Study on the Route-Selection Behavior of Bus Passengers Based on an Evolutionary Game under the Condition of Information Guidance

Abstract

Bus guidance information supplied by advanced traveler information systems provide a reference for route selection by bus passengers. This study analyzes the interaction between bus passengers’ route-choice behavior and the relationship in a game. The game revenue of the passengers waiting on the bus platform is quantified by parameters such as the number of passengers waiting on the platform, the number of passengers on the bus, and the passenger capacity of the bus. Based on the characteristics of passenger-bounded rationality, an evolutionary game model of bus choice behavior under information-induced conditions is established. Through the evolutionary game model, the distribution of the proportion of passengers receiving the guidance strategy is obtained when the evolution is stable. Finally, according to the distribution results, the passenger guidance strategy is discussed to achieve a win-win situation between passengers and bus operators.

1. Introduction

With the development of intelligent transportation, advanced traveler information systems can provide bus passengers with real-time traffic information, allowing them to make reasonable travel plans. Providing this information to bus passengers will affect their travel time and route choice, thereby improving their travel efficiency. However, whether bus passengers will accept this induced information ultimately depends on their route selection behavior. Therefore, the travel route selection behavior of bus passengers under the information-induced condition has attracted the attention of many researchers.

Shi et al. [1] assembled data on ridership and stop amenities, including Real-Time Information System (RTIS), shelters, lighting, litter receptacles, benches, and bike hoops. Additionally, the impact of bus stop facilities on transit ridership through the above data was analyzed. Study results have shown that RTIS have a significant impact on transit ridership. From the perspective of travelers, Fonzone et al. [2] analyzed the characteristics of passengers’ bus choice behavior under RTI based on revealed-preference and stated-preference survey data. Wu et al. [3] refined intelligent bus information into bus route information, estimated arrival time information, in-vehicle congestion information, total travel information, and fare information; the researchers then analyzed the impact of the above information on passenger travel paths. These studies have shown that bus itinerary information and in-vehicle congestion information have a significant impact on passengers’ travel route choices. Trozzi et al. [4] analyzed the impact of bus congestion information on passengers’ bus-route selection and obtained the characteristics of passengers’ bus selection behavior under different congestion levels. From this, they proposed a route selection model that considers bus congestion information. Drabicki et al. [5] improved an existing bus control strategy by providing passengers with real-time bus congestion information (RTCI) to reduce the aggregation effect of bus passengers. Yu et al. [6] discussed the bus-route selection behavior of passengers when they saw two buses arriving at their destination and knew the real-time congestion information of the bus in advance. The authors then established a related model based on the obtained data to improve the bus route schedule.

The above studies show that real-time bus information has a significant impact on passengers’ travel behavior. Moreover, they demonstrated that this information can be used to establish corresponding route-selection or bus-scheduling models based on the behavioral characteristics of passengers to improve the travel efficiency of passengers.

Solutions from the realm of game theory have been proposed for various transportation topics, such as traffic signal control [7], lane-changing behavior in a connected environment [8], and the nature of equilibrium [9] to name a few. In terms of traffic choice behavior, Guan Hongzhi et al. [10] established a traveler’s traffic choice behavior model using evolutionary game theory based on the assumption that travelers have incomplete road network information and bounded rationality of choice behavior. This study theoretically proved the applicability of evolutionary game theory in the analysis of traffic choice behavior. Corman [11] quantified the trade-off between the minimization of train delays and passenger travel times and the performance, stability, and convergence of the equilibrium point based on game theory. The proposed game theoretical approach is able to easily consider information and interdependence of the actions of multiple stakeholders. Through relevant experiments and game theory, Bouman [12] studied the interaction of route choices among passengers and their final decision-making behavior under the condition that travelers obtain public transportation congestion information.

At present, most studies have shown that real-time bus information, such as arrival time and in-vehicle congestion information, has a significant impact on passengers’ bus-route selection. These studies have combined relevant theories and models to analyze passengers’ route selection behavior under information-induced conditions. However, in the research process, these studies mainly focus on the path selection behavior of individuals, and the research content from the same group level is less. In addition, although the application of game theory models in the analysis of transportation choice behavior is becoming increasingly mature, this research mainly focuses on transportation mode selection and vehicle route selection. In contrast, few studies concentrate on passenger bus-route selection.

Therefore, this study attempts to analyze the internal interaction of bus passengers by applying evolutionary game theory to study the time evolution law of bus passengers’ travel route selection behavior under the condition of information induction. In addition, the study aims to clarify the dynamic adjustment results of bus passenger group behavior. From this, the goal is to propose guidance strategies and bus scheduling adjustments to achieve a win-win for passengers and bus operating companies.

2. Basic Problem Description

As participants of an intelligent bus system, passengers receive intelligent bus information services during travel, and use the same limited bus resources, including bus stations, electronic bus stop boards, and buses. Under the system’s information service, each passenger can choose the most suitable bus travel plan—such as the route with the shortest travel time, least number of transfers, lowest travel cost, or least congestion—to minimize their costs or maximize their own interests during travel. However, as part of the bus system, the bus choices of individual passengers will impact one another, and they could even cause passengers to lose interest. A representative of this is the impact of the perception of congestion. Take the bus selection among passengers in Figure 1 as an example. Assuming that the existing buses, k₁ and k₂, can meet the travel needs of passengers at bus station i and that, for passengers, the above two vehicles have no significant impact on their travel except for the crowding factor. Passengers obtain congestion information for each bus through the information provided by the intelligent bus system, and the congestion degree of k₁ is higher than that of k₂. The possible combinations of bus choices that passengers can make based on this information include (k₁, k₁), (k₁, k₂), (k₂, k₁), and (k₂, k₂). If all passengers choose to travel by bus k₁, the congestion degree of bus k₁ is likely to be higher than that of k₂, which is called the “congestion drift” phenomenon in road congestion research. Another extreme situation is that platform passengers all take into account the “congestion drift” situation, so they choose bus k₂ to ensure their own travel comfort, which results in bus k₂ becoming more crowded. In fact, the most probable situation is that most passengers will choose to travel on bus k₁ according to the bus guidance information, and a small number will choose bus k₂ because they believe that k₁ might become more crowded as it is the choice of most people. There is a game relationship between passengers, and passengers consider each other’s bus choices to determine their own bus choices. Therefore, game theory can be used to analyze the bus choice behavior of waiting passengers.

3. Basic Assumptions

Bus passengers aim to maximize their own benefits. However, under the conditions of a lack of awareness of other passengers and dynamic changes in the external environment, it is impossible for travelers to always have perfect information to pursue their greatest interests, but they always hope to benefit. Passengers obtain bus information through mobile phones, electronic bus stop boards, and other sources. Considering the negative impact of low-quality bus information on passengers’ bus choices, an intelligent bus system should always release information with high accuracy, and travelers will not refuse to refer to bus information to select bus options because of distrust. Based on the above analysis, the basic assumptions for modeling are as follows:

(1)

Passenger-bounded rationality.

(2)

All passengers choose to travel by bus k₁ or bus k₂ as their travel strategy.

(3)

Passengers reserve and select bus vehicles through smart terminal devices. In addition, for buses with different shifts on the same line, passengers can only reserve the bus closest to them.

(4)

According to the information on the number of people waiting for reservations in the bus system, passengers consider the benefits of personal travel choices, and finally determine their bus ride plans.

(5)

Taking the congestion degree on k₁ or k₂ as the standard to measure passenger benefit, the lower the congestion degree, the greater the benefit.

(6)

Passenger and vehicle information is updated at a certain frequency.

4. Static Game Analysis

4.1. Game Representation

Taking the above passenger bus selection behavior as the research object of the game model proposed here, the passenger group starts from bus station i, and buses k₁ and k₂ can meet their travel needs. From time t₀, passengers at station i obtain in-vehicle information from buses k₁ and k₂, which are about to arrive at the station, through the intelligent bus system. The system shows that bus k₁ is less crowded, and it induces passengers to choose bus k₁ to travel. The induction effect will last until time t₁; that is, the next time the system center releases the bus information. Passengers need to choose between buses k₁ and k₂, but they do not know the final choice of other passengers, and they need to consider the impact of others’ choices on themselves. Therefore, each passenger plays a game with other passengers. In this game, each passenger is a player. Thus, the game between the passenger groups can be understood as the game between two people who are randomly matched in the group, and the income of one person represents the income of the group that person belongs to. In sum, the three basic elements of the game model in this section are expressed as follows.

(1) Player set F, two randomly matched waiting passengers on the platform, are the players in the game, represented by F₁ and F₂, respectively.

$$F=\left\{F_O\right\},O=1,2$$

(1)

(2) The strategy set S_o of players F_o is to select or give up taking bus k₁.

$$S_O=\left\{\mathrm{Select}\left(S\right),\mathrm{Give} \mathrm{up}\left(G\right)\right\},O=1,2$$

(2)

(3) The expression of the player’s profit, U_o, is shown in Formula (3).

$$U_O=\frac{1}{C_{k_n,o}}=\frac{E_{k_n}}{q_{k_n,i-1}^{\mathrm{on}}+q_{k_n,i}^{\mathrm{in}}},n=1,2,$$

(3)

where $C_{k_n,o}$ is the congestion degree of bus $k_n$, taken by player o, expressed as bus loading rate. $q_{k_n,i-1}^{\mathrm{on}}$ is the number of people on bus $k_n$ before the bus arrives at station i. $q_{k_n,i}^{\mathrm{in}}$ is the number of people getting on bus $k_n$ at station i, and $E_{k_n}$ is the rated passenger capacity of vehicle $k_n$.

The normative expression of this game is shown in Equation (4).

$$G=\left(F_o,S_o,U_o\right),o=1,2$$

(4)

4.2. Game Profit Calculation

Players F₁ and F₂ finally select or give up taking bus k₁ under the influence of the bus information, which generates four strategy combinations, as shown in

Table 1. Game strategy player combinations.

		F1
		S	G
F2	S	P1(S,S)	P3(S,G)
	G	P4(G,S)	P2(G,G)

.

Under different strategy combinations, players F₁ and F₂ have different profits. The profit values of F₁ and F₂ under each strategy combination is calculated according to Formula (4).

(1) P₁(S,S), that is, both F₁ and F₂ choose bus k₁. The expressions of their profits, U₁ and U_2, are shown in Equation (5).

$$U_1=U_2=\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}},$$

(5)

where $q_i^{\mathrm{wait}}$ denotes the passengers waiting for the bus at platform i.

The corresponding profit combination of P₁(S,S) is: ($\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$,$\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$).

(2) P₂(G,G), that is, both F₁ and F₂ choose to travel on bus k₂. The expressions of their profits, U₁ and U_2, are shown in Equation (6).

$$U_1=U_2=\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$$

(6)

The corresponding profit combination of P₂(G,G) is ($\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$,$\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$).

(3) P₃(S,G), that is, F₁ chooses bus k₁ and F₂ chooses bus k₂. Because the congestion perception of an individual on a bus is equal to the crowd perception of the group on the bus, the congestion perception of the players on the bus can be obtained by calculating the congestion degree on the bus. Assume that the proportion of passengers who finally choose bus k₁ among the waiting passengers on platform i is α. Therefore, the number of people who finally take bus k₁ is $\alpha \cdot q_i^{\mathrm{wait}}$, and F₁ belongs to this group. The number of passengers who give up choosing bus k₁ is $(1-\alpha )\cdot q_i^{\mathrm{wait}}$, and F₂ belongs to this group. The expressions of their profits, U₁ and U_2, are shown in Equations (7) and (8).

$$U_1=\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\alpha \cdot q_i^{\mathrm{wait}}}$$

(7)

$$U_2=\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+(1-\alpha )\cdot q_i^{\mathrm{wait}}}$$

(8)

The corresponding profit combination of P₃(S,G) is ($\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\alpha \cdot q_i^{\mathrm{wait}}}$,$\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+(1-\alpha )\cdot q_i^{\mathrm{wait}}}$).

(4) P₄(G,S), that is, F₁ chooses to travel on bus k₂ whereas F₂ chooses bus k₁. Similar to the profit calculation method for P₃(S,G), it is assumed that the proportion of passengers who finally give up bus k₁ is β. Finally, the expressions of profits U₁ and U₂ for players F₁ and F₂ are obtained by Equations (9) and (10).

$$U_1=\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\beta \cdot q_i^{\mathrm{wait}}}$$

(9)

$$U_2=\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+(1-\beta )\cdot q_i^{\mathrm{wait}}}$$

(10)

The corresponding profit combination of P₄(G,S) is ($\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\beta \cdot q_i^{\mathrm{wait}}}$,$\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+(1-\beta )\cdot q_i^{\mathrm{wait}}}$).

The profit combinations of P₃(A,R) and P₄(R,A) vary with α or β. Considering the convenience of subsequent analysis, the parameters α and β are treated with the idea of average earnings, so the processing results will not lose the research significance. In fact, the calculated profits of P₁(S,S) and P₂(G,G) are also average profits. Treating α and β with the idea of average earnings can be directly attributed to calculating the mathematical expectations of α and β, denoted by E(α) and E(β), respectively. The calculation process of E(α) is as follows.

α obeys a uniform distribution in the interval (0, 1) in the calculation of game profits. Therefore, the probability density function of α is shown in Formula (11).

$$f(\alpha )=\left\{\begin{array}{l}1, \alpha \in (0,1)\\ 0, \mathrm{otherwise}\end{array}\right.$$

(11)

Then, $E(\alpha )={\displaystyle {\int }_{-\infty }^{+\infty }\alpha f(}\alpha )d\alpha ={\displaystyle {\int }_0^1\alpha }d\alpha =\frac{1}{2}$. Similarly, $E(\beta )=\frac{1}{2}$.

Therefore, the final profit combination of P₃(S,G) obtained after processing is ($\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$,$\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$), and the final profit combination of P₄(G,S) is ($\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$,$\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$).

Finally, the game profit matrix of two randomly paired passengers, F₁ and F₂, is shown in

Table 2. Profit matrix of game.

		F2
		S	G
F1	S	${\mathrm{P}}_1(\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$$,\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$)	${\mathrm{P}}_3(\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$$,\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$)
	G	${\mathrm{P}}_4(\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$$,\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}$)	${\mathrm{P}}_2(\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$$,\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}$)

.

5. Research on Evolutionary Game

Passengers will make corresponding bus riding strategies according to bus information and information on other passengers every time they travel, and they will constantly adjust their choices according to the benefits obtained, that is, congestion perception, in order to maximize travel benefits. This process is a dynamic, continuous trial-and-error elimination, learning, and adjustment process. Successful strategies will be imitated, and some specific patterns or rules will be generated as passenger behavior standards. From individuals to groups, the strategy distribution of the final passenger group will gradually evolve in a certain direction [13]. Evolutionary game theory is a theory that combines the above game process and the dynamic evolution process.

For game G = (F_o,S_o,U_o), U₂(s₁,s₂) = U₁(s₂,s₁); that is, profit U₁ of F₁ when player F₁ adopts strategy S₂ and F₂ adopts strategy S₁ is equal to profit U₂ of F₂ when F₁ adopts strategy S₁ and F₂ adopts strategy S₁. At the same time, the game model also satisfies U₂(s₂,s₁) = U₁(s₁,s₂), U₂(s₁,s₁) = U₁(s₁,s₁), and U₂(s₂,s₂) = U₁(s₂,s₂), so the game between the two passengers, F₁ and F_2, is a two-person symmetric game [14]. A two-player symmetric game is the basic object of evolutionary game analysis, and it is suitable for studying the evolutionary game problems of intra-group games without essential differences. The analytical framework of this evolutionary game is a random-paired repeated game among members of a single large group. According to the above proposition, it is certain that the game between two randomly paired passengers, F₁ and F₂, is suitable for studying the evolutionary game problem within the traveler group.

5.1. Duplicative Dynamic

Suppose U₂(s₁,s₂) = U₁(s₂,s₁) = g, U₂(s₂,s₁) = U₁(s₁,s₂) = h, U₂(s₁,s₁) = U₁(s₁,s₁) = e, and U₂(s₂,s₂) = U₁(s₂,s₂) = f. The proportion of group members adopting the S₁ strategy is x, and the proportion adopting the S₂ strategy is 1 − x. $E\left(U_{s_1}\right)$, $E\left(U_{s_2}\right)$, and $E\left(U\right)$ are used to represent the expected profit of the S₁ strategy, the expected profit of the S₂ strategy, and the average expected profit, respectively, as shown in Equations (12)–(14).

$$E\left(U_{s_1}\right)\hspace{0.33em}=x\cdot e+\left(1-x\right)\cdot h$$

(12)

$$E\left(U_{s_2}\right)=x\cdot g+\left(1-x\right)\cdot f$$

(13)

$$E\left(U\right)=x\cdot E\left(U_{s_1}\right)+\hspace{0.33em}\left(1-x\right)\cdot E\left(U_{s_2}\right)$$

(14)

The evolutionary game will eventually reach a stable state, and the process of reaching a stable state is reflected by the duplicative dynamic equation. The duplicative dynamic is used to describe the dynamic strategy adjustment process of the players—the core of which is that the individuals who adopt the more successful strategies in the group will gradually increase. Duplicative dynamic equations are often represented by dynamic differential equations. Therefore, the duplicative dynamic equation of the above evolutionary game model can be expressed as Equation (15).

$$\begin{array}{ll}\frac{\mathrm{d}x}{\mathrm{d}t}& =x\cdot \left[E\left(U_{s_1}\right)-E\left(U\right)\right]=x\cdot \left[E\left(U_{s_1}\right)-x\cdot E\left(U_{s_1}\right)-\left(1-x\right)E\left(U_{s_2}\right)\right]\\ & =x\cdot (1-x)\cdot \left[x\cdot (e-g)+(1-x)\cdot (h-f)\right]\end{array}$$

(15)

Bringing the expressions of g, h, e, and f into Equation (15), the final duplicative dynamic equation of the evolutionary game is obtained as shown in Equation (16).

$$\frac{\mathrm{d}x}{\mathrm{d}t}=x^2\cdot (1-x)\cdot (\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}-\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}})+x\cdot {(1-x)}^2\cdot (\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+\frac{1}{2}\cdot q_i^{\mathrm{wait}}}-\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}})$$

(16)

The equation reflects the time evolution process of the proportion of passenger group members adopting the induced strategy, revealing the dynamic adjustment process of group behavior.

Let $\frac{\mathrm{d}x}{\mathrm{d}t}=0$, and the three stable states $x_1^{*}$, $x_2^{*}$, and $x_3^{*}$ of the duplicative dynamic are solved by Equations (17)–(19)

$$x_1^{*}=0$$

(17)

$$x_2^{*}=1$$

(18)

$$x_3^{*}=\frac{\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}-\frac{2E_{k_1}}{2q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}}{\frac{E_{k_1}}{q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}-\frac{2E_{k_2}}{2q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}+\frac{E_{k_2}}{q_{k_2,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}-\frac{2E_{k_1}}{2q_{k_1,i-1}^{\mathrm{on}}+q_i^{\mathrm{wait}}}}$$

(19)

The steady state of the duplicative dynamic suggests that the proportion of the passenger population receiving the induction strategy may stabilize at $x_1^{*}$, $x_2^{*}$, or $x_3^{*}$.

5.2. Evolutionary Stability Strategies (ESS)

When the evolution is stable, the proportion of passenger group members adopting the induction strategy is $x^{*}$. Because of the stability of ESS with small fluctuations, there is a neighborhood in which, when $x<x^{*}$,$\frac{\mathrm{d}x}{\mathrm{d}t}>0$, x can be further increased to $x^{*}$ and, when $x>x^{*}$, $\frac{\mathrm{d}x}{\mathrm{d}t}<0$, x can be further reduced to $x^{*}$ [15]. Therefore, when $x=x^{*}$ the slope of $\frac{\mathrm{d}x}{\mathrm{d}t}$ at $x^{*}$ is negative; that is, when the proportion of passengers adopting the induction strategy is in a stable state, the conditions of $\frac{\mathrm{d}x}{\mathrm{d}t}=0$ and ${\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)}^{\prime }<0$ must be satisfied. The approximate phase diagram of the duplicative dynamic game equation is shown in Figure 2.

The three intersection points of the phase curve and the horizontal axis in the phase diagram are the steady state points $x_1^{*}$, $x_2^{*}$, and $x_3^{*}$. The determination of the specific location needs to be discussed according to the following three situations: $x_3^{*}<0$, $0<x_3^{*}<1$, and $x_3^{*}>1$. The following results are obtained.

(1)

When $\frac{q_{k_2,i-1}^{\mathrm{on}}}{E_{k_2}}-\frac{q_{k_1,i-1}^{\mathrm{on}}}{E_{k_1}}<\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}$, $x_3^{*}<0$ and ${\left(\frac{\mathrm{d}{x}_1^{*}}{\mathrm{d}t}\right)}^{\prime }<0$, so $x^{*}=x_1^{*}=0$.

(2)

When $\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}<\frac{q_{k_2,i-1}^{\mathrm{on}}}{E_{k_2}}-\frac{q_{k_1,i-1}^{\mathrm{on}}}{E_{k_1}}<\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, $0<x_3^{*}<1$ and ${\left(\frac{\mathrm{d}{x}_3^{*}}{\mathrm{d}t}\right)}^{\prime }<0$, so $x^{*}=x_3^{*}$.

(3)

When $\frac{q_{k_2,i-1}^{\mathrm{on}}}{E_{k_2}}-\frac{q_{k_1,i-1}^{\mathrm{on}}}{E_{k_1}}>\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, $x_3^{*}>1$, and ${\left(\frac{\mathrm{d}{x}_2^{*}}{\mathrm{d}t}\right)}^{\prime }<0$, so $x^{*}=x_2^{*}=1$.

In the above results, $\raisebox{1ex}{$q_{k_1,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_1}$}\right.$ and $\raisebox{1ex}{$q_{k_2,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_2}$}\right.$ represent the loading rate of k₁ and k₂, respectively, at time t₀. $\raisebox{1ex}{$q_{k_2,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_2}$}\right.-\raisebox{1ex}{$q_{k_1,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_1}$}\right.$ represents the difference between the loading rate of k₂ and k₁ at time t₀. Its meaning can be understood as the difference in the degree of congestion in the two vehicles. The greater the difference, the greater the difference in the degree of congestion, which is represented by $\Delta C$. The bus system center induces passengers to travel by bus k₁, so the condition that $\Delta C$ is greater than 0 should be satisfied. That is, the congestion degree of bus k₁ should meet the condition that it is not greater than the congestion degree of k₂.

Through result (1), it can be determined that, when $\Delta C<\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}$, the proportion of passenger group members receiving induction in 0 is ESS. In other words, during the period from t₀ to t₁, with the passage of time, the members of the passenger group will gradually give up choosing bus k₁ as their travel plan.

Through result (2), it can be found that, when $\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}<\Delta C<\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, the proportion of passenger group members receiving induction in $x_3^{*}$ is ESS. In other words, from t₀ to t₁, with the passage of time, the proportion of passengers choosing bus k₁ will gradually maintain $x_3^{*}$. This proportion is related to the rated passenger capacity of buses k₁ and k₂, the number of passengers waiting at the platform from t₀ to t₁, and the number of passengers on buses k₁ and k₂ at t₀.

Through result (3), it can be concluded that, when $\Delta C>\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, the proportion of passenger group members receiving induction in 1 is ESS. In other words, during the period from t₀ to t₁, with the passage of time, the members of the passenger group will gradually come to choose bus k₁.

The above evolutionary game process can be illustrated by a simple applied numerical example, the basic assumptions are shown in Section 3, and the following parameters are entered: $E_{k_1}=\hspace{0.33em}{E}_{k_2}=\hspace{0.33em}75$; $q_i^{\mathrm{wait}}=\hspace{0.33em}24$; $q_{k_1,i-1}^{\mathrm{on}}=\hspace{0.33em}30$; $q_{k_1,i-1}^{\mathrm{on}}=\hspace{0.33em}\hspace{0.33em}40$, so $\Delta C\hspace{0.33em}=\hspace{0.33em}2/15$. According to the calculation method of game profit in Section 4, the game profit matrix of two randomly paired passengers, F₁ and F₂, is shown in

Table 3. Value of game profit matrix.

		F2
		S	G
F1	S	${\mathrm{P}}_1(\frac{40}{27}$$,\frac{40}{27}$)	${\mathrm{P}}_3(\frac{40}{21}$$,\frac{20}{13}$)
	G	${\mathrm{P}}_4(\frac{20}{13}$$,\frac{40}{21}$)	${\mathrm{P}}_2(\frac{5}{4}$$,\frac{5}{4}$)

.

According to the profit matrix, $x_3^{*}\hspace{0.33em}=\hspace{0.33em}\hspace{0.33em}90.3\%$ is obtained by duplicative dynamic equation, that is the distribution of the proportion of passengers receiving the guidance strategy. Therefore, it is expected that 3 of the 25 waiting passengers will finally choose the bus k₂, and the rest will be induced to choose k₁.

According to the above analysis results, when the evolution is stable, the distribution of the proportion $x^{*}$ of passengers receiving the guidance strategy is shown in Figure 3. Based on the results regarding the dynamic adjustment of passenger-group behavior, recommendations for implementing induction strategies can be made.

5.3. Induction Strategy Suggestions

The induction strategy proposed in this paper mainly aimed at travel information induction for the two alternative bus lines, and the basic idea of the trip induction strategy under multiple alternative bus lines is the same. The induction strategy is suggested as follows.

(1) The intelligent public transportation system center does not always need to release induction information to the passengers waiting on the platform, and it can determine whether to induce according to the distribution of the evolutionary stability strategy.

Introduce $\Delta C=\raisebox{1ex}{$q_{k_2,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_2}$}\right.-\raisebox{1ex}{$q_{k_1,i-1}^{\mathrm{on}}$}\!\left/ \!\raisebox{-1ex}{$E_{k_1}$}\right.$ and compare its numerical relationship with $\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}$ and $\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$. When $\Delta C<\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}$, if the inducement information is released, the proportion of passengers who accept the inducement will converge to 0, which will lead to a decrease in the function of the information inducement system and make the information inducement meaningless. However, the profits of passengers who accept the induction decrease will directly affect the passengers’ trust in the information induction system.

(2) The time of re-publishing the induced information was adjusted according to the convergence rate of the proportion of the passenger population who accept the induced strategy when the evolution is stable.

The above analysis results show that when $\frac{q_i^{\mathrm{wait}}}{2E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{E_{k_2}}<\Delta C<\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, the proportion of passengers who accept the induction strategy will converge to $x_3^{*}$, and when $\Delta C>\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, the proportion of passengers who accept the induction strategy will converge to 1. The convergence rate of $x^{*}$ is related to the degree of difference between $\Delta C$ and $\frac{q_i^{\mathrm{wait}}}{E_{k_1}}-\frac{q_i^{\mathrm{wait}}}{2E_{k_2}}$, and the greater the degree of difference, the greater the convergence rate, as shown in Figure 4.

To ensure the effectiveness of induction, the difference degree should be greater than a certain proportion. In this case, the benefit of induction is good. However, if the proportion is too large, the convergence rate of $x^{*}$ will be too fast. This would cause many passengers to choose to travel by bus k₁ within a short time, which may lead to the phenomenon of “congestion drift.” Therefore, when the convergence rate of $x^{*}$ is too fast, guidance information, such as the number of people on the bus and the reservation situation, can be updated and released to passengers in advance to ensure the guidance information does not lose meaning.

6. Conclusions

This study quantifies the game profits of passengers waiting at the same bus station. It also clarifies the relationship between the game profits of passengers and the number of passengers waiting for the bus at station i from t₀ (when the bus guidance information is released) to t₁ (when the next guidance information is released), the number of passengers on the bus before the bus arrives at station i, and the passenger capacity of the vehicle. The evolutionary game model of passengers’ bus-route selection behavior under the condition of information induction is established, and the evolutionary stability strategy is analyzed. This model overcomes the shortcomings of traditional learning models such as reinforcement learning and belief learning. During the game process, participants understand each other’s behavior characteristics and decision-making habits through the results of the game, so that in the future game, they can not only master the opponent’s future selection strategies according to the game history, but also adjust their actions according to the obtained information to maximize their own benefits. At the same time, the evolutionary game analysis pays attention to the benefits of all parties and the determination of the final stable state point, which has certain advantages in terms of stability and computational cost. Therefore, the evolutionary game model of passengers’ bus-route selection behavior proposed in this paper clearly reveals the regular pattern of time evolution of passengers’ bus choice behavior under the condition of information induction, and the result of dynamic adjustment to the passengers’ group behavior, and based on the above research results, suggestions for implementing induction strategies are put forward.

This study only considers a situation in which passengers are faced with two alternatives and only the congestion degree of bus is taken as the player’s profit. When there are multiple alternatives in the bus selection problem, and considering the different prices of different transport alternatives, the two-party evolutionary game model in this paper will become a multi-party evolutionary game model, and its basic analysis steps and principles are consistent with the two-party evolutionary game model. However, when conducting multi-party evolutionary game analysis, it is necessary to further define the analysis scenarios, such as the number and category of waiting passengers. At the same time, more stable state points will appear in the model results, so the induction strategy will also be determined according to more scenarios. In addition, considering the continuous iteration of information, performing multiple repeated games should also be considered in subsequent research.