Solitaire Winning Probability Environment

This environment is designed to analyze and predict winning probabilities in various solitaire-style games using both theoretical mathematical analysis and empirical simulation.

Overview

The system combines two approaches to determine game winning probabilities:

1. Theoretical Analysis: Uses AI to derive mathematical formulas for exact probability calculations
2. Empirical Simulation: Runs Monte Carlo simulations to verify theoretical predictions

Key Components

GamePredictor Class

The core component that handles:

• AI-powered probability analysis
• Mathematical formula evaluation
• Game simulation
• Probability comparison between theoretical and empirical results

Features

• AI Analysis: Uses LLM to analyze game mechanics and derive mathematical formulas
• Formula Evaluation: Supports complex mathematical expressions including:
  • Factorials
  • Combinations (C(n,r))
  • Permutations (P(n,r))
  • Standard mathematical operations
• Simulation Engine: Runs multiple game simulations to verify theoretical predictions
• QA Dataset Generation: Creates training data for AI models by generating question-answer pairs

Reward Function

The environment implements a sophisticated reward function that evaluates the quality of probability predictions:

1. Base Reward Calculation:

   • Compares the predicted probability with the ground truth probability
   • Calculates the relative error: 1 - min(abs(gt - predicted) / gt, 2)
   • Adds a small bonus of 0.2 for valid predictions
   • Clips the final reward between -1 and 1

2. Length Penalty:

   • Applies a length-based penalty for responses that exceed 50% of the maximum token length
   • No penalty for responses under the threshold
   • Linear scaling of penalty based on response length
   • Helps encourage concise and efficient solutions

3. Validation Checks:

   • Verifies proper formula formatting and syntax
   • Ensures responses contain valid mathematical expressions
   • Handles edge cases and invalid responses gracefully

4. Quality Metrics:

   • Tracks percentage of correct predictions
   • Monitors response lengths and quality
   • Provides feedback for model improvement

Usage

# Initialize the predictor
predictor = GamePredictor(openai_api_key, openai_api_base)

# Define games to analyze
games = {
    'game_name': game_function,
    # ... more games
}

# Get predictions for all games
results = await predictor.predict_games(games)

# Generate QA dataset
await predictor.generate_qa_csv(games, n_simulations, "output.csv")

Output Format

The system provides comprehensive analysis for each game:

• AI's mathematical reasoning
• Derived probability formula
• Calculated theoretical probability
• Simulated empirical probability
• Comparison assessment between theory and simulation

Supported Games

The environment includes several example games:

• Easy games (1-4)
• Card matching games (2-4 cards)
• Odd card game

Requirements

• Python 3.x
• OpenAI API access
• Required packages:
  • openai
  • asteval
  • asyncio

Purpose

This environment serves multiple purposes:

1. Educational: Demonstrates probability theory in practical game scenarios
2. Research: Provides a framework for analyzing game mechanics
3. AI Training: Generates datasets for training AI models in probability analysis
4. Verification: Validates theoretical probability calculations through simulation

Contributing

New games can be added by implementing game functions that return a boolean indicating win/loss. The system will automatically analyze and provide probability predictions for any valid game implementation.