evolution of x(t) before and after the implementation of DPCOC policy under different bus travel times

Fig.9 Evolution of y(t) before and after the implementation of DPCOC policy under different bus travel times

Figure 7 shows the dynamic process of control function 𝛩(𝑡) under different bus travel times. The curve of 𝛩(𝑡) becomes more steep with the increase of bus travel time, showing a gradual increase of the control intensity. This result indicates that DPCOC policy takes effect during the whole trip. Figure 8 and 9 compare the effect of the fixed parking fee

policy and DPCOC policy under the influence of different bus travel time. After the implementation of DPCOC policy, the proportion of private car is significantly lower and converges to the equilibrium state at a faster speed. Obviously, DPCOC policy can alter evolutionary equilibrium points, reduce car trip and successfully induce bus trip under all possible traffic conditions. 4.2.2 Disturbance effect of base price on commute choice evolution Let bus travel time B= 25, let base price q vary from 15 to 22, use the same method in 3.2.1 and the same parameters in 4.2.1.The following results can be reached.

Fig.10 Disturbance effect on trajectory optimal control trajectory by different base prices

Fig.11 Evolution of x(t) before and after the implementation of DPCOC policy under different base prices

Fig.12 Evolution of y(t) before and after the implementation of DPCOC policy under different base prices

Figure 10 shows the dynamic trend of the control function 𝛩(𝑡) under different base prices. The 𝛩(𝑡) shows an obvious growth trend with the passage of time. This means DPCOC policy takes effect during the whole process of commuters’ behavior evolution. In addition, the control intensity increases with the reduction of base price. This means DPCOC policy can adjust the control intensity according to different base prices, showing a strong adaptability. Figure 11 and 12 demonstrate the comparison of fixed parking fee policy and DPCOC policy under different base prices. Obviously, the proportion of private car choice shows a downward trend for both types of commuters, and the evolution speed increases. In all, by comparison analysis shown in figure 8~9 and figure 11~12, we reach the following conclusions. (1) DPCOC policy can significantly induce public transit and reduce the private car trip, thus it can effectively alleviate

traffic congestion. (2) DPCOC policy has significant advantage over fixed parking fee policy in bus trip induction under all traffic situations. (3) The advantage of DPCOC policy over fixed parking fee policy is more significant with the decrease of the base price or the increase of bus travel time. (4) DPCOC policy can improve the reliability and predictability for the entire transportation system by keeping the evolution path at a stable state. 4.2.3 Comparison analysis of total social cost under different conditions The total social cost is calculated according to (21) and (22) and the results are as follows.

Fig.13 Comparison of total social cost before and after the implementation of DPCOC policy

Figure 13 (Left panel) shows the comparison result of the effects of fixed parking fee policy and DPCOC policy under different base prices when bus travel time is 25. Firstly, under all kinds of situations, DPCOC policy can significantly reduce the total social cost and the effect is more significant when the base price is smaller. This proves the advantage of DPCOC policy over fixed parking fee policy. In addition, figure 13 (left panel) also shows that under fixed parking fee policy, the total social cost significantly increases with the decrease of the base price. This result shows a high sensitivity of total social cost to the base price, and to some extent explains why it is difficult for the transportation management department to set an appropriate parking charge price by fixed parking fee policy. Comparatively, after the implementation of DPCOC policy, the variation range of social cost decreases dramatically and shows a smooth trend. In addition, the total social cost is much lower than fixed parking fee policy. These results show that compared to the fixed parking fee policy, DPCOC policy has stronger adaptability and flexibility to different base prices, which makes it much easier for the transportation management department to set the appropriate park charge price in a real-time dynamic manner. Figure 13 (right panel) shows the comparison of the effects of fixed parking fee policy and dynamic optimal control policy under different bus travel times when base price is 15. Under fixed parking fee policy, the social cost increases significantly, especially when it is larger than 13, the social cost demonstrates an obvious upward trend, showing high sensitivity; after the implementation of DPCOC policy, the total social cost is also greatly reduced, and the effect of cost reduction is getting more significant as the bus travel time increases. This result shows that the compared to the fixed parking fee policy, DPCOC policy can not only reduce the total social cost to large extent, but also reduce the sensitivity of social cost to bus travel time, thus improve the stability and flexibility of the entire transportation system. To sum up, DPCOC policy has the following two advantages. (1) It can minimize the total social cost under all traffic situations. (2) It can effectively keep down the sensitivity of total social cost to different transportation parameters and enhance its ability of responding to emergencies. The results by this research can be applied in the appropriate setting of

parking charge price and the effective induction of commuters’ bus trip mode.

5. Conclusions In this research we study the dynamic evolutionary game of commuters’ trip mode choice behavior and discuss the influence of optimal control policy on commuters’ mode choice behavior. We firstly classify the commuter group into two different types, namely urban commuters and suburban commuters. We then establish the model of dynamic trip mode choice behavior under asymmetric conditions using evolutionary game theory. We further analyze the evolution pattern and evolutionary stable strategy of trip modes under different traffic conditions. Due to the limitation of fixed parking fee policy, we introduce the idea of optimal dynamic parking charge in view of real-time decision making and evolutionary game theory. The analytical properties are proved and solution algorithms are given using Pontryagin Maximum principle. Also, the influence on evolutionary path and evolutionary stable strategy is discussed. Finally, the correctness and validity are both proved by numerical examples, comparison analysis is made between fixed parking fee policy and DPCOC policy and following results are obtained. (1) Under fixed parking fee policy, the evolutionary path and evolutionary stable strategy are completely decided by the travel time of each trip mode and the only management method for public transit induction is to increase the parking charge price. (2) The dynamic optimal control policy demonstrates significant advantage over fixed parking fee policy in multiple aspects. Firstly, it effectively turns the evolutionary path and evolutionary stable strategy to a better situation. Secondly, it minimizes the total social cost. Thirdly, it reduces the sensitivity of trip mode choice behavior to traffic congestion and improves the ability to resist interferences and emergencies. Finally, it is able to control the mode trip choice to a stable state and make the behavior more predictable for the transportation management department. The research results provide theoretical references and practical insights for the effective implementation of bus induction, dynamic parking pricing and congestion tolls in view of minimizing the total social cost. In future study, we intend to consider control policies for more complex and real situations such as HOV, bus pricing, dynamic information induction and traffic mixed flow. The study results can further enrich the research content of trip mode choice and real time traffic induction policy.

Acknowledgements This research work has been supported by the ShangHai Natural Science Foundation of China (Grant No. 15ZR1429200). Natural Science Foundation of China (Grant No. 71601118).The Doctoral Fund of University of Shanghai for Science and Technology (Grant No. BSQD201407)

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Highlights 

Consider commuters' mode choice behavior in view of evolutionary game theory.

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Discuss the stability of evolution process under different traffic situations.

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Establish a dynamic parking optimal control model.

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Proves that the dynamic parking optimal control charge policy is an effective method.