Forum Thread

OneLastGambit

agarber wrote:

Aes, thanks for nice job! Though I have a correction from math point of view. These

aesthetocyst wrote:

b. Each of a character's 3 colors are completely separate possibilities.[/b] That is, separate database items. I have seen players ask from time to time whether, when a token is opened, the character is selected by RNG, and then the color determined by a 2nd RNG. Similarly, players often ask why we can't have colorless covers, or convert one color to another. Well, only the devs know the code involved and the database structure, but distributions of pulls indicates that each color of each character is a completely separate item. In any sizable dataset, plotting the distribution of individual character colors by quantity pulled yields a normal distribution. Plotting the distribution of characters by quantity pulled (that is, disregarding colors entirely) yields a non-normal distribution.

actually are both "non-normal" distributions though one (or both) can look like that.

TL;DR: Math is easy.

The normal distribution (or bell curve which is density of the normal distribution) come from the Central Limit Theorem https://en.wikipedia.org/wiki/Central_limit_theorem

In simple words it is the following. If X1, X2, ... , Xn are equally distributed independent random variables, then their average is a normal distribution with certain parameters.

What does that mean in case of LT pulls? Let's say we have chosen one 4* - IW, and we are pulling LT 100 times. Then each random variable will give us value 0 (if we did not pull her), or 1 (yes, one more IW). Then we take the average of 0's and 1's after 100 pulls, and this will give us an outcome for one experiment for average of 100 random variables. Now if you take 100 average outcomes and plot them, you will get something close to a normal distribution (actually not really, as rate of convergence could be not that fast).

In your experiment, first value was the average of measured random variables for IW, second - for Flaptain, and so on. The problem is that they are not independent. If you pulled 50 IWs in 100 LTs, there is no way you could pull 51 Flaptains from the same 100 LTs, and independent random variables give non-zero probability for that.

Now come my speculations, as I am not an expert in statistics or probability

So, why "almost" normal distributions appear? Reason for that is the following. Though the measurements you have plotted are not independent, if you throw away small part of them, then remaining will be "almost" independent because pulling each 4* has small expectation, so large deviation from average for each is also almost impossible, and thus it is highly unlikely to be affected by the boundary "you can't get more than 100 4*+5* from 100 pulls".

In your case for colored chars you have 93 outcomes (right?), so these "additional" chars that bring dependency problem are not visible. In case of 31 uncolored chars, they are more visible and that is why you can see non-normal plot.

In the examples listed above was it considered that the pool tokens from which to pull is essentially infinite? The pool of tokens we have is not the pool of tokens available in game and this the published odds won't correspond to your accumulated tokens as the published odds apply to the total in game economy of tokens (which has no finite limit.)

So in your flaptain/IW example above where you state that if you pull 50 IW in a row the. You cannot pull 51 flaptains next is only true when applied to your pool of tokens the same cannot be said when applying it to the total pool of tokens available.

dsds

OneLastGambit wrote:

dsds wrote:

Sorry i didn't mean to sound unappreciative. I appreciate all the effort you have put in. No doubt that is a pretty huge project to do.

We all have confirmation bias whether it be for or against or whatever. I am not saying that it couldn't happen to me, I could very well be bias as well. I am just doing some constructive criticisms here. Like anything scientific. You publish your findings and other people will try to find something wrong with it. If everyone says yes that is correct and it's undisputable, then there would be no progress. I mean we are all human after all.

Here's my own experience though. I am pretty tired of the grind recently, so there was one new 4star release that I didn't do moon knight. And it so happens that I get that character a few weeks after his pve. Then there's the other user that I agreed with that got more new characters as well. So that's at least two people here that see this. It is such a small sample size, it's very likely a coincidence, but this is something that can be considered. There is a motive for this. 1000hp per roster slot is quite a bit. If it gets even 1% of the mpq population to buy a new slot, it's quite a bit of money.

If you dismiss everyone's criticism, there would be no progress and we would further from the truth.

Please don't take it personally, This is just how scientific progress works. If you don't think the theory is good, then don't try to disprove it and leave it at that. I am not calling anyone out to disprove it. I am just stating a criticism that is constructive with no ill will.

The highlighted section is the problem with your method. Any scientific analysis cannot be done on a personal experience. It needs quanitifiable data and opinion is not reliably quantifiable. I think everyone involved here would be perfectly fine with you providing data analysis from a significant and representative data set which disproves or questions the theory, but that's not what you are doing.
Provide evidence and people will agree but simply stating your experience is different is not a valid enough point to be considered scientific constructive criticism.

Here's the thing with the proof. It's even harder to do than what aes has done with his data set. It would require a lot more people than what aes had. This is because the theory is based on the fact that at 0 covers, you are at a higher chance to get the cover at 1. But once the cover is at 1 or in the queue of the player, the probability goes back to normal. So in order to prove this, you would have to take the first 10-20 draws from a player who is relatively new and then repeat until there is a reasonable sample size. Not only is there a problem with the number of players that you would have to involve in this undertaking, but also it's less likely that the types of players required for the research would have the draws required. For instance a new player who has less than 50 days isn't likely to have that many LTs. I mean we could see it only for standard tokens and maybe heroics.

Anyway, I am not trying to push this theory or say it is fact (yes wrong choice of words, I admit earlier). It is simply a criticism or suggestion about the results that is likely not going to be proven. I understand that and will leave it at that.

It's not even a conspiracy theory because if you look at it. Yes, it gets people to buy roster slots, but it also allows people to participate and compete in more pves since they are more likely to get the required characters needed to be competitive if the theory was true.

mpqr7

There are many long, detailed posts like OP about how to improve mpq. So you can't say "all you do is complain" on the forums. Clearly we are a group of people with a strong emotional investment and desire to improve the game. You don't invest this much energy if you don't love the game and want it to be better!

Dayv

stochasticism wrote:

I lost some of the more depressing numbers we came up with from that formula but you can quite quickly learn things like 720cp for the last cover you need for a 5* is actually a good deal and is a better deal as more 5*s get added to the pool.

It's a good deal only if your sole goal is that one 5* cover of a specific color. If every other cover you pull is simply being sold for ISO, then you'll get that specific cover faster (on average) at the cost of 35 covers worth of ISO.

However, the more other covers you can use, the more wasteful this becomes. Champion a significant portion of 4*, and work on building or champing other 5* as well, and the opportunity cost becomes both really hard to calculate and probably quite high. 4* champs generate a not-insignificant amount of LTs and CP over time, which further increase the number of pulls you get to do and make it a self-feeding system.

So my advice is this:

1) Don't focus on a single character to the exclusion of all others.

2) Consider hoarding ISO and tokens/CP until you have enough ISO to make it worthwhile to pull covers until a pattern emrges for where you should use the ISO. (Ideally, enough to champion any 4* or 5*.) (I do not follow this advice because I like instant gratification.

3) Don't let D3 off the hook on fixing the unusable cover problem.

agarber

OneLastGambit wrote:

In the examples listed above was it considered that the pool tokens from which to pull is essentially infinite? The pool of tokens we have is not the pool of tokens available in game and this the published odds won't correspond to your accumulated tokens as the published odds apply to the total in game economy of tokens (which has no finite limit.)

So in your flaptain/IW example above where you state that if you pull 50 IW in a row the. You cannot pull 51 flaptains next is only true when applied to your pool of tokens the same cannot be said when applying it to the total pool of tokens available.

It was considered that the pool is infinite, but the random variables are still independent, since we already pulled those tokens. Let me try to describe that with similar Flaptain/TW example dealing with infinite pool.

Random variable IW1: Number of IW pulled from 100 tokens.
Random variable F1: Number of Flaptains pulled from other 100 tokens.

These two variables are independent (we assume the RNG is honest), as each next pull is not affected by previous pulls, and both have Binomial Distribution. If we create similar variables for all other chars, and measure them by pulling hundreds of tokens counting IW in the first 100, Flaptain in second 100, Cho in 3rd 100 and so on, they should be plotted in approximately binomial way, or bell curve if you like as binomial converges to normal.

In the experiment Aes ran, we have other random variables (numbers would be different, but idea is like that):
Random variable IW2: Number of IW pulled from first 100 tokens.
Random variable F2: Number of Flaptains pulled from the same first 100 tokens.

Both are again Binomially distributed, but not independent. Again, we can measure similar variables on the data of 100 pulls, but they should not give us binomial picture on the plot, as this 50-51 impossible case I mentioned before.

I would conjecture that the plot will converge (in certain sense) to uniform distribution. Something like expected value of dispersion goes to 0 and pretty fast or something like that. Don't have a reasonable idea how to prove that yet... Must be known actually

stochasticism

agarber wrote:

OneLastGambit wrote:

In the examples listed above was it considered that the pool tokens from which to pull is essentially infinite? The pool of tokens we have is not the pool of tokens available in game and this the published odds won't correspond to your accumulated tokens as the published odds apply to the total in game economy of tokens (which has no finite limit.)

So in your flaptain/IW example above where you state that if you pull 50 IW in a row the. You cannot pull 51 flaptains next is only true when applied to your pool of tokens the same cannot be said when applying it to the total pool of tokens available.

It was considered that the pool is infinite, but the random variables are still independent, since we already pulled those tokens. Let me try to describe that with similar Flaptain/TW example dealing with infinite pool.

Random variable IW1: Number of IW pulled from 100 tokens.
Random variable F1: Number of Flaptains pulled from other 100 tokens.

These two variables are independent (we assume the RNG is honest), as each next pull is not affected by previous pulls, and both have Binomial Distribution. If we create similar variables for all other chars, and measure them by pulling hundreds of tokens counting IW in the first 100, Flaptain in second 100, Cho in 3rd 100 and so on, they should be plotted in approximately binomial way, or bell curve if you like as binomial converges to normal.

In the experiment Aes ran, we have other random variables (numbers would be different, but idea is like that):
Random variable IW2: Number of IW pulled from first 100 tokens.
Random variable F2: Number of Flaptains pulled from the same first 100 tokens.

Both are again Binomially distributed, but not independent. Again, we can measure similar variables on the data of 100 pulls, but they should not give us binomial picture on the plot, as this 50-51 impossible case I mentioned before.

I would conjecture that the plot will converge (in certain sense) to uniform distribution. Something like expected value of dispersion goes to 0 and pretty fast or something like that. Don't have a reasonable idea how to prove that yet... Must be known actually

Let's introduce a third variable to each scenario.

O1 = # of other 4*s pulled on a third set of 100 tokens
O2 = # of other 4*s pulled on the same set of 100 tokens as F2 and IW2

Then, the following holds.

IW1, F1 and O1 are all binomially distributed with n=100 and probability p=p_f, p_iw or p_o depending on the which class of characters you are talking about.

The tuple (IW2, F2, O2) follow a multinomial distribution (c.f. http://www.math.uah.edu/stat/bernoulli/Multinomial.html). An important property of the multinomial is that the marginal distributions (i.e. the distribution of Iw2 F2 and O2 individually) are themselves are binomially distributed random variables. This means that if we just focus on the count of pulls from each group then each will converge to a normal distribution. It is also true that these marginals are correlated, but that only really comes into play if you are looking at some function of these quantities, like F2+ IW2. Regardless of the fact that they are correlated, they still produce binomial distributed marginals.

agarber

stochasticism wrote:

Let's introduce a third variable to each scenario.

O1 = # of other 4*s pulled on a third set of 100 tokens
O2 = # of other 4*s pulled on the same set of 100 tokens as F2 and IW2

Then, the following holds.

IW1, F1 and O1 are all binomially distributed with n=100 and probability p=p_f, p_iw or p_o depending on the which class of characters you are talking about.

The tuple (IW2, F2, O2) follow a multinomial distribution (c.f. http://www.math.uah.edu/stat/bernoulli/Multinomial.html). An important property of the multinomial is that the marginal distributions (i.e. the distribution of Iw2 F2 and O2 individually) are themselves are binomially distributed random variables. This means that if we just focus on the count of pulls from each group then each will converge to a normal distribution. It is also true that these marginals are correlated, but that only really comes into play if you are looking at some function of these quantities, like F2+ IW2. Regardless of the fact that they are correlated, they still produce binomial distributed marginals.

Right. The Aes's approach was slightly different. He measured those random variables IW2, F2, Cho2 etc. on certain amount of tokens (hundreds). And then plotted the values. If I got that correctly for each k he plotted how many variables got the value k. All variables are binomial, so if they would be independent, then he should get something like plot of the density of binomial distribution. Since they are not independent, he got something else.

The question is. How the thing he got is called?!

Pylgrim

mpqr7 wrote:

There are many long, detailed posts like OP about how to improve mpq. So you can't say "all you do is complain" on the forums. Clearly we are a group of people with a strong emotional investment and desire to improve the game. You don't invest this much energy if you don't love the game and want it to be better!

This, very much. Devs keep away from the forums nowadays and blame all the rude bullies that bruise their delicate feelings. However, there's a good number of threads like this one that merit a frank conversation and that could be kept in mature politeness and constructive cooperation. Moreover, such open communication would encourage more people to be positive and constructive, fostering even more and better discourse. Would it stop the bullies and ever-whiners from being rude? No, they are an expected constant that needs to be disregarded, not feared. Silence, on the other hand, only encourages negativism and pesimism... and allows the bullies and whiners to thrive and believe they're in the right.

I'm pretty sure that if IceIX was still around he'd have already dropped by in this thread with at least some hints, or at least to acknowledge Aes' hard work and love for the game.

chill21genlee

simonsez wrote:

aesthetocyst wrote:

Hoarding is a way to get ahead, to partially "defeat" RNG via the brute force of sheer quantity

This isn't correct. Hoarding is not a way to get ahead. If we assume that the RNG is fair, and the internal probability tables are constant, it doesn't matter if you open 1 token a day for a month, or hoard for a month and open 30 in one day. Your expected outcome of the 30 pulls is the same either way.

Hoarding only helps if you're waiting for a character to get added to the token pool, or if you don't have ISO and don't want to risk pulling extra covers you'd have to sell.

Agreed very well written post but the rng is applied PER PULL..which should mean that how many you open, good and bad streaks, all of that is conpletely incidental to the point..
When you open a token, what you get is governed by the stated percentages applied on a 1 per basis, and that's it end of story.

We've all been frustrated with bad runs and delighted with good runs or getting covers we actually need, but at the end of the day its all anecdotal.

Whether the devs monitor this at a metadata level to ensure their percentages are accurate and the rng is functioning as designed is another matter entirely.
My guess would be NO to both considering how so many other aspects od development and ongoing maintenence seem to be handled, but hey i dont work there so who knows.
If i did i would darn sure stop adding yet more and more versions of hulk spidey wolvie and iron man when there is an almost bottomless cast of chars to choose from.