Care Plan Generation for Underserved Patients Using Multi-Agent Language Models: Applying Nash Game Theory to Optimize Multiple Objectives

This study demonstrates that a Nash bargaining-based multi-agent language model framework significantly improves the safety and efficiency of care plans for underserved Medicaid patients compared to a single-model baseline, while highlighting that equity requires explicit design attention rather than emerging automatically from multi-objective optimization.

The Gist

Imagine you are a doctor managing a caseload of hundreds of patients who are struggling with complex health issues. These patients aren't just dealing with diabetes or heart disease; they are also fighting hunger, homelessness, and lack of transportation.

In the real world, there aren't enough doctors to spend hours with every single patient. So, hospitals are trying to use Artificial Intelligence (AI) to help write "Care Plans"—roadmaps that tell patients what to do next.

But here's the problem: AI is like a single person trying to juggle three balls at once, and it usually drops one.

• If the AI tries to be Safe (covering every possible medical risk), the plan becomes a 50-page novel that no one can read.
• If the AI tries to be Efficient (short and quick), it might forget to mention that the patient can't afford their medicine.
• If the AI tries to be Fair (addressing social needs), it might miss a critical medical warning.

This paper asks: Can we build a team of AI "experts" who argue with each other to find the perfect middle ground?

The Solution: A "Boardroom" of AI Agents

Instead of one AI trying to do everything, the researchers built a system with four specialized AI agents working together, using a mathematical concept called Nash Bargaining (think of it as a very fair, logical negotiation).

Here is how the team works, using a Restaurant Kitchen analogy:

1. The Head Chef (Generator): This AI looks at the patient's file and cooks up a first draft of the care plan.
2. The Safety Inspector (Safety Agent): This agent tastes the dish and yells, "Wait! You forgot to mention the patient's high blood pressure! That's dangerous!"
3. The Time Manager (Efficiency Agent): This agent looks at the menu and says, "This recipe is too long. If we give the patient a 20-page plan, they won't read it. Let's cut the fluff."
4. The Community Liaison (Equity Agent): This agent checks the ingredients and says, "The plan says 'go to the gym,' but this patient has no shoes and lives in a food desert. We need to change the plan to fit their reality."

The "Nash Bargaining" Step:
Usually, the Head Chef would just pick one person's advice. But in this study, the four agents sit down at a table. They don't just vote; they negotiate. They use a special math formula (Nash Bargaining) that forces them to find a solution where no one loses out. It finds the "sweet spot" where the plan is safe enough, short enough, and fair enough all at the same time.

The Experiment: The "Taste Test"

The researchers tested this "Team AI" against a "Solo AI" (a single AI trying to critique its own work, like a chef tasting their own soup over and over). They did this with 200 real Medicaid patients from Virginia and Washington.

The Rules of the Game:

• Privacy First: All the thinking happened on computers inside the hospital. No patient data ever left the building (a strict rule for medical privacy).
• Fair Comparison: Both the Team AI and the Solo AI were given the exact same amount of computer power to work with.

The Results: Who Won?

The researchers had a "Judge" (another AI) rate the care plans on a scale of 0 to 1 for Safety, Efficiency, and Fairness.

• Safety & Efficiency: The Team AI (Nash) won clearly. The plans were safer (fewer missed risks) and more efficient (easier to read and act on) than the Solo AI. It's like the Team AI served a meal that was both nutritious and easy to eat.
• Fairness (Equity): This is the twist. The Team AI did not do better than the Solo AI at addressing social needs. The score was exactly the same.

Why did the Team AI fail at Fairness?
The authors realized that simply having a "Fairness Agent" isn't enough if the rules of the game don't force the AI to actually solve the problem. It's like having a nutritionist on the team who points out the patient is hungry, but the kitchen doesn't have any food to give them. The AI could see the problem, but the system wasn't designed to fix the deep-rooted social issues.

The Big Takeaway

This study proves that AI teams are better than AI individuals when it comes to balancing safety and speed. By letting specialized agents "negotiate" using math, we can create care plans that are practical and safe for patients.

However, the study also warns us: You can't just "add" fairness to an AI and expect it to work. If you want AI to be truly fair to underserved patients, you have to design the system specifically to solve those problems, not just hope the math will fix it automatically.

In short:

• Old Way: One AI tries to do everything and ends up doing a mediocre job at all three.
• New Way: A team of specialized AIs negotiates to create a plan that is safe, short, and smart.
• The Catch: Even a great team needs a better "menu" (system design) to truly solve deep social inequalities.

Technical Summary

1. Problem Statement

Care management programs for Medicaid patients face a structural bottleneck: clinicians must simultaneously address clinical safety (risk coverage), resource efficiency (concise, actionable plans), and social equity (addressing social determinants of health).

• The Conflict: Single Large Language Model (LLM) approaches typically optimize for one objective at the expense of others. For example, a model maximizing safety may produce verbose, unactionable plans; one maximizing efficiency may omit critical social barriers; and one maximizing equity representation may lack clinical specificity.
• The Gap: Existing multi-agent frameworks often use simple aggregation (averaging or voting), which fails to find a mathematically optimal balance between conflicting objectives. Furthermore, most AI health interventions rely on cloud-based APIs, violating data-residency requirements common in Medicaid programs where patient data cannot leave local infrastructure.

2. Methodology

The study employed a two-part approach using a cohort of 5,148 activated Medicaid patients from Virginia and Washington. All inference was performed on locally hosted open-source models (DeepSeek-R1 8B and Llama 3.1 8B) to ensure data privacy.

A. Study Design

1. Retrospective Evaluation: Applied five deterministic strategies to the full cohort to map the multi-objective trade-offs:
   • Control (Template)
   • Single-Agent strategies (Safety-only, Efficiency-only, Equity-only)
   • Multi-Agent (Nash) strategy.
2. Controlled Paired Experiment (N=200): A pre-registered experiment where each of the 200 patients received two plans:
   • Intervention: A Nash-orchestrated multi-agent plan.
   • Baseline: A compute-matched sequential self-critique plan (a single model revising its own output multiple times).

B. The Nash Orchestration Framework

The core innovation is the application of Nash Bargaining Theory to synthesize outputs from role-specialized agents:

1. Generation: A generator model creates an initial draft.
2. Specialized Critique: Three distinct agents (Safety, Efficiency, Equity) review the draft and produce role-specific revisions.
3. Synthesis: Instead of averaging, the system applies a Nash bargaining solution. This mathematical framework identifies the unique Pareto-optimal point that maximizes the product of gains for all three objectives above a "disagreement baseline." This ensures no single objective dominates the others.
4. Judging: A separate "Judge" model scores the final plans on a 0–1 scale for Safety, Efficiency, Equity, and a Composite score (mean of the three).

C. Metrics

• Safety: Risk coverage and guideline concordance.
• Efficiency: Conciseness and prioritization of impactful interventions.
• Equity: Integration of social determinants of health (SDOH) and feasibility.
• Composite: Arithmetic mean of the three.

3. Key Results

Retrospective Findings

• Single-agent strategies maximized their specific domain but severely degraded the others (e.g., Safety Agent Only scored 0.843 on Safety but only 0.507 on Efficiency).
• The Multi-Agent (Nash) strategy achieved the highest simultaneous scores across all domains (Safety: 0.916, Efficiency: 0.650, Equity: 0.900, Composite: 0.822), demonstrating it successfully resolved the trade-off.

Controlled Paired Experiment (N=200)

The Nash-orchestrated plans significantly outperformed the compute-matched sequential self-critique baseline:

Metric	Nash Mean	Baseline Mean	Paired Difference	p-value	Cohen's d
Safety	0.813	0.797	+0.017	$6.65 \times 10^{-6}$	0.327 (Small-Med)
Efficiency	0.706	0.684	+0.022	$7.28 \times 10^{-13}$	0.543 (Med-Large)
Equity	0.746	0.746	0.000	0.987	-0.001 (Null)
Composite	0.755	0.742	+0.013	$6.20 \times 10^{-6}$	0.328 (Small-Med)

• Safety & Efficiency: The improvements were statistically significant with confidence intervals excluding zero. The effect size for Efficiency was notably large (0.543), suggesting multi-agent heterogeneity is particularly effective for tasks requiring prioritization and compression.
• Equity: The result was null (difference = 0.000). Both methods performed identically on equity metrics.

4. Key Contributions

1. Theoretical Application: Successfully adapted Nash Bargaining Theory from economics to LLM orchestration, proving it can generate Pareto-optimal solutions for conflicting clinical objectives better than simple averaging or self-critique.
2. Privacy-Preserving AI: Demonstrated that high-quality, multi-objective care planning is achievable using locally hosted, 8B-parameter open-source models, satisfying strict Medicaid data-residency constraints without relying on proprietary cloud APIs.
3. Empirical Evidence: Provided the first controlled evidence that role-specialized multi-agent negotiation yields measurably better care plan quality (Safety and Efficiency) than sequential self-critique in a real-world Medicaid cohort.
4. Equity Insight: Highlighted that multi-objective optimization alone does not solve equity issues. The null result on equity suggests that fairness requires explicit design constraints and rubric definitions beyond just weighting objectives; it cannot be assumed to emerge automatically from balancing other goals.

5. Significance and Implications

• Clinical Workflow: This approach offers a scalable solution to reduce clinician burnout by automating the complex task of balancing safety, efficiency, and social needs in care plans.
• Responsible AI Deployment: The study establishes a blueprint for deploying AI in sensitive government programs (like Medicaid) where data sovereignty is non-negotiable.
• Future Directions: The authors emphasize that while Nash orchestration improves safety and efficiency, equity requires explicit, upstream design attention (e.g., how needs are assessed and what interventions are available) rather than just post-hoc text generation optimization. Future work must focus on developing equity-specific rubrics and linking these AI-generated plans to actual patient health outcomes.