3.3. Stability Analysis of the Model
Based on the previous analysis, the equilibrium solutions of the evolutionary game model between Local Government 1 and Local Government 2 can be analyzed through a two-dimensional dynamic system formed by their copying dynamic equations [
52]. Let
, solving this system of equations leads to four pure-strategy equilibrium solutions,
,
,
, and
, and one mixed strategy solution,
, where
.
However, these five equilibrium points are not necessarily evolutionarily stable strategies for the system. According to the method proposed by Friedman (1991), the evolutionary stability of a two-dimensional dynamic system can be deduced through a local stability analysis of the system’s Jacobian matrix [
53]. To achieve this, partial derivatives of
and
are taken to obtain the Jacobian matrix
:
where
Simultaneously, the trace
and determinant
of the Jacobian matrix are given by the following:
When the trace condition
and the determinant condition
are satisfied, it can be determined that the equilibrium solutions of the two-dimensional dynamic system correspond to evolutionarily stable strategies. Further inference from the trace and determinant conditions indicates that the equilibrium solutions for evolutionarily stable strategies must satisfy
and
. By further solving the Jacobian matrix
, the values of the five local equilibrium solutions at
,
,
, and
are obtained, as shown in
Table 3.
From
Table 3, it can be observed that
has values of 0 at both
and
, which clearly does not satisfy the conditions
and
. Therefore, this point is not a stable point for the evolution of the system’s strategy. The stability of the other four pure-strategy equilibrium solutions can be discussed in the following four specific cases.
- (1)
Scenario One: When the conditions
and
are satisfied, the stability of the four pure-strategy equilibrium solutions is as shown in
Table 4.
From
Table 4, it can be observed that the evolutionarily stable strategy point for the game between the two local governments is
. This signifies a non-cooperation strategy where both local governments choose to favor industrial land supply for industrial takeover competition. For any local government, if its growth preference is strong, it will not adhere to the market mechanism of land supply and demand but will opt for the non-cooperation strategy, favoring industrial land supply. Even for a local government with a relatively weak growth preference, if the benefits from industrial takeover competition outweigh the losses it incurs and are greater than the compensation for the losses of the other party, it will still choose the non-cooperation strategy favoring industrial land supply. In this scenario, both players in the game have heterogeneous preferences satisfying
and
, and income heterogeneity satisfies
and
. This indicates that both players in the game have growth preferences stronger than a certain threshold, while income heterogeneity is strictly confined within a specific threshold by the industrial takeover. Therefore, once the growth preferences of both sides exceed a certain threshold, both will choose a land supply policy favoring industrial land for industrial takeover competition. An example of this situation is the industrial takeover competition among provinces in the central region of China.
According to
Table 4, the evolutionarily stable strategy points in the game between the two local governments are when both choose the non-cooperation strategy favoring industrial land supply for industrial takeover competition. For either local government, if its preference for growth is strong, it will not adhere to the market mechanism of land supply and demand but will choose the non-cooperation strategy favoring industrial land supply. Even for a local government with a relatively weak preference for growth, if the benefits from industrial takeover competition outweigh the losses after offsetting its own losses, it will still choose the non-cooperation strategy favoring industrial land supply. In this case, both sides of the game have heterogeneous preference satisfaction in terms of growth and income, indicating that both sides have a growth preference stronger than a certain threshold, while the heterogeneity in income is strictly limited within a certain threshold by industrial takeover.
Therefore, when the growth preference of both sides surpasses a certain threshold and the heterogeneity in government income is strictly limited within a certain threshold by industrial takeover, both sides will choose the non-cooperation strategy favoring industrial land supply for industrial takeover competition. An example of this situation is the industrial takeover competition among provinces in the central region of China.
Based on this, Proposition 1 can be formulated as follows:
Proposition 1. When the growth preference of both sides in the game is greater than a certain threshold, and the heterogeneity in government income is strictly limited within a certain threshold by industrial takeover, both sides will choose the non-cooperation strategy.
- (2)
Scenario Two: The stability of the four pure-strategy equilibrium solutions is shown in
Table 5 when
and
.
From
Table 5, it can be observed that the evolutionarily stable strategy point in the game between the two local governments is
. In this scenario, Local Government 1 chooses the cooperation strategy of freely supplying residential and industrial land according to market mechanisms, while Local Government 2 chooses the non-cooperation strategy, leaning towards industrial land. For Local Government 1, the compensation provided by the other party outweighs the benefits of choosing to compete for industrial land, so it will choose the cooperative strategy of supplying land according to market mechanisms. On the other hand, for Local Government 2, the benefits of adopting a strategy favoring industrial land competition, after subtracting the losses generated by this strategy and the compensation given to the other party, still result in a positive income. Therefore, it will choose the non-cooperation strategy.
In this situation, both parties in the game prefer heterogeneity, satisfying and . The income of Local Government 2 satisfies , indicating that the growth preferences of both parties are constrained by two threshold values in opposite directions. The party constrained by the positive threshold will have its income strictly limited within a certain range by industrial takeover. Therefore, regardless of the strategy chosen by the other party, the non-cooperation strategy of favoring industrial land supply for industrial takeover competition will be the dominant strategy. Through long-term game evolution, the other party will effectively identify this dominant strategy, and considering maximizing its own interests, it will eventually abandon the non-cooperation strategy of competition and choose the cooperation strategy of freely supplying residential and industrial land according to market mechanisms. An example of this situation is the industrial takeover competition between developed central regions and underdeveloped peripheral areas in some provinces in central and western China.
Thus, Proposition 2 can be formulated as follows:
Proposition 2. When the growth preferences of both parties in the game are constrained by two threshold values in opposite directions, and the government income of the party constrained by the positive threshold is strictly limited within a certain range by industrial takeover, the party with growth preferences constrained by the positive threshold will choose the non-cooperation strategy, while the party with growth preferences constrained by the negative threshold will choose the cooperation strategy.
- (3)
Scenario Three: When
and
, the stability of the four pure-strategy equilibrium solutions is as shown in
Table 6.
From
Table 6, it can be observed that the evolutionarily stable strategy point in the game between the two local governments is
. In other words, Local Government 1 chooses the non-cooperation strategy, leaning towards industrial land supply, while Local Government 2 opts for the cooperation strategy of supplying land according to market mechanisms. For Local Government 1, the benefits from adopting a strategy favoring industrial land for industrial development, after deducting the losses incurred by this strategy and compensating the other party, still result in positive net gains. Therefore, it chooses the non-cooperation strategy. On the other hand, for Local Government 2, the rational choice is to adopt the cooperation strategy of supplying land according to market mechanisms because the loss compensation offered by the other party exceeds the benefits of choosing to compete for industrial land development. In this scenario, both players in the game exhibit heterogeneous preferences, satisfying
and
. The income of Local Government 1 meets the condition
. This indicates that the growth preferences of both players in the game are constrained by two threshold values in opposite directions. Moreover, the income of the player constrained in the positive direction is strictly limited within a certain range by the industrial development activities. This symmetry contrasts with the situation described in the second scenario.
- (4)
Scenario Four: When
and
, the stability of the four pure-strategy equilibrium solutions is presented in
Table 7.
From
Table 7, it can be observed that the evolutionarily stable strategy point in the game between the two local governments is
, meaning both local governments choose the cooperation strategy of supplying land according to market mechanisms. For either player in the game, adopting the non-cooperation strategy favoring industrial land for industrial development results in a growth effect that, after offsetting losses and compensating the other party, is less than zero. Therefore, the rational choice for both players is to adopt the cooperation strategy of supplying land according to market mechanisms. In this scenario, both players in the game exhibit heterogeneous preferences, satisfying
and
. The income heterogeneity is characterized by
and
. This indicates that both players in the game have government incomes exceeding a certain threshold, and the gap in heterogeneous preferences is strictly confined within a specific threshold. Consequently, a compensation system linked to government income will eliminate the intrinsic motivation for both sides to engage in industrial land competition. The cooperative relationship between the two sides is thus very stable. For example, industrial land competition among some developed cities in the eastern coastal areas of China is relatively uncommon. Therefore, Proposition 3 can be stated as follows:
Proposition 3. When the government income of both players in the game exceeds a certain threshold, and the growth preference is strictly confined within a specific threshold, both sides will choose the cooperation strategy.