3.2.1. Basic Model
At the primary stage, supposing that the proportion of developers choosing to construct LCBs is
x (
), the ratio of developers who choose to build CBs is 1 −
x. Similarly,
y (
) is the ratio of homebuyers buying LCBs, and 1 −
y is the proportion of homebuyers purchasing CBs. During the game process, players will adjust their strategies when their payoffs are under the average; thus,
x and
y are changing. According to the above relationships between two players and the basic assumptions, the payoff matrix is depicted in
Table 2.
Let
UD1 and
UD2 represent the expected profits of “Construct LCB” and “Construct CB” for developers, respectively. Based on
Table 2, these profits can be expressed as follows:
The average profit of developers is denoted as
which is obtained as follows:
Let
UH1 and
UH2 represent the expected profits of “Buy LCB” and “Buy CB” for homebuyers, respectively. Based on
Table 2, these profits can be expressed as follows:
The average profit of homebuyers is written as
which is obtained as follows:
According to Friedman et al. [
36] and Xiao et al. [
37], the replicator dynamic equations of the constructing LCBs strategy selected by developers (
F(
x)) and the buying LCBs strategy adopted by homebuyers (
F(
y)) respectively, are obtained as follows:
3.2.2. Model Analysis
The adjustment speeds of developers and homebuyers’ behavioral strategies are expressed by Equations (14) and (15), respectively. The game will reach a relatively stable equilibrium state when these replication factor dynamic equations are zero. Let
and
; we obtain four fixed equilibrium points: (0,0), (1,1), (0,1) and (1,0). When
,
and
, that is,
and
, we obtain the fifth equilibrium point (
x0,
y0):
The stability of equilibrium points can be analyzed by the Jacobian matrix. The type of equilibrium points can be judged by calculating the determinant
det(
J) and trace
tr(
J) of the matrix [
36]. When the equilibrium point of the combination of Equations (13) and (14) satisfies the condition
and
, the equilibrium point is an ESS; when
and
, it is an unstable point; when
, and
or uncertainty, it is a saddle point. The Jacobian matrix
J composed of the above replicator dynamic equations is as follows:
Substituting the above five equilibrium points into the Jacobian matrix,
Table 3 shows the calculations of their
det(
J) and
tr(
J).
Based on Equations (4)–(7),
; thus, the kinds of equilibrium points depend on the symbols
,
and
. There are six situations, and
Table 4 analyses the local stability of the equilibrium points of the game in different situations.
Situation 1:
, and
.
Figure 1a displays the evolutionary process of developers and homebuyers’ behavior strategies.
Table 4 and
Figure 1a show that (1,1) is the ESS of the game; (Construct LCB, Buy LCB) is the ESS of the game between developers and homebuyers.
Situation 2:
, and
.
Table 4 and
Figure 1b also show that (1,1) is the ESS of the game in condition 2, which means that (Construct LCB, Buy LCB) is the ESS.
Situation 3:
, and
. Regarding the game in situation 3,
Table 4 and
Figure 1c show that the point (0,1) is the ESS, namely (Construct CB, Buy LCB) is the ESS.
Situation 4:
, and
.
Table 4 and
Figure 1d also show that (0,1) is the ESS of the game in situation 4, which means that (Construct CB, Buy LCB) is the ESS.
Situation 5:
, and
.
Table 4 and
Figure 1e show the dynamic evolutionary process. Under this situation, the ESS of the game is (1,0), namely (Construct LCB, Buy CB).
Situation 6:
, and
.
Table 4 and
Figure 1f display the two-dimensional coordinate graphs of the relationship between the two game parties and the evolutionary path. Under situation 6, there is no ESS, and the evolutionary path appears as a closed-loop.
Table 4 and
Figure 1 show that under the situation of
, and
, all the ESS are (Construct LCB, Buy LCB). It is an ideal state. Increasing the subsidies, the carbon tax, the external environmental benefits of LCBs and consumers’ awareness of the LCB concepts, and decreasing the cost of low-carbon technologies will have a positive impact on the low-carbon behavioral choices of developers and consumers. The players will actively make the decisions to choose LCBs when they obtain sufficiently high additional benefits from LCBs. In situations 3 and 4, for both
and
, provided that
is satisfied, (Construct CB, Buy LCB) is the ESS. This result means that regardless of whether consumers are interested in LCBs, developers always choose to build ordinary houses. However, based on the above assumption, once developers construct CBs, homebuyers have to buy them. For this situation, it can be understood that LCBs have not yet been accepted and fully recognized by most developers in the real world.
Under situation 5, (Construct LCB, Buy CB) is the only ESS. In contrast to the previous situations, homebuyers tend to buy CBs, but they accept LCBs. This result indicates that consumers’ low-carbon awareness is not strong. In reality, developers and consumers are likely to encounter cost-benefit mismatches when choosing LCBs; specifically, the dwellers enjoy all benefits during the use phase while developers pay all the upfront costs [
38]. In addition, the transaction game between developers and homebuyers is complicated. The players’ strategies are affected by various factors, such as information asymmetry, the severe inequality in the bargaining power of developers and home buyers during the market transaction [
39], and many external uncertainty factors.
There is no ESS in the last situation; the four saddle points and a central point are obtained. The central point (
x0,
y0) satisfies the elementary condition of the ESS, so it is the Lyapunov stability. The Lyapunov stability has two stable attributes: indifferent and asymptotic stabilities. If it is asymptotically stabled, point (
x0,
y0) is the ESS. We further discuss its stability. Consider the point (
x0,
y0), the Jacobian matrix
J′ is valued as:
The Eigenvalues of Jacobian matrix
J′ can be deduced as:
where
λ1 and
λ2 are virtual characteristic roots, so the point (
x0,
y0) is not asymptotically stabilities. Therefore, in this situation, this system does not have an ESS, and any subtle changes may have a significant impact on the behaviors of developers and homebuyers. The government should put forward relevant measures to change this situation. The last situation is discussed in depth below.