Impact of the Pity system on Special Valiant crystals
DNA3000
Member, Guardian Guardian › Posts: 20,311 Guardian
According to the latest stream, Kabam is experimenting with a form of pity system in the context of special Valiant featured crystals. According to the stream, the system will work as follows:
If you pull four crystals in a row that are not 6* rarity or higher, the next one will be at least a 6* champ
If you pull 29 crystals in a row that are not 7* rarity, the next one will be a 7* champ
If you pull 199 crystals in a row that are not the 7* featured champ, the next one will be the 7* featured champ
This is not precisely how it was explained, but I believe this more clearly represents the intent Kabam was describing. If you miss four in a row, the game still rolls the next one, but if that fifth roll is a 5* champ it "forces" it to be a 6* champ to break the streak of misses, it doesn't just automatically make the fifth drop a 6* (because if it did, that might cost you a 7* champ if that roll just happened to be a lucky pull).
Given those mechanics, how does the pity system alter the effective drop rates of the crystals? Since every time the pity system triggers the crystal drops more 6s, 7s, and 7* featured champs than the drop odds would ordinarily produce, the average number of those drops goes up. By how much?
There's two ways to calculate this: solve a bunch of recursive equations, or ask a computer to do it. Computer?
Measured drop rates:
5* rarity: 67.06 %
6* rarity: 27.82 %
7* rarity (not featured): 4.42 %
7* featured rarity: 0.7 %
Compared to the published odds of the Featured Valiant:
5* rarity: 73 %
6* rarity: 24 %
7* rarity (not featured): 2.64 %
7* featured rarity: 0.36 %
(Note: the published odds lists 7* drop rates as 3% but that includes the 7* featured champ odds of 0.36%, so I've separated them here for comparison purposes).
Because almost one in four drops is a 6* anyway, implementing a miss streakbreaker of five doesn't improve those odds dramatically, but it does improve them slightly. The bigger changes are with 7* rarity and 7* featured drops. On average, ordinarily about 38 drops are 7* champs (not featured) so breaking miss streaks at 30 has a substantial impact on the overall rate of return of the crystal: the crystal will drop about 67% more 7* champs with the pity system than without it. And the largest impact is on the featured 7* champion drop rate. About one in 278 crystals drops the featured 7* champ, and a pity system that streakbreaks at 200 increases the drop rate to almost double its initial rate.
Now, this is all separate from the fact that these streakbreaking pity drops are *predictable*. In theory if you keep track of all of your drops, you can actually predict in some cases when your next drop is guaranteed to be a 6*, a 7*, or even a 7* featured champ. If you open 199 crystals and none of them are the 7* featured, the next one will be. And since the pity system is (according to the stream) global across all special featured valiant crystals, if there's a featured you want among the special valiants that are announced, you could simply stop opening those crystals until the champ you want is released, and then the first crystal of that champion you open will be guaranteed to drop the featured champ.
As far as I am aware, you don't have to stop buying crystals, you just have to stop opening them. So if you really want a particular champ and you keep track of your openings and you know #200 is coming up, you can continue to buy featured Valiants and just hold them until the champ you want gets released, open that one, then go back and open all the other crystals you've stockpiled. I'm not sure to what degree anyone intends to play those kinds of games, but it is theoretically possible.
For everyone else, the above numbers estimate how the pity system alters the effective rate of return on the crystals. You obviously need to buy at least 30 to see the drop rate improvements on 7* champs and 200 to see the improvement for 7* featured, but for anyone thinking about buying that many, that's the overall drop rate improvements you're statistically going to see (random being random, everyone will see different overall rates of return).
If you pull four crystals in a row that are not 6* rarity or higher, the next one will be at least a 6* champ
If you pull 29 crystals in a row that are not 7* rarity, the next one will be a 7* champ
If you pull 199 crystals in a row that are not the 7* featured champ, the next one will be the 7* featured champ
This is not precisely how it was explained, but I believe this more clearly represents the intent Kabam was describing. If you miss four in a row, the game still rolls the next one, but if that fifth roll is a 5* champ it "forces" it to be a 6* champ to break the streak of misses, it doesn't just automatically make the fifth drop a 6* (because if it did, that might cost you a 7* champ if that roll just happened to be a lucky pull).
Given those mechanics, how does the pity system alter the effective drop rates of the crystals? Since every time the pity system triggers the crystal drops more 6s, 7s, and 7* featured champs than the drop odds would ordinarily produce, the average number of those drops goes up. By how much?
There's two ways to calculate this: solve a bunch of recursive equations, or ask a computer to do it. Computer?
Measured drop rates:
5* rarity: 67.06 %
6* rarity: 27.82 %
7* rarity (not featured): 4.42 %
7* featured rarity: 0.7 %
Compared to the published odds of the Featured Valiant:
5* rarity: 73 %
6* rarity: 24 %
7* rarity (not featured): 2.64 %
7* featured rarity: 0.36 %
(Note: the published odds lists 7* drop rates as 3% but that includes the 7* featured champ odds of 0.36%, so I've separated them here for comparison purposes).
Because almost one in four drops is a 6* anyway, implementing a miss streakbreaker of five doesn't improve those odds dramatically, but it does improve them slightly. The bigger changes are with 7* rarity and 7* featured drops. On average, ordinarily about 38 drops are 7* champs (not featured) so breaking miss streaks at 30 has a substantial impact on the overall rate of return of the crystal: the crystal will drop about 67% more 7* champs with the pity system than without it. And the largest impact is on the featured 7* champion drop rate. About one in 278 crystals drops the featured 7* champ, and a pity system that streakbreaks at 200 increases the drop rate to almost double its initial rate.
Now, this is all separate from the fact that these streakbreaking pity drops are *predictable*. In theory if you keep track of all of your drops, you can actually predict in some cases when your next drop is guaranteed to be a 6*, a 7*, or even a 7* featured champ. If you open 199 crystals and none of them are the 7* featured, the next one will be. And since the pity system is (according to the stream) global across all special featured valiant crystals, if there's a featured you want among the special valiants that are announced, you could simply stop opening those crystals until the champ you want is released, and then the first crystal of that champion you open will be guaranteed to drop the featured champ.
As far as I am aware, you don't have to stop buying crystals, you just have to stop opening them. So if you really want a particular champ and you keep track of your openings and you know #200 is coming up, you can continue to buy featured Valiants and just hold them until the champ you want gets released, open that one, then go back and open all the other crystals you've stockpiled. I'm not sure to what degree anyone intends to play those kinds of games, but it is theoretically possible.
For everyone else, the above numbers estimate how the pity system alters the effective rate of return on the crystals. You obviously need to buy at least 30 to see the drop rate improvements on 7* champs and 200 to see the improvement for 7* featured, but for anyone thinking about buying that many, that's the overall drop rate improvements you're statistically going to see (random being random, everyone will see different overall rates of return).
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Comments
What happen to milestone store offer? That was better value wise…
When nafaria came out on web store he was I believe $500,600 spending in the store to get him?
And $400,$500 for 7* jacko lantern in webstore also..
So why have this system in first place?
I feel like this pity crystal is a disguise to make you spend more because people thought spending that much for 7* nefaria and jack was a lot…
Not we have these.
I just totally stopped buying these crystals and just use my money and units for big sales for 4th and cyber.
200 featured crystal’s is just way to much.
I believe the last 7* I went for was for werewolf by night and got incredibly lucky
Crazy how werewolf is not even talked about much nowadays.
If you open three hundred crystals and do not get the 7* featured, you are literally no closer to getting it than when you started. Statistically speaking, you're still approximately 280 crystals away, because that's how independent odds work. You never get closer with more crystals. With the pity system, that's no longer true. You *might* get lucky and get them sooner, but it is now impossible to go 300, 500, or even a thousand crystals deep and have bad luck mean you still didn't get the champ you're chasing, and are still potentially hundreds of crystals away from getting it.
Presumably, this change is intended to make the crystal more attractive to players already buying them, and act to convince players who are on the fence about them to give them a try. This change is unlikely to change the opinions of players who have no intention of buying them and expect much larger returns on their investment than that. Kabam is not chasing after those players, nor should they.
However, I'm intrigued, and I know it's been requested. I'll keep an open mind.
you sure it works this way?
I think this system reset for every featured set.
While trying to answer a slightly different question, I ran into a small bug in my calculations which mostly affect the lower rarities. For comparison these are the numbers I reported originally:
Measured drop rates:
5* rarity: 67.06 %
6* rarity: 27.82 %
7* rarity (not featured): 4.42 %
7* featured rarity: 0.7 %
These are the corrected numbers:
Measured drop rates:
5* rarity: 64.87 %
6* rarity: 29.92 %
7* rarity (not featured): 4.52 %
7* featured rarity: 0.7 %
Things are slightly better for pulling 6* rarity (and thus slightly worse for pulling 5* rarity) but the 7* drop rates are basically identical.
When you open lots and lots of crystals, you will find the numbers above in the long run match your returns. However, what happens if you open a smaller number of crystals? Well, for most crystals, the answer is the same. Opening lots of crystals and measuring the drop rates is a way to estimate the actual return. But if a crystal has, say, 3 % chance to drop something, then that will always be true whether you open one, a hundred, or a billion crystals. Your actual *results* will vary, because random chance means sometimes you will get more and sometimes less, but your *expected* average return will always be 3%. Opening lots of crystals is just a way to make the measured drop rates more accurate.
That's not true with the pity crystals. Your expected rate of return on one crystal is not the same thing as your expected rate per crystal over thousands of crystals. That's because the pity system only triggers after so many misses. The very first crystal you open cannot benefit from the pity system for the simple reason it is impossible to have missed 6* champs five times in a row, or 7* champs thirty times in a row, or 7* featured 200 times in a row after opening only one crystal. The expected return on that crystal is actually identical to the base odds of the featured Valiant, because for all intents and purposes the pity system doesn't exist.
So if a player were to open 200 Featured Valiant crystals with the announced pity system, what would their average expected return be? Well, to measure that I have to redo the calculations but instead of opening millions of crystals and averaging, I have to open them in batches of 200 and reset the pity timers after every 200 crystals. When I do that, I get:
Measured drop rates:
5* rarity: 64.99 %
6* rarity: 29.94 %
7* rarity (not featured): 4.47 %
7* featured rarity: 0.6 %
Compared to the numbers I posted above, the 6* rarity drop rates and the 7* rarity drop rates are very close, close enough to be practically identical. That's because after 200 crystals, there have been plenty of opportunities for the pity timer to trigger and add drops, such that opening lots and lots more wouldn't change those rates much. But the 7* featured number changes significantly. It is still much higher than the base rate (0.3%) but it is noticeably lower than the long term average return (0.7%). And that's because every single crystals out of that batch of 200 is essentially immune to the pity timer. Nothing about them changes, or can change. It is only the 200th crystal that has any chance of being affected by the pity timer. If all the others miss, this one is the guaranteed hit.
interestingly, The expected return jumps from 0.3% to 0.6% after opening just a single crystal that is inside the pity timer for 7* featured. That is still most of the benefit of the pity timer. And then if you continue to open crystals, that rate improves slowly and converges on 0.7%. So if you are shooting for 7* featured drops, open at least 200 - if you can of course. Anything less than that, and you're no better (and no worse) off than opening regular old featured Valiants without the pity timer.
For 7* normal drops, though, the drop rates do improve substantially after much fewer crystals. But you need to open at least thirty to get pity timer benefits for 7* basic drops, for analogous reasons to the 7* featured.
For those interested, and can speak Portuguese, MystiKay's video that covers this is here:
I do not speak Portuguese, but for those that do, I believe this video covers the same ground I'm posting here.