Mathematical Analysis

Deep dive into the algorithms and patterns

Last updated: December 2025

This analysis is based on real calculator outputs from testing various scenarios. All findings are derived from actual algorithm results, not theoretical assumptions.

Package Efficiency Analysis

Not all RP packages are created equal. Here's the efficiency (RP per Euro) of each package in the EUR region:

Package RP Price RP per € Bonus vs Smallest
Smallest 575 RP €4.99 115.2 -
€10.99 Package 1380 RP €10.99 125.6 +9.0%
€21.99 Package 2800 RP €21.99 127.3 +10.5%
€34.99 Package 4500 RP €34.99 128.6 +11.6%
€49.99 Package 6500 RP €49.99 130.0 +12.8%
€99.99 Package 13500 RP €99.99 135.0 +17.2%
€244.99 Package 33500 RP €244.99 136.8 +18.8%
Largest 60200 RP €429.99 140.0 +21.5%
Surprising Finding: Despite having poor efficiency, the €10.99 package appears frequently in specific scenarios! Testing revealed it's used heavily when you need small amounts of RP (see below).

The €10.99 Package Mystery - Solved!

Test Results:

Pattern Discovered: The €10.99 package is actually optimal when you need to "top up" a small amount. When you already have significant RP, buying a €99.99 package would be massive overkill. The €10.99 package fills the gap perfectly!

Example from testing: €10.99 -> €34.99 -> €10.99 -> €10.99 -> €10.99 was optimal for 10000 starting RP with 50 pulls needed.

The Two Optimization Strategies

1. Optimal Strategy (Minimize Expected Cost)

This strategy considers the probability of getting the item early. It uses the formula:

Expected Cost = Σ(Package Cost × Probability You Need It)

Key insights:

2. Most Bonus RP Strategy (Minimize Cost at Pity)

This strategy assumes worst-case: you need all pulls until pity. It optimizes for:

Minimum Total Cost to reach Pity RP

Key insights:

Real Patterns From Testing

Finding #1: Strategy Convergence at Low Drop Rates

Test Case: 0.2% drop rate (very low), 80 pulls needed

Why: With only 0.2% drop rate, there's an 85% chance of hitting pity. The probability of early drops is so low that optimizing for expected cost converges to optimizing for worst-case.

Finding #2: The €49.99 Spam Strategy

Test Case: 2% drop rate (high), 80 pulls needed

Discovery: With high drop rates, the optimal strategy prefers medium packages repeatedly! This is because there's a 78% chance you'll drop it before needing all 5 packages. Buying €49.99 packages means you'll likely only spend €99.98-€149.97, while the pity strategy commits to €239.96.

Finding #3: The €244.99 Package is Nearly Useless

Testing 6 different scenarios, the €244.99 package appeared only ONCE (in expensive pulls scenario with pity strategy). It's almost never optimal because:

Finding #4: Small Packages First is Real

Standard scenario (80 pulls, 0.5% drop):

The optimal strategy starts with TWO €21.99 packages despite their lower efficiency. If you get lucky and drop early, you save money. The pity strategy jumps straight to large packages because it assumes worst-case.

Finding #5: €99.99 Dominance

Across all 6 test scenarios, the €99.99 package appeared in 5 out of 6 strategies (all except the "few pulls needed" case). It's the sweet spot of efficiency and cost.

Algorithm Performance Stats

Real Computation Data

From our test runs, here's what the algorithm actually does:

Observation: The number of strategies grows with RP needed, not just pulls. Expensive pulls (1000 RP each) generated the most strategies because more RP is needed total.

Unique Solutions Found

Interesting pattern - the algorithm finds many strategies with identical costs:

Counterintuitive Discoveries

Discovery #1: Order Sometimes Doesn't Matter

In the standard test, these had nearly identical expected costs (within €0.03):

But only the first one is truly optimal! The algorithm found 5 top strategies all within €0.06 of each other.

Discovery #2: High Drop Rate Drastically Changes Strategy

The 2% drop rate case showed the most dramatic shift:

The paradox: The optimal strategy has HIGHER worst-case cost but MUCH lower expected cost. You pay more if unlucky, but save significantly on average.

Discovery #3: The €4.99 Package Has a Purpose

The smallest package appeared in pity strategies more than optimal ones! It's used as a "filler" to minimize leftover RP:

Discovery #4: Strategy Diversity Increases With Drop Rate

Higher drop rates create more differentiation between strategies because order matters more when you're likely to drop early.

Final Insight: This calculator evaluates thousands of possible package combinations in milliseconds. The optimal strategy is often counterintuitive - starting small, using "inefficient" packages in the right context, and carefully balancing expected vs worst-case costs. The mathematics of gacha optimization is far more nuanced than "just buy the biggest package."