mpc-game-controllers-misspecification

name: mpc-game-controllers-misspecification description: "Stability and sensitivity analysis for objective misspecifications among Model Predictive Game (MPG) controllers. Multi-agent control design with game-theoretic solution concepts and heterogeneous controller analysis. Use when designing multi-agent MPC with game-theoretic predictions, analyzing stability under model misspecifications, or quantifying sensitivity to game parameters. Activation: model predictive games, MPG controllers, multi-agent game control, objective misspecification, heterogeneous controllers, game-theoretic control." arxiv: "2604.08303v1" author: "Ada Yildirim, Bryce L. Ferguson" date: "2026-04-09" categories: ["math.OC", "cs.MA", "cs.SY"]

Stability and Sensitivity Analysis for MPC Game Controllers

Problem Statement

Multi-Agent Control with Game-Theoretic Models

When multiple agents implement Model Predictive Game (MPG) controllers, each agent:

• Possesses a model of other agents' behavior
• Uses game-theoretic solution concepts to predict collective behavior
• Iteratively solves finite-horizon games to synthesize control actions

Objective Misspecification Problem

Misspecification Source:

• Inaccurate estimates of other agents' objectives
• Conjectures about game parameters
• Heterogeneous models across agents

Result: Prediction misalignments affecting system behavior

Core Framework

Model Predictive Game Structure

Each agent i at time t:
┌─────────────────────────────────────────────────────────────┐
│ 1. Observe: Current state x(t)                              │
│ 2. Predict: Other agents' behavior using game model         │
│ 3. Solve: Finite-horizon game Gi(x(t))                      │
│ 4. Apply: First control action ui(t)                        │
│ 5. Repeat: At next time step                                │
└─────────────────────────────────────────────────────────────┘

Heterogeneous Controller Configuration

Agent 1: Controller C₁ with game model G₁(θ₁)
Agent 2: Controller C₂ with game model G₂(θ₂)
   ⋮
Agent N: Controller Cₙ with game model Gₙ(θₙ)

Where θᵢ are potentially different parameter estimates

Main Results

1. Stability Criteria

Theorem (Stability Under Misspecification):

The multi-agent system with heterogeneous MPG controllers is stable if:

‖∂Vᵢ/∂θⱼ‖ ≤ Lᵢⱼ for all i, j

where Vᵢ is the value function for agent i's game

Key Insights:

• Stability depends on Lipschitz constants of value functions
• Bounded sensitivity to parameter variations
• Coupling between agents' prediction errors

2. Sensitivity Quantification

Sensitivity of Equilibria:

∂x*/∂θᵢ = -[∇²ₓₓU]⁻¹ · ∂/∂θᵢ(∇ₓU)

where:
- x*: equilibrium state
- U: joint utility function
- θᵢ: agent i's game parameters

Interpretation:

• Jacobian of equilibrium with respect to parameters
• Measures how parameter errors propagate to system behavior
• Provides design guidelines for robust controller tuning

Methodology

Sensitivity Analysis Steps

def analyze_mpg_sensitivity(controllers, equilibrium):
    """
    Analyze sensitivity of MPG equilibrium to parameter misspecifications
    
    Args:
        controllers: List of MPG controller instances
        equilibrium: Current equilibrium state
    
    Returns:
        sensitivity_matrix: ∂x*/∂θ for each agent
        stability_margin: Bounds on tolerable misspecification
    """
    n_agents = len(controllers)
    sensitivity = {}
    
    for i, controller in enumerate(controllers):
        # Compute value function gradient
        grad_V = compute_value_gradient(controller, equilibrium)
        
        # Compute Hessian of joint utility
        hessian_U = compute_joint_hessian(controllers, equilibrium)
        
        # Sensitivity: ∂x*/∂θᵢ
        sensitivity[i] = -np.linalg.inv(hessian_U) @ grad_V
    
    # Stability margin from Lipschitz constants
    lipschitz_bounds = [compute_lipschitz_constant(c) for c in controllers]
    stability_margin = min(lipschitz_bounds)
    
    return sensitivity, stability_margin

Prediction Alignment Analysis

def measure_prediction_alignment(agent_i, agent_j, state):
    """
    Measure prediction misalignment between two agents
    
    Returns:
        alignment_error: ‖predictionᵢ - predictionⱼ‖
    """
    pred_i = agent_i.predict_other_behavior(state)
    pred_j = agent_j.predict_other_behavior(state)
    
    return np.linalg.norm(pred_i - pred_j)

Design Guidelines

Controller Configuration

1. Parameter Estimation Bounds:

   Set conservative bounds: |θ̂ᵢ - θⱼ| ≤ δ_max
   where δ_max derived from stability analysis
   

2. Robust Game Formulation:

   Incorporate uncertainty sets in game models:
   min_uᵢ max_θ∈Θ Jᵢ(uᵢ, u₋ᵢ; θ)
   

3. Adaptive Parameter Learning:

   Online estimation of other agents' parameters:
   θ̂(t+1) = θ̂(t) + α·[observed - predicted]
   

Tuning Recommendations

Applications

Autonomous Vehicle Coordination

• Scenario: Multiple AVs at intersection
• Challenge: Each AV models others differently
• Solution: Robust MPG with sensitivity bounds

Distributed Robotics

• Scenario: Collaborative manipulation
• Challenge: Heterogeneous controller designs
• Solution: Stability-certified parameter ranges

Smart Grid Control

• Scenario: Multiple prosumers trading energy
• Challenge: Unknown cost functions
• Solution: Adaptive MPG with online learning

Mathematical Background

Game-Theoretic MPC

Standard MPC:

min_u J(x, u) s.t. x⁺ = f(x, u)

Game-Theoretic MPC:

Each agent i:
min_{uᵢ} Jᵢ(x, uᵢ, u₋ᵢ)
s.t. x⁺ = f(x, uᵢ, u₋ᵢ)
u₋ᵢ determined by game solution concept (Nash, Stackelberg, etc.)

Sensitivity Analysis Fundamentals

Implicit Function Theorem Application:

If F(x*, θ) = 0 defines equilibrium, then:
∂x*/∂θ = -(∂F/∂x)⁻¹ · (∂F/∂θ)

Implementation Considerations

Computational Complexity

• Per-agent cost: O(n³) for n-dimensional game
• Total system: O(N·n³) for N agents
• Can be parallelized: Each agent solves independently

Communication Requirements

• Minimal: Only state observations needed
• No explicit coordination: Emerges from game solution
• Robust to delays: Finite-horizon prediction absorbs latency

References

• Paper: "Stability and Sensitivity Analysis for Objective Misspecifications Among Model Predictive Game Controllers" (arXiv:2604.08303v1, 2026)
• Authors: Ada Yildirim, Bryce L. Ferguson
• Categories: math.OC, cs.MA, cs.SY

Related Skills

• discounted-mpc-robust-control: For MPC under plant-model mismatch
• density-driven-optimal-control: For multi-agent coverage control
• decentralized-stochastic-momentum-admm: For distributed optimization

Activation Keywords

• model predictive games
• MPG controllers
• multi-agent game control
• objective misspecification
• heterogeneous controllers
• game-theoretic MPC
• multi-agent stability analysis