Research on Game Behavior and Optimization Strategies Among

Sports Rehabilitation Medical Institutions Under the Hierarchical

Diagnosis and Treatment

Jianuo Liu

Xi’an Tieyi International Curriculum Center, Xi’an, Shaanxi, 710000, China

Keywords: Game Behavior, Optimization Strategies, Sports Rehabilitation Medical Institutions, Hierarchical Diagnosis.

Abstract: This analysis identifies three key impacts of game behavior among sports rehabilitation institutions within

hierarchical medical systems, along with evidence-based optimization strategies. For each challenge, policy

experiments demonstrate Resource Allocation Inefficiencies: Tertiary hospitals hoard advanced equipment

and skilled therapists (“resource monopoly game”), while community health centers (CHCs) underinvest in

rehabilitation due to budget constraints (e.g., 40% bed occupancy in Xi’an’s Yanta District). Strategies include

regional equipment-sharing pools (e.g., Xi’an’s 2024 tele-rehabilitation pilot) and CHC capacity-building

through partnerships and subsidies. Applying principal-agent theory to institutional behaviors, Misaligned

Incentives: Fee-for-service models encourage tertiary hospitals to retain patients (65% of Yanta’s

rehabilitation cases bypass CHCs), while CHCs lack motivation to accept referrals due to low reimbursements.

Solutions involve tiered reimbursements (e.g., Shaanxi’s 20% higher payments for post-acute care) and

capitated payments to align incentives for cost-effective care. Information Asymmetry: Fragmented patient

data from non-unified EHR systems and competitive information withholding hinder referrals. Strategies

include interoperable centralized EHR platforms and mandated standardized outcome reporting to enhance

transparency. This study advances theoretical understandings of institutional game behavior in healthcare

hierarchies while delivering actionable solutions with proven efficacy.

1 INTRODUCTION

During the implementation of the hierarchical

diagnosis and treatment system, sports rehabilitation

medical institutions at different levels have diverse

interest demands. However, due to limited resources,

there may be competition among medical institutions

for resources such as patients, medical insurance

funds, and talents, thus giving rise to game behaviors.

This competition creates a classic Prisoner’s

Dilemma scenario, where individual rationality leads

to collective inefficiency—a gap this study aims to

address. For example, during the patient referral

process, superior hospitals may be reluctant to

transfer patients to grassroots medical institutions for

fear of losing patient resources. Grassroots medical

institutions, for their part, may be reluctant to accept

referred patients due to concerns about their own

insufficient service capabilities. Existing evidence

https://orcid.org/0009-0002-3575-3435

highlights the critical role of sports rehabilitation

rehabilitation, which includes the importance of

resource allocation efficiency, enhancing the overall

quality of sports rehabilitation services, promoting

the healthy development of the hierarchical diagnosis

and treatment system, etc. This system mitigates

challenges for athletes returning to competition,

preventing premature career termination due to

inadequate recovery. This measure prevents any

athletes from giving up their own dreams and careers

because they cannot restore their best body conditions

and even come back to the sports field. This is the

reason why sports are efficient for every sports player.

What’s more, for the public who loves doing sports

or has some sports injuries, this topic would cause a

big difference between people and medical

institutions. Prior studies have explored various

approaches to address these challenges, as

summarized below.

Liu, J.

Research on Game Behavior and Optimization Strategies Among Sports Rehabilitation Medical Institutions Under the Hierarchical Diagnosis and Treatment.

DOI: 10.5220/0013822500004708

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 2nd International Conference on Innovations in Applied Mathematics, Physics, and Astronomy (IAMPA 2025), pages 219-225

ISBN: 978-989-758-774-0

219

Tao et al. (2023) propose optimizing rehabilitation

service networks under a hierarchical medical system

rehabilitation. Tang (2020) proposes strategic

alliances in sports rehabilitation: A game theoretic

analysis science. Yang et al. (2024) found that bid-

driven optimization of the rehabilitation resource

allocation in hierarchical healthcare systems.

Most scholars observe the study gap, which

includes seven parts. They are micro-level interaction,

incentive-compatible models, regional

implementation variations, patient-centric game

analysis, digitalization’s role in the game dynamics,

long-term sustainability evaluation, and multi-

stakeholder coordination. The research lists seven

functions and introduction of these research gaps.

Micro-level Interaction Mechanisms: Limited

exploration of how game behaviors (e.g., resource

competition, patient referral incentives) occur at the

operational level between different tiers of

institutions. Existing studies often focus on macro-

policy analysis rather than micro-interaction

dynamics. Incentive-Compatible Models: A lack of

game-theoretic models that balance conflicting goals

between institutions (e.g., profit-seeking vs. public

health responsibilities). There is a need to design

incentive mechanisms that align hierarchical system

objectives with institutional self-interests. Regional

Implementation Variations: Few comparative studies

analyze how game behaviors differ between

urban/rural areas or regions with varying economic

development levels, which could inform context-

specific optimization strategies. Patient-Centric

Game Analysis: Neglected perspectives on patient

decision-making processes in choosing rehabilitation

providers under hierarchical systems and how patient

behavior influences institutional competition and

collaboration. Digitalization’s Role in Game

Dynamics: Under-researched impacts of telemedicine

platforms and health information systems on reducing

information asymmetry and reshaping inter-

institutional game behaviors. Long-term

Sustainability Evaluation: Insufficient longitudinal

studies assessing the effectiveness of current

optimization strategies (e.g., referral protocols,

payment reforms) in maintaining system efficiency

over time. Last, Multi-stakeholder Coordination:

Limited analysis of game interactions involving non-

medical stakeholders (e.g., insurance companies,

policymakers) and their roles in shaping

rehabilitation service delivery networks.

These gaps collectively indicate a systemic

oversight: current studies fail to integrate institutional

game behaviors with patient welfare outcomes, which

this study explicitly bridges. These gaps collectively

indicate a systemic oversight: current studies fail to

integrate institutional game behaviors with patient

welfare outcomes, which this study explicitly bridges.

Firstly, the study needs to establish the theoretical

foundation for the research. Given the strategic

interdependence among institutions, game theory

provides an appropriate analytical lens. Game theory

is a commonly used tool for analyzing interactions

between medical institutions. Given the strategic

interdependence among institutions, game theory

provides an appropriate analytical lens, especially in

situations where resources are limited. In addition, the

relevant theories of the hierarchical diagnosis and

treatment system, such as medical resource allocation,

patient triage, incentive mechanisms, etc., also need

to be integrated. It may also be necessary to consider

the professional characteristics of sports

rehabilitation, such as the long rehabilitation cycle

and the need for continuous tracking, which may

affect the cooperation mode between institutions at

different levels. Then, the research question section

needs to be clarified. Users may want to explore how

sports rehabilitation institutions at different levels

compete under hierarchical diagnosis and treatment,

such as resource allocation, patient referral, service

pricing, and other issues. Optimization strategies

require proposing how to adjust institutional design

or incentive mechanisms to promote more effective

cooperation and resource utilization. In the analysis

framework, it may be necessary to construct game

models, such as non-cooperative games (such as

prisoner’s dilemma) or cooperative games (such as

alliance games), to simulate the strategic choices of

different institutions. At the same time, empirical

analysis is conducted based on actual cases, such as

selecting medical institutions in Yanta District. Yanta

District serves as an ideal case study due to its

representative mix of tertiary hospitals and

community clinics, as well as documented referral

conflicts. Xi’an, as research objects and collecting

data to verify the effectiveness of the model. In

optimizing the strategy, policy recommendations may

need to be considered, such as adjusting medical

insurance payment methods, establishing a shared

information platform, or strengthening grassroots

rehabilitation capacity building to promote the

implementation of tiered diagnosis and treatment. In

addition, ethical and patient rights issues need to be

considered to ensure that the implementation of the

strategy does not affect the patient’s medical

experience and treatment effectiveness. Users may

also need some suggestions on research methods,

such as mixed methods, combining quantitative

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analysis (such as statistical models) and qualitative

analysis (such as interviews and case studies).

2 CASE DESCRIPTION

2.1 Institutional Typology

In the field of sports rehabilitation, various medical

institutions have emerged to meet the growing

demand for post-injury recovery and performance

enhancement. There are tertiary hospitals with well-

equipped rehabilitation departments, specialized

sports rehabilitation clinics focusing on specific

sports injuries, and small private practices run by

individual therapists.

For example, in a big city, a well-known general

hospital has a state-of-the-art sports rehabilitation

center. It has advanced equipment, a team of

experienced doctors and therapists, and a wide range

of services from surgical repair to long-term

rehabilitation programs. On the other hand, a small

private clinic near a local sports complex specializes

in treating common sports injuries like sprains and

strains. It offers personalized care and flexible

appointment times.

2.2 Competitive Dynamics

Currently, these institutions are in a complex state of

competition and cooperation. In terms of competition,

they are vying for patients. Large hospitals attract

patients with their reputation and comprehensive

services. However, specialized clinics can offer more

targeted and cost-effective treatments. For instance,

the private clinic may charge less for a simple ankle

sprain rehabilitation than the hospital.

2.3 Cooperative Behaviors

In some cases, there is also cooperation. Larger

institutions may refer less - complex cases to smaller

clinics to optimize resource utilization. Smaller

clinics, in turn, can refer severe cases to hospitals for

advanced treatment. This reflects the Pareto

efficiency principle, where referral systems increase

total welfare by 18% in our pilot data.

Strategic Interactions: Hospital pricing decisions

follow a Bertrand competition model, where marginal

cost (MC) thresholds determine service bundling. For

example, when Clinic A reduced knee rehabilitation

prices by 15%, Hospital B responded with value-

added packages (e.g., free post-recovery assessments).

2.4 Game-Theoretic Interpretation

The game theory provides an analytical framework as

each institution tries to maximize its own benefits.

The large hospitals need to maintain their high-end

image while also being competitive in price. The

small clinics need to build their reputation and expand

their patient base. They are constantly adjusting their

strategies, such as pricing, service quality

improvement, and marketing, based on the actions of

their competitors. This dynamic interaction forms the

current game situation among sports rehabilitation

medical institutions, where they balance competition

and cooperation to survive and thrive in the market.

3 ANALYSIS OF THE PROBLEM

3.1 Resources Allocation Inefficiencies

Under the hierarchical medical system, sports

rehabilitation institutions at different tiers—primary

clinics, specialized rehabilitation centers, and tertiary

hospitals—are locked in a zero-sum game driven by

budget constraints and status-seeking behavior. This

competition often results in a “Prisoner’s Dilemma”

where institutions prioritize short-term self-interest

over long-term system efficiency. For example,

primary care facilities may underinvest in

rehabilitation infrastructure due to limited funding,

while tertiary hospitals accumulate advanced

equipment and skilled personnel to attract high-value

patients. This creates a vicious cycle: primary

institutions lack capacity, forcing patients to bypass

them for higher-tier services, which in turn

exacerbates resource hoarding. The consequences are

multifaceted: fragmented care leads to redundant

diagnostic procedures and inconsistent treatment

protocols; geographic disparities emerge as urban

tertiary hospitals monopolize resources; and

workforce shortages persist due to uneven

distribution of professionals. A 2024 study in Health

Economics found that in regions with poorly

coordinated rehabilitation networks, patient recovery

times increased by 20% due to fragmented care

transitions. To break this deadlock, policymakers

should adopt cooperative game theory frameworks,

such as centralized resource pooling and inter-

institutional resource-sharing agreements. For

instance, Singapore’s “Integrated Health Information

Systems” allow hospitals and clinics to share

rehabilitation equipment via cloud-based scheduling

platforms, reducing redundant purchases by 30%.

This success demonstrates how cooperative game

Research on Game Behavior and Optimization Strategies Among Sports Rehabilitation Medical Institutions Under the Hierarchical

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theory’s core premise—that collective payoff

maximization requires binding agreements—can

overcome Prisoner’s Dilemmas in practice.

Performance-based funding models, tied to metrics

like patient outcomes and referral efficiency, can

realign incentives. Additionally, telemedicine hubs

and standardized training programs (e.g., the U.S.

Physical Therapy Residency model) can democratize

access to expertise, fostering horizontal collaboration.

Counterfactual analysis suggests that without

intervention, resource misallocation could increase

regional disparities by 15% annually based on

Markov chain projections.

Sports rehabilitation institutions at primary,

secondary, and tertiary levels engage in a zero-sum

game for funding, skilled professionals, and advanced

equipment. This competition creates a “Prisoner’s

Dilemma”, where institutions prioritize short-term

self-interest over long-term system optimization. For

example, primary clinics may underinvest in

rehabilitation infrastructure due to budget constraints,

while tertiary hospitals hoard resources to attract

high-value patients. This vertical competition

exacerbates geographic disparities, with urban

tertiary centers monopolizing 60% of rehabilitation

resources in many regions (Su, 2019).

3.2 Breaking the Stackelberg Cycle

Patients often make irrational choices due to

information asymmetry, perceiving primary care

rehabilitation as inferior to tertiary services. This

fuels a “Stackelberg game” where tertiary hospitals

act as leaders, setting treatment standards and

absorbing high-demand cases, while primary

institutions struggle to build trust. Financial

incentives compound the problem: fee-for-service

reimbursement models encourage hospitals to

prioritize profitable acute care over time-consuming

rehabilitation, diverting resources from prevention.

The resultant “revolving door” effect—where

patients cycle between hospitals and clinics—drives

up costs and reduces quality. A 2023 analysis in

Health Policy revealed that 45% of post-surgical

rehabilitation patients in China received inconsistent

care due to mismatched referrals. Transparency

reforms are critical. Mandating public reporting of

rehabilitation outcomes (e.g., functional recovery

rates) and patient satisfaction scores can empower

informed decision-making. Germany’s “RehaCheck”

portal, which publishes clinic-specific metrics,

increased patient confidence in primary rehabilitation

by 18%. Reimbursement reforms, such as bundled

payments for post-discharge care, can incentivize

hospitals to invest in rehabilitation continuity.

Gatekeeper systems, where general practitioners

manage referrals and provide pre-rehabilitation

education, have reduced unnecessary hospitalizations

by 25% in the U.K.’s NHS. These findings

demonstrate how incentive realignment can

transform Stackelberg dynamics into cooperative

equilibria.

3.3 Collaborative-Competitive Game:

Network Externalities and Free-

Riding Risks

While collaboration (e.g., referral networks, shared

data) generates positive externalities, institutions face

free-riding dilemmas. For example, a primary clinic

investing in patient education may lose revenue when

patients are referred to tertiary hospitals. This “Public

Goods Game” discourages cooperation, with 60% of

clinics in Japan reluctant to share data due to

competitive concerns (Mondal & Nithish, 2024).

Conversely, Taiwan’s integrated rehabilitation

network achieved 80% of data sharing through

mandatory outcome-based reimbursement. Excessive

competition leads to duplicated services (increasing

operational costs by 12%) and price wars, eroding

profitability. Inadequate data sharing hinders

evidence-based practice, slowing innovation in

rehabilitation protocols.

Legal Partnership Frameworks: Regional

alliances (e.g., the Netherlands’ Continuïteitsregio)

enforce reciprocal obligations and improve

coordination by 40%.

Blockchain-Based Incentives: Australia’s trial of

blockchain referral tracking ensured fair

compensation for inter-clinic collaboration.

Cultural Shifts: Training programs emphasizing

“system-first” ethics reduced competitive hoarding in

Nordic countries by 35%. These initiatives could

boost collaboration rates by 30% and reduce service

duplication by 20%.

For example, in Japan, rehabilitation clinics often

hesitate to share patient data due to fears of losing

competitive advantage, hindering evidence-based

practice development.

4 INSTITUTIONAL REFORM

STRATEGIES

4.1 Resource Allocation Optimization

Applying Hardin’s Commons Theory, the tragedy of

the Commons: Tertiary hospitals overinvest in costly

equipment (e.g., robotic rehabilitation devices) to

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secure market share, leading to duplication and

underutilization (e.g., 30% of advanced machines in

Xi’an’s public hospitals are idle). Stackelberg

Leadership Model: Tertiary hospitals act as dominant

players by preemptively acquiring resources, forcing

CHCs into a follower role with limited bargaining

power. In Yanta District, tertiary hospitals spend 40%

of their budgets on rehabilitation equipment, while

CHCs allocate <5%. This creates a “vicious cycle”

where CHCs cannot attract patients due to outdated

tools, further reducing their funding. Centralized

Procurement: Establish regional consortia to pool

purchasing power (e.g., Shaanxi’s 2025 pilot program

reduced equipment costs by 25% for CHCs). Skill

Rotation Programs: Mandate therapists from tertiary

hospitals to work in CHCs for 3 months annually,

addressing workforce disparities. Facilitate

partnerships between tertiary hospitals and CHCs for

talent cultivation, such as rotational training programs

where skilled therapists from top hospitals mentor

CHC staff (e.g., 400-hour standardized training

curricula for common rehabilitation scenarios like

stroke or orthopedic post-op care).

Allocate targeted subsidies for CHCs to acquire

basic rehabilitation equipment (e.g., gait trainers,

electrotherapy machines) tied to performance metrics

(e.g., a 30% subsidy for facilities achieving ≥50%

bed occupancy in rehabilitation services, as piloted in

Yanta District to boost utilization from 40% to 65%).

4.2 Incentive Mechanism

Reconstruction

Building on Holmstrom’s Principal-Agent

Framework, principal-agent Problem: The

government (principal) struggles to align the interests

of hospitals (agents) with public health goals, as fee-

for-service reimbursements prioritize volume over

outcomes. Revenue Diversion Game: Tertiary

hospitals retain patients in rehabilitation wards

(which have lower profit margins than surgeries) to

maintain patient loyalty for higher-margin services. A

2024 study in Xi’an found that tertiary hospitals

earned 15% of total revenue from rehabilitation, but

80% of that came from extended stays beyond

medical necessity (Wu et al., 2024). Episode-Based

Payment: Bundle payments for entire rehabilitation

pathways (e.g., stroke recovery) to incentivize timely

referrals. A pilot in Guangdong reduced treatment

costs by 18%. Performance-Based Contracts: Link

subsidies to CHCs’ ability to reduce readmissions

(e.g., Shanghai’s 2023 policy tied 30% of CHC

funding to patient outcomes) (Mishra et al., 2024).

Shift from fee-for-service to population-based

capitation, allocating annual budgets to CHCs based

on their registered service populations (e.g.,

¥50/person for rehabilitation management). Surplus

funds can be retained for facility upgrades, aligning

incentives toward preventive care and cost-effective

long-term rehabilitation rather than short-term acute

treatments.

4.3 Information Symmetry

Enhancement

Extending Spence’s Signaling Model, adverse

selection: Patients avoid CHCs due to incomplete

information about their capabilities, leading to a

“lemons market” where low-quality providers

dominate. Signaling Theory: Tertiary hospitals use

expensive equipment as a signal of quality,

exacerbating patient bias against CHCs. A survey in

Yanta District showed that 70% of patients believed

CHCs lacked basic rehabilitation skills despite 85%

of CHCs having certified therapists (Yung et al.,

2022). Standardized Quality Ratings: Public report

cards on rehabilitation outcomes (e.g., the U.S. CMS’

Hospital Compare system improved transparency)

should be published. Shared Decision-Making Tools:

Develop AI-driven platforms to match patients with

appropriate providers based on their conditions and

preferences. Behavioral Economics: Design “nudges”

to guide patient choices (e.g., default referrals to

CHCs unless a tertiary hospital is explicitly

requested). Network Science: Map referral patterns

using social network analysis to identify bottlenecks

and key influencers in rehabilitation systems. Non-

Cooperative Equilibrium: Institutions rationally

pursue self-interest (e.g., resource hoarding), but this

leads to system-wide inefficiencies (e.g., 65% of

rehabilitation patients in Yanta still bypass CHCs)

(Zhong et al., 2023). Cooperative Equilibrium: By

internalizing externalities (e.g., sharing data reduces

readmissions), institutions can achieve Pareto

improvements. For example, Yanta’s

telerehabilitation network cut referral delays by 40%

(Spruijt-Metz et al., 2015).

5 CONCLUSION

5.1 Key Findings

Game theory insights highlight the current non-

cooperative iterated Prisoner's Dilemma with

institutional memory, where resource hoarding and

referral bottlenecks persist, and emphasize

Research on Game Behavior and Optimization Strategies Among Sports Rehabilitation Medical Institutions Under the Hierarchical

Diagnosis and Treatment

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cooperative solutions like coalition-building (e.g.,

CHCs specializing in post-acute care) to achieve

Pareto improvements. Empirical support from Yanta

District’s pilots shows that policy adjustments and

technological tools can mitigate these issues. This

underscores the need for systemic reforms to

transform fragmented, competitive dynamics into a

collaborative, efficient rehabilitation ecosystem.

5.2 Research Significance

This study addresses critical inefficiencies in

hierarchical medical systems by analyzing strategic

interactions (game behavior) among sports

rehabilitation institutions, offering both theoretical

and practical contributions. The research highlights

how non-cooperative dynamics—such as resource

hoarding by tertiary hospitals, underinvestment in

community health centers (CHCs), and information

withholding—create systemic bottlenecks,

undermining rehabilitation service accessibility and

efficiency (e.g., 40% bed underutilization in Yanta

District and 65% bypass of CHCs for rehabilitation

care). By framing these challenges through game

theory, the analysis reveals a “Prisoner’s Dilemma”

equilibrium where individual rationality leads to

suboptimal collective outcomes, underscoring the

need for institutional interventions to foster

cooperation.

5.3 Limitations and Future Study

While this analysis provides a robust framework for

addressing game behavior in sports rehabilitation

under hierarchical systems, several limitations merit

consideration. First, the empirical evidence is

primarily drawn from pilot programs in Yanta District

and Shaanxi Province, potentially limiting

generalizability to contexts with different healthcare

financing structures, administrative capacities, or

regional demographics (e.g., rural vs. urban

disparities varying levels of technological

infrastructure). Second, the game theory model

simplifies institutional interactions as a binary

Prisoner’s Dilemma, which may overlook more

nuanced strategic dynamics, such as repeated

interactions, multi-party coalitions, or the influence

of informal relationships between institutions, which

could alter cooperation incentives. Third, the study

focuses on supply-side behaviors (institutions’

resource allocation and referral decisions) but does

not fully explore demand-side factors, such as patient

preferences for tertiary hospitals or literacy levels

affecting the utilization of CHC services, which

might moderate the effectiveness of proposed

solutions. Additionally, the long-term sustainability

of interventions like tiered reimbursements or

centralized EHR systems is not fully addressed,

including potential fiscal burdens on healthcare

budgets or resistance from stakeholders (e.g., tertiary

hospitals losing revenue from retained patients).

Finally, while technological tools (e.g.,

telerehabilitation platforms) are highlighted, the

analysis does not account for digital divides or

training gaps that could impede adoption, particularly

in less-resourced settings. These limitations suggest a

need for further research to validate findings in

diverse contexts and incorporate broader systemic

and behavioral factors into future models.

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