Demand-Aware Subsidy Optimization for Community-Based Elderly Care Services: A Data-Driven Stackelberg Game Model

Abstract

Efficient subsidy allocation for heterogeneous community-based elderly care services requires identifying latent service demand and optimizing service providers’ responses under fiscal constraints. To address this problem, this study develops a data-driven Stackelberg game framework for demand-aware subsidy optimization. The framework transforms unstructured user-generated reviews into empirical demand-priority weights through Latent Dirichlet Allocation (LDA) topic modeling and incorporates these data-derived parameters into a budget-constrained Stackelberg game to optimize differentiated subsidy decisions. A total of 11,257 elderly care-related reviews from e-commerce platforms were collected and refined into 3907 valid samples. Six major service needs were identified: Daily Living Care, Meals and Nutrition, Smart Safety Monitoring, Medical Assistance, Professional Nursing, and Home Rehabilitation. These demand weights were incorporated into the optimization model to characterize heterogeneous service priorities and determine equilibrium subsidy decisions. The numerical experiments show that differentiated subsidies substantially improve demand-weighted service coverage compared with demand-unaware allocation under the same fiscal constraint. At a budget of RMB 300,000, the demand-aware model achieves a demand-weighted coverage of 0.8328, compared with 0.7309 for the demand-unaware model. Budget sensitivity analysis further shows diminishing marginal returns to fiscal expansion, with full potential demand reached at the model-based threshold of RMB 479,500. Robustness analysis indicates that the main allocation results remain stable under perturbations of the empirically derived demand-priority weights.

1. Introduction

China is experiencing rapid population aging, accompanied by a growing number of disabled and semi-disabled older adults and the continuous decline of traditional family-based care due to shrinking household size and increasing numbers of empty-nest families. Against this background, community-based elderly care has become a fundamental component of China’s elderly care system, while smart community elderly care driven by digital technologies, big data, and artificial intelligence has emerged as an important direction for service transformation. Existing studies have investigated smart elderly care from the perspectives of policy systems, digital platforms, and service mechanisms, demonstrating that effective implementation depends not only on digital infrastructure but also on policy support, organizational coordination, and user capabilities [1,2]. Meanwhile, advances in artificial intelligence, machine learning, and sensor technologies have further promoted smart elderly care toward intelligent sensing, health monitoring, and data-driven decision-making, providing a technological foundation for more precise elderly care services [3].

Despite these developments, substantial supply–demand mismatches remain a major challenge for community-based elderly care. On the supply side, elderly care services are often homogeneous and insufficiently adapted to diverse user needs, while on the demand side, service preferences and resource requirements vary considerably across service types and user groups. Spatial analyses have revealed significant disparities in the distribution of elderly care facilities, particularly in suburban and underdeveloped areas [4]. Moreover, the adoption of smart elderly care services is influenced by factors such as age, health status, and economic capability, resulting in heterogeneous service demands [5]. Recent studies have employed big data techniques to examine elderly welfare service utilization and its influencing factors, providing new perspectives for demand identification [6]. However, identifying heterogeneous demand alone does not directly resolve the subsequent resource allocation problem. A key challenge is therefore how to translate observed user demand into quantitative parameters that can inform subsidy decisions under limited fiscal resources.

Existing research provides valuable insights into both elderly care demand identification and subsidy optimization, but these two streams have rarely been connected through a unified quantitative framework. On the demand side, most studies rely on questionnaires, interviews, and other structured data sources, which may have limited ability to capture fine-grained and naturally expressed service preferences. Recently, natural language processing and text mining have been introduced into smart community research to identify public concerns from online textual information [7]. Topic modeling and generative artificial intelligence have also been applied to healthcare review analysis and online health consultation services [8]. Several studies have further attempted to identify elderly care service demands using text mining [9,10]. However, these studies mainly focus on demand classification or descriptive analysis, and the resulting demand information is rarely converted into quantitative parameters for subsequent supply-side decision-making. Thus, although these approaches provide increasingly detailed evidence on what users demand, their implications for quantitative resource allocation remain limited.

On the optimization side, game-theoretic approaches, particularly Stackelberg models, have been widely used to examine elderly care subsidy design, market-oriented elderly care transformation, and smart elderly care supply chain coordination. These studies demonstrate that government subsidies can influence service provision and improve social welfare [11,12]. Nevertheless, most existing optimization models formulate demand-related parameters based on theoretical assumptions or conventional statistical information, rather than explicitly deriving them from actual user-generated data. As a result, the optimization literature provides mechanisms for determining how subsidies should be allocated, but generally pays less attention to how empirically observed and heterogeneous user demands can be incorporated into these allocation decisions. Thus, a gap remains between what users demand and how limited public resources should be optimally allocated.

This gap leads to the following central research question: How can heterogeneous service demands inferred from unstructured user-generated data be quantitatively incorporated into a subsidy optimization model under limited fiscal resources?

To address this question, this study develops a data-driven, demand-aware subsidy optimization framework based on a budget-constrained Stackelberg game. The framework first transforms unstructured online reviews into empirical demand-priority weights through Latent Dirichlet Allocation (LDA) topic modeling and then incorporates these data-derived parameters into the Stackelberg game to characterize government subsidy decisions and service providers’ responses under fiscal constraints. Based on the resulting equilibrium, uniform and demand-weighted differentiated subsidy strategies are compared to examine their effects on service coverage and social welfare.

The main contributions of this study are threefold.

(1)

Data-driven demand parameterization. Rather than simply combining text mining with game-theoretic optimization, this study develops a parameterization mechanism that transforms unstructured user-generated content into empirical demand-priority weights and explicitly embeds these weights into both the provider-side demand-matching benefit and the government’s demand-coverage objective. This creates a quantitative linkage between observed demand signals and endogenous subsidy decisions.

(2)

Demand-aware Stackelberg optimization. This study incorporates empirically derived demand heterogeneity, service-specific costs, and fiscal constraints into a Stackelberg game to optimize differentiated subsidy decisions while accounting for service providers’ strategic responses.

(3)

Policy and sensitivity analysis. This study provides policy-oriented evidence on how fiscal constraints, demand heterogeneity, and service cost structures influence optimal differentiated subsidies and social welfare. Sensitivity analyses identify the conditions under which demand-aware subsidy policies remain effective, offering practical guidance for adjusting subsidy intensity and allocation across service categories.

The empirical application to community-based elderly care illustrates how real-world user-generated data can be transformed into actionable demand parameters for budget-constrained subsidy optimization. More broadly, the framework provides a data-driven approach for integrating heterogeneous user demand into strategic resource allocation, with community-based elderly care serving as an important application context.

2. Data-Driven Demand Parameterization from User-Generated Content

This section develops a data-driven demand parameterization procedure that transforms unstructured online reviews into quantitative inputs for the subsequent subsidy optimization model. Rather than interpreting topic proportions as population-level demand prevalence, this study uses them to characterize the relative prominence of different elderly care needs reflected in user-generated content. Specifically, Latent Dirichlet Allocation (LDA) is employed to identify latent demand themes from online reviews, after which the relative prevalence of the identified topics is normalized and used to construct empirical demand-priority weights. These weights serve as behavioral indicators of relative demand salience and are subsequently incorporated into the Stackelberg game as demand-related parameters.

The analytical procedure consists of four steps: data collection and preprocessing, latent demand identification, construction of demand-priority weights, and robustness validation. This procedure establishes the link between unstructured user-generated information and the quantitative subsidy optimization model developed in Section 3.

2.1. Data Collection and Preprocessing

The data for this study were sourced from user reviews on the Taobao and Tmall e-commerce platforms, covering products and services in the elderly care sector, including smart assistive devices (such as smart home monitoring systems and alarms for the elderly), home care products (such as multifunctional beds and wheelchairs), health monitoring devices (such as blood pressure monitors and blood glucose meters), and in-home care services. For each product category, the top three best-selling items were selected, and a total of 11,257 raw reviews were collected using the Octopus crawler over a period of nearly three years.

The top three best-selling products within each category were selected primarily to ensure sufficient review volume and textual richness for topic modeling. This sampling strategy provides a consistent rule across product categories and reduces the risk of sparse review records. Nevertheless, because high-selling products may differ systematically from less popular products, this design may introduce product-level selection bias. The resulting demand-priority weights should therefore be interpreted as empirical indicators of demand salience within the sampled review corpus rather than population-representative estimates. Because the current dataset is restricted to the predefined top-three product sample, alternative product-sampling strategies could not be directly evaluated in this study and are left for future research.

To ensure data quality, this study employed Python 3.12 to systematically clean the raw reviews. A seven-step cleaning process was designed; the specific steps and results are shown in Table 1.

Table 1. Steps and results of comment data cleaning.

Step	Operation	Before	After	Removed
1	Remove comments with empty values	11,257	11,255	2
2	Filter out very short comments (<5 words)	11,255	10,639	616
3	Remove completely duplicate comments	10,639	10,516	123
4	Filter out comments with no Chinese content	10,516	10,508	8
5	Filter out advertising/fake review comments	10,508	10,421	87
6	Filter out comments unrelated to senior care	10,421	4099	6322
7	Filter out comments with empty text after tokenization	4099	3907	192

As shown in Table 1, the raw data contained a significant number of duplicate, overly brief, and empty comments, as well as some fake reviews created for advertising purposes. To remove reviews unrelated to the elderly care domain, this study developed a domain-specific keyword dictionary and regular-expression filtering rules covering terms such as “elderly,” “care,” “rehabilitation,” “monitoring,” and “safety.” Based on these predefined elderly care relevance criteria, 6322 reviews that did not satisfy the required domain-related criteria were excluded, accounting for 56.2% of the original total. This filtering step was conducted before tokenization and topic modeling to prevent unrelated reviews from affecting the identified latent topic structure. After the complete data-cleaning process, 3907 valid elderly care-related reviews were ultimately retained, representing a retention rate of 34.7%.

Chinese word segmentation was performed using the Jieba library with part-of-speech (POS) tagging. A three-stage filtering strategy was applied: (1) POS filtering retained only content words, including nouns, verbs, and adjectives; (2) stop-word filtering employed an enhanced stop-word list containing 199 entries; and (3) word-length filtering removed all single-character tokens. In addition, a custom dictionary comprising 84 domain-specific terms related to elderly care was constructed, in which expressions such as “smart elderly care” and “disability care” were incorporated as unified lexical entries. During corpus construction, the dictionary was further refined by setting the minimum document frequency to $no\text{\_}below = 5$ and the maximum document frequency to $no\text{\_}above = 0.50$, resulting in a final vocabulary consisting of 922 feature terms. The resulting corpus is used to identify latent demand themes rather than to estimate the population-level prevalence of elderly care needs.

2.2. Latent Demand Identification via LDA Topic Modeling

LDA is an unsupervised generative probabilistic topic model based on the assumption [13] that each document can be represented as a mixture of multiple latent topics, with each topic corresponding to a probability distribution over words. In this study, the $c_v$ topic coherence score was adopted as the primary evaluation criterion. The candidate number of topics was set as $k \in \{3, 4, 5, 6\}$, and an LDA model was trained for each candidate k value with the following configurations: $random\text{\_}state = 100$, $passes = 15$, and adaptive settings for both alpha and eta parameters.

The results presented in Table 2 indicate that the $c_v$ score reaches its highest value of 0.4943 at $k=6$, although the difference between $k=5$ (0.4934) and $k=6$ is relatively small. Therefore, the final topic number was not determined by coherence alone. To further assess the robustness of the two candidate solutions, the LDA models with $k=5$ and $k=6$ were repeatedly estimated using five different random seeds (100, 200, 300, 400, and 500), while keeping the other model configurations unchanged. The six-topic solution achieved a higher mean coherence score (0.5020 vs. 0.4793) and a lower standard deviation (0.0100 vs. 0.0125) than the five-topic solution. It also showed slightly higher cross-run topic similarity and top 10 keyword overlap than the five-topic solution. These results provide additional support for the relative robustness of the six-topic solution. Moreover, the six-topic solution yields six substantively distinguishable demand themes, namely Daily Living Care, Meals and Nutrition, Smart Safety Monitoring, Medical Assistance, Professional Nursing, and Home Rehabilitation. Considering the coherence results, robustness across different random seeds, and the interpretability of the resulting demand themes jointly, $k=6$ was ultimately selected.

Table 2. LDA topic number selection results.

Number of Topics k	Coherence Value (${\mathit{c}}_{\mathit{v}}$)	Log Perplexity	Remarks
3	0.4596	−6.8606	—
4	0.4684	−6.9106	—
5	0.4934	−6.9246	Significant improvement in Coherence
6	0.4943 (highest)	−6.9049	Ultimately selected

2.3. Construction of Demand-Priority Weights

The topic proportions obtained from LDA are not interpreted as the actual prevalence of elderly care needs in the general population. Because the data were collected from e-commerce reviews, the observed topic distribution may be affected by platform selection, product purchasing behavior, and users’ digital engagement. Therefore, the resulting weights are defined as data-derived relative demand-priority weights, which capture the relative salience of different demand themes in the observed user-generated content.

Let $N$ denote the total number of valid reviews, and let ${\theta }_{di}$ denote the LDA topic proportion of review $d$ for topic $i$, where $d=1,\dots ,N$ and $i=1,\dots ,M$. To obtain a single primary demand category for each review, the topic with the highest topic proportion is assigned as the primary topic $z_d=\arg \underset{i}{\mathrm{m}\mathrm{a}\mathrm{x}}{\theta }_{di}$.

Let $n_i$ denote the number of reviews whose primary topic is $i$. It is therefore calculated as $n_i={\displaystyle {\sum }_{d=1}^N}1\left(z_d=i\right)$, where $1\left(\cdot \right)$ is the indicator function. The empirical demand-priority weight of service category $i$ is then defined as $w_i={\displaystyle \frac{n_i}{N}},i=1,\dots ,M.$

In this study, six demand categories are identified, so $M=6$. Each valid review is assigned to one primary demand category, as shown in Equation (1), and consequently, ${\displaystyle {\sum }_{i=1}^6}{w}_i=1,w_i\ge 0.$

$${\displaystyle \sum _{i=1}^6}{n}_i=N$$

(1)

A larger $w_i$ indicates that the corresponding demand theme appears more prominently in the observed user-generated content and is therefore assigned a higher priority in the subsequent demand-aware subsidy optimization model. Importantly, these weights should not be interpreted as unbiased estimates of population-level service demand. Instead, they provide an empirical representation of the relative demand salience observed in the collected online reviews.

Based on the semantic interpretation of the high-frequency keywords and the practical context of smart community-based elderly care, the six LDA topics are mapped to the following service categories: Daily Living Care, Meals and Nutrition, Smart Safety Monitoring, Medical Assistance, Professional Nursing, and Home Rehabilitation. Labels were assigned based on the semantic consistency of the highest-weighted keywords and their correspondence with recognized elderly care service functions. Their corresponding empirical demand-priority weights are $0.2838$, $0.2117$, $0.2012$, $0.1305$, $0.1065$, and $0.0663$, respectively.

As a robustness check, an alternative set of demand-priority weights was calculated using the full topic-probability distribution. Although the weight magnitudes differ from those obtained using the dominant-topic approach, the ranking of all six demand themes remains unchanged, indicating that the relative demand-priority structure is robust to the weighting method. The dominant-topic approach is retained because it provides mutually exclusive service categories and a clear mapping to the subsequent subsidy optimization model. These weights are subsequently incorporated into the Stackelberg model as demand-side priority parameters. The results are shown in Table 3.

Table 3. Top 10 keywords and their document proportions for each theme.

Topic ID	Topic Label	Documents	Weight	Top 10 Keywords (Weight)
Topic 1	Professional Nursing	416	10.65%	Caregiving (0.056); Convenience (0.052); Recovery (0.050); Hygiene (0.038); Relief (0.036); Stress (0.033); Eating (0.030); Waste (0.030); Prevention (0.020); Gentle (0.019)
Topic 2	Medical Assistance	510	13.05%	Health (0.085); Comfort (0.082); Hospital (0.031); Pressure Sores (0.026); Air Mattress (0.021); Medication (0.021); Doctor (0.021); Mood (0.020); Voice (0.018); Large Print (0.018)
Topic 3	Home Rehabilitation	259	6.63%	Rehabilitation (0.091); Living Alone (0.052); Moderate (0.044); Elderly (0.042); Security (0.035); Elders (0.033); Weak (0.032); Kitchen (0.031); Stable (0.028); Care (0.018)
Topic 4	Smart Safety Monitoring	786	20.12%	Safety (0.113); Satisfaction (0.029); Monitoring (0.028); Different (0.027); Sleep (0.026); Functionality (0.024); Light (0.023); Going Out (0.022); Automatic (0.021); One-Touch (0.021)
Topic 5	Meals and Nutrition	827	21.17%	Nutrition (0.089); Practical (0.074); Formula (0.049); Fresh (0.037); Taste (0.029); Reliable (0.026); Health Preservation (0.025); Medical Care (0.024); Mother (0.021); Diet (0.018)
Topic 6	Daily Living Care	1109	28.38%	Home Delivery (0.052); Bedridden (0.043); Comfort (0.031); Getting Up (0.027); Flavor (0.027); Portion Size (0.026); Older Adults (0.026); Effective (0.025); Family (0.024); Non-Slip (0.023)

These data-derived relative demand-priority weights provide the empirical demand-side input for the Stackelberg subsidy optimization model developed in Section 3.

2.4. Validation and Robustness of Demand Weights

To examine whether the subsequent subsidy allocation is sensitive to moderate uncertainty in the empirically derived demand-priority weights, a perturbation-based robustness analysis is conducted. Specifically, each baseline weight $w_i$ is independently perturbed by a random factor, as shown in Equation (2):

$$w_i^{\prime }{=w}_i\left(1+{ϵ}_i\right)$$

(2)

where ${ϵ}_i~U(-\delta ,\delta )$.

Because the perturbed weights no longer necessarily sum to one, they are normalized, as shown in Equation (3):

$${\tilde{w}}_i={\displaystyle \frac{w_i^{\prime }}{{\sum }_{j=1}^M{w}_j^{\prime }}}$$

(3)

The normalized vector $\tilde{w}$ is then used as the demand-priority vector in the subsequent Stackelberg model. To maintain consistency between the demand-priority weights and the convex cost specification, the category-specific marginal cost coefficients are recalibrated according to Equation (4):

$${\tilde{\gamma }}_i={\displaystyle \frac{\alpha {\tilde{w}}_i}{D_i}}$$

(4)

where $\alpha$ is the demand-matching benefit coefficient, and $D_i$ denotes the potential demand upper bound for service category $i$.

Thus, the perturbed demand weights affect the equilibrium through both the demand-priority parameters and the corresponding convex cost coefficients.

For each perturbed weight vector, the provider’s optimal supply and the government’s optimal subsidy allocation are recalculated under the same fiscal budget. The stability of the optimized service coverage and subsidy allocation is then evaluated across the resulting equilibrium solutions.

Three perturbation levels are considered $\delta \in \left\{0.05,0.10,0.20\right\}$, corresponding to $\pm 5\%$, $\pm 10\%$, and $\pm 20\%$ deviations from the baseline weights. For each perturbation level, repeated random simulations are conducted under the same fiscal budget, and the stability of the optimized service coverage and subsidy allocation is evaluated across the resulting equilibrium solutions.

This analysis does not attempt to establish the statistical representativeness of the online review data. Rather, it examines whether the model-based numerical results remain stable when the empirically derived demand-priority weights are subject to moderate uncertainty. A relatively stable equilibrium allocation and service coverage across perturbation scenarios would provide evidence that the proposed demand-aware subsidy framework is not overly sensitive to small variations in the estimated demand-priority weights.

3. Demand-Aware Subsidy Optimization: A Stackelberg Game Model

Building on the data-derived relative demand-priority weights identified in Section 2, this section develops a demand-aware Stackelberg game model for differentiated elderly care subsidies. The government acts as the leader and determines category-specific unit subsidies, while service providers act as followers and choose their service quantities. A convex variable cost function is introduced to capture increasing marginal costs and ensure a unique provider best response. The model is solved by backward induction: the provider’s optimal supply response is first derived, followed by the government’s budget-constrained demand-coverage optimization problem. The resulting Stackelberg equilibrium establishes an analytical link between online review-based demand information and differentiated subsidy decisions.

3.1. Problem Definition

This section formulates the elderly care subsidy allocation problem as a two-stage Stackelberg game between the government and elderly care service providers. Consider elderly care service categories identified from the LDA analysis in Section 2. Let $w_i$ denote the data-derived relative demand-priority weight of service category $i$, $s_i$ denote the unit government subsidy for category $i$, and $q_i$ denote the annual service supply quantity, where $i=1,\dots ,6$.

The government and elderly care service providers have different objectives. The government seeks to maximize the overall demand-weighted coverage of elderly care services while satisfying a limited fiscal budget. Service providers, in contrast, seek to maximize their operating profits by determining the quantity of each service category supplied in response to the subsidy policy. The game consists of two sequential stages.

Stage 1: Government decision. The government determines the differentiated subsidy vector $s=s_1,\dots ,s_M$, where $s_i\ge 0$ represents the subsidy provided for each unit of service category $i$.

Stage 2: Provider decision. After observing the subsidy vector, elderly care service providers determine the corresponding supply vector $q=q_1,\dots ,q_M$, where $q_i$ denotes the annual supply quantity of service category $i$.

The government anticipates the provider’s optimal response when selecting the subsidy vector. Therefore, the equilibrium is obtained through backward induction: the follower’s optimal response is first derived, followed by the leader’s subsidy optimization problem.

To capture the endogenous limitation of service expansion, a convex variable cost function is introduced. The convex cost structure generates increasing marginal operating costs and ensures a unique optimal supply response for each service category. In addition, the potential demand scale $D_i$ is introduced as an upper bound on effective service supply, preventing the model from generating service quantities beyond the potential demand of a category.

3.2. Model Assumptions and Notation Definition

The Stackelberg model is developed based on assumptions concerning demand heterogeneity, provider cost structure, subsidy mechanisms, potential demand capacity, and government fiscal constraints. These assumptions provide the behavioral and economic foundations for the subsequent derivation of the provider’s response function and the government’s optimal subsidy policy.

Assumption 1.

Rational decision-making.

The government and elderly care service providers are assumed to be rational decision makers. Each player chooses its decision variable to maximize its respective objective given the relevant model parameters and the decision of the other player.

Assumption 2.

Independent service categories.

The six elderly care service categories are modeled separately. The supply decision for one service category does not directly constrain the supply quantity of another category. Cross-category substitution and complementarity effects are not explicitly considered. This assumption is adopted to maintain model tractability and isolate the effect of demand heterogeneity on subsidy allocation.

Assumption 3.

Demand heterogeneity.

The relative importance of each service category is represented by the data-derived relative demand-priority weight $w_i$ obtained from the LDA analysis in Section 2. These weights satisfy $w_i\ge 0, {\displaystyle {\sum }_{i=1}^6}{w}_i=1$. The weights measure the relative prominence of the corresponding demand themes in the observed online review corpus rather than the actual prevalence of these needs in the overall elderly population.

Assumption 4.

Increasing marginal service costs.

The variable cost of providing service category $i$ increases with the supply quantity. The total cost of the provider is specified, as shown in Equation (5):

$$C\left(q\right)=F+{\sum }_{i=1}^6(c_i{q}_i+{\displaystyle \frac{1}{2}}{\gamma }_i{q}_i^2), {\gamma }_i>0$$

(5)

where $F$ is the annual fixed operating cost of elderly care service providers, $c_i$ is the unit variable supply cost of service category $i$, and ${\gamma }_i>0$ is the convex cost coefficient. The marginal variable cost of service category $i$ is shown in Equation (6). Because ${\gamma }_i>0$, the marginal operating cost increases with the supply quantity.

$$M{C}_i\left(q_i\right)=c_i+{\gamma }_i{q}_i$$

(6)

Assumption 5.

Demand-matching benefit.

Service provision in a category with a higher demand priority generates a greater demand-matching benefit, represented by ${\alpha w}_i{q}_i$. Here, $w_i$ is dimensionless and represents the relative demand priority of category $i$, while $\alpha$ converts this demand signal into a benefit per service unit with the same monetary unit as $p_i$ and $c_i$. Thus, ${\alpha w}_i{q}_i$ represents a reduced-form demand-matching benefit in the provider’s profit function rather than a direct government payment or observed market revenue. The parameter is set to $\alpha = 0.5$ with reference to the calibration results [14]. This value serves as a benchmark calibration for the present model rather than an empirically estimated structural parameter for the current dataset.

Assumption 6.

Fiscal budget constraint.

The government has a fixed annual subsidy budget $B$. Therefore, total subsidy expenditure cannot exceed the available fiscal resources as ${\displaystyle {\sum }_{i=1}^6}{s}_i{q}_i\le B$.

Assumption 7.

Potential demand scale and parameter calibration.

Let $D_i$ denote the potential market demand scale or benchmark demand capacity for service category $i$. The effective supply of each service category cannot exceed its potential demand as $0\le q_i\le D_i,i=1,\dots ,6.$

To ground the convex cost parameter ${\gamma }_i$ in the potential service capacity, it is assumed that when the unit government subsidy compensates for the unit operating deficit, $s_i=c_i-p_i$, the provider’s optimal supply quantity reaches the potential demand scale $D_i$.

This establishes the parameter calibration identity, ${\gamma }_i={\displaystyle \frac{\alpha w_i}{D_i}}$. This calibration links the convex cost coefficient to both the observed demand-priority weight and the potential service capacity.

The parameter settings are summarized in Table 4.

Table 4. Core model parameters and variable definitions.

Type	Symbol	Definition and Description	Basis for Parameter Setting
Demand Parameters	$w_i$	Data-derived relative demand-priority weight of the $i$-th elderly care service category ($i = 1, 2, \dots , 6$)	Results of the LDA model presented in Section 2
	$D_i$	Potential market demand scale of the $i$-th elderly care service category (service instances/year)	Estimated based on community elderly population size and relative demand structure represented by $w_i$
Cost Parameters	$p_i$	Unit market price of the $i$-th elderly care service category (RMB/hour or RMB/service instance)	Referenced from the State Administration for Market Regulation and Standardization Administration of the People’s Republic of China [15] and local civil affairs departments’ service pricing guidelines
	$c_i$	Unit variable supply cost of the $i$-th elderly care service category (RMB/hour or RMB/service instance)	Referenced from the elderly care service cost parameters reported by Guo and Wang [16]
	${\gamma }_i$	Convex cost coefficient of the $i$-th elderly care service category, capturing increasing marginal supply costs	Calibrated via ${\gamma }_i={\displaystyle \frac{\alpha w_i}{D_i}}$ based on the potential demand scale $D_i$ and demand-matching coefficient $\alpha$
	$F$	Annual fixed operating cost of elderly care service providers (RMB 10,000/year)	Determined based on the fixed cost parameter range [16] using the median value
Matching Parameter	$\mathsf{A}$	Demand-matching benefit conversion coefficient	Referenced from the calibration results of the dynamic game model for elderly care services [14]
Decision Variable	$s_i$	Unit government subsidy for the $i$-th elderly care service category (RMB/service instance)	Government decision variable determined through equilibrium analysis
	$q_i$	Annual service supply quantity of the $i$-th elderly care service category provided by elderly care institutions (service instances/year)	Institutional decision variable determined through equilibrium analysis
Budget Parameter	$\mathrm{B}$	Total annual government fiscal budget for elderly care service subsidies (10,000 RMB/year)	Estimated based on the 2024 project standards [17] (3000 RMB/person) and the estimated service coverage scale

Remark on Parameter Calibration: The benchmark calibration ${\gamma }_i={\displaystyle \frac{\alpha w_i}{D_i}}$ anchors the convex cost parameter to potential capacity. Under full coverage ($q_i^{*}=D_i$), the optimal subsidy simplifies to $s_i^{*}=c_i-p_i$. Economically, this implies that when fiscal resources are sufficient to satisfy 100% of potential demand across all categories, the government must fully cover the operating deficit ($c_i-p_i$) of each service. In this extreme equilibrium state, the relative demand weight $w_i$ no longer dictates the subsidy level. However, under constrained fiscal budgets ($B<B_{full}$), $w_i$ governs the optimal shadow prices, priority ranks, and resource allocation trajectories across categories.

3.3. Provider’s Best-Response Function

The Stackelberg game model is solved through backward induction. First, the follower’s optimization problem is solved to obtain the elderly care service provider’s optimal response to the government subsidy. Subsequently, this response function is incorporated into the government’s subsidy optimization problem.

Given the subsidy vector $s=s_1,\dots ,s_M$ determined by the government, elderly care service providers choose the optimal supply quantity for each service category. Under the proposed convex cost formulation, the provider’s profit is given by Equation (7):

$$\pi \left(q,s\right)={\displaystyle \sum _{i=1}^6}[\left(p_i+s_i+\alpha w_i-c_i\right)q_i-{\displaystyle \frac{1}{2}}{\gamma }_i{q}_i^2]-F$$

(7)

where $p_i{q}_i$ represents service revenue, $s_i{q}_i$ represents government subsidy revenue, ${\alpha w_{i}q}_i$ represents the demand-matching benefit associated with service provision, $c_i{q}_i$ is linear variable operating costs, ${\displaystyle \frac{1}{2}}{\gamma }_i{q}_i^2$ is increasing marginal operating costs, and F represents the provider-level fixed operating cost.

For a given subsidy $s_i$, the first-order derivative of provider profit with respect to $q_i$ is shown in Equation (8):

$${\displaystyle \frac{\partial \pi }{\partial q_i}}=p_i+s_i+\alpha w_i-c_i-{\gamma }_i{q}_i$$

(8)

The second-order derivative is shown in Equation (9):

$${\displaystyle \frac{{\partial }^2\pi }{\partial q_i^2}}=-{\gamma }_i<0$$

(9)

Therefore, the provider’s profit function is strictly concave in $q_i$, ensuring a unique optimal supply quantity for each service category.

Solving the first-order condition in Equation (8) gives the unconstrained optimal quantity in Equation (10):

$$q_i={\displaystyle \frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}}$$

(10)

Considering both the non-negativity constraint and the potential demand capacity constraint $q_i\ge 0$, the provider’s best-response function is shown in Equation (11):

$$q_i^{*}\left(s_i\right)=\min \left\{D_i,\max \left[0,{\displaystyle \frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}}\right]\right\}$$

(11)

Equation (11) indicates that the provider’s supply response has three regions. First, when $p_i+s_i+\alpha w_i-c_i\le 0$, the provider supplies zero units. Second, when $0<{\displaystyle \frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}}<D_i$, the provider supplies an interior quantity that increases continuously with the subsidy. Third, when the unconstrained optimal quantity exceeds $D_i$, supply is capped at the potential demand capacity $q_i^{*}\left(s_i\right)=D_i$. Positive service supply occurs when $p_i+s_i+\alpha w_i-c_i>0,$ or equivalently, $s_i>c_i-p_i-\alpha w_i.$

Therefore, the minimum subsidy required to induce a positive supply is $\overline{s_i}=c_i{-p}_i-\alpha w_i$.

The value ${\overline{s}}_i$ represents the zero-supply threshold rather than a full-capacity supply condition. Once the subsidy exceeds this threshold, supply increases continuously with the subsidy until the potential demand capacity $D_i$ is reached. For an active service category operating in the interior region, the subsidy responsiveness is ${\displaystyle \frac{\partial q_i^{*}}{\partial s_i}}={\displaystyle \frac{1}{{\gamma }_i}}>0$.

Thus, a larger ${\gamma }_i$ implies a stronger increase in marginal operating cost and consequently a smaller supply response to a given subsidy.

Specifically, when the government subsidy compensates for the unit operating deficit, $s_i=c_i-p_i$, the provider’s optimal response in Equation (11) becomes $q_i^{*}={\displaystyle \frac{\alpha w_i}{{\gamma }_i}}$. Aligning this benchmark supply response with the community’s potential demand scale $D_i$ yields the calibration formula ${\gamma }_i={\displaystyle \frac{\alpha w_i}{D_i}}$. This ensures that the convex cost parameter reflects both the demand priority of each service category and its potential service capacity.

3.4. Government’s Optimal Subsidy Problem

Anticipating the providers’ best-response functions, the government determines the differentiated subsidy vector $s$ to maximize overall demand-weighted service coverage subject to its fiscal budget.

For service category $i$, the realized service coverage ratio is defined as ${\rho }_i={\displaystyle \frac{q_i}{D_i}},0\le {\rho }_i\le 1.$

The demand-weighted overall service coverage is therefore defined, as shown in Equation (12):

$$DC\left(q\right)={\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i}{D_i}}$$

(12)

Because ${\displaystyle {\sum }_{i=1}^6}{w}_i=1$ and $0\le q_i\le D_i,$ the demand-weighted coverage satisfies $0\le DC\left(q\right)\le 1.$ A higher value of $DC\left(q\right)$ indicates a higher level of demand fulfillment. In particular, $DC\left(q\right)=1$ indicates that the potential demand of all six service categories is fully covered.

The use of $w_i$ in Equation (12) allows the government to assign greater importance to service categories with higher demand priority. Therefore, the LDA-derived demand-priority weights directly enter the government decision problem rather than serving only as descriptive statistics.

The government’s optimization problem is formulated as Equation (13):

$$\underset{s,q}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i}{D_i}}$$

(13)

$$\begin{gathered} s.t. {\displaystyle \sum _{i=1}^6}{s}_i{q}_i\le B,0\le q_i\le D_i, i=1,\dots ,6, \\ {s}_i\ge 0,i=1,\dots ,6 \end{gathered}$$

(14)

where $B$ denotes the government’s annual subsidy budget. Constraints in Equation (14) ensure that total subsidy expenditure does not exceed the available fiscal resources and prevent effective service supply from exceeding the potential demand capacity of each category, while also imposing the non-negativity of government subsidies.

The government does not directly determine $q_i$. Instead, service quantities are endogenously determined by the providers’ best-response functions. Substituting Equation (11) into the government’s objective yields the reduced-form problem as Equation (15):

$$\underset{s_i\ge 0}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{\min \left\{D_i,\max \left[0,\frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}\right]\right\}}{D_i}}$$

(15)

$$s.t. {\displaystyle \sum _{i=1}^6}{s}_i\min \left\{D_i,\max \left[0,{\displaystyle \frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}}\right]\right\}\le B$$

(16)

These equations explicitly establish the leader–follower relationship in the model: the government chooses subsidies while anticipating the endogenous supply responses of providers.

For an active service category operating in the interior region, $0<{\displaystyle \frac{p_i+s_i+\alpha w_i-c_i}{{\gamma }_i}}<D_i,$

Equation (11) can be rearranged as Equation (17):

$$s_i=c_i-p_i-\alpha w_i+{\gamma }_i{q}_i$$

(17)

Accordingly, the subsidy expenditure for category $i$ can be expressed as $s_i{q}_i=\left(c_i-p_i-\alpha w_i\right)q_i+{\gamma }_i{q}_i^2$.

Therefore, for interior active categories, the government’s optimization problem can equivalently be represented in terms of service quantities as Equation (18):

$$\begin{gathered} \underset{q}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i}{D_i}} \\ s.t. {\displaystyle \sum _{i=1}^6}\left[\left(c_i-p_i-\alpha w_i\right)q_i+{\gamma }_i{q}_i^2\right]\le B, 0\le q_i\le D_i \end{gathered}$$

(18)

Equation (18) shows that the government allocates its limited fiscal resources according to the demand-weighted service coverage generated by the subsidy-induced supply response.

To rigorously characterize the optimal subsidy vector under both interior and boundary solutions ($q_i=0$ or $q_i=D_i$), we formulate the complete Lagrangian of the government decision problem including the box constraints $0\le q_i\le D_i$ as Equation (19).

$$\mathcal{L}\left(q,\mu ,{\lambda }^{-},{\lambda }^{+}\right)={\displaystyle \sum _{i=1}^M}{w}_i{\displaystyle \frac{q_i}{D_i}}-\mu \left({\displaystyle \sum _{i=1}^M}\left[\left(c_i-p_i-\alpha w_i\right)q_i+{\gamma }_i{q}_i^2\right]-B\right)+{\displaystyle \sum _{i=1}^M}{\lambda }_i^{-}{q}_i+{\displaystyle \sum _{i=1}^M}{\lambda }_i^{+}\left(D_i-q_i\right)$$

(19)

where $\mu \ge 0$ is the multiplier for the fiscal budget constraint, and ${\lambda }_i^{-}\ge 0,{\lambda }_i^{+}\ge 0$ are the Lagrange multipliers associated with the lower bound ($q_i\ge 0$) and upper bound ($q_i\le D_i$), respectively.

The Karush–Kuhn–Tucker (KKT) first-order necessary conditions for optimality are shown in Equation (20), along with the complementary slackness conditions, including $\mu \left({\displaystyle {\sum }_{i=1}^M}{s}_i{q}_i^{*}-B\right)=0, {\lambda }_i^{-}{q}_i^{*}=0, {\lambda }_i^{+}\left(D_i-q_i^{*}\right)=0, \mu \ge 0,\hspace{0.33em}{\lambda }_i^{-}\ge 0,\hspace{0.33em}{\lambda }_i^{+}\ge 0$:

$${\displaystyle \frac{\partial \mathcal{L}}{\partial q_i}}={\displaystyle \frac{w_i}{D_i}}-\mu \left(c_i-p_i-\alpha w_i+2{\gamma }_i{q}_i^{*}\right)+{\lambda }_i^{-}-{\lambda }_i^{+}=0,\hspace{1em}\forall i=1,\dots ,M$$

(20)

Based on these KKT conditions, the equilibrium service supply $q_i^{*}$ falls into three distinct regimes:

• Inactive Regime ($q_i=0$): Occurs when ${\displaystyle \frac{w_i}{D_i}}+{\lambda }_i^{-}=\mu \left(c_i-p_i-\alpha w_i\right)$. The marginal demand-weighted coverage gain is insufficient to offset the marginal fiscal cost. The required subsidy $s_i^{*}=0$.
• Interior Active Regime ($0<q_i<D_i$): Here, ${\lambda }_i^{-}=0$ and ${\lambda }_i^{+}=0$. The optimality condition simplifies to Equation (21) corresponding to equalizing the marginal coverage efficiency per fiscal dollar across active categories. The unit subsidy is $s_i=c_i-p_i-\alpha w_i+{\gamma }_i{q}_i^{*}$.

  $${\displaystyle \frac{w_i/D_i}{c_i-p_i-\alpha w_i+2{\gamma }_i{q}_i^{*}}}=\mu$$

  (21)
• Full Capacity Regime ($q_i=D_i$): Occurs when ${\lambda }_i^{+}\ge 0$. The unconstrained supply exceeds $D_i$, so supply is capped at $q_i^{*}=D_i$. The unit subsidy is $s_i^{*}=c_i-p_i-\alpha w_i+{\gamma }_i{D}_i$.

Because the reformulated government problem is a convex optimization problem, the KKT conditions are also sufficient for global optimality.

Equation (21) indicates that, at an interior optimum, the government allocates fiscal resources such that the marginal increase in demand-weighted coverage per additional unit of subsidy expenditure is equalized across active service categories. Thus, the model indicates that categories with higher demand-priority weights tend to receive greater subsidy priority, while categories with larger cost gaps or stronger marginal cost growth require more fiscal resources to generate additional service coverage.

The complementary-slackness conditions associated with the government budget constraint are $\mu \ge 0,{\displaystyle {\sum }_{i=1}^6}{s}_i{q}_i\le B,$ and $\mu \left(B-{\displaystyle {\sum }_{i=1}^6}{s}_i{q}_i\right)=0.$ If the government budget is binding, then ${\displaystyle {\sum }_{i=1}^6}{s}_i{q}_i=B$, and therefore $\mu >0.$ If the budget is not fully utilized, then $\mu =0.$

It should be noted that the provider-level fixed operating cost $F$ is retained in the provider profit function in Equation (7), but it does not enter the government’s demand-coverage objective in Equation (13). This is because the government objective focuses on the allocation of subsidy resources to improve effective service coverage, whereas $F$ represents a provider-level operating cost that does not directly change the marginal supply response conditional on provider participation.

3.5. Stackelberg Equilibrium

The Stackelberg equilibrium is obtained through backward induction. In the second stage, elderly care providers observe the subsidy vector determined by the government and choose their optimal service quantities according to the best-response function in Equation (11). In the first stage, the government anticipates these responses and chooses the subsidy vector that maximizes demand-weighted service coverage subject to the fiscal budget constraint.

For a given subsidy vector $s$, the provider’s optimal response is determined by the best-response function in Equation (11).

Accordingly, the equilibrium supply vector is expressed as Equation (22):

$$q^{*}\left(s\right)=\left(q_1^{*},\left(s_1\right),q_2^{*},\left(s_2\right)\dots q_6^{*}\left(s_6\right)\right)$$

(22)

The government anticipates this response and chooses the subsidy vector, as shown in Equation (23):

$$s^{*}\in \arg \underset{s\ge 0}{\mathrm{m}\mathrm{a}\mathrm{x}}\left\{{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i^{*}\left(s_i\right)}{D_i}}:{\displaystyle \sum _{i=1}^6}{s}_i{q}_i^{*}\left(s_i\right)\le B\right\}$$

(23)

The corresponding equilibrium supply vector is $q^{*}=q^{*}\left(s^{*}\right)$. Therefore, the Stackelberg equilibrium is defined as the pair $\left(s^{*},q^{*}\right)$ such that $s^{*}$ is the government’s optimal subsidy policy, and $q^{*}$ is the elderly care service providers’ optimal supply response to that policy.

The equilibrium satisfies three fundamental conditions.

(1) Provider optimality

For a given equilibrium subsidy vector $s^{*}$, each provider chooses the unique optimal supply quantity expressed as Equation (24):

$$q_i^{*}=\min \left\{D_i,\max \left[0,{\displaystyle \frac{p_i+s_i^{*}+\alpha w_i-c_i}{{\gamma }_i}}\right]\right\}$$

(24)

Because ${\displaystyle \frac{{\partial }^2\pi }{\partial q_i^2}}=-{\gamma }_i<0,$ the provider’s objective is strictly concave in $q_i$, and the best response is unique.

(2) Government optimality

Given the provider response function, the government chooses, as shown in Equation (25):

$$s^{*}\in \arg \underset{s\ge 0}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i^{*}\left(s_i\right)}{D_i}}$$

$$s.t. {\displaystyle \sum _{i=1}^6}{s}_i{q}_i^{*}\left(s_i\right)\le B$$

(25)

Thus, no feasible unilateral change in the subsidy vector can increase the government’s demand-weighted service coverage.

(3) Fiscal feasibility

The equilibrium subsidy allocation must satisfy ${\displaystyle {\sum }_{i=1}^6}{s}_i^{*}{q}_i^{*}\le B.$ If the fiscal budget is fully utilized, then ${\displaystyle {\sum }_{i=1}^6}{s}_i^{*}{q}_i^{*}=B.$ For an active service category operating in the interior region, the equilibrium subsidy can be recovered from the provider’s best-response condition, as shown in Equation (26):

$$s_i^{*}=c_i-p_i-\alpha w_i+{\gamma }_i{q}_i^{*}$$

(26)

Considering the non-negativity constraint on subsidies, the corresponding general expression is represented by Equation (27):

$$\begin{gathered} s_i^{*}=\max \left\{0,\hspace{0.17em}{c}_i-p_i-\alpha w_i+{\gamma }_i{q}_i^{*}\right\} \\ s.t. 0\le q_i^{*}\le D_i \end{gathered}$$

(27)

These equations jointly characterize the Stackelberg equilibrium of the proposed demand-aware subsidy allocation model.

Through this backward-induction procedure, the model establishes an endogenous linkage between data-derived demand priorities, differentiated subsidies, provider supply responses, and demand coverage. Specifically, the LDA model provides the empirical demand-priority weights $w_i$, which determine the relative importance of the six elderly care service categories. These weights enter the providers’ demand-matching benefit and the government’s demand-coverage objective. Given the government’s subsidy policy, providers determine their optimal service quantities according to their market prices, operating costs, demand-matching benefits, and increasing marginal costs. Within the proposed model, the government then allocates its limited fiscal budget across service categories to maximize demand-weighted service coverage.

The resulting equilibrium therefore provides a quantitative mechanism for translating unstructured online review information into differentiated public subsidy policies. The numerical analysis in Section 4 subsequently evaluates the equilibrium subsidy allocation, service supply, demand coverage, fiscal expenditure, and robustness under alternative subsidy schemes and parameter settings.

4. Numerical Experiments

This section evaluates the proposed demand-aware Stackelberg subsidy model through numerical experiments. The experiments are designed to examine the effectiveness of differentiated subsidies, the influence of fiscal budget constraints, the robustness of the optimization results to variations in demand-priority weights, and the contribution of demand-aware optimization to subsidy allocation. Unless otherwise stated, the numerical experiments use the six elderly care service categories and the parameter settings introduced in Section 3.

4.1. Baseline Parameter Settings

The six elderly care service categories identified from the LDA analysis are used as the basic decision units. Their demand-priority weights are treated as data-derived relative demand-priority weights rather than population-level demand probabilities. The weights are normalized and satisfy ${\sum }_i{w}_i=1$.

The baseline parameters are summarized in Table 4. The unit service revenues $p_i$, unit linear costs $c_i$, and potential demand $D_i$ are specified for each service category. The convex cost coefficient is calibrated according to ${\gamma }_i={\displaystyle \frac{\alpha w_i}{D_i}},$ where $\alpha =0.5$ is the demand-matching benefit coefficient.

This specification links the curvature of the provider’s operating cost to the observed relative demand priority while preserving category-specific demand capacity. The baseline parameter values are shown in Table 5.

Table 5. Baseline parameter values for numerical simulation.

Category $\mathit{i}$	Service Category Name	Demand Weight ${\mathit{w}}_{\mathit{i}}$	Market Price ${\mathit{p}}_{\mathit{i}}$ (RMB)	Variable Cost ${\mathit{c}}_{\mathit{i}}$ (RMB)	Potential Demand ${\mathit{D}}_{\mathit{i}}$ (Inst.)	Convex Cost Coeff. ${\mathit{\gamma }}_{\mathit{i}}$
1	Daily Living Care	0.2838	40	52	5000	$2.8380\times {10}^{-5}$
2	Meals and Nutrition	0.2117	30	38	4000	$2.6463\times {10}^{-5}$
3	Smart Safety Monitoring	0.2012	60	85	4500	$2.2356\times {10}^{-5}$
4	Medical Assistance	0.1305	80	110	3000	$2.1750\times {10}^{-5}$
5	Professional Nursing	0.1065	70	100	3500	$1.5214\times {10}^{-5}$
6	Home Rehabilitation	0.0663	80	120	2000	$1.6575\times {10}^{-5}$

The annual government subsidy budget is set to $B=\mathrm{479,500} \mathrm{RMB}.$ The total potential demand across the six categories is ${\displaystyle {\sum }_{i=1}^6}{D}_i=\mathrm{22,000}.$ For the provider, the effective marginal incentive before considering the convex cost is $p_i+s_i+\alpha w_i-c_i.$

Under the no-subsidy condition $s_i=0$, this term is negative for all six categories. Specifically, the values range from $-7.8942$ RMB for Meals and Nutrition to $-39.9669$ RMB for Home Rehabilitation. Therefore, providers have no economic incentive to supply any of the six services in the absence of government subsidies. The demand-coverage indicator used in the experiments is defined as Equation (12). For a given subsidy vector, the provider’s supply response is determined by Equation (11).

The experiments therefore jointly evaluate subsidy expenditure, service supply, and demand-weighted coverage rather than considering subsidy levels alone.

4.2. Optimal Uniform vs. Optimal Differentiated Subsidies

To examine the value of demand-aware differentiated subsidies, three policy scenarios are compared under a common fiscal budget of RMB 300,000. Setting the same budget for the uniform and differentiated subsidy policies ensures that the comparison isolates the effect of subsidy allocation rather than differences in fiscal expenditure.

Scenario 1: No subsidy.

The government provides no subsidy for any service category. Since the effective marginal incentive before considering the convex cost is negative for all six categories, service providers do not supply any elderly care services. Therefore, the total service supply and demand coverage are both zero.

Scenario 2: Optimal uniform subsidy.

The government provides the same unit subsidy $s^U$ to all six service categories, such that $s_1=s_2=\cdots =s_6=s^U.$

Unlike the fixed RMB 20 uniform subsidy used in the original comparison, the uniform subsidy level is re-optimized under the common fiscal budget of RMB 300,000. Specifically, $s^U$ is determined such that the resulting subsidy expenditure satisfies $s^U{\displaystyle \sum _{i=1}^6}{q}_i\left(s^U\right)=\mathrm{300,000}.$

Using the provider response function in Equation (11), the resulting uniform subsidy is $s^U=24.9668$ RMB per service unit. The corresponding equilibrium supply vector is $q^U=\left(5000,\hspace{0.25em},4000,\hspace{0.25em},3015.95,\hspace{0.25em}0\hspace{0.25em}0\hspace{0.25em}0\right).$

The resulting total service supply is 12,015.95 service instances, and the demand coverage is 0.6303.

Scenario 3: Optimal differentiated subsidy.

The government determines category-specific subsidies by solving the proposed Stackelberg optimization model under the same fiscal budget of RMB 300,000. The optimal subsidy vector is $s^D=\left(12.0000,\hspace{0.25em},8.0000,\hspace{0.25em},25.0000,\hspace{0.25em}30.0000\hspace{0.25em}29.9495\hspace{0.25em}0\right),$ which generates the equilibrium supply vector $q^D=\left(5000,\hspace{0.25em},4000,\hspace{0.25em},4500,\hspace{0.25em}3000\hspace{0.25em}183.64\hspace{0.25em}0\right).$

The resulting total service supply is 16,683.64 service instances, while the demand coverage reaches 0.8328. The corresponding subsidy expenditure is exactly RMB 300,000.

The comparison among the three scenarios is presented in Table 6. Under the same fiscal expenditure, the differentiated subsidy policy increases demand coverage from 0.6303 under the optimal uniform subsidy to 0.8328, representing an improvement of approximately 32.13%. It also increases total service supply from 12,015.95 to 16,683.64 service instances. These results indicate that, given the same fiscal resources, allocating subsidies according to heterogeneous demand priorities and service cost structures can achieve substantially higher demand coverage than applying a uniform subsidy rate.

Table 6. Comparison of subsidy policies.

Policy	Subsidy Vector (RMB/Unit)	Supply Vector	Total Supply	Demand Coverage $\mathit{D}\mathit{C}$	Subsidy Expenditure (RMB)
No subsidy	(0, 0, 0, 0, 0, 0)	(0, 0, 0, 0, 0, 0)	0	0	0
Uniform subsidy	(24.9668, 24.9668, 24.9668, 24.9668, 24.9668, 24.9668)	(5000, 4000, 3015.95, 0, 0, 0)	12,015.95	0.6303	300,000
Differentiated subsidy	(12, 8, 25, 30, 29.9495, 0)	(5000, 4000, 4500, 3000, 183.64, 0)	16,683.64	0.8328	300,000

4.3. Budget Sensitivity

To examine the effect of fiscal capacity on the optimal subsidy allocation, the government budget is varied from RMB 100,000 to RMB 479,500 while all other model parameters remain unchanged. For a given supply quantity $q_i$, the subsidy required to induce the provider to supply that quantity is as in Equation (14).

Accordingly, the government expenditure associated with a supply vector $q$ is expressed as Equation (28):

$$E\left(q\right)={\displaystyle \sum _{i=1}^6}\left[\left(c_i-p_i-\alpha w_i\right)q_i+{\gamma }_i{q}_i^2\right]$$

(28)

The government therefore solves Equation (29):

$$\begin{gathered} \underset{q}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{w}_i{\displaystyle \frac{q_i}{D_i}} \\ s.t.E\left(q\right)\le B,0\le q_i\le D_i \end{gathered}$$

(29)

Five budget levels are considered $B=\mathrm{100,000}$, $\mathrm{200,000}$, $\mathrm{300,000}$, $\mathrm{400,000}$, and $\mathrm{479,500}$ RMB. The resulting optimal supply allocations, subsidy levels, total supply, and demand-weighted coverage are reported in Table 7.

Table 7. Budget sensitivity of the optimal differentiated subsidy policy.

Government Budget $\mathit{B}$ (RMB)	${\mathit{q}}_1^{\mathit{*}}$	${\mathit{q}}_2^{\mathit{*}}$	${\mathit{q}}_3^{\mathit{*}}$	${\mathit{q}}_4^{\mathit{*}}$	${\mathit{q}}_5^{\mathit{*}}$	${\mathit{q}}_6^{\mathit{*}}$	Total Supply	$\mathit{D}\mathit{C}$	Optimal Subsidy Vector $\left({\mathit{s}}_1^{\mathit{*}},{\mathit{s}}_2^{\mathit{*}},{\mathit{s}}_3^{\mathit{*}},{\mathit{s}}_4^{\mathit{*}},{\mathit{s}}_5^{\mathit{*}},{\mathit{s}}_6^{\mathit{*}}\right)$
100,000	5000	4000	321.20	0	0	0	9321.20	0.5099	(12.00, 8.00, 24.91, 0, 0, 0)
200,000	5000	4000	4320.69	0	0	0	13,320.69	0.6887	(12.00, 8.00, 25.00, 0, 0, 0)
300,000	5000	4000	4500	3000	183.64	0	16,683.64	0.8328	(12.00, 8.00, 25.00, 30.00, 29.95, 0)
400,000	5000	4000	4500	3000	3500	12.51	20,012.51	0.9341	(12.00, 8.00, 25.00, 30.00, 30.00, 39.97)
479,500	5000	4000	4500	3000	3500	2000	22,000.00	1.0000	(12.00, 8.00, 25.00, 30.00, 30.00, 40.00)

Notes: $q_i^{*}$ denotes the optimal supply quantity for service category $i$, $s_i^{*}$ denotes the corresponding optimal unit subsidy, and $DC={\displaystyle {\sum }_{i=1}^6}{w}_i{\displaystyle \frac{q_i^{*}}{D_i}}$ denotes demand-weighted service coverage. Categories with zero optimal supply receive zero subsidy. Minor deviations in the reported quantities and subsidy levels may arise from numerical rounding.

The results reveal a clear and nonlinear relationship between fiscal capacity and demand-weighted service coverage. Under the smallest budget of RMB 100,000, the model fully activates Daily Living Care and Meals and Nutrition, while only 321.20 units of Smart Safety Monitoring are supplied. The resulting demand-weighted coverage is 0.5099. When the budget increases to RMB 200,000, the supply of Smart Safety Monitoring expands to 4320.69 units, increasing demand-weighted coverage to 0.6887.

With a budget of RMB 300,000, Smart Safety Monitoring reaches its full potential demand of 4500 units, while Medical Assistance is also fully activated and Professional Nursing begins to receive partial coverage. Total supply increases to 16,683.64 units and demand-weighted coverage rises to 0.8328. When the budget reaches RMB 400,000, the first five service categories reach their potential demand levels, while 12.51 units of Home Rehabilitation are supplied. Demand-weighted coverage consequently increases to 0.9341.

Finally, at the baseline budget of RMB 479,500, all six categories reach their potential demand levels $q^{*}=(5000, 4000, 4500, 3000, 3500, 2000)$, yielding $D{C}^{*}=1.0000$.

The corresponding subsidy vector is $s^{*}=\left(12,\hspace{0.25em},8,\hspace{0.25em},25,\hspace{0.25em}30\hspace{0.25em}30\hspace{0.25em}40\right)$ and total subsidy expenditure is exactly ${\displaystyle {\sum }_{i=1}^6}{s}_i^{*}{q}_i^{*}=\mathrm{479,500} \mathrm{RMB}$.

The budget sensitivity results demonstrate that the proposed model does not simply distribute additional fiscal resources proportionally across service categories. Instead, resources are progressively directed toward categories according to their marginal contribution to demand-weighted coverage and their subsidy-induced supply response. High-priority categories tend to be activated earlier, whereas categories with lower demand priority and higher economic barriers receive resources only after the more valuable opportunities for coverage expansion have been sufficiently exploited.

An important implication is that the marginal benefit of additional fiscal expenditure decreases as the budget approaches the full-coverage threshold. The increase in demand-weighted coverage from RMB 100,000 to RMB 200,000 is substantial, whereas the incremental improvement becomes smaller as the budget approaches RMB 479,500. This indicates diminishing returns to subsidy expansion and provides a model-based quantitative basis for evaluating the trade-off between additional fiscal expenditure and incremental elderly care service coverage.

4.4. Robustness and Monte Carlo Analysis

To assess the robustness of the proposed subsidy optimization model to uncertainty in the empirically derived demand-priority weights, a Monte Carlo perturbation analysis is conducted. The analysis recognizes that the demand-priority weights obtained from LDA are data-derived indicators and may be subject to sampling variation and potential fluctuations in the underlying review data. Therefore, the stability of the optimal subsidy policy is examined under different levels of perturbation in the demand-priority weights.

Let the perturbed demand-priority weight for service category $i$ be defined as Equation (30).

$${\tilde{w}}_i=w_i\left(1+{\epsilon }_i\right)$$

(30)

where ${\epsilon }_i\sim U\left(-,\delta \delta \right).$

Because the demand-priority weights must sum to one, the perturbed weights are normalized as in Equation (31):

$$w_i^{\prime }={\displaystyle \frac{w_i\left(1+{\epsilon }_i\right)}{{\displaystyle {\sum }_{j=1}^6}{w}_j\left(1+{\epsilon }_j\right)}}$$

(31)

Three perturbation levels are considered, namely $\delta =0.05$, $0.10$, and $0.20$, corresponding to$\pm 5\%, \pm 10\%, \mathrm{a}\mathrm{n}\mathrm{d}\pm 20\%$ random deviations from the baseline demand-priority weights. For each perturbation level, 1000 independent Monte Carlo simulations are conducted. The government budget is fixed at $B=\mathrm{300,000}$ MB, consistent with the intermediate-budget scenario in Section 4.3. All other model parameters remain unchanged.

Because the demand-priority weights enter both the provider’s demand-matching benefit and the convex cost calibration, the perturbed convex cost coefficients are recalculated consistently as in Equation (32):

$${\gamma }_i^{\prime }={\displaystyle \frac{\alpha w_i^{\prime }}{D_i}}$$

(32)

The corresponding provider response under the perturbed parameters is therefore as in Equation (33):

$$q_i^{*}\left(s_i\right)=\min \left\{D_i,\max \left[0,{\displaystyle \frac{p_i+s_i+\alpha w_i^{\prime }-c_i}{{\gamma }_i^{\prime }}}\right]\right\}$$

(33)

For each Monte Carlo replication, the government optimization problem is then solved using the perturbed parameter vector $w_i^{\prime }$ and ${\gamma }_i^{\prime }$, subject to the same budget constraint of RMB 300,000.

To isolate the impact of demand weight uncertainty from changes in the cost structure, a secondary “fixed-${\gamma }_i$” Monte Carlo experiment is conducted alongside the coupled calibration model. In this setup, $w_i$ is perturbed by $\delta \in \left\{\pm 5\%,\pm 10\%,\pm 20\%\right\}$, while the convex cost coefficients ${\gamma }_i$ are strictly held constant at their baseline values listed in Table 5. The results are presented in Table 8.

Table 8. Monte Carlo robustness analysis under demand-weight perturbations.

Perturbation Level $\mathit{\delta }$	Experiment Type	Mean $\mathit{D}\mathit{C}$	Standard Deviation	5th Percentile	95th Percentile
$\pm 5\%$	Fixed-${\gamma }_i$	0.8327	0.0028	0.8281	0.8372
	Coupled-${\gamma }_i$	0.8325	0.0035	0.8266	0.8380
$\pm 10\%$	Fixed-${\gamma }_i$	0.8328	0.0056	0.8235	0.8418
	Coupled-${\gamma }_i$	0.8326	0.0070	0.8213	0.8441
$\pm 20\%$	Fixed-${\gamma }_i$	0.8330	0.0112	0.8145	0.8512
	Coupled-${\gamma }_i$	0.8332	0.0139	0.8096	0.8561

As shown in Table 8, holding ${\gamma }_i$ fixed yields even lower standard deviations across all perturbation thresholds. This confirms that the demand-aware optimization model is inherently robust to sampling variation in $w_i$, independent of cost parameter recalibration.

The stability of the mean coverage indicates that the optimization framework does not depend critically on a particular realization of the demand-priority weights. Instead, the allocation mechanism continues to direct fiscal resources toward service categories that generate relatively high demand-weighted coverage under the perturbed demand structure. The increasing standard deviation under larger perturbations is also intuitive, as greater uncertainty in the empirical demand structure creates greater variation in the relative marginal value of subsidizing different service categories.

The robustness analysis therefore provides evidence that the main conclusion is not driven by small fluctuations in the estimated demand-priority weights. Even under relatively substantial $\pm 20\%$ perturbations, the optimized solution maintains a demand-weighted coverage level centered around approximately 0.83 at the RMB 300,000 budget level. This suggests that the numerical results of the proposed demand-aware subsidy model are reasonably stable when the empirical demand-priority weights are subject to uncertainty.

4.5. Demand-Unaware Ablation and Equal-Priority Benchmark

To further evaluate the contribution of demand-aware optimization, an ablation study is conducted by removing the demand-priority information from the government’s subsidy allocation objective. The purpose of the ablation analysis is to determine whether incorporating the empirically derived demand-priority weights $w_i$ improves the effectiveness of subsidy allocation relative to a demand-unaware allocation mechanism.

In the full model, the government maximizes demand-weighted service coverage, $\underset{q}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle {\sum }_{i=1}^6}{w}_i{\displaystyle \frac{q_i}{D_i}}$.

In the demand-unaware ablation model, the differentiated demand-priority weights are removed from the government objective and replaced by equal weights, $w_i^A={\displaystyle \frac{1}{6}},i=1,\dots ,6$.

The ablation objective therefore becomes $\underset{q}{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \sum _{i=1}^6}{\displaystyle \frac{1}{6}}{\displaystyle \frac{q_i}{D_i}}$.

The budget constraint and supply bounds remain unchanged, ${\displaystyle {\sum }_{i=1}^6}\left[\left(c_i-p_i-\alpha w_i\right)q_i+{\gamma }_i{q}_i^2\right]\le B$, subject to $0\le q_i\le D_i$.

Importantly, the provider-side parameters continue to use the empirically derived demand-priority weights. This design isolates the contribution of demand-aware government optimization rather than removing demand information from the entire model. Both models are evaluated under the same budget of RMB 300,000.

To address potential circularity concerns when evaluating the model solely via the demand-weighted coverage metric ($DC$), an unweighted baseline coverage metric ($UC$) is introduced as an additional evaluation criterion in Equation (34):

$$UC\left(q\right)={\displaystyle \frac{{\displaystyle {\sum }_{i=1}^M}{q}_i}{{\displaystyle {\sum }_{i=1}^M}{D}_i}}$$

(34)

While $DC$ evaluates how effectively the allocation fulfills prioritized market needs, $UC$ measures the aggregate volumetric service rate across all service categories, completely independent of the LDA-derived weights $w_i$.

The resulting allocation and performance are presented in Table 9.

Table 9. Ablation comparison between the full and demand-unaware models.

Model	${\mathit{q}}_1^{\mathit{*}}$	${\mathit{q}}_2^{\mathit{*}}$	${\mathit{q}}_3^{\mathit{*}}$	${\mathit{q}}_4^{\mathit{*}}$	${\mathit{q}}_5^{\mathit{*}}$	${\mathit{q}}_6^{\mathit{*}}$	Total Supply	Demand-Weighted Coverage $\mathit{D}\mathit{C}$	Unweighted Coverage $\mathit{U}\mathit{C}$
Full model	5000.00	4000.00	4500.00	3000.00	183.64	0.00	16,683.64	0.8328	0.7583
Demand-unaware ablation	5000.00	4000.00	0.00	3000.00	1268.06	2000.00	15,268.06	0.7309	0.6940

Notes: For both models, performance is evaluated using the original empirical demand-priority weights $w_i$, rather than the equal weights used in the ablation objective.

The results show a clear performance difference between the two models. Under the same RMB 300,000 budget, the full demand-aware model achieves superior performance under both metrics. Specifically, it improves $DC$ from 0.7309 to 0.8328 (+13.9%) and $UC$ from 0.6940 to 0.7583 (+9.3%). This demonstrates that the advantage of the proposed model is not merely an artifact of the evaluation weighting scheme, but stems from a more cost-effective alignment between fiscal subsidies, service costs, and supply capacities.

The difference arises primarily from the allocation of limited fiscal resources across service categories. Under the full model, Daily Living Care, Meals and Nutrition, and Smart Safety Monitoring are prioritized, with the first three categories receiving full coverage. Medical Assistance also reaches its full potential demand, while the remaining budget is allocated to Professional Nursing. In contrast, the demand-unaware model treats all service categories as equally important in the government’s objective. As a result, it allocates resources to Home Rehabilitation and Professional Nursing despite their relatively lower empirical demand-priority weights, while Smart Safety Monitoring receives no supply.

This difference is particularly important because the six service categories have substantially different demand-priority weights. For example, Daily Living Care has a demand-priority weight of 0.2838, whereas Home Rehabilitation has a weight of only 0.0663. An allocation mechanism that ignores these differences may achieve a relatively balanced distribution of service coverage but does not necessarily maximize the coverage of the most prominent demand themes identified from the review data.

The ablation results therefore provide direct evidence for the value of integrating empirical demand information into the subsidy optimization process. The advantage of the full model does not simply come from increasing total supply; rather, it comes from allocating the limited fiscal budget toward service categories with greater demand-weighted marginal contributions. Consequently, the demand-priority weights extracted from unstructured user-generated content play a substantive role in translating heterogeneous elderly care needs into differentiated subsidy policies.

Overall, the ablation study confirms the central mechanism of the proposed framework: data-derived demand priorities improve the government’s ability to target scarce fiscal resources toward service categories with greater empirical demand importance. This finding supports the integration of LDA-based demand identification with Stackelberg subsidy optimization and indicates that the demand-aware component contributes directly to the model-based allocation performance rather than serving a purely descriptive role.

4.6. Sensitivity to Cost Calibration

To verify that our findings are not driven by this specific calibration identity, we conducted an alternative calibration test where the convex cost parameter is completely decoupled from $w_i$ as in Equation (35):

$${\gamma }_i^{alt}={\displaystyle \frac{\theta }{D_i}}$$

(35)

Setting $\theta =0.1$ and $B=\mathrm{RMB} \mathrm{300,000}$, the optimization is re-executed under the alternative calibration. The resulting optimal subsidy vector is (11.9581, 7.9942, 24.9994, 30.0348, 29.9521, 0.0000) RMB per service unit, which generates the equilibrium supply vector (5000.00, 4000.00, 4500.00, 3000.00, 188.01, 0.00). The corresponding subsidy expenditure is exactly RMB 300,000, and the demand-weighted coverage is 0.8329. The solution satisfies the provider best-response function and the government’s optimality conditions. The allocation hierarchy remains broadly consistent with the baseline results, indicating that the main allocation pattern and the superiority of the demand-aware policy are not driven by the benchmark cost calibration.

5. Discussion and Managerial Implications

5.1. Discussion of Main Findings

This study integrates LDA-based demand identification with a Stackelberg game to develop a demand-aware subsidy optimization framework for smart community elderly care. The results provide three main findings.

Elderly care service needs are heterogeneous, and this heterogeneity should be considered in subsidy allocation. The six service categories identified from user-generated content have substantially different demand-priority weights. The ablation analysis provides numerical evidence for the value of incorporating these differences: under the same RMB 300,000 budget, the full model achieves a demand-weighted coverage of 0.8328, compared with 0.7309 for the demand-unaware model. This indicates that demand-aware optimization can improve allocation performance within the proposed model under a limited fiscal budget.

The effect of fiscal expansion is nonlinear. As the budget increases from RMB 100,000 to RMB 479,500, demand-weighted coverage rises from 0.5099 to 1.0000. However, the incremental improvement becomes smaller as the budget approaches the full-coverage threshold. This suggests that additional subsidies have diminishing marginal effects once high-priority service categories have been sufficiently covered.

The optimal subsidy policy reflects both demand priority and economic conditions. Although high-priority categories tend to receive resources earlier, subsidy allocation is also affected by service prices, costs, potential demand, and providers’ marginal supply costs. Therefore, the optimal subsidy vector cannot be determined by demand weights alone.

The robustness analysis further shows that the main results remain relatively stable under $\pm 5\%, \pm 10\%, \mathrm{a}\mathrm{n}\mathrm{d} \pm 20\%$ perturbations of the demand-priority weights. This supports the reliability of the proposed allocation mechanism despite uncertainty in empirically derived demand information.

5.2. Managerial Implications

The findings provide several practical implications for policymakers. First, governments can move from uniform subsidies toward demand-aware allocation. Empirical demand information obtained from user-generated content can be periodically updated and incorporated into subsidy decisions, allowing policies to better reflect changing service priorities. In practice, public agencies could implement the proposed framework through a periodic policy cycle. Recent user-generated content and, where available, administrative or service-utilization data could be used to update the demand-priority weights and recalibrate the relevant demand and cost parameters. The updated parameters could then be incorporated into the subsidy optimization model under the available fiscal budget, with the resulting subsidy levels reviewed periodically as new demand information becomes available. However, these weights should be interpreted as relative demand-priority indicators in the observed online data rather than population-level demand proportions. Second, limited fiscal resources should be allocated progressively. Under constrained budgets, high-priority categories can be supported first, while additional categories can be gradually covered as fiscal capacity increases. The budget sensitivity results provide a quantitative reference for evaluating the trade-off between fiscal expenditure and service coverage. Third, policymakers should consider provider-side economic conditions. Subsidies affect service supply through providers’ marginal costs and economic incentives. Therefore, subsidy design should jointly consider demand importance and supply-side feasibility rather than relying solely on demand rankings.

6. Conclusions and Limitations

6.1. Conclusions

This study develops a demand-aware subsidy optimization framework by integrating LDA-based demand identification with a government–provider Stackelberg game. Six elderly care service categories are identified from user-generated reviews, and their relative prominence is incorporated into the subsidy optimization model as data-derived relative demand-priority weights.

The numerical results show that differentiated subsidies can improve allocation performance within the proposed model under a RMB 300,000 budget. Budget sensitivity analysis further shows that service coverage increases with fiscal capacity but exhibits diminishing marginal returns, reaching full potential demand at the model-based budget threshold of RMB 479,500. The robustness analysis indicates that the main conclusions remain stable under substantial perturbations of the demand-priority weights.

Overall, the study provides numerical evidence that integrating empirical demand information with strategic subsidy optimization can provide a more targeted approach to elderly care resource allocation. The main contribution is not the specific subsidy values, but the proposed framework for translating unstructured user information into quantitative and economically constrained subsidy decisions.

6.2. Limitations and Future Research

Several limitations should be acknowledged. First, the demand-priority weights are derived from Taobao/Tmall user-generated reviews and therefore represent the relative prominence of demand themes in the observed online data rather than population-level demand prevalence. In addition, selecting the top three best-selling products within each category may overrepresent popular products and digitally active consumers, potentially introducing product-level selection bias. The current study does not externally validate these weights against independent survey, administrative, or service utilization data. Future research could combine online reviews with surveys, demographic statistics, and service utilization data, as well as broader product samples, to improve representativeness and provide external validation. Second, the current model considers a government–provider interaction within a single decision period. Future research could incorporate multiple stakeholders, regional heterogeneity, service quality, provider competition, and dynamic changes in demand and fiscal conditions. In addition, the model assumes that the six service categories are independent, while cross-category substitution, complementarity, and shared resource constraints may exist in practice. Future research could incorporate these interdependencies into a more comprehensive subsidy optimization framework.