An attempt to build a deterministic Physics Engine for RTP Control

Background

This invention relates to computer-implemented gaming systems, specifically to methods and systems for creating deterministic physics-based games of chance with controlled Return to Player (RTP) percentages through pre-computed physics simulations and probability distribution modeling.

Traditional online games of chance rely on random number generators (RNGs) to determine outcomes, which can lead to unpredictable variance in RTP percentages over time. While these systems can achieve target RTPs over very large sample sizes, they cannot guarantee precise RTP control for smaller gaming sessions or specific time periods.

Physics-based games of chance, such as Plinko, roulette, or ball-drop games, typically use real-time physics engines to simulate realistic ball movement and collision dynamics. However, these real-time simulations are computationally expensive and cannot guarantee specific outcome distributions required for precise RTP control.

Existing systems either sacrifice realistic physics for RTP control (using pure RNG with predetermined outcomes) or sacrifice RTP precision for realistic physics (using real-time physics engines with unpredictable outcomes).

There exists a need for a system that combines realistic physics-based gameplay with precise RTP control while maintaining computational efficiency and provable fairness.

Summary

The present invention provides a novel method and system for creating deterministic physics-based games of chance with precise RTP control through pre-computed physics simulations and probability distribution modeling.

The system comprises:

• A physics simulation engine that pre-computes realistic game outcomes using accurate physics modeling.
• An RTP calculation and adjustment algorithm that modifies natural probability distributions to achieve target RTP percentages.
• A drop generation system that creates the exact number of physics simulations needed for each outcome bin according to the adjusted probability distribution.
• A storage and retrieval system for managing large datasets of pre-computed physics simulations.
• A runtime selection system that chooses appropriate pre-computed simulations during gameplay while maintaining the appearance of real-time physics.

The invention enables precise RTP control (within 1–3% of target values) while maintaining realistic physics-based animations and ensuring provable fairness through auditable pre-computed outcomes.

Detailed Description

System Architecture

The deterministic physics engine system consists of several interconnected components:

1. Physics Simulation Engine. Utilizes a high-fidelity physics library (e.g., Matter.js, Box2D) to create realistic simulations. For Plinko-style games, this includes:
   • Ball physics with realistic mass, velocity, and collision properties.
   • Peg collision detection and response with appropriate restitution coefficients.
   • Gravity simulation and air resistance modeling.
   • Bin detection and outcome recording.
   Captures complete trajectory data for each simulation, including frame-by-frame position coordinates, velocity vectors, collision events, and final outcome bin determination.
2. RTP Calculation and Adjustment Algorithm. Determines the required probability distribution for each game configuration to achieve target RTP:
   1. Calculate natural probability distribution using binomial theory or empirical testing.
   2. Compute natural RTP: Natural RTP = Σ_i P_natural,i × Multiplier_i.
   3. Determine adjustment factor: Adjustment_factor = Target_RTP / Natural RTP.
   4. Apply bin-specific adjustment function f(Adjustment_factor, i).
   5. Calculate drop counts: Total_drops × Adjusted_probabilities[i].
3. Drop Generation System. Generates the required number of simulations per outcome bin:
   • Adaptive initial positioning based on target bin probability.
   • Multi-attempt persistence with increasing position variance.
   • Extreme positioning strategies for rare outcomes.
   • Duplication algorithms as fallback for unachievable targets.
   • Validation and storage of trajectory data for each simulation.
4. Storage and Retrieval System. Manages large datasets of pre-computed simulations:
   • Hierarchical file organization by configuration and risk level.
   • JSON format with compressed trajectory data and metadata (timestamps, achieved RTP, checksums).
   • File naming convention: drops-{configuration}-{risk_level}.json.
5. Runtime Selection System. Selects and plays back simulations during gameplay:
   1. Determine target outcome bin from adjusted distribution.
   2. Load the appropriate simulation file.
   3. Randomly select a pre-computed trajectory.
   4. Play back trajectory for smooth, real-time animation.
   5. Optimize performance via smart caching, background preloading, and memory management.

Implementation Examples

The following TypeScript code examples demonstrate the core patentable algorithms and methods of the deterministic physics engine system.

RTP Calculation and Adjustment Algorithm

The mathematical foundation of the patent lies in the RTP calculation and probability adjustment algorithm:

interface BinDistribution {
  binIndex: number;
  naturalProbability: number;
  adjustedProbability: number;
  dropCount: number;
  multiplier: number;
}

export const TARGET_RTP = {
  [RiskLevel.LOW]: 0.9,     // 90% RTP
  [RiskLevel.MEDIUM]: 0.85,  // 85% RTP
  [RiskLevel.HIGH]: 0.8,     // 80% RTP
} as const;

/**
 * Core RTP adjustment algorithm - the heart of the patent
 */
export function adjustProbabilitiesForTargetRTP(
  rowCount: number,
  riskLevel: RiskLevel,
  totalDrops: number = 5000
): BinDistribution[] {
  const naturalProbs = calculateNaturalProbabilities(rowCount);
  const multipliers = binPayouts[rowCount][riskLevel];
  const targetRTP = TARGET_RTP[riskLevel];
  
  const naturalRTP = calculateNaturalRTP(naturalProbs, multipliers);
  const adjustmentFactor = targetRTP / naturalRTP;
  
  // Apply bin-specific adjustment function
  const adjustedProbs = naturalProbs.map((prob, binIndex) => {
    return prob * getBinAdjustmentFactor(adjustmentFactor, binIndex, naturalProbs.length);
  });
  
  // Normalize probabilities to sum to 1
  const probSum = adjustedProbs.reduce((sum, prob) => sum + prob, 0);
  const normalizedProbs = adjustedProbs.map(prob => prob / probSum);
  
  // Calculate exact drop counts
  return normalizedProbs.map((prob, binIndex) => ({
    binIndex,
    naturalProbability: naturalProbs[binIndex],
    adjustedProbability: prob,
    dropCount: Math.round(totalDrops * prob),
    multiplier: multipliers[binIndex]
  }));
}

Physics Simulation Engine with Trajectory Capture

The physics simulation engine captures complete trajectory data for pre-computation:

interface DropTrajectory {
  id: string;
  targetBin: number;
  actualBin: number;
  frames: TrajectoryFrame[];
  collisions: CollisionEvent[];
  duration: number;
  isValid: boolean;
}

export class DeterministicDropGenerator {
  private engine: Matter.Engine;
  private world: Matter.World;
  
  /**
   * Generate physics simulation for specific target bin
   * Core method for creating deterministic outcomes
   */
  async generateDropForBin(
    targetBin: number, 
    maxAttempts: number = 1000
  ): Promise<DropTrajectory | null> {
    
    for (let attempt = 0; attempt < maxAttempts; attempt++) {
      const trajectory = await this.simulateSingleDrop(targetBin, attempt);
      
      if (trajectory && trajectory.actualBin === targetBin && trajectory.isValid) {
        return trajectory;
      }
    }
    
    // Apply extreme positioning strategy for difficult bins
    return await this.generateWithExtremePositioning(targetBin);
  }
  
  /**
   * Simulate single drop with trajectory capture
   */
  private async simulateSingleDrop(
    targetBin: number, 
    attempt: number
  ): Promise<DropTrajectory> {
    
    // Adaptive initial positioning based on target bin
    const initialX = this.calculateInitialPosition(targetBin, attempt);
    const ball = this.createBall(initialX);
    
    const trajectory: DropTrajectory = {
      id: `drop_${targetBin}_${attempt}_${Date.now()}`,
      targetBin,
      actualBin: -1,
      frames: [],
      collisions: [],
      duration: 0,
      isValid: true
    };
    
    // Simulation loop with frame capture
    let frameCount = 0;
    const maxFrames = 300; // 5 seconds at 60fps
    
    while (frameCount < maxFrames) {
      Matter.Engine.update(this.engine, 1000 / 60);
      
      // Capture frame data
      trajectory.frames.push({
        timestamp: frameCount * (1000 / 60),
        position: { x: ball.position.x, y: ball.position.y },
        velocity: { x: ball.velocity.x, y: ball.velocity.y },
        rotation: ball.angle
      });
      
      // Check if ball reached bottom
      if (ball.position.y > this.getBinLevel()) {
        trajectory.actualBin = this.determineBin(ball.position.x);
        break;
      }
      
      frameCount++;
    }
    
    // Validate trajectory realism
    trajectory.isValid = this.validateTrajectory(trajectory);
    
    return trajectory;
  }

Deterministic Runtime Selection System

The runtime system selects pre-computed simulations while maintaining unpredictability:

export class DeterministicPlinkoEngine {
  private dropDatabase: PrecomputedDropDatabase = {};
  private binDistribution: BinDistribution[] = [];
  
  /**
   * Core deterministic selection method
   * Uses RTP-adjusted probabilities to select target bin, then chooses simulation
   */
  selectDeterministicDrop(): DropTrajectory {
    // Step 1: Determine target bin using RTP-adjusted probabilities
    const targetBin = this.selectTargetBinWithRTPControl();
    
    // Step 2: Select random simulation from target bin
    const availableDrops = this.getDropsForBin(targetBin);
    
    if (availableDrops.length === 0) {
      throw new Error(`No drops available for bin ${targetBin}`);
    }
    
    const randomIndex = Math.floor(Math.random() * availableDrops.length);
    return availableDrops[randomIndex];
  }
  
  /**
   * RTP-controlled target bin selection
   * Maintains unpredictability while ensuring RTP adherence
   */
  private selectTargetBinWithRTPControl(): number {
    const random = Math.random();
    let cumulativeProbability = 0;
    
    for (const binData of this.binDistribution) {
      cumulativeProbability += binData.adjustedProbability;
      
      if (random <= cumulativeProbability) {
        return binData.binIndex;
      }
    }
    
    // Fallback to last bin
    return this.binDistribution[this.binDistribution.length - 1].binIndex;
  }
}

Adaptive Generation with Persistence Algorithms

Advanced generation techniques for difficult-to-achieve outcomes:

export class AdaptiveDropGenerator extends DeterministicDropGenerator {
  
  /**
   * Generate drops with persistence for difficult bins
   */
  async generateDropsForBinWithPersistence(
    targetBin: number,
    requiredCount: number
  ): Promise<DropTrajectory[]> {
    
    const generatedDrops: DropTrajectory[] = [];
    const maxAttemptsPerDrop = this.calculateMaxAttempts(targetBin);
    
    for (let i = 0; i < requiredCount; i++) {
      let drop: DropTrajectory | null = null;
      let attempts = 0;
      
      // Standard generation attempts
      while (!drop && attempts < maxAttemptsPerDrop) {
        drop = await this.generateDropForBin(targetBin, 100);
        attempts += 100;
      }
      
      // Extreme positioning if standard fails
      if (!drop) {
        drop = await this.generateWithExtremePositioning(targetBin);
      }
      
      // Duplication with variation as last resort
      if (!drop && generatedDrops.length > 0) {
        drop = this.duplicateWithVariation(generatedDrops[0], targetBin);
      }
      
      if (drop) {
        generatedDrops.push(drop);
      }
    }
    
    return generatedDrops;
  }
  
  /**
   * Calculate maximum attempts based on bin difficulty
   */
  private calculateMaxAttempts(targetBin: number): number {
    const binCount = this.rowCount + 1;
    const centerBin = (binCount - 1) / 2;
    const distanceFromCenter = Math.abs(targetBin - centerBin);
    
    // Edge bins are exponentially more difficult
    const difficultyMultiplier = Math.pow(2, distanceFromCenter);
    return Math.min(100000, 1000 * difficultyMultiplier);
  }
}

Trajectory Validation and Quality Control

Validation algorithms ensure physics realism and prevent exploitation:

export class TrajectoryValidator {
  
  /**
   * Comprehensive trajectory validation
   */
  static validateTrajectory(trajectory: DropTrajectory): boolean {
    return (
      this.validatePhysicsRealism(trajectory) &&
      this.validateTemporalConsistency(trajectory) &&
      this.validateCollisionLogic(trajectory) &&
      this.validateOutcomePlausibility(trajectory)
    );
  }
  
  /**
   * Validate physics realism (velocity, acceleration constraints)
   */
  private static validatePhysicsRealism(trajectory: DropTrajectory): boolean {
    const maxVelocity = 1000; // pixels per second
    const maxAcceleration = 2000; // pixels per second squared
    
    for (let i = 1; i < trajectory.frames.length; i++) {
      const frame = trajectory.frames[i];
      const prevFrame = trajectory.frames[i - 1];
      const dt = (frame.timestamp - prevFrame.timestamp) / 1000;
      
      // Check velocity constraints
      const velocityMagnitude = Math.sqrt(
        frame.velocity.x ** 2 + frame.velocity.y ** 2
      );
      
      if (velocityMagnitude > maxVelocity) {
        return false;
      }
      
      // Check acceleration constraints
      const acceleration = {
        x: (frame.velocity.x - prevFrame.velocity.x) / dt,
        y: (frame.velocity.y - prevFrame.velocity.y) / dt
      };
      
      const accelerationMagnitude = Math.sqrt(
        acceleration.x ** 2 + acceleration.y ** 2
      );
      
      if (accelerationMagnitude > maxAcceleration) {
        return false;
      }
    }
    
    return true;
  }
}

Claims