4.1. Stable State Analysis
Smith introduced the concept of the evolutionarily stable strategy (ESS) in 1982 [
39], which is a strategy adopted by a population in a given environment [
40]. Each ESS corresponds to a Nash equilibrium solution, but not all Nash equilibrium solutions belong to the ESS. The probability of a participant choosing a strategy will reach a stable level when the participant cannot obtain higher benefits by changing the strategy, and this strategy is called ESS.
According to evolutionary game theory, the equilibrium point must satisfy the replication dynamic equations , , and , , . The equilibrium points containing strategies of 0, 1 and not 0, 1 are included in the integrated game system of BSC, MC, and CT. The solution process of the equilibrium state is analyzed as follows.
By Equation (6), it is known that x = 0 and x = 1 are the two roots of . There are other roots of that are not 0 and 1.
Case 1: When y = 0, there exists Z1 (see Equation (11)), which makes . If z = Z1 > 0, any decision is in a steady state for the BSC system. If z ≠ Z1, when , there will be and , implying that BSC chooses to cooperate and MC chooses not to cooperate as ESS. When , there will be , , implying that both BSC and MC choose not to cooperate as ESS.
Case 2: When y = 1, there exists Z2 (see Equation (12)), which makes . If , any decision is in a steady state for the BSC system. If , when , there will be and , implying that BSC and MC choose to cooperate as ESS. When , there will be and , implying that BSC chooses not to cooperate while MC chooses to cooperate as ESS.
Case 3: When z = 0, there exists Y1 (see Equation (13)), which makes . If , for the BSC system, any decision is in a steady state. If , when , there will be , , implying that BSC chooses to cooperate while CT chooses not to cooperate as ESS. When 0 < y < Y1, there will be , , implying that neither BSC nor CT choosing not to cooperate as ESS.
Case 4: When
z = 1, there exists Y
2 (see Equation (14)), which makes
. If
, for the BSC system, any decision is in a steady state. If
, when
, there will be
,
, implying that it is ESS when both BSC and CT choose to cooperate. When 0 <
y <
Y2, there will be
,
, implying that the state is ESS when BSC chooses to cooperate while CT chooses not to cooperate.
By Equation (8), it is known that y = 0 and y = 1 are the two roots of . There are other roots of that are not 0 or 1.
Case 5: When = 0, there exists (such as Equation (15)), which makes . If , for the MC system, any decision is ESS. If , when , there will be , , implying that BSC chooses to cooperate while MC chooses not to cooperate is ESS. When , there will be , , implying that the state is ESS when neither BSC nor MC chooses not to cooperate.
Case 6: When x = 1, there exists (such as Equation (16)), which makes . If , any decision is in a steady state for the MC system. If , when , there will be , , implying that it is ESS when BSC and MC choose to cooperate. When , there will be , , implying that the state is ESS when BSC chooses to cooperate while MC does not.
Case 7: When , there exists (such as Equation (17)), which makes . If , any decision is in a steady state for the MC system. If , when , there will be , , implying that it is ESS when MC chooses to cooperate while CT does not. When , there will be , , implying that the state is ESS when neither MC nor CT choose not to cooperate.
Case 8: When
, there exists
(such as Equation (18)), which makers
. If
, any decision is in a steady state for the MC system. If
, when
, there will be
,
, implying that it is ESS when both MC and CT choose to cooperate. When
, there will be
,
, implying that the state is ESS when CT chooses to cooperate while MC does not.
According to Equation (9), we know that and are the two roots of . There are other roots of that are not 0 or 1.
Case 9: When , there exists (as Equation (19)), which makers . If , any decision is in a steady state for the MC system. If , when , there will be , , implying that it is in a steady state when the BSC chooses to cooperate while CT chooses not to cooperate; when , there will be , , implying that the state is stable when neither BSC nor CT choose not to cooperate.
Case 10: When x = 1, there exists Y4 (as Equation (20)), which makers . If y = Y4 > 0, any decision is in a steady state for the MC system. If y ≠ Y4, when 1 > y > Y4, there will be , , implying that it is ESS when the BSC and CT choose to cooperate. When , there will be , , implying that the state is ESS when BSC chooses to cooperate while CT does not.
Case 11: When y = 0, there exists X3 (as Equation (21)), which makers . If , any decision is in a steady state for the MC system. If , when , there will be , , implying that it is ESS when CT chooses to cooperate while MC does not; when , there will be , , implying that the state is ESS when neither MC nor CT chooses not to cooperate.
Case 12: When
y = 1, there exists X
4 (as Equation (22)), which makers
. If
, any decision is in a steady state for the MC system. If
, when
, there will be
,
, implying that it is ESS when both MC and CT choose to cooperate. When
, there will be
,
, implying that the state is ESS when MC chooses to cooperate while CT does not.
In addition, eight pure strategy equilibrium points
,
,
,
,
,
,
,
can be obtained. Since the equilibrium point is not necessarily the optimal ESS, we use the Jacobian matrix to judge the stability of the above points. According to the theory proposed by Taylor and Jonker, the equilibrium point is considered as the ESS of the game if all the eigenvalues of the Jacobian matrix are all negative real parts [
41]. The Jacobian matrix of the evolutionary game system in this paper is as follows.
The eigenvalues and stability conditions of the Jacobian matrix at each equilibrium point are shown in
Table 4. Each equilibrium point can evolve into a stable strategy, and results can be obtained by adding constraints.
- (1)
When the three participants reach the equilibrium point , it means that the benefit of BSC and MC choosing integrated cooperation is greater than the total cost, and the cost of CT choosing the integrated mode is less than the cost of not choosing integrated. Since , , , equilibrium points and , , cannot reach a steady state at the same time.
- (2)
When the three participants reach the equilibrium point , it means that the benefit of BSC and MC choosing integrated cooperation is less than the total cost, and the cost of CT choosing the integrated mode is greater than the cost of not choosing integrated. Since , , , equilibrium points , , and cannot reach a steady state at the same time.
- (3)
When the three participants reach the equilibrium point , it means that the benefit of BSC and MC choosing integrated cooperation is less than the total cost, and the travel cost of CT choosing the integrated mode is less than the cost of not choosing integrated. Since , , equilibrium points , and cannot reach a steady state at the same time.
- (4)
When the three participants reach the equilibrium point , it means that the benefit of BSC choosing integrated cooperation is less than the total cost. The benefit of MC choosing integrated cooperation is greater than the total cost, and the travel cost of CT choosing the integrated mode is greater than the cost of not choosing integrated. Since , , equilibrium points , and cannot reach a steady state at the same time.
- (5)
When the three participants reach the equilibrium point , it means that the benefit of BSC choosing integrated cooperation is greater than the total cost. The benefit of MC choosing integrated cooperation is less than the total cost, and the travel cost of CT choosing the integrated mode is greater than the cost of not choosing integrated travel. Since , , equilibrium points , and cannot reach a steady state at the same time.
- (6)
When the three participants reach the equilibrium point , it means that the benefit of BSC choosing integrated cooperation is greater than the total cost. The benefit of MC choosing integrated cooperation is greater than the total cost, and the travel cost of CT choosing the integrated mode is greater than the cost of not choosing the integrated mode.
- (7)
When the three participants reach the equilibrium point , it means that the benefit of BSC choosing integrated cooperation is greater than the total cost. The benefit of MC choosing integrated cooperation is less than the total cost, and the travel cost of CT choosing the integrated mode is less than the cost of not choosing the integrated mode.
- (8)
When the three participants reach the equilibrium point , it means that the benefit of BSC choosing integrated cooperation is less than the total cost. The benefit of MC choosing integrated cooperation is greater than the total cost, and the travel cost of CT choosing the integrated mode is less than that of not choosing the integrated mode.
4.3. Active Regulation Effect Based on Policy Factors
To further validate and test the proposed model, this section uses the empirical data to reveal the evolutionary mechanism of the integration system based on the system dynamics simulation method.
The bicycle–metro integration model contains eighteen parameters
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
. The initial value settings of the parameters are shown in
Table 6.
is the average value of shared bicycles connecting metro stations in Beijing. We assume that the maximum connecting surge volume
is 500 passengers because of the integration mode. If both BSC and MC are willing to choose integrated cooperation, the surge of connecting passenger flow is
. If BSC chooses integrated cooperation while MC does not, the surge connecting passenger flow is
. If BSC does not choose integrated cooperation while MC chooses integrated cooperation, the surge of connecting passenger flow is
. If BSC and MC do not choose integrated cooperation while CT chooses integrated cooperation, the surge of connecting passenger flow is
.
Since the emergence of sharing bicycles, its development has experienced several booms and fallbacks. From the enormous incentives at the beginning until today, users of sharing bicycles have gradually stabilized. Recently, the sharing bicycle system has set up a punishment mode for irregular parking. Still, this feature has not been commonly enabled due to some disputes caused by inaccurate positioning. Therefore, this section has great significance on the operation of sharing bicycle systems by exploring the reward and punishment penalty policy of parking rules. We set the initial value of irregular parking punishment as 0.5 RMB/person and the initial value of regular parking reward as 0.2 RMB/person.
Due to financial pressures, the subsidies of the funding units will gradually decrease as integration is achieved. The system’s evolution is presented below, in which the subsidy of the three participants decreases with the increase in travelers’ willingness to choose integration cooperation.
Case 1: Let
update the replication dynamics equation. The BSC system received a decreasing subsidy as the probability of CT choosing the integration increased. The initial integration intentions of the three participants were set as
= 0.3,
= 0.4,
= 0.5. As shown in
Figure 2a–c, with the phasing out of
, the dynamic evolutionary trajectory of the three participants shows spiral convergence and the existence of a Nash equilibrium. According to
Figure 2d, the dynamic evolutionary trajectory’s fluctuation range gradually decreased and then tended to be stable, but not in the optimal state. The travel willingness under this equilibrium state was around 0.61.
Case 2: Let
. According to
Figure 3, the dynamic evolutionary trajectories of the three participants eventually reached equilibrium. The travel willingness of MC under this equilibrium state was 0, and the willingness probability of CT and BSC reached 1.
Case 3: Let
. As shown in
Figure 4, as the willingness of CT to choose integrated cooperation increased, the willingness of BSC and MC to choose integrated cooperation would be increased and would eventually reach equilibrium. The willingness of each player to choose integrated cooperation was 1, which was the same as the expectations.
Therefore, under the mechanism of phasing out the
subsidy policy gradually, we focused on how to increase the probability of the ideal event occurring by optimizing the parameter design. According to
Figure 5, as the subsidy
for BSC decreased, CT did not eventually reach optimal equilibrium. According to
Figure 6, as the subsidy
for MC decreased, the willingness of BSC to choose integration changed slightly, but when
fell to 450, its willingness to choose integration gradually evolved to 0. According to
Figure 7, as the subsidy
for CT decreased, the integration willingness of MC changed slightly, but when
decreased to 200, the cooperation willingness of CT tended to decrease, increase in the initial stage, and finally reach equilibrium. This indicates that there were critical values of
,
, and
to promote the system equilibrium.
In addition, in terms of the unit price of sharing bicycle and metro trips, we found that the change in the unit price of sharing bicycles had a greater impact on the system equilibrium than the change in metro, by comparing
Figure 8a,b. This indicates that the unit price of the sharing bicycle was an important factor affecting the CT’s transportation connection. If the degree of subsidy for CT could dispel their concerns due to the price issues, then the evolutionary game of the three participants would also reach equilibrium. As the irregular parking ratio g
1 increased, the CT system reached a steady state that was willing to cooperate more and more slowly through
Figure 9a. The continued increase in g
1 would eventually lead to the cooperation willingness of CT decreasing to 0. From
Figure 9a,b, the change in the irregular parking ratio had little impact on the evolutionary pattern of metro evolution, and the reward and punishment mechanism of irregular parking had more impact on the CT.