3.1. Basic Assumptions
Scholars and local governments in the Chinese transportation sector have put forward many suggestions for the development of CVIS, including subsidies for RIVs manufacturers’ research and development (R&D), construction of supporting facilities and infrastructure, cooperation with vehicle enterprises to purchase the computing power and storage unit of ADVs, and prioritizing RIVs license plate number.
There are two possible forms of autonomous driving: connected vehicles that communicate with and rely on sensors and devices in the road infrastructure, and autonomous vehicles that are fully independent but cost more due to the extra sensor technology installed on the vehicles. To allow the connected vehicles to work, government must install the necessary road infrastructure. Thus, the manufacturers that satisfying the production qualification of RIVs and ADVs will face two alternative production strategies: RIVs or ADVs. In this paper we suppose that there is no essential difference between RIVs and ADVs in terms of vehicle performance except for the intelligence level. ADVs can realize intelligent functions such as autonomous driving through the equipment on vehicles, while RIVs can realize the same intelligent functions as ADVs only with the support of roadside infrastructure. The unit production costs of RIVs and ADVs are
and
, respectively. Due to the high cost of onboard equipment and high R&D costs, current ADVs are significantly more expensive than RIVs, thus in this paper we assume
. The sales of RIVs and ADVs are
and
, respectively. From the perspective of consumers, they are more willing to enjoy autonomous driving at a lower price, so in this paper we assume
. The intelligence levels of RIVs and ADVs are
and
, respectively. Compared with RIVs, ADVs have a great advantage in intelligence level when there is no CVIS, then we assume
.
Table 2 summarizes the meaning of the parameters and abbreviations used in this paper.
3.2. Profit Function of RIVs and ADVs
The Hotelling model is often used in economics to study the impact of market structure on the efficiency of resource allocation. When different goods are similar substitutes, since each manufacturer wants to maximize its own interests, it is necessary to consider the behavior of other competitors. In this case, the manufacturer’s pricing model is called the Hotelling model [
27]. Under normal circumstances, the Hotelling model needs to meet the following three basic assumptions:
- (1)
The products have the same material properties or can be similar substitutes for each other;
- (2)
The sum of market percent rate of products is 1;
- (3)
Consumers are evenly distributed in the [0, 1] range and have unit demand.
Based on the description of RIVs and ADVs in
Section 3.1, the Hotelling model can be used for the profit analysis of RIVs and ADVs.
Figure 1 depicts the hoteling line of RIVs and ADVs. The RIVs and ADVs is located at 0 and 1, respectively. Consumers are evenly distributed from 0 to 1, RIVS and ADVs compete for marginal consumers. Each consumer is only willing to buy one type of vehicle. If consumers choose RIVs, then RIVs will bring them the utility value of
. Each consumer is faced with a travel cost (T) that will reduce the utility. And when consumers are located at
, they will spend TX to meet their needs. In this case, net utility specification of RIVs consumers (
) and ADVs consumers (
could be deduced:
Suppose there is a marginal consumer located at
who does not care about buying any type of vehicle. In other words, for this kind of consumer, RIVs and ADVs bring the same utilities, then we have:
By solving Equation (3), we can get
. Therefore, the market share of RIVs and ADVs could be clearly determined. Vehicle buyers distributed in
are more inclined to buy RIVs, while customers in
will buy ADVs. Then, the profit functions of the manufacturers of RIVs and ADVs can be obtained as follows:
3.3. Establishment of Evolutionary Game Model
In this paper, we construct an evolutionary game model to discuss the tradeoffs between public and private investment in autonomous driving and investigate how investment decisions of government and manufacturer can ultimately shape the outcome. Thus, the investment policy strategy space of governments and manufacturers can be defined as and , respectively. and indicate that the governments built or not built the CVIS, and and denote the production of RIVs or ADVs.
When governments started to build CVIS and vigorously construct the roadside infrastructure, government needs to bear the cost of the construction, development, and maintenance of roadside infrastructure while obtaining the benefit from CVIS. In this case, when manufacturers choose to produce RIVs, they can receive a certain subsidy from the government. On the contrary, if they choose to produce ADVs, they can also get income by selling the computing power and storage units to the governments.
In this paper we use (0) and to represent the ratio of RIVs manufacturers and ADV manufacturers, respectively. Where, x = 0 indicates that no manufacturers are willing to produce RIVs, while x = 1 signifies that all manufacturers are inclined to produce RIVs. And we use (0) and to denote the proportion of governments that choose to build (B) and not build (NB) CVIS, respectively. Similarly, y = 0 denotes that no government is willing to invest in the construction of CVIS, and y = 1 indicates full governmental investment in CVIS. The values of x and y are flexible and vary according to real-world circumstances. Next, we will separately establish the income matrix and replicator dynamic equation for the local governments and manufacturers, and then analyze their mutual strategies.
3.3.1. Investment Strategy Analysis of Manufacturers
The decision space matrix
contains the combination of different strategies of manufacturers and local governments. Thus the decision space matrix of manufacturers can be defined as
according to the investment policy decision space
and
:
For each strategy combination, the income of manufacturers will be different, as shown in the income matrix
below:
In the case that manufacturers decide to adopt
production strategy, their expected utility can be written as:
Based on Equation (7), the average expected utility of manufacturers can be written as:
According to the expression of replicator dynamic equation, the growth rate of
RIV production by automakers is equal to the difference between
and
. Thus, we can get the replicator dynamic equation of manufactures:
3.3.2. Investment Strategy Analysis of Governments
The decision space matrix of governments
can be expressed as follow:
Same as the income matrix
of the manufacturers, the income matrix of government has the specific expression as below:
And when local governments adopt strategy
their expected utility can be defined as:
The average expected utility of local governments can be denoted as the following:
Similarly, the replicator dynamic equation of local governments can be shown as follows:
3.4. System Stability Analysis
According to the description of Equations (10) and (15), we can calculate the equilibrium points of the bilateral evolutionary game, as shown in Equation (16).
According to Equation (16), it is easy to find that there are four balance points in the system (i.e.,
). These special points will form the boundaries of the evolutionary game domain
. In addition to these four obvious equilibrium points, there is another equilibrium point in the system according to Equation (16), which must satisfy the following equation:
As mentioned above, five balance points exists in the system, but it is worth noting that we cannot be sure whether these five equilibrium points are all evolutionary stable strategies of the system. To investigate the stability trend of the evolutionary game between local governments and automakers, according to Equation (16), the Jacobian matrix (
) can be deduced:
On the basis of previous studies [
24,
29], the properties of the Jacobian matrix of the system will determine the stability performance of the equilibrium point. In other words, the stability is related to the values of the detersminant (
) and trace (
) of the matrix. An evolutionarily stable strategy (ESS) exists only when the value of
and
. Based on Equation (18),
and trace
on each equilibrium point can be obtained, as shown in
Table 3.
Scenario 1: The primary goal of Scenario 1 is to explore how the construction of the CVIS can maximize benefits for local governments in terms of traffic congestion relief, accident prevention, and environmental improvements. Thus, the profit
obtained by the local governments through the construction of CVIS is greater than the sum of the cost of the building, developing, and maintaining of roadside infrastructure (
), subsidiaries for RIVs manufacturers (
) and the purchase of ADVs computing power and storage unit (
). Namely,
in this scenario. At the same time, the profit of RIVs
is higher than that of ADVs
. Under this premise, we can judge whether the determinant
and trace
on each equilibrium point are greater than 0 or less than 0. Then, the stability performance of each equilibrium point can be obtained, as shown in
Table 4. The symbol “+” represents
or
are positive (
), and “−” denotes the values are negative (
).
According to the determination method of the stable point described above, we can know that in this scenario, the final evolutionary stability point will appear at (1,1). In other words, the system will eventually reach an equilibrium state in which the manufacturer chooses RIV production strategy and the government adopts the strategy of building the CVIS (i.e., B strategy).
Scenario 2: Scenario 2 aims to assess the system’s response to potential future shifts in the cost-effectiveness of onboard equipment, which might enhance the profitability of ADVs relative to RIVs. Compared with Scenario 1, the numerical relationship between
and
changes. With the development of technology, the cost of onboard equipment may be greatly reduced in the future, which will greatly increase the profitability of the manufacturers who produce ADVs. In this case, it is assumed that
<
. That is, for manufacturers, the profit of ADVs
is much greater than that of RIVs
. However, the government’s subsidies for RIVs can bridge the gap between them. Thus, the stability performance of the system at each equilibrium point in this scenario can be obtained, as shown in
Table 5.
From the results in
Table 4, we can find that this scenario is consistent with scenario 1, and the final stable point will also appear at the point (1,1) even if some conditional assumptions of model parameters change. The system will also eventually reach a steady state of “
RIV” production strategy and “
B” strategy. This is because both parties will choose the investment strategy that is most profitable for themselves.
Scenario 3: Different from the first two scenarios, Scenario 3 aims to understand the conditions under which the government and manufacturers might opt out of CVIS and RIV strategies due to economic constraints, predicting stability at a point where neither RIV production nor CVIS construction is pursued. That is, the cost of the building, developing, and maintaining of roadside infrastructure is higher than the dividend brought by CVIS to the local governments (i.e.,
). Meanwhile, the profit of RIVs
is smaller than that of ADVs
, but government subsidies can fill this gap, that is,
and
.
Table 6 presents the judgment results of the stability performance of the system at each equilibrium point.
Under the above premise, we can find that in this scenario the system will eventually reach stability at (0,0), that is, “ADV” production strategy for manufacturer and “NB” strategy for government will be the strategies that make the system stable.
To help local governments and manufacturers make strategic choices, we adopt the analysis method of phase diagram, which is presented in
Figure 2.
Figure 2a depicts the phase diagram of scenario 1 and 3, the area in this figure is cut into four parts by five different equilibrium points, and the point
is the saddle point of the evolutionary game. Equilibrium points are labeled as:
,
,
,
, and
. As described in scenarios 1 and 3,
and
are the stable points of these two scenarios, respectively. Therefore, in terms of these two scenarios, the system will continue to converge from an unstable initial state
to the final stable state
or
, as shown by the black arrow in
Figure 2a. For scenario 2,
is a non-system local equilibrium point, the phase diagram in this scenario is shown in
Figure 2b, the evolutionary trend shows a convergence from point
to
.
From the above analysis, it can be concluded that different assumptions about the model parameters will affect the final evolutionary process results. Therefore, in the next section, we will conduct a numerical simulation analysis for different parameters based on the current situation of China’s transportation industry and the specific situation of China’s domestic automobile manufacturers.