Nash Equilibrium Validation Using AI Agent Digital Twins

Experiment Overview

This project validates a game-theoretic Nash equilibrium solution through simulation using AI agents as digital twins. We model participants in a federated insight system where individual AI agents make autonomous economic decisions, mimicking real human behavior under different incentive structures.

The Experiment: 100 AI agents receive personalized prompts describing their economic situation, game rules, and strategic options. Each agent independently decides whether to act honestly or maliciously based on their perceived incentives. The simulation tests whether our theoretical Nash equilibrium holds when faced with autonomous decision-making agents.

Key Findings: The AI agent simulation confirmed equilibrium stability under fair rotation with 3 temporary validators per round, validating the mathematical model with behavioral evidence.

Player Definitions & Game Flow

Player Types (Dynamic Roles)

All 100 participants are identical at start but receive temporary role assignments each round:

🔄 Insight Generators (IGs) - 97 per round

Role: Generate local insights, vote on submissions
Economics: Earn $0.00008/round, pay $0.0004 honest cost, stake $4.84
Behavior: Choose honest (quality insights) vs malicious (low effort/gaming)
AI Agent Decisions: "Do I invest effort in quality insights or try to game the system?"

⚖️ Harmonizers (H) - 3 per round (randomly selected)

Role: Aggregate IG submissions, create final proposals, do mandatory IG work
Economics: Earn ~$0.0047/round, pay $0.0001 honest cost, stake $161.28
Constraints: Must spend 25% time on IG work (get 40x multiplier)
AI Agent Decisions: "Do I fulfill my IG work quota honestly or try to skimp?"

Round Flow (Each of 100 Rounds)

1. Role Assignment 🎯

System selects 3 participants as temporary H validators based on:
- Reputation scores (higher = more likely)
- Fairness rotation (avoid consecutive selections)
- Byzantine status (malicious agents penalized)
Remaining 97 become IGs for this round

2. IG Phase 💡

Each IG receives personalized prompt with:
- Current balance, reputation, role history
- Game rules, costs, rewards, detection probabilities
- Strategic temptations and consequences
AI Agent Decision: Vote honestly (quality assessment) vs maliciously (random/biased)
IGs submit votes on candidate submissions
Pay voting costs, gain reputation if honest

3. H Validator Phase ⚖️

Each of 3 H validators receives prompt with:
- IG vote tallies and submission quality
- Their mandatory IG work requirements (25% quota)
- Economic incentives and penalties
AI Agent Decision: Make honest aggregate decision vs game the system
H validators must do IG work (get 40x multiplier)
Create collective final decision

4. Economic Settlement 💰

Distribute rewards based on performance:
- IGs: Base rate if honest, penalties if caught cheating
- Hs: Higher rate + IG work bonuses if quota met
Update balances, reputation scores
Apply detection/penalties for malicious behavior (98% catch rate)

5. Metrics Collection 📊

Track individual agent decisions and outcomes
Monitor role distribution, earnings, reputation
Validate Nash equilibrium conditions each round

AI Agent Behavioral Models

🎭 Persona Assignment (using NVIDIA Nemotron-Personas dataset):

Each of the 100 participants is assigned a unique persona from the Nemotron dataset
Personas include rich demographic, professional, and personality information
Examples: "disciplined and organized" retiree, "ambitious and competitive" professional, "creative and imaginative" artist
Personality traits influence decision-making tendencies (e.g., risk-averse personas are more likely to choose safe strategies)

🧠 Decision Factors (each agent considers):

Current economic position vs others
Probability of detection if malicious
Long-term reputation consequences
Fair rotation opportunities (when will I be H again?)
Recent system events (Byzantine catches, punishments)

📚 Memory & Learning:

Personal history of past roles and outcomes
Observations of other agents' successes/failures
System-wide trends (detection rates, earnings)
Strategic adaptation over 100 rounds

This creates a realistic behavioral laboratory where economic theory meets autonomous agent decision-making under uncertainty.

Problem Statement

In federated insight systems with two participant types:

Insight Generators (IG): Create local insights, earn $0.00008/round, stake $4.84
Harmonizers (H): Aggregate insights & create knowledge proposals, earn variable amounts, stake $161.28

Without proper incentive design, this creates a massive profit differential causing everyone to abandon IG work → system collapse.

Solution: Fair Rotation + Two-Tier Incentive System

Our solution implements:

1. Fair Rotation System

3 temporary H validators selected per round from all 100 participants
Each participant serves as H validator ~3 times over 100 rounds
Selection based on reputation + fairness (avoiding repeat selections)
Nash equilibrium: π_H = 3/100 = 0.03 selection probability per round

2. Two-Tier Incentive Structure

IG Work Multiplier: 40x rewards for harmonizers doing IG work (β_IG = 40)
Minimum Quota: 25% of harmonizer time must be IG work (θ = 0.25)
Performance Bonuses: $0.002 for meeting quotas (B_perf)
Adjusted Costs: Reduced H honest cost to $0.0001 for fair rotation equilibrium

Results

Profit ratio: 58x (down from 312x) while maintaining validator incentives
Nash equilibrium: ✅ All conditions satisfied (IGs prefer honest, Hs prefer honest)
Fair participation: All 100 participants can serve as H validators over time
Behavioral validation: AI agents consistently chose honest strategies

Quick Start

# Step 1: Generate Nash equilibrium configuration
python nash_formulation.py 100 100  # Creates simulation_params_n100.json

# Step 2: Run simulation with AI agents (config file required)
python nash_simulation.py --config simulation_params_n100.json

# Optional: Run without LLM for faster testing
python nash_simulation.py --config simulation_params_n100.json --no-llm

# Optional: Specify custom output file
python nash_simulation.py --config simulation_params_n100.json --output my_results.json

# Test LLM connection
python test_llm_connection.py

Core Files

File	Description
`formulation.tex`	Mathematical Nash equilibrium proof with experimental validation
`nash_formulation.py`	Nash parameter calculation for fair rotation system
`nash_simulation.py`	AI agent simulation with temporary role assignments
`game_rules_prompt.py`	LLM prompts for agent decision-making
`persona_manager.py`	Loads and manages personas from NVIDIA Nemotron dataset
`simulation_params_n100.json`	Validated configuration for 100 participants, 100 rounds
`nash_results_*.json`	Simulation results with agent behavioral data

AI Agent Architecture

Agent Decision Process

Each AI agent receives:

Personal State: Balance, reputation, stake locked, role history
Game Rules: Costs, rewards, detection probabilities, consequences
Current Context: Round number, recent events, other participants' actions
Strategic Options: Honest vs malicious behavior with expected outcomes

Behavioral Modeling

Persona-Based Variation: Each agent's decisions are influenced by their assigned Nemotron persona
- Competitive/ambitious personas: ~60% chance of risky behavior
- Creative/young personas: ~50% chance of risky behavior
- Disciplined/cautious personas: ~15% chance of risky behavior
Memory: Each agent maintains history of past actions and outcomes
Temptation Modeling: Agents consider short-term gains vs long-term reputation
Byzantine Simulation: ~15% of agents are configured as Byzantine (malicious)

Validation Metrics

Decision Consistency: Do agents choose Nash equilibrium strategies?
Response to Incentives: Do behavior changes match theoretical predictions?
Fair Rotation: Do all participants get equal opportunities over time?
Equilibrium Stability: Does the system remain stable over 100 rounds?

Mathematical Foundation

The fair rotation Nash equilibrium ensures:

π_H = 3/100 = 0.03 (selection probability per participant per round)
E[U_IG(honest)] = $0.00008 > E[U_IG(malicious)] ✓
E[U_H(honest)] = $0.0047 > E[U_H(malicious)] = $0.0036 ✓
Profit ratio = 58x (manageable vs 312x problematic)

Key insight: With fair rotation, each participant gets selected as H validator ~3 times over 100 rounds, creating realistic promotion opportunities while maintaining system balance.

Experimental Results