Introduction

Urban agglomerations are fundamental to China’s industrialization and urbanization, providing critical frameworks for economic growth, regional coordination, and global competitiveness (Fu and Zhang, 2020). Acknowledging this importance, the Chinese government emphasizes coordinated development among cities of different sizes, positioning urban agglomerations as a central theme in regional economic research.

Urban agglomerations constitute a highly integrated spatial framework, transitioning from pure competition to a balanced interplay of competition and cooperation among cities (Peng et al. 2020). Human capital accumulation within these agglomerations drives industrial clustering (Pan et al. 2024a), with agglomeration economies being key drivers in their formation (Giuliano et al. 2019). Industrial clusters draw population inflows, expand the urban scale, and stimulate urbanization (Zhang et al. 2022), thereby further advancing the development of cities and urban agglomerations (Pan et al. 2024b; Zeng and Zong, 2023). Building on Redding and Rossi-Hansberg’s (2017) analysis of the uneven distribution of economic activities through quantitative spatial models and a canonical urban model, research on spatial sorting and selection has predominantly relied on quantitative spatial approaches. The integration of evolutionary game theory into spatial sorting and selection studies, as opposed to traditional quantitative spatial models, offers insights into how firms adapt strategies based on prior experience. This framework facilitates a more nuanced analysis of interactions and strategic evolution among firms of different sizes, illustrating their entry into cities (spatial sorting) and eventual displacement or movement toward city centers (spatial selection). Duranton and Puga (2023) contend that quantitative research anchored in robust micro-foundations and consistent with the fundamental principles governing cities and the broader economy is the most reliable. Furthermore, industrial clustering generates diffusion effects that attract labor, stimulate investment, and ultimately drive economic growth (Ahmad et al. 2021; Qi et al. 2024).

The interplay between spatial sorting and selection is fundamental to understanding agglomeration phenomena from a micro-level perspective. Sorting theory posits that large cities attract high-productivity firms, enhancing efficiency despite elevated operational costs (Gaubert, 2018). Micro-level entities regularly evaluate and adjust their locations based on productivity advantages (Eeckhout and Kircher, 2018). Spatial selection theory, in contrast, emphasizes the decision-making processes of firms in evaluating their operational sustainability within specific urban scales, explaining industry relocation or consolidation in core cities, which ultimately enhances regional productivity (Fajgelbaum and Gaubert, 2020). Regulatory frameworks and policies, such as Special Economic Zones, enhance firm survival in core cities (Zhang and Xu, 2024), whereas stricter environmental standards may drive relocations to evade compliance costs (Lu and Ouyang, 2024).

Current research on spatial sorting and selection within urban agglomerations predominantly employs econometric models to validate these phenomena. Forslid and Okubo (2021) applied a new economic geography model to examine the relationship between capital intensity, productivity, and firm spatial sorting. Kaplan et al. (2022) employed a single variance index model to measure spatial sorting, whereas Silva and Azzoni (2022) utilized a two-step estimation method to analyze the effects of agglomeration and spatial sorting on wages. Forslid and Okubo (2024) employed a long-term general equilibrium model to investigate human capital agglomeration and spatial selection in developed and developing countries. They observed that low-skilled labor exhibits low mobility, and spatial selection in poorer countries achieves equilibrium rapidly. Bhattacharjee et al. (2023) employed a spatial Durbin model to examine how spatial agglomeration affects firm survival during spatial selection, revealing that firm innovation significantly improves survival prospects in urban agglomerations. Giuliani et al. (2024) applied a survival regression model to investigate how spatial externalities influence startup survival during spatial selection over time. In summary, existing literature predominantly relies on qualitative analyses, conventional economic models, and spatial econometric techniques to confirm the existence of spatial sorting and selection.

Empirical tests of spatial sorting and selection have yielded inconsistent results (Au and Henderson, 2006; Forslid and Okubo, 2015), indicating instability in micro-level behavior patterns. Behrens et al. (2014) observed that most theoretical models of spatial sorting and selection extend beyond general equilibrium frameworks, complicating efforts to directly link heterogeneous entities to city size. The absence of specific functional models has further impeded definitive conclusions on spatial selection. Scholars like Behrens and Robert-Nicoud (2015) argue that deriving robust empirical conclusions on spatial sorting and selection is challenging, with related evidence hardly convincing. This paper asserts that existing spatial sorting and selection models predominantly rely on standard econometric methods. The inconsistencies in empirical results highlight limitations in these models, which fail to sufficiently capture the complex interactions of diverse entities in urban environments. Existing research has not adequately examined how firms adapt and interact over time within cities, or how micro-level entities dynamically adjust spatial positioning under the dual pressures of sorting and selection. The application of evolutionary game theory provides a novel perspective, facilitating detailed analysis of firm interactions and adaptive processes in urban agglomerations as evolving strategic dynamics. This approach offers a nuanced understanding of how firms adapt, expand, or relocate in response to evolving urban economic conditions, emphasizing the key determinants of these decisions.

Research hypotheses and evolutionary game model construction

This study provides an in-depth analysis of spatial sorting and selection by employing a tripartite evolutionary game model, uncovering a strong interdependence between the two phenomena. Scholars suggest that “spatial sorting” is closely related to “spatial selection” among heterogeneous agents, as evidenced by the clustering of highly productive, skilled individuals in larger cities. The key distinction, however, is that “spatial sorting” involves the migration of individuals or firms to urban centers, whereas “spatial selection” often results in their exclusion or relocation.

Spatial sorting and selection constitute two sequential stages in the firm decision-making process. In the “pre-entry” phase, firms evaluate whether to enter a city based on its scale and resources, whereas in the “post-entry” phase, they reassess whether to remain or exit based on operational compatibility. Spatial sorting and selection form a dynamic continuum: firms in peripheral cities evaluate whether to move to central cities based on perceived compatibility, while firms already located in central cities reassess their presence based on ongoing evaluations of compatibility.

In this study, the tripartite evolutionary game model classifies participants into three distinct categories: high-skill firms, medium-skill firms, and low-skill firms. These firms initially operate in peripheral cities and engage in production activities. To determine their optimal strategies within market dynamics, each type of firm engages in complex games and competitive processes to decide whether to enter the central city. As this study examines industrial spatial agglomeration within urban clusters, “micro-individuals” primarily refer to firms. Urban centers, functioning as collective entities, play a pivotal role in spatial sorting and selection through mechanisms that include higher-capability firms (complementary mechanism) and exclude less capable ones (survival of the fittest). This inclusion-exclusion dynamic is reflected in the competitive interactions among firms of varying skill levels within urban environments.

Consistent with prior research, spatial sorting is rarely a firm’s “final decision.” Firms continuously evaluate the alignment between their capabilities and the urban scale, adjusting strategies as needed. This study introduces a game participant framework to emphasize the strategic choices firms face when mismatches occur between their capabilities and urban scale. Additionally, market forces influence firm selection, as firms tend to prefer larger markets to maximize benefits when prices remain constant. In equilibrium, prices generally decline over time, allowing only firms with lower marginal costs to sustain their positions in larger markets, while others relocate to smaller ones.

The study examines the tendency of lower-skilled firms to move toward central urban areas, often motivated by seemingly irrational desires. This trend reinforces the self-sustaining growth of central cities but also exacerbates congestion effects, raising entry costs. As the number of participants increases, competition intensifies, and only firms with the most favorable profit-to-cost ratios are likely to thrive. By constructing a tripartite game model, this study aims to offer a comprehensive understanding of the relationship between industrial spatial agglomeration and urban scale distribution, focusing on the evolution of these dynamics through spatial sorting and selection. The conceptual framework of the dynamic changes in spatial selection and sorting is shown in Fig. 1.

Fig. 1
figure 1

The conceptual research framework.

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Accordingly, this study develops an evolutionary game model encompassing high-skill, medium-skill, and low-skill firms and proposes the following hypotheses, based on the preceding discussion of the mechanisms underlying spatial sorting and selection:

Hypothesis 1: This study posits that the evolutionary game takes place within an urban agglomeration consisting of one central city and multiple smaller peripheral cities. Within this urban agglomeration, three distinct types of participants, all characterized by bounded rationality, are involved: high-skill firms, medium-skill firms, and low-skill firms. These firms engage in producing a homogeneous final product (CES) but differ in their production efficiency.

Hypothesis 2: The strategic options available to the high, medium, and low-skilled firms are as delineated below:

\({\alpha}_{{enterprise}}=\left({\alpha}_{1},{\alpha}_{2}\right)=\left({\rm{enter}}\; {\rm{central}}\; {\rm{city}},{\rm{do}}\; {\rm{not}}\; {\rm{enter}}\; {\rm{central}}\; {\rm{city}}\right)\)

\({\beta}_{{enterprise}}=\left({\beta }_{1},{\beta}_{2}\right)=\left({\rm{enter}}\; {\rm{central}}\; {\rm{city}},{\rm{do}}\; {\rm{not}}\; {\rm{enter}}\; {\rm{central}}\; {\rm{city}}\right)\)

\({\gamma}_{{enterprise}}=\left({\gamma}_{1},{\gamma}_{2}\right)=\left({\rm{enter}}\; {\rm{central}}\; {\rm{city}},{\rm{do}}\; {\rm{not}}\; {\rm{enter}}\; {\rm{central}}\; {\rm{city}}\right)\)

Hypothesis 3: We assume that the production of each firm in the game requires both labor and capital, with complete factor mobility within the urban agglomeration. Given the homogeneous nature of the final product produced by the firms, those with higher skill levels can produce more homogeneous products at lower costs, thereby obtaining greater efficiencies. The benefits obtained by the three types of firms are denoted as \({Q}_{\alpha }\), \({Q}_{\beta }\), and\({Q}_{\gamma }\).

Hypothesis 4: This paper assumes that F denotes the locational advantages gained by firms in the central city, including factors such as resource endowment and the diversity of industries and functions within the city, while D represents the additional geographic transportation costs faced by firms in peripheral cities. Within the framework of evolutionary game theory, it is assumed that when F > D, firms are more inclined to enter the central city to obtain locational advantages and reduce transportation costs, thereby achieving a higher level of utility. Conversely, when F ≤ D, firms may choose to remain in peripheral cities or may lack a clear incentive to migrate spatially.

Hypothesis 5: Beyond the foundational locational factors, the augmentation in urban dimensions is predominantly ascribed to the genesis of agglomeration economies. We posit that in scenarios where the central city harbors more than two firms, a resultant productivity augmentation ensues, denoted as \({\mathbb{A}}\). This augmentation encapsulates the increasing returns to scale attributable to agglomeration economies, accruing to the firms. \({\mathbb{A}}\) embodies the aggregate factor productivity emanating from all comparative advantages inherent to the central city, and given the increasing returns to urban scale,\({\mathbb{A}}\) assumes a value greater than one (\({\mathbb{A}} > 1)\). As there are differences in skills among firms, it always holds that \({{\mathbb{A}}{\rm{Q}}}_{{\rm{\alpha }}} > {{\mathbb{A}}{\rm{Q}}}_{{\rm{\beta }}}{\mathbb{ > }}{\mathbb{A}}{{\rm{Q}}}_{{\rm{\gamma }}}\). In addition to the benefits derived from scale, firms also incur additional external costs due to the negative externalities of agglomeration economies, such as traffic congestion, resource competition, environmental pollution, and rising rents and land prices. These are collectively referred to as urban costs \(C\). Additionally, due to the limited resources of central cities, when at least two types of firms coexist in the same city, this exacerbates the competitive pressures on central cities, thereby triggering congestion effects. These effects typically manifest as competition for resources, increased pressure on infrastructure, and a decline in quality of life. These effects have also been incorporated into the consideration of urban cost C Table 1.

Table 1 Main parameter settings and meanings.
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Incorporating the aforementioned assumptions, the mixed strategy game matrix for high-skilled, medium-skilled, and low-skilled firms is systematically presented in Table 2. Within this matrix, ①represents the strategic choices for high-skilled firms, ②represents the strategic choices for medium-skilled firms, and ③represents the strategic choices for low-skilled firms.

Table 2 Mixed Strategy Game Matrix for High, Medium, and Low-skilled firms.
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Dynamic evolution analysis

The expected benefits and average expected benefits of high-skilled firms choosing to enter or exit are \({E}_{11},{E}_{12},\bar{{E}_{1}}\).

$$\,\left\{\begin{array}{c}{\rm{\& }}{E}_{11}={yz}{{\mathbb{(}}{\mathbb{A}}Q}_{\alpha }+F-C)+y(1-z)({{\mathbb{A}}Q}_{\alpha }+F-C)\,\\ \,+\left(1-y\right)z({{\mathbb{A}}Q}_{\alpha }+F-C)+\left(1-y\right)(1-z){(Q}_{\alpha }+F)\\ {\rm{\& }}{E}_{12}={yz}\left({Q}_{\alpha }-D\right)+y(1-z)\left({Q}_{\alpha }-D\right)\\ \,+\left(1-y\right)z\left({Q}_{\alpha }-D\right)+\left(1-y\right)\left(1-z\right)\left({Q}_{\alpha }-D\right)\\ {\rm{\& }}\bar{{E}_{1}}=x{E}_{11}+\left(1-x\right){E}_{12}\end{array}\right.$$
(1)

The replicator dynamic equation for the strategy choices of high-skilled firms is:

$${\rm{F}}\left({\rm{x}}\right)=\frac{{\rm{dx}}}{{\rm{dt}}}={\rm{x}}\left({{\rm{E}}}_{11}-\bar{{{\rm{E}}}_{1}}\right)={{\rm{x}}(1-{\rm{x}})({\rm{E}}}_{11}-{{\rm{E}}}_{12})$$
(2)

Concurrently, the first derivative of \(x\), a critical variable in the equation, is determined as:

$$\frac{{dF}(x)}{{dx}}=(1-2x){(E}_{11}-{E}_{12})$$
(3)

Assuming:

$$\begin{array}{c}G\left(y\right)={E}_{11}-{E}_{12}\\ =-{yz}\left[({{\mathbb{A}}{\mathbb{-}}1)Q}_{\alpha }-C\right]+y[{{\mathbb{(}}{\mathbb{A}}{\mathbb{-}}1)Q}_{\alpha }-C]\\ +z[{{\mathbb{(}}{\mathbb{A}}{\mathbb{-}}1)Q}_{\alpha }-C]+F+D\end{array}$$
(4)

When \({\rm{G}}\left({\rm{y}}\right)=0\), then,

$${{\rm{y}}}^{* }=\frac{z[{{\mathbb{(}}{\mathbb{A}}{\mathbb{-}}1)Q}_{\alpha }-C]+{\rm{F}}+{\rm{D}}}{({\rm{z}}-1)[({{\mathbb{A}}{\mathbb{-}}1)Q}_{\alpha }-C]}$$
(5)

Based on the principles of stability in differential equations, the likelihood of high-skill firms opting to establish in central cities reaches a state of equilibrium when it satisfies the condition \(F\left(x\right)=0\) alongside \({dF}(x)/{dx} < 0\). Here, \(x\) represents the local maximum of the function \(F\left(x\right)\). In this context, Eqs. (2), (3), (4), and (5) must align with one of the following scenarios:

Scenario One: Characterized by \(G\left(y\right)=0\), implying that at \(y={y}^{* }\), the derivative \({dF}(x)/{dx}=(1-2x)G(y)\equiv 0\). This condition indicates the absence of an Evolutionarily Stable Strategy (ESS).

Scenario Two: At the juncture where \(x=0\), satisfying \({dF}(x)/{dx}=G(y) < 0\) necessitates \(G\left(y\right) < 0\). Consequently, for \(y > {y}^{* }\), this criterion delineates the ESS pertinent to high-skill firms.

Scenario Three: At \(x=1\), fulfilling the condition \({dF}(x)/{dx}=-G(y) < 0\) requires \(G\left(y\right) > 0\). Therefore, for \(y < {y}^{* }\), this scenario corresponds to the ESS for high-skill firms.

Given that \(\partial G\left(y\right)/\partial y={(1-{\rm{z}})({\mathbb{A}}{\rm{Q}}}_{{\rm{\alpha }}}-{\rm{C}}-{{\rm{Q}}}_{{\rm{\alpha }}})\) within the model constraints of \({\rm{z}}\le 1\), a stable equilibrium point is absent when \({{\mathbb{A}}Q}_{\alpha }-{\rm{C}}-{{\rm{Q}}}_{{\rm{\alpha }}}\) is greater than zero. Conversely, when \({{\mathbb{A}}Q}_{\alpha }-{\rm{C}}-{{\rm{Q}}}_{{\rm{\alpha }}}\) is less than zero, \(\partial G\left(y\right)/\partial y\) becomes negative, and \(G\left(y\right)\) is a decremental function of \(y\). Under these circumstances, the probability for high-skill firms to enter central cities, denoted as V\({A}_{1}\), is:

$${A}_{1}={{\iint}_{0}}^{1}\frac{{z}[{({\mathbb{A}}-1){Q}}_{\alpha }-{C}]+{\rm{F}}+{\rm{D}}}{({\rm{z}}-1)[({{\mathbb{A}}-1}){Q}_{\alpha}-{C}]}{\rm{dzdx}}$$
(6)

Corollary: The probability of high-skill firms entering central cities exhibits a positive correlation with factors such as total factor productivity (\({\mathbb{A}}\)), locational advantages of the central city (\({\rm{F}}\)), and transportation costs (\({\rm{D}}\)). Conversely, it is inversely related to urban costs (\({\rm{C}}\)).

Proof: Deriving the first-order partial derivatives of the probability \({A}_{1}\) for high-skill firms entering central cities, it is evident that \(\partial {A}_{1}/\partial \left({{\mathbb{A}}Q}_{\alpha }-{Q}_{\alpha }\right) > 0\), \(\partial {A}_{1}/\partial {\rm{F}} > 0\), \(\partial {A}_{1}/\partial D > 0\), and \(\partial {A}_{1}/\partial C < 0\). This implies a direct relationship of \({{\mathbb{A}}Q}_{\alpha }-{Q}_{\alpha }\), \({\rm{F}}\), and \(D\), with \({A}_{1}\), and an inverse relationship of \(C\) with \({A}_{1}\).

Corollary: Primarily, the greater the agglomeration benefits provided by the central city, the more pronounced are the total factor productivity and locational advantages for high-skill firms, incentivizing their migration from peripheral to central cities. Secondly, transportation costs to peripheral cities compel high-skill firms to relocate to central cities, where consumer markets and productivity are higher. Lastly, urban costs significantly inhibit the migration of high-skill firms to central cities. When the production of high-skill firms in central cities cannot offset these urban costs, such firms may choose not to enter or to leave central cities.

Subsequently, the expected benefits and average expected benefits of medium-skilled firms choosing to enter or exit are \({E}_{21},{E}_{22},\bar{{E}_{2}}\):

$$\left\{\begin{array}{c}{E}_{21}={xz}\left({{\mathbb{A}}Q}_{\beta }+F-C\right)+x\left(1-z\right)\left({{\mathbb{A}}Q}_{\beta }+F-C\right)\\ \,+\left(1-x\right)z\left({{\mathbb{A}}Q}_{\beta }+F-C\right)+\left(1-x\right)\left(1-z\right)\left({Q}_{\beta }+F\right)\\ {E}_{22}={xz}\left({Q}_{\beta }-D\right)+x\left(1-z\right)\left({Q}_{\beta }-D\right)\,\\ \,+\left(1-x\right)z\left({Q}_{\beta }-D\right)+\left(1-x\right)\left(1-z\right)\left({Q}_{\beta }-D\right)\\ \bar{{E}_{2}}=y{E}_{21}+\left(1-y\right){E}_{22}\,\end{array}\right.$$
(7)

The replicator dynamic equation for the strategy choices of medium-skilled firms is:

$$F\left(y\right)=\frac{{dy}}{{dt}}=y\left({E}_{21}-\bar{{E}_{2}}\right)={y(1-y)(E}_{21}-{E}_{22})$$
(8)

The probability of medium-skill firms entering central cities, denoted as \({B}_{1}\), is:

$${B}_{1}={{\iint}_{0}}^{1}\frac{x[{\mathbb{}}({\mathbb{A}}-1){Q}_{\beta}-{C}]+F+D}{x\left[\left({\mathbb{A}}-1\right){Q}_{\beta}-{C}\right]-[\left({\mathbb{A}}-1\right){Q}_{\beta}-{C}]}{d}{\rm{ydx}}$$
(9)

This elucidates that the likelihood of medium-skill firms remaining in central cities is positively associated with total factor productivity (\({\mathbb{A}}\)), central city locational advantage (\({\rm{F}}\)), and transportation costs (\({\rm{D}}\)), while being negatively associated with urban costs (\(C\)).

In a similar vein, the expected benefits and average expected benefits of low-skilled firms choosing to enter or exit are \({E}_{31},{E}_{32},\bar{{E}_{3}}\):

$$\,\left\{\begin{array}{c}{\rm{\& }}{E}_{31}={xy}{\mathbb{(}}{\mathbb{A}}{Q}_{\gamma }+F-C)+x(1-y{\mathbb{)}}{\mathbb{(}}{\mathbb{A}}{Q}_{\gamma }+F-C)\\ \,+\left(1-x\right)y\left({\mathbb{A}}{Q}_{\gamma }+F-C\right)+\left(1-x\right)\left(1-y\right)\left({Q}_{\gamma }+F\right)\\ {\rm{\& }}{E}_{32}={xy}\left({Q}_{\gamma }-D\right)+x(1-y)\left({Q}_{\gamma }-D\right)\\ \,+\left(1-x\right)y\left({Q}_{\gamma }-D\right)+\left(1-x\right)\left(1-y\right)\left({Q}_{\gamma }-D\right)\\ {\rm{\& }}\bar{{E}_{3}}=z{E}_{31}+\left(1-z\right){E}_{32}\end{array}\right.$$
(10)

The replicator dynamic equation for the strategy choices of low-skilled firms is:

$$F\left(z\right)=\frac{{dz}}{{dt}}=z\left({E}_{31}-\bar{{E}_{3}}\right)={z(1-z)(E}_{31}-{E}_{32})$$
(11)

The probability of low-skill firms choosing to enter central cities, denoted as \({C}_{1}\), is:

$${C}_{1}={{\iint}_{0}}^{1}\frac{y[({\mathbb{A}}-1){Q}_{\gamma }-{C}]+{\rm{F}}+{\rm{D}}}{(y-1)[({\mathbb{A}}-1){Q}_{\gamma}-{C}]}{\rm{dydz}}$$
(12)

This indicates that the probability of low-skill firms sustaining their presence in central cities is positively influenced by total factor productivity (\({\mathbb{A}}\)), locational advantage (F), and transportation costs (D), and negatively influenced by urban costs (C).

In summary, key determinants such as total factor productivity (\({\mathbb{A}}\)), locational advantage (F), transportation costs (D), potential business losses (L), and urban costs (C), are instrumental in deciding the feasibility of firm operations in central cities. The initial skill disparities among the participants in the evolutionary game necessitate a comprehensive analysis of the system’s equilibrium in the three-party evolutionary game to establish a stable equilibrium.

Following a comprehensive analysis of the strategic stability of each participant within the evolutionary game framework, this manuscript delves into an in-depth stability examination of the equilibrium points inherent in the tripartite evolutionary game system. Typically, the equilibrium points in evolutionary games are characterized as strictly Nash equilibrium points showcasing asymptotic stability, as well as being representative of pure strategy Nash equilibriums. Ritzberger and Weibull (1995) advanced the proposition that under the conditions \(F\left(x\right)={dx}/{dt}=0\), \(F\left(y\right)={dy}/{dt}=0\) and \(F\left(z\right)={dz}/{dt}=0\), one could deduce the existence of the equilibrium points within the tripartite evolutionary game system as \({E}_{1}=\left(\mathrm{0,0,0}\right)\), \({E}_{2}=\left(\mathrm{1,0,0}\right)\), \({E}_{3}=\left(\mathrm{0,1,0}\right)\), \({E}_{4}=\left(\mathrm{0,0,1}\right)\), \({E}_{5}=\left(\mathrm{1,1,0}\right)\), \({E}_{6}=\left(\mathrm{1,0,1}\right)\), \({E}_{7}=\left(\mathrm{0,1,1}\right)\) and \({E}_{8}=\left(\mathrm{1,1,1}\right)\). At this juncture, the Jacobian matrix pertinent to the tripartite evolutionary game synergy is attainable (refer to Eq. 13).

$$J=\frac{\partial \left({\rm{F}}\left({\rm{x}}\right),F\left(y\right),F\left(z\right)\right)}{\partial \left(x,y,z\right)}\,=\left[\begin{array}{ccc}\frac{\partial F\left(x\right)}{\partial x} & \frac{\partial F\left(x\right)}{\partial y} & \frac{\partial F\left(x\right)}{\partial z}\\ \frac{\partial F\left(y\right)}{\partial x} & \frac{\partial F\left(y\right)}{\partial y} & \frac{\partial F\left(y\right)}{\partial z}\\ \frac{\partial F\left(z\right)}{\partial x} & \frac{\partial F\left(z\right)}{\partial y} & \frac{\partial F\left(z\right)}{\partial z}\end{array}\right]$$
(13)

Invoking the principles of Lyapunov stability, an equilibrium point is classified as asymptotically stable if and only if all the eigenvalues of the Jacobian matrix are negative real numbers. Conversely, the presence of positive real numbers among these eigenvalues indicates an unstable equilibrium. Instances where the Jacobian matrix’s eigenvalues are a mixture of zero and negative real numbers represent a critical state, casting ambiguity on the equilibrium point’s stability. The equilibrium point、eigenvalues and signs of the Jacobian matrix are shown in Table 3.

Table 3 Equilibrium Points Eigenvalues and signs of Jacobian Matrix.
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Our analysis of the evolutionary game model reveals that the sign of the eigenvalue is influenced by multiple factors. Further examination indicates that the sign is strongly associated with the balance between benefits and costs incurred by firms operating in the central city. Variations in costs and benefits across firms of different skill levels lead to diverse outcomes in the evolutionary game. Therefore, exploring the evolutionary stable strategy (ESS) requires a detailed and context-specific analysis.

Condition 1:\({({\mathbb{A}}-1){Q}}_{\alpha }+F+D > C\) and \({({\mathbb{A}}-1){Q}}_{\beta }+F+D < C\), \({E}_{2}(\mathrm{1,0,0})\) constitutes an evolutionary stable strategy (ESS), where high-skill firms enter the central city, while medium- and low-skill firms stay in peripheral cities. In this scenario, the net benefits high-skill firms gain in the central city derived from their comparative advantage\({({\mathbb{A}}-1)Q}_{\alpha }\), additional profits\(F\), and saved transportation costs\(D\)-surpass the urban and congestion costs\(C\). However, for medium-skill firms, the benefits are insufficient to cover their costs in the central city, and for low-skill firms, the net gains are minimal due to their comparative disadvantages. High-skill firms prefer the central city as they can leverage their production efficiency and market advantages for higher economic returns. Conversely, medium- and low-skill firms, facing higher costs, choose to operate in peripheral cities to avoid the central city’s congestion costs and competitive pressures.

Condition 2:\({({\mathbb{A}}-1){Q}}_{\beta }+F+D > C\) and \({({\mathbb{A}}-1){Q}}_{\gamma }+F+D < C\), \({E}_{5}(\mathrm{1,1,0})\) constitutes an evolutionary stable strategy (ESS), where both high-skill and medium-skill firms opt to produce in the central city, while low-skill firms stay in peripheral cities due to cost-benefit considerations. Medium-skill firms enter the central city because their net benefits, including comparative advantage, additional profits\(F\), and saved transportation costs D, outweigh the urban and congestion costs\(C\). However, the benefits for low-skill firms in the central city do not compensate for their losses. Due to their superior comparative advantages and production efficiency, high-skill firms achieve significantly higher net benefits than medium-skill firms. This equilibrium allows high-skill and medium-skill firms to remain in the central city, benefiting from the agglomeration economies and market size, which enhance their production efficiency and innovation capacity. Meanwhile, low-skill firms, facing high costs, intense competition, and congestion in the central city, prefer to stay in peripheral cities where costs are lower and competition is less intense, seeking a more favorable environment for survival and growth.

Condition 3, \({({\mathbb{A}}-1){Q}}_{\gamma }+F+D > C\), \({E}_{8}(\mathrm{1,1,1})\) constitutes an evolutionary stable strategy (ESS), with all three types of firms producing in the central city. In this case, the benefits for low-skill firms entering the central city exceed the urban costs, the benefits for high-skill and medium-skill firms also exceed their respective costs, making production in the central city a stable strategy for all firms. This outcome suggests that even with urban costs, the central city’s market size and agglomeration effects are sufficient to support firms of all skill levels, providing a conducive environment for their development.

The eigenvalues and signs of the Jacobian matrix at this specific stage are shown in Table 4:

Table 4 Signs of Eigenvalues and Conclusions of Jacobian Matrix.
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The analysis of the three ESS outcomes reveals that, regardless of conditions, high-skill firms inevitably enter the central city for production, confirming the presence of spatial sorting. However, these scenarios do not definitively confirm the existence of spatial selection. Under Condition 3, all firms, regardless of skill level, produce in the central city, indicating that all firms are capable of operating successfully in this urban setting. This outcome underscores why spatial selection is often considered empirically inconsistent in existing research. The phenomenon is closely tied to firms’ production skills; when even the lowest-skilled firms in an industry can operate in the central city, spatial selection among micro-level entities becomes less pronounced. This observation partially explains the tendency of the service industry to concentrate in central cities, while manufacturing often clusters in peripheral areas. In contrast, Conditions 1 and 2 illustrate the coexistence of spatial sorting and spatial selection.

The locational choices of high-skill firms (entry) and low-skill firms (non-entry) exemplify the “ex-ante entrance” and “complementarity mechanisms” of spatial sorting. These mechanisms necessitate that micro-entities align their location decisions with their skill levels and urban scale. Spatial sorting imposes more stringent selection criteria in larger cities. Due to high urban costs in larger cities, some firms—such as medium-skill and low-skill firms that have entered and are already operating—must reassess their viability, embodying the “ex-post entrance” and “survival of the fittest” principles of spatial selection. The game concludes with high-skill firms remaining in central cities, while medium-skill and low-skill firms continue to operate in peripheral areas. However, this is not the end of economic activity. The influx of high-skill firms enhances productivity and specialization in central cities, expanding their scale and triggering subsequent rounds of spatial sorting. New firms must assess their competitiveness relative to existing high-skill firms and decide whether to enter central urban areas, thereby initiating a new cycle of the game. This establishes a symbiotic and cyclical relationship between spatial sorting and spatial selection, perpetually enhancing specialization and agglomeration economies in central cities. This iterative process attracts higher-quality micro-entities, intensifies competition, expands urban scale, and reinforces the clustering of top-tier firms in larger urban centers.

Simulation analysis

In the earlier sections of this paper, we identified that the evolutionary game model can yield three stable equilibrium points under different conditions. To verify the model’s stability and the accuracy of our analysis, this section will set specific parameters based on the previous assumptions, aligned with actual economic scenarios. We will use MATLAB 2022a software to simulate how the model’s stable equilibrium points perform under various constraints.

We have established three sets of parameter arrays, each corresponding to a specific condition. For Condition 1, parameter array 1 is defined as

$$\mathrm{int}1=\{{{\mathbb{A}}=1.5,{\rm{Q}}}_{{\rm{\alpha }}}=500,{{\rm{Q}}}_{{\rm{\beta }}}=300,{{\rm{Q}}}_{{\rm{\gamma }}}=100,C=350F=100,D=50\}.$$

For Condition 2, parameter array 2 is defined as

$$\mathrm{int}2=\{{{\mathbb{A}}=1.5,{\rm{Q}}}_{{\rm{\alpha }}}=500,{{\rm{Q}}}_{{\rm{\beta }}}=300,{{\rm{Q}}}_{{\rm{\gamma }}}=100,C=275F=100,D=50\}.$$

For Condition 3, parameter array 3 is defined as

$$\mathrm{int}3=\{{{\mathbb{A}}=1.5,{\rm{Q}}}_{{\rm{\alpha }}}=500,{{\rm{Q}}}_{{\rm{\beta }}}=300,{{\rm{Q}}}_{{\rm{\gamma }}}=100,C=100F=100,D=50\}.$$

The initial simulation tests which strategies each of the three types of firms will adopt under these different conditions. Building on this, we will delve deeper into analyzing how variations in factors such as initial enterprise willingness(\(x,y,z\)), skill differences (\({{{\rm{Q}}}_{{\rm{\alpha }}},{\rm{Q}}}_{{\rm{\beta }}},{{\rm{Q}}}_{{\rm{\gamma }}}\)), and urban costs \(C\) influence the evolutionary game process and its final results. Through these simulation experiments, we aim to gain a comprehensive understanding of the model’s dynamic behavior and evaluate its potential applications in real-world economic activities.

Numerical experiment and simulation

Each of the three parameter sets was evolved 50 times from different initial values, resulting in the tripartite evolutionary game strategy interaction simulation shown in Fig. 2. Figure 2a–c) respectively display the evolutionary game simulation results under the influence of Conditions 1, 2, and 3. Figure 2 indicates that the Jacobian matrix results of the tripartite evolutionary game are robust. Under the specific parameter combination satisfying Condition 1, the model’s dynamic evolution tends toward the equilibrium point \({E}_{2}\left(1,0,0\right)\), suggesting that in this condition, the strategy combination results in high-skill firms choosing to enter the central city, medium-skill and low-skill firms opting not to enter the central city. For the parameter settings satisfying Condition 2, the model’s evolutionary trajectory points toward an equilibrium point \({E}_{5}\left(1,1,0\right)\), where the strategy combination is: high-skill firms and medium-skill firms enter the central city, while low-skill firms still do not enter the central city. Further, under the parameter settings of Condition 3, the model shows that all three types of firms tend to engage in production activities in the central city, i.e., the evolution leads to an equilibrium point \({E}_{8}\left(1,1,1\right)\).This simulation result aligns closely with the analytical solutions obtained using the evolutionary game theory model, providing strong empirical support for the robustness of the tripartite evolutionary game model.

Fig. 2: Stable strategy choices under different conditions in tripartite evolutionary games.
figure 2

a Under Condition 1, firms with different skill levels stabilize at strategy choices (1,0,0). b Under Condition 2, the stable strategy choices are (1,1,0) across skill levels. c Under Condition 3, all firms converge to the strategy choice (1,1,1).

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Impact of changes in initial participation willingness on spatial selection and spatial sorting

Figure 3 illustrates the impact of changes in the initial willingness of the three types of firms on the evolutionary results, with all other parameters held constant. It is assumed that the initial willingness probability of the three types of firms varies from 0.2 to 0.5, and then to 0.8. The analysis of Fig. 3 indicates that, despite significant changes in the initial willingness probabilities, their impact on the model’s convergence to an equilibrium strategy is not substantial. The detailed analysis is as follows:

Fig. 3: Impact of changes in initial willingness on stable strategy choices.
figure 3

a Under Condition 1, the stable strategy choices of firms with different skill levels after initial willingness increases from 0.2 to 0.5 and 0.8. b Under Condition 2, the stable strategy choices of firms with different skill levels after initial willingness increases from 0.2 to 0.5 and 0.8. c Under Condition 3, the stable strategy choices of firms with different skill levels after initial willingness increases from 0.2 to 0.5 and 0.8.

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Under Condition 1, high-skill firms, due to their higher production efficiency, have expected returns from entering the central city that exceed the associated costs. As a result, the evolutionary rate increases as initial willingness rises. In contrast, the returns for medium-skill and low-skill firms do not cover their costs. Although the evolutionary process of medium-skill firms shows some fluctuations, it does not alter their final strategy of exiting the central city.

Under Condition 2, where the expected returns for medium-skill firms entering the central city exceed the costs, an increase in initial willingness significantly accelerates the convergence of all three types of firms toward the equilibrium strategy.

Under Condition 3, all firms have expected returns from production in the central city that exceed the costs, leading all of them to choose to enter the central city. High-skill firms, due to their skill advantages, enter the central city at the fastest pace, followed by medium-skill firms. Although low-skill firms converge at a slower rate, they ultimately reach the same equilibrium state as the other two types of firms.

Impact of skill differences on spatial selection and spatial sorting

To explore the sensitivity of the three types of firms to skill differences in their strategic behavior, this paper designs simulation experiments focusing on changes in skill differences. Keeping other conditions constant, we simulated three scenarios with large (\({Q}_{\alpha }=500,{Q}_{\beta }=300,{Q}_{\gamma }=100\)), medium (\({Q}_{\alpha }=400,{Q}_{\beta }=300,{Q}_{\gamma }=200\)), and small (\({Q}_{\alpha }=350,{Q}_{\beta }=300,{Q}_{\gamma }=250\)) skill differences. By adjusting the values of the parameters \({{{\rm{Q}}}_{{\rm{\alpha }}},{\rm{Q}}}_{{\rm{\beta }}},{{\rm{Q}}}_{{\rm{\gamma }}}\), we analyzed the specific impact of these changes on the strategic choices of firms. The results are shown in Fig. 4.

Fig. 4: Impact of skill differences on stable strategy choices.
figure 4

a Under Condition 1, the stable strategy choices of firms with different skill levels as skill differences increase from small to significant. b Under Condition 2, the stable strategy choices of firms with different skill levels as skill differences increase from small to significant. c Under Condition 3, the stable strategy choices of firms with different skill levels as skill differences increase from small to significant.

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Analysis of Fig. 4 shows that under Condition 1, the impact of skill differences on different firms is relatively small. Under Condition 2, the skill differences have a more significant impact on firms that choose not to enter the central city. This may suggest that under certain market conditions, firms with lower skill levels are more inclined to avoid direct competition with high-skill firms in central cities. In the scenario of Condition 3, the size of skill differences does not significantly affect the decisions of high-skill and medium-skill firms, but it has a more pronounced impact on low-skill firms. As skill differences increase, the evolution speed of low-skill firms toward the equilibrium point slows down. This phenomenon may arise from the fact that greater skill differences make it more difficult for low-skill firms to find a competitive advantage in the central city market when facing high-skill competitors, thereby slowing their strategy adjustment and market positioning processes.

Impact of urban cost variations on spatial selection and spatial sorting

Figure 5 presents a detailed analysis of how changes in urban costs influence the outcomes of the evolutionary game. Under Condition 1, where high-skill firms derive benefits from entering the central city that outweigh their costs, while medium-skill firms incur costs exceeding their benefits, changes in urban costs minimally affect the evolutionary speed and outcomes for high-skill and low-skill firms. In contrast, the impact on medium-skill firms is more pronounced; an increase in urban costs accelerates their decision not to enter the central city. because higher urban costs amplify the relative disadvantages of medium-skill firms, prompting quicker market exit decisions.

Fig. 5: Impact of urban cost changes on stable strategy choices.
figure 5

a Under Condition 1, the stable strategy choices of firms with different skill levels as urban costs increase from low to high. b Under Condition 2, the stable strategy choices of firms with different skill levels as urban costs increase from low to high. c Under Condition 3, the stable strategy choices of firms with different skill levels as urban costs increase from low to high.

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Under Condition 2, where medium-skill firms’ benefits from producing in the central city exceed their costs, while low-skill firms incur costs exceeding their benefits, urban cost changes significantly influence the evolutionary speed of both medium-skill and low-skill firms. Specifically, rising urban costs slow the entry of medium-skill firms into the central city. This is likely because higher costs erode the net benefits of entering the central city, requiring medium-skill firms to take additional time to devise cost-control strategies to remain competitive. Conversely, rising urban costs prompt low-skill firms to abandon plans for entering the central city more quickly. This is likely because higher urban costs raise entry barriers for low-skill firms, encouraging them to operate in lower-cost peripheral markets instead.

Under Condition 3, where all firm types generate positive returns in the central city, high-skill and medium-skill firms quickly reach equilibrium. However, rising urban costs noticeably slow the rate at which low-skill firms reach equilibrium. This is likely because rising urban costs undermine the competitiveness of low-skill firms, weakening their foothold in central city markets and requiring more time for strategic adjustments.

Conclusions

This study analyzes an evolutionary game model to reveal the spatial dynamics of firms with varying skill levels within urban clusters. The analysis identifies three stable equilibrium points under different conditions. To validate the model’s stability and analytical accuracy, specific parameters aligned with real-world economic scenarios were set based on prior assumptions, followed by simulations. The results show that the model performs consistently with theoretical predictions under various constraints, confirming the robustness of the tripartite evolutionary game model.

Numerical experiments further reveal that while initial participation willingness, skill differences, and urban costs influence the evolutionary process and outcomes, these factors do not fundamentally change firms’ ultimate strategic choices. The study finds that the effect of skill differences on firms’ entry and exit decisions varies by condition: Under Condition 1, the impact on all firm types is relatively small. Under Condition 2, skill differences significantly affect low-skill firms that opt out of entering the central city, prompting them to make quicker strategic decisions. Similarly, under Condition 3, skill differences strongly influence the entry speed of low-skill firms, with larger skill gaps leading to slower entry rates.

These findings have important implications for urban planning and policy-making. First, high-skill firms tend to cluster in large cities, making it challenging for non-central cities to achieve their industrial relocation goals. Instead, these cities often attract low-skill firms displaced by competition in larger cities. This outcome not only hinders regional coordination but also exacerbates disparities between cities. Therefore, non-central governments should prioritize enhancing firm quality over quantity by attracting high-quality firms to drive long-term economic growth.

Second, current household registration policies, such as China’s HUKOU system, partially obstruct the spatial selection mechanism. The study reveals that central cities have strong spatial selection capabilities, yet the high barriers of the HUKOU policy discourage high-skill labor from entering, hindering their high-quality development. Central cities should reform HUKOU policies by relaxing restrictions and lowering barriers to basic public services, promoting labor mobility and intercity integration. Improving resource allocation efficiency could foster more balanced urban development and enhance overall city competitiveness.

Finally, the dynamic interplay between spatial sorting and spatial selection suggests a cyclical, symbiotic relationship within urban economies. The continuous entry of high-skill firms not only boosts productivity and specialization in central cities but also expands these cities’ scale, thereby attracting new firms and intensifying competition. This iterative process reinforces the agglomeration economies and innovation potential of high-skill firms in larger urban centers. Therefore, policymakers should fully consider the diverse needs and competitive dynamics of firms with different skill levels when designing and implementing economic policies. In particular, while promoting central cities as innovation hubs for high-skill firms, it is also crucial to provide support mechanisms for medium- and low-skill firms, such as developing infrastructure in peripheral areas or offering targeted subsidies, to foster regional coordinated development.