Basic Statistical Analysis of Snowbreak Rate Up and Guaranteed Gacha Banners
The information contained in this study has been confirmed to match banner details on 1.8 release
Introduction
This brief study attempts to replicate the gacha system of Snowbreak and compare the currently existing Rate Up banner with the upcoming Guaranteed banner with spark. The study begins with a list of assumptions and explanations for each type of banner's rules, followed by details on the simulation setup. An example analysis and explanation is provided for one scenario. Readers familiar with the topic might wish to skip straight to the graphs comparing the required numbers of rolls to reach M0/M1/M2/M5 Operative and T1/T2 Weapon under the two banner types. Finally some limitations are declared, small conclusions drawn and TL;DR customarily provided.
Assumptions
| - | Rate Up Operative | Guaranteed Operative | Rate Up Weapon | Guaranteed Weapon |
|---|---|---|---|---|
| Base rate | 0.7% | 0.1% | 0.7% | 0.1% |
| Increment after soft pity | 5% | 0.75% | 6% | 5% |
| Soft pity | 62 | 70 | 45 | 60 |
| Hard pity | 80 | 100 | 60 | 80 |
| Spark | - | 165 | - | 130 |
| Consolidated rate including guarantee | 1.86% | 1.85% | 2.37% | 2.35% |
Rate Up Banner
Under the Rate Up banner, the base rate of any standard or banner 5-star appearing is 0.7% from the first roll up to and including the soft pity at 62 rolls for operative and 45 rolls for weapon, after which the rates increment by 5% for operative and 6% for weapon, until the hard pity at 80 rolls for operative and 60 rolls for weapon, when the rates are set at 100% and a 5-star drop is guaranteed.
Getting a 5-star drop at any point resets the pity count. The pity count carries over between banners of the same type.
When a 5-star drops, there is a 50% chance that it will be the banner 5-star. If a standard 5-star and not a banner 5-star drops this time, the next time a 5-star drops is guaranteed to be the banner 5-star.
Guaranteed Banner with Spark
Under the Guaranteed banner, the base rate of a banner 5-star appearing is 0.1% from the first roll up to and including the soft pity at 70 rolls for operative and 60 rolls for weapon, after which the rates increment by 0.75% for operative and 5% for weapon, when the rates are set at 100% and a banner 5-star drop is guaranteed.
Getting a banner 5-star drop at any point resets the pity count. The pity count carries over between banners of the same type.
Unlike the Rate Up banner, there is no standard 5-star drop in the Guaranteed banner. Instead, a Random Standard Manifestation Box is given every 165 rolls (tentatively called spark for illustrative purposes, not official translation) on the Guaranteed Operative banner and a Random Standard Weapon Box is given every 130 rolls on the Guaranteed Weapon banner. These boxes can then be redeemed for 60 extracts of a random standard Manifestation and a copy of a standard weapon.
The spark count is independent of the pity counts and carries over between all banners of the same type.
Simulation Setup
Simulations each consisting of 100 million rolls are conducted with the above assumptions for 4 types of banners, Rate Up Operative, Guaranteed Operative, Rate Up Weapon and Guaranteed Weapon. Every time a banner 5-star drops, the roll count is recorded, until the target has been reached. The number of rolls required to reach each target is then graphed.
A probability density function (PDF) is graphed for each type of banners, showing the density approximating the probability that a target number of banner 5-star drops at each number of rolls. A cumulative distribution function (CDF) is graphed to show the cumulative proportion of the sample population achieving a target number of 5-star drops at each number of rolls.
These two graphs are displayed side by side, with samples for Rate Up and Guaranteed banners overlaid for comparison. Comparisons are made for Operative and Weapon banners with different roll targets: Operative M0, M1, M2 and M5, and Weapon T1 and T2.
Example Analysis and Explanation
Comparison of Rate Up and Guaranteed Banners targeting Operative M0
Rate Up Operative (targeting M0 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1858800
Number of standard 5-star: 619894 (33.3491% of total number of 5-star)
Number of banner 5-star: 1238906 (66.6509% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8588%, Banner 1.2389%, Standard 0.6199%
Key Statistics
Mean: 81, Mode: 67, Median: 70, Standard deviation: 37, Interquartile range: 53
Guaranteed Operative with Spark (targeting M0 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1845970
Number of standard 5-star: 606060 (32.8315% of total number of 5-star)
Number of banner 5-star: 1239910 (67.1685% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8460%, Banner 1.2399%, Standard 0.6061%
Key Statistics
Mean: 81, Mode: 82, Median: 83, Standard deviation: 15, Interquartile range: 11
Consolidated Rate Including Guarantee
The rate at which 5-star drops if there are infinite rolls on the banner. This is used to quantify the payoff for each banner, and is one of the metrics the developer can choose to declare as part of their legal obligation. In Snowbreak's case, this is what was declared.
Here we can verify that the statistical model closely replicates the declared consolidated rates of 1.86% for Rate Up banner and 1.85% for Guaranteed banner, and that the rate for Banner 5-star remains the same at 1.24%. Assuming the statistical model is correct, there is not enough evidence to suggest the banner 5-star rate has changed. The difference in overall consolidated rates can be attributed to the new spark system rewarding a standard 5-star every 165 rolls, which equates to a theoretical 1/165=0.6061%, which corroborates the declared 0.61% consolidated rate for standard 5-star under Guaranteed banner, as opposed to 0.62% under Rate Up banner.
Key Statistics
Mean - The average number of rolls required to reach a target. More useful when describing a normally distributed population, ie all values centred around a target or average value like a bell curve. We can see the average number of rolls it takes to get a banner 5-star drop is exactly the same on both Rate Up and Guaranteed banner.
Mode - The number that most frequently appears as the required number of rolls to reach a target. On the Guaranteed banner the highest chance a banner 5-star shows up is at 82 rolls. It can be misleading in the case of Rate Up banner because it is multimodal, ie there are two common values where 5-star drops most frequently occur, at 67 and at 135.
Median - The number below which 50% of the population resides, ie half of the people rolling will require fewer than this number of rolls. By 70 rolls on the Rate Up and 83 rolls on the Guaranteed banner, 50% of people will have had a banner 5-star drop.
Standard deviation - Equals to the square root of variance, measures how much the samples deviate from the average, more useful for normal distribution. Larger standard deviation means it can take much fewer or much more rolls depending on luck.
Interquartile range - The range below which 25% of the population resides, and above which another 25% of the population resides, ie half of the people rolling will require a number of rolls within this range, similar to standard deviation but more useful when the distribution is skewed, eg all guaranteed banners.
The graphs above attempt to compare the number of rolls required to get one copy of the banner operative under the current Rate Up banner and under the upcoming Guaranteed banner.
The horizontal axes show the number of rolls required to reach the target, which in this case is M0 Operative.
On the left, the probability density graph's vertical axis shows the density, ie the proportion of the sample population requiring a certain number of rolls, which can be intepreted as the probability that a certain number of rolls is required. For example, roughly 5% of the sample population rolled M0 Operative at 67 rolls under Rate Up banner, and 82 rolls under Guaranteed banner. The graph shows very rare chances of getting a 5-star before soft pity and a concentration of 5-star drops between soft pity and hard pity as expected, with some people winning the 50% Rate Up requiring fewer rolls than the Guaranteed Banner, while some losing the 50% Rate Up requires much more roll as they need to reach a second soft pity.
On the right, the cumulative distribution graph's vertical axis shows the cumulative density, ie the cumulative proportion of the sample population requiring a certain number of rolls. The cumulative density goes from 0, ie no one gets M0 at 0 rolls, to 1, ie everyone will have had M0 by roughly 150 rolls. It can be more intuitive to compare the two banner types here, as it can be seen that up until around 70 rolls, about 57% of the population will have gotten M0 under Rate Up banner, compared to about 6% under Guaranteed Banner. At about 80 rolls both banners are roughly the same, and if the 50% Rate Up is not won at this point, the Guaranteed banner can save a person about 40 rolls, because the whole population gets M0 at 100 rolls, whereas the Rate Up banner will require rolling until 130-140 for a second soft pity.
Comparison of Rate Up and Guaranteed Banners for Selected Targets
Operative M0
Rate Up Operative (targeting M0 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1858800
Number of standard 5-star: 619894 (33.3491% of total number of 5-star)
Number of banner 5-star: 1238906 (66.6509% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8588%, Banner 1.2389%, Standard 0.6199%
Key Statistics
Mean: 81, Mode: 67, Median: 70, Standard deviation: 37, Interquartile range: 53
Guaranteed Operative with Spark (targeting M0 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1845970
Number of standard 5-star: 606060 (32.8315% of total number of 5-star)
Number of banner 5-star: 1239910 (67.1685% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8460%, Banner 1.2399%, Standard 0.6061%
Key Statistics
Mean: 81, Mode: 82, Median: 83, Standard deviation: 15, Interquartile range: 11
Operative M1
Rate Up Operative (targeting M1 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1859871
Number of standard 5-star: 619163 (33.2906% of total number of 5-star)
Number of banner 5-star: 1240708 (66.7094% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8599%, Banner 1.2407%, Standard 0.6192%
Key Statistics
Mean: 161, Mode: 134, Median: 158, Standard deviation: 53, Interquartile range: 70
Guaranteed Operative with Spark (targeting M1 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1846409
Number of standard 5-star: 606060 (32.8237% of total number of 5-star)
Number of banner 5-star: 1240349 (67.1763% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8464%, Banner 1.2403%, Standard 0.6061%
Key Statistics
Mean: 161, Mode: 166, Median: 165, Standard deviation: 21, Interquartile range: 16
Operative M2
Rate Up Operative (targeting M2 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1858567
Number of standard 5-star: 619071 (33.3090% of total number of 5-star)
Number of banner 5-star: 1239496 (66.6910% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8586%, Banner 1.2395%, Standard 0.6191%
Key Statistics
Mean: 242, Mode: 269, Median: 242, Standard deviation: 65, Interquartile range: 85
Guaranteed Operative with Spark (targeting M2 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1845963
Number of standard 5-star: 606060 (32.8316% of total number of 5-star)
Number of banner 5-star: 1239903 (67.1684% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8460%, Banner 1.2399%, Standard 0.6061%
Key Statistics
Mean: 242, Mode: 250, Median: 248, Standard deviation: 26, Interquartile range: 22
Operative M5
Rate Up Operative (targeting M5 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1858919
Number of standard 5-star: 619602 (33.3313% of total number of 5-star)
Number of banner 5-star: 1239317 (66.6687% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8589%, Banner 1.2393%, Standard 0.6196%
Key Statistics
Mean: 726, Mode: 705, Median: 726, Standard deviation: 112, Interquartile range: 153
Guaranteed Operative with Spark (targeting M5 Operative) Simulation Results
Number of rolls: 100000000
Number of 5-star: 1846037
Number of standard 5-star: 606060 (32.8303% of total number of 5-star)
Number of banner 5-star: 1239977 (67.1697% of total number of 5-star)
Consolidated rate including guarantee: Combined 1.8460%, Banner 1.2400%, Standard 0.6061%
Key Statistics
Mean: 726, Mode: 750, Median: 735, Standard deviation: 45, Interquartile range: 59
Weapon T1
Rate Up Weapon (targeting T1 Weapon) Simulation Results
Number of rolls: 100000000
Number of 5-star: 2372192
Number of standard 5-star: 791310 (33.3578% of total number of 5-star)
Number of banner 5-star: 1580882 (66.6422% of total number of 5-star)
Consolidated rate including guarantee: Combined 2.3722%, Banner 1.5809%, Standard 0.7913%
Key Statistics
Mean: 63, Mode: 49, Median: 53, Standard deviation: 27, Interquartile range: 47
Guaranteed Weapon with Spark (targeting T1 Weapon) Simulation Results
Number of rolls: 100000000
Number of 5-star: 2351062
Number of standard 5-star: 769230 (32.7184% of total number of 5-star)
Number of banner 5-star: 1581832 (67.2816% of total number of 5-star)
Consolidated rate including guarantee: Combined 2.3511%, Banner 1.5818%, Standard 0.7692%
Key Statistics
Mean: 63, Mode: 64, Median: 65, Standard deviation: 10, Interquartile range: 4
Weapon T2
Rate Up Weapon (targeting T2 Weapon) Simulation Results
Number of rolls: 100000000
Number of 5-star: 2372471
Number of standard 5-star: 790773 (33.3312% of total number of 5-star)
Number of banner 5-star: 1581698 (66.6688% of total number of 5-star)
Consolidated rate including guarantee: Combined 2.3725%, Banner 1.5817%, Standard 0.7908%
Key Statistics
Mean: 126, Mode: 148, Median: 127, Standard deviation: 39, Interquartile range: 52
Guaranteed Weapon with Spark (targeting T2 Weapon) Simulation Results
Number of rolls: 100000000
Number of 5-star: 2350649
Number of standard 5-star: 769230 (32.7242% of total number of 5-star)
Number of banner 5-star: 1581419 (67.2758% of total number of 5-star)
Consolidated rate including guarantee: Combined 2.3506%, Banner 1.5814%, Standard 0.7692%
Key Statistics
Mean: 126, Mode: 130, Median: 130, Standard deviation: 13, Interquartile range: 6
Limitations and Conclusions
While the simulations managed to replicate the displayed consolidated rates for all cases, the assumed rates used in the simulations are still estimates and might not accurately replicate how the gacha actually works in the codes of the game. Even if the numbers came from a dream.
The study employs a theoretical approach and would greatly benefit from verification by empirical studies, ie recording rolls from everyone and check if they check out with these numbers.
Even with such limitations, some conclusions can reasonably be drawn:
- Two third of all 5-star ever rolled will be the banner 5-star (a tiny bit more under Guaranteed because of the reduced standard 5-star rate)
- The payoffs are exactly the same when comparing Rate Up and Guaranteed banner (1.24% banner operative and 1.58% banner weapon)
- The spreads are smaller under Guaranteed (requires more rolls to drop, but less rolls if unlucky) compared to Rate Up
In simpler terms:
- In the long run, they are the same
- Guaranteed banners will almost always require at least soft pity and even closer to hard pity for 5-star drop
- Rate Up banner is better if the 50% is won, otherwise Guaranteed banner is better