Slot Statistics - Critiques Requested

[binary128]

Jufo,

First of all, I can't thank you enough for all of the work that you have done here.

You're right - If I had done this work through my DB samples analysis, well ... I'd still be doing this work.

Personally, I find this most recent distribution chart, the one with the 10,000 samples, each sample representing 10,000 rounds, to be the most informative.

I've spent a lot of time in an assortment of wiki files, trying hard to get a handle on all of this. I realized that I have never, to my knowledge anyway, worked with non-normal data, at least not to the extent which your work has uncovered. To me, SD has always been a plus or minus around some mean.

This second distribution chart made the actual distribution visually obvious. The actual mean may equal the expected mean, but their modes are completely different. The skewness is also a significant issue, with the right hand tail being the most significant.

My goal in all of this was to create a page which provides "workable data" to the Player, something of value for the Player. However, as both you and I have stated, and wrestled with, repeatedly throughout this thread, the information (both data and words) has to be technically accurate.

I've read and re-read and re-read your recent posts, spent time wandering through an assortment of wiki files, thought and puzzled and re-thought ...

I've concluded that, right now anyway, I do not know how to present the words for the SD data in a manner that is both understandable and accurate.

I still think that it is in there somewhere. Maybe I can find, use or have one of our people create a Flash chart that will present a curve for any selected slot. I use something like this on our current Certified Fair Gambling page, but this chart is just a simple run chart. What we're talking about here is something significantly different, significantly more complicated.

What I've done - I've simply removed Standard Deviation from the Definition of Terms section. Both of the remaining bits (Variance and Line Pay Hit Frequency) are OK.

Again, Jufo, thank you very much for all of this work. With these SD words removed from the file, I believe the file can now be deployed.

I've gained a great deal of understanding about the statistics of slots. So, for me anyway, this journey has been of tremendous value.

Chris

 

[nisosbar]

I'm reading and enjoying this, but haven't got much to add. I can't play on Galewind or MG and so my interest is more limited than those who can deposit and play on that software. But I still enjoy reading about the mechanics behind slots.

 

[Jufo]

Chris, I am happy that I could provide you insight to this topic. When you asked me for input to this thread, I thought, well he is going to have perhaps more than he bargained for

The topic of slots variance and how the results actually distribute has been quite an uncharted territory until now, so I took this opportunity to gain deeper knowledge on the subject, and I explored and studied things too as we went along. In the process I made that simulator to try to reach deeper level of understanding.

I've spent a lot of time in an assortment of wiki files, trying hard to get a handle on all of this. I realized that I have never, to my knowledge anyway, worked with non-normal data, at least not to the extent which your work has uncovered. To me, SD has always been a plus or minus around some mean.

This second distribution chart made the actual distribution visually obvious. The actual mean may equal the expected mean, but their modes are completely different. The skewness is also a significant issue, with the right hand tail being the most significant.

Yeah, you can think of it like this: since there is that heavy right tail (high RTPs), you have to remove "probability mass" from somewhere else for the RTP to hold. One might think that the left tail (low RTPs) is heavier as well to negate the effect of the right tail, but it's not so: the low RTPs are in fact under a tighter envelope than what the normal distribution predicts. So what happens is that the median and mode of the distribution shift to the left to account for the added probability mass in the right tail.

I've concluded that, right now anyway, I do not know how to present the words for the SD data in a manner that is both understandable and accurate.

I still think that it is in there somewhere. Maybe I can find, use or have one of our people create a Flash chart that will present a curve for any selected slot. I use something like this on our current Certified Fair Gambling page, but this chart is just a simple run chart. What we're talking about here is something significantly different, significantly more complicated.

Agreed, the whole topic is way to complicated to be able to fit in a few sentences, so your task was next to impossible. However, I really like the idea of displaying the payout curve for each slot. Players could then see how the distribution curve would look like for given number of spins and would see where they lie on the curve. Really great idea, but probably challenging to implement.

I've gained a great deal of understanding about the statistics of slots. So, for me anyway, this journey has been of tremendous value.

Hehe, you're welcome. Enzo is probably reading this and laughing at his beard, "I already knew all of this".

 

[Jufo]

I'm reading and enjoying this, but haven't got much to add. I can't play on Galewind or MG and so my interest is more limited than those who can deposit and play on that software. But I still enjoy reading about the mechanics behind slots.

The "mechanics" described here are universal statistical concepts so they apply to all slots (and to some extent all games with large payouts), so the choice of software wouldn't matter.

 

[binary128]

Jufo,

Well, it took me 7 months after I posted the Game RTP thread to gather, calculate, organize and publish this next level of data. So, it might take me another 7 months to bring it to the next level.

But yes, I think the only effective way to "explain" this to the Player is through the use of distribution charts. I could add some words as necessary, but the charts will really carry the message in an understandable way.

I've given it a little thought. I think that the easiest way to do it is just using static graphic files rather than the dynamic tool that is Flash. This will make it easier to deploy.

Now the tough part is generating the data that will be needed for the charts. Ugghh. That's what is going to take the 7 months.

Again, thank you so much for your help in all of this.

Chris

 

[binary128]

Well, the final version of this Help file has now been distributed throughout all of our development, demo and production products.

I decided that to expose this file to the Players I would add a link labeled "Slot Statistics" to the main Help page for each of the slots. Thus, anyone that is playing a slot and clicks the Help button will launch a window in which is a list of subjects. "Slot Statistics" is now one of those subjects.

Even though I ultimately needed to remove some information associated with the Standard Deviation, I still think that this is an amazingly powerful, interesting, and helpful file. The statistics for each of the slots is really good stuff. The chart at the bottom of the page is also really good stuff.

I'd like to thank zap987 for finding the documentation error for the Gems of Isis game. (No Wild symbol in reel 1 of a Free Spin game.)

I'd like to thank ChopleyIOM for his critique of our modification to the documentation for the Gems of Isis file.

I'd like to thank GrandMaster and Enzo for jumping on to the errors in my presentation of the Standard Deviation.

Jufo, I owe you big time for all of your help.

It has occurred to me to hire you as a consultant to assist in generating the data that we will need to populate those distribution graphics we've discussed. It might prove to be a lot of work. (Play a couple hundred rounds in Lucky Lanterns to see what I mean.) But, we pay well, there is no deadline, and we're pretty easy to work with.

Anyway, just as with the previous Game RTPs file, this one, for the moment anyway, is now in the "all done" column, and I view it as a great success.

For those CM members that have been reading this thread, I hope that you got out of it as much useful and interesting information on slots as I did.

Chris

 

[Jufo]

I am happy to hear that the page is finally complete. I feel like pretty much everything that there was to be said about the topic was discussed in this thread, so overall job well done. Perhaps the one thing that had to be left "in the air" was categorizing slots into Low, Med and High variance groups, but it is understandable because there is very little infromation available on how they are categorized within other softwares.

I would gladly help with generating those distribution graphs. I'll be looking forward to working on it.

 

[Casinomeister]

Admin note: exploiting the board

Please don't use the forum to recruit members to consult or "play" at your casino. That's crossing the line of exploiting the board. Offers like this should be vetted by me. Thank you.

 

[binary128]

@Casinomeister: Woops. Gotcha.

Right now, this "next level" project to which I referred is tucked into the freezer. Over time it will move from the freezer to the fridge to some small pot on a back burner.

The ultimate result will be another publication in our Help system.

What I'd like to address, and hopefully clarify, for the Player in this new Help file are a couple of (what I feel are) points of confusion which I have seen in an assortment of other threads/posts throughout the Forum.

1. Theoretical RTP is some number that is so far off in the distance that they would need to play for months and months and months in order to even approach it.

2. The implication in these types of statements is that the Players approach to that number will always be from the negative side. That is, their game play results will always be negative, and after those months and months of play they will eventually be approaching the loss specified in the Theoretical RTP.

Jufo made a post in which he documented that, for 10,000 players, each playing 2,000 rounds, 36.5%, or 3,651 players, actually won money. That is, they not only exceeded the Theoretical RTP, they exceeded a 100% return.

Jufo did not report on the percentage of these 10,000 Players whose 2,000-round RTP exceeded the Theoretical RTP. That would have been an interesting number, because that would be the number of Players that are approaching the Theoretical RTP not from the negative side but from the positive side of that value.

Providing this type of information, using properly formatted charts to facilitate their understanding, would be, I think, of tremendous interest to the "average" CM member. In addition, what we have seen in this thread is that the "standard distribution" for a slot is not a normal distribution. Its Median and Mode are shifted left, with a significant skew to the right side of the curve. I think that the details of this would be of great interest to the "statistically minded" CM members.

Again to reference Jufo, this has been "uncharted territory" (no pun intended) until now.

Jufo's simulator requires an exact reel loading table, and the only tables that I have are for the Galewind Software slots. In addition, my database tables are also only for the Galewind Software slots. Therefore, the only slots that we have to work with in order to pursue this topic are ... Galewind's. I don't really see any way around that.

At any length, when this publication is at such a state that it is presentable for discussion, I will be sure and provide you with a link to the file, and ask for your approval prior to starting a thread.

Chris

 

[Jufo]

Jufo did not report on the percentage of these 10,000 Players whose 2,000-round RTP exceeded the Theoretical RTP. That would have been an interesting number, because that would be the number of Players that are approaching the Theoretical RTP not from the negative side but from the positive side of that value.

I checked the numbers from the simulation and they were (for 2000 spins):

50.0% of players had RTP of 96.69% or lower (so 96.69% is the Median, or most "typical" player).
3.0% of players were between Median and Average (T-RTP) (ie. between 96.69% and 97.48%)
47.0% of players were at RTP (97.48%) or above it.

For 10,000 spins the above frequencies were nearly identical.

Providing this type of information, using properly formatted charts to facilitate their understanding, would be, I think, of tremendous interest to the "average" CM member. In addition, what we have seen in this thread is that the "standard distribution" for a slot is not a normal distribution. Its Median and Mode are shifted left, with a significant skew to the right side of the curve. I think that the details of this would be of great interest to the "statistically minded" CM members.

Yes I agree. People usually like to see how their results compare with other players. And in this case "other players" would be a very large sampled population.

What I'd like to address, and hopefully clarify, for the Player in this new Help file are a couple of (what I feel are) points of confusion which I have seen in an assortment of other threads/posts throughout the Forum.

1. Theoretical RTP is some number that is so far off in the distance that they would need to play for months and months and months in order to even approach it.

2. The implication in these types of statements is that the Players approach to that number will always be from the negative side. That is, their game play results will always be negative, and after those months and months of play they will eventually be approaching the loss specified in the Theoretical RTP.

A couple of notes about this:

It's true that the longer you play, the closer your RTP will get to the T-RTP, and like you wrote you can get closer to it either from negative side or positive side. The reason why people usually assume you are reaching it from the negative side, might be 1) because that's the side you are on most often, ie. it's not 50-50 (see my frequencies above), and 2) if you are on the right-hand-side of the RTP you might be inclined to stop playing to not lose your winnings, that is, if people are above RTP, they don't usually think: sh*t I am ahead, I need to play more to drop back closer to the T-RTP. So when people chase T-RTP they do it most often from the negative side.

But there is another key concept which justifies the view of approaching T-RTP from the negative side and it has to with the Median.

Below is yet another simulation chart from the Hot Pepper slot. This time it shows the distribution after only 100 spins.



Note that the X-axis is plotted up to 200% but distribution continues to the right up to 1700%. 241 samples out of 10,000 had more than 200% RTP with the highest being 1700%.

The plot shows how after 100 spins, the win distribution is very far from normal distribution. But note especially that the Mode of the distribution (the most likely value with the highest peak) is at around ~76% and the Median is at 88.05% RTP. Median means the player who is in the middle (exactly half of all players will have RTP lower than him, half of them will have higher). So this "most middle" player, in 100 spins, sees RTP of only 88.05%.

Now from the previous sims, the Median for 2000-spin case was 96.65% and for 10000-spin case it was 97.13%. Therefore:

As the number of spins increases, the Median approaches the T-RTP and from the negative side.

So, perhaps the reason why it is often assumed that one approaches T-RTP from the negative side is because the Median (and also Mode) approaches it from that side and this kind of describes the result of "the most average player".

UPDATE: I also checked that for 100 spins, the ratio of players who were at T-RTP (97.48%) or above it was 40.6%. (for 2000 spins this number was 47.0%)

 

[GrandMaster]

A remark for the mathematically minded on the Berry-Esséen theorem (

You do not have permission to view link Log in or register now.

) I mentioned earlier:
In the notation of the Wikipedia page, C is just slightly less than 0.5 and rho<M sigma^2, where M is the maximum payout. (rho is the expectation of the cube of the absolute deviation from the mean, while sigma^2 is the expectation of its square. As the terms corresponding to the highest payouts already dominate sigma^2, this estimate is probably not off by more than a factor of 2.)
Hence the difference between the actual cumulative distribution function and the normal estimate is less than M/(2*sigma*sqrt(n)) at any point, where n is the number of spins.

For a typical slot, M is several hundreds or even a thousand, sigma is in the 5 to 10 range, so M/(2*sigma) can be around 100. To be certain that the error is at most 1%, n would have to be 100000000, for an error of 0.1%, n would have to be 10000000000. The actual numbers may be a magnitude or two smaller, because the theorem is a universal estimate valid for any probability distribution and for any deviation from the mean, nevertheless my feeling is that you probably need to play millions of spins to get within 1% of the normal distribution everywhere.

 

[Jufo]

For a typical slot, M is several hundreds or even a thousand, sigma is in the 5 to 10 range, so M/(2*sigma) can be around 100. To be certain that the error is at most 1%, n would have to be 100000000, for an error of 0.1%, n would have to be 10000000000. The actual numbers may be a magnitude or two smaller, because the theorem is a universal estimate valid for any probability distribution and for any deviation from the mean, nevertheless my feeling is that you probably need to play millions of spins to get within 1% of the normal distribution everywhere.

Yeah, that statement is along the same lines as my simulation of sets of 10,000 spins and failing the normality tests for the data (it wasn't even close). What I also tried, was to remove the largest 5% of payouts from the data and see if the distribution is normal enough then but the normality tests still failed because the lower end results (low RTPs) were too far from normal distribution as well.

So what you wrote verified my observation that normal distribution just doesn't work with slots for any such number of spins that a single player might ever play, and so it was a wise decision for Chris to remove this part from his help file altogether.

 

[binary128]

Here is what I am envisioning for this "Standard Deviation Player Chart" system.

We start with a distribution (line chart rather than bar chart, since it is really hard to follow the bar charts) that represents some "really large sample size". I don't want this "really large sample size" to be associated with billions, or even hundreds of millions, because that size of number becomes "unreal" to the viewer.

So, that would be the first thing to figure out - what is a good choice for this "really large sample size" for each of the slots.

And then under this chart is a series of numbers which, when clicked, overlays the distribution of the slot for that number of spins.

Example - 500, 1000, 2500, 5000, 10000, 50000

When the viewer clicks one of these numbers, then a second chart (also a line chart) is overlayed on top of the first "really large sample size" chart.

The viewer will be able to see what their possible RTP might be given their selected sample size (number of rounds).

So, that would be the second thing to figure out - what is a good choice for this set of numbers.

Although the Player might be interested in as small a number as 50, I think that the result would be a little silly. IMO, 100 is also too small, which is why I listed 500 above.

This issue might make for a good Poll. What do the Players think is a good number of rounds for them to get an accurate "feel" for the slot?

The trouble with this Poll is that the "feel" of the slot that we are trying to get at is the slot's payout. But a slot's "feel factor" for the Player is also affected by so many other things - the graphics, the sounds, the animation, the variance.

I don't think that the "average slot player" would be able to differentiate between all of these things in responding to the Poll.

@ChopleyIOM - If you would like to publish that final data that I sent you on your game play experience at Pinnacle, then I think this would be a really good place to post it. You might also be able to give a good opinion on what minimum game count you think is valid.

Chris

 

[ChopleyIOM]

Comment #1 - Auto Play settings. That one goes onto our To Do List.

Boo! This still hasn't been changed!

I've deposited with Pinnacle for this evening's slotting as I fancy a change of pace, and also that fab 97.5% RTP goodness.

I am however still having to enter my autoplay settings every time I start it off

Do you have an ETA on the autoplay changes Chris?

(It's a minor irritation rather than any sort of show-stopper, hence the smilies )

 

[binary128]

Boo! This still hasn't been changed!

I've deposited with Pinnacle for this evening's slotting as I fancy a change of pace, and also that fab 97.5% RTP goodness.

I am however still having to enter my autoplay settings every time I start it off

Do you have an ETA on the autoplay changes Chris?

(It's a minor irritation rather than any sort of show-stopper, hence the smilies )

ChopleyIOM,

I returned to this thread to get some reference material for that "slot charts" project, and saw your comment.

We've just started to throw around the Auto-Play memory system. We reached some decision points, so I thought that I would ask you for your input/opinion.

I understand that when you are in a game (Gems of Isis for instance) and you've set up the Auto-Play options, you want those options to be remembered each time you click the Auto Play button.

Would you want the system to use the same Auto Play settings for each game? That is, if you leave Gems of Isis and enter Golden Goal, would you want your Gems of Isis Auto Play settings to follow you? Or would you want each game to load with the default, and you then configure settings for each game?

We've figured that if you then re-enter Gems of Isis the previous Auto Play settings will be remembered. That is, once you've configured a game it will be remembered throughout that session.

Personally, I'm leaning towards a game-specific setting. But, then again, I'm not a big slot player.

Chris

P.S. I hope your Pinnacle session was a success.

 

[ChopleyIOM]

Personally, I'm leaning towards a game-specific setting. But, then again, I'm not a big slot player.

Chris

P.S. I hope your Pinnacle session was a success.

Yes Chris that would be ideal, game-specific settings, not least because different slots will have different lines/stakes etc.

I don't know what level of persistence you'll be able to manage, but to give you an example, Microgaming casinos, 3Dice and Jackpot Party (Jackpot Party don't have a client BTW) remember game-specific settings indefinitely, not just for the current session.

As for the success of the session, unfortunately it was a losing session, the lanterns weren't being that lucky for me

Still, there is always next time!

 

[binary128]

Yes Chris that would be ideal, game-specific settings, not least because different slots will have different lines/stakes etc.

I don't know what level of persistence you'll be able to manage, but to give you an example, Microgaming casinos, 3Dice and Jackpot Party (Jackpot Party don't have a client BTW) remember game-specific settings indefinitely, not just for the current session.

ChopleyIOM,

OK, game-specific settings.

Re: "remember game-specific settings indefinitely". Yeah, we can do that with a persistent cookie. Personally, I'm against persistent cookies (invasion of privacy), so we don't have them anywhere in the app. But it sounds like you're OK with them, or some of them anyway. I'd guess that most people are, so this might turn out to be our first persistent cookie.

As for the success of the session, unfortunately it was a losing session, the lanterns weren't being that lucky for me

Still, there is always next time!

I just checked. Yeah, your last session was grim. In total you're not doing so bad I guess - about 10,000 games with a total RTP of 98.5%. You're still on the plus side of the RTP curve, but no longer on the plus side of the win/lose curve. (The Casino giveth, and the Casino taketh away.)

Thanks for the feedback.

Chris

 

[Jasminebed]

I think a consistency to what is default for a slot is good, at least at your casino. There is a learning curve for players, so don't set it to something wild like $125 dollars a spin.

Max bet should be max bet for the coin size selected.

Slot bet size should definitely be remembered within a session if you leave a game and come back.

It is the player's responsibilty the check the bet before pressing spin. But we are here to have a bit of fun, don't make it too hard for us.

Recently played a software that every time I came into a game, default was 5 coins at 1 cent a line. If I didn't check, I'm betting $1 to $1.50 a spin, a mistake I could live with.

And please please please don't put the max bet button too close to the spin button.

I do expect max bet to bet the maximum number of coins for the coin size, not to change from a penny a line to $2 for all 25 lines.

We give so much trust to the casinos we play with, I'd trust you with cookies.

 

[binary128]

I think a consistency to what is default for a slot is good, at least at your casino. There is a learning curve for players, so don't set it to something wild like $125 dollars a spin.

Max bet should be max bet for the coin size selected.

Slot bet size should definitely be remembered within a session if you leave a game and come back.

It is the player's responsibilty the check the bet before pressing spin. But we are here to have a bit of fun, don't make it too hard for us.

Recently played a software that every time I came into a game, default was 5 coins at 1 cent a line. If I didn't check, I'm betting $1 to $1.50 a spin, a mistake I could live with.

And please please please don't put the max bet button too close to the spin button.

I do expect max bet to bet the maximum number of coins for the coin size, not to change from a penny a line to $2 for all 25 lines.

We give so much trust to the casinos we play with, I'd trust you with cookies.

Jasminebed,

Thanks for the input.

All of our slots load with the minimum chip amount, the minimum chip count, and the max lines bet.

However, where you say "Slot bet size should definitely be remembered within a session if you leave a game and come back." - we don't do that. If you play a slot with a different setting, leave that slot, then return to it in the same session, it will still load with the minimum chip amount, the minimum chip count, and the max lines bet. (I'd rather that the Player experienced an "oh no" moment than an "OH NO" moment.)

We also don't have a "Max Bet" button in any of our slots. I always found that the differences between what these buttons do from machine to machine extremely annoying. So, we just don't have them.

*** There's a question for any slot players that might be reading this - is our lack of a Max Bet button a significant problem with our games? Is the functionality of a Max Bet button outweighed by the potential confusion?

Re: persistent cookies. Our internal debate continues. I'm a child of the sixties, so for me anything that smacks of invasion of privacy is the devil incarnate.

And finally Jasminebed, where you say "I think a consistency to what is default for a slot is good, at least at your casino." (emphasis mine) - what do you mean when you say "at least at your casino".

Chris

 

[Jasminebed]

Well, I merely meant that your casino is the only one you have any control over, and the only one that is asking my opinion.

While I think it's ideal as MG does to remember last bet size, if there is consistency in a slot being reset to minimum, players will learn after a few "oh no" moments.

I have no problem with no Max Bet button. The accidental Max Bet is usually a groaner, and a big OH NO NO NO at Intercasino where you are playing for 20 or 25 cents and suddenly find a bet of $125 taking your whole balance.

3Dice introduced a totally customizable feature with what happens when you hit max bet and autoplay functions.

 

[Jasminebed]

oops, double posting

 

[binary128]

ChopleyIOM,

I remember somewhere back in this (now really old) thread (I'm too lazy to go back through the posts), I indicated that I was very careful about what I put onto our "To Do List" because once it was on that list it was "carved in stone". It may take us awhile to get it done, but if it was on the list then getting it done was guaranteed.

When you first checked out our Casino, you came back with a handful of suggestions. One of those suggestions was to have a "memory system" for the auto-play feature in the slots (and Keno). I thought that was a really good idea, so I added it to the "To Do List".

Well, finally, it is done. It took us so long because it got bundled into a whole lot of other stuff.

1.) The popup window now resizes itself based on your screen resolution. (The max resize dimension is 1020 X 680 pixels, the min is 540 X 360 pixels.)

2.) Because of the new resize system (the current popup has for 8 years been fixed at 780 X 520), we needed to re-render every graphic, and then re-build every one of the 34 games to maintain graphic quality at the maximum 1020 X 680 dimensions. (That work truly, truly sucked.)

3.) We added a new "Play Log" system to the Play-for-Free product (which, for a number of boring reasons, proved to be a bit of a tricky business).

4.) We added a lot of "manual window resize" handler code. Once the popup has launched and re-sized itself, you can increase the size of the window, but NOT the Flash frame. (The Flash frame centers itself within its manually resized "parent window", and a background graphic fills the rest of the space.) You can, however, decrease the size of the window AND the Flash frame all the way down to the minimum 540 X 360 pixels, below which you just start eating into the Flash frame. (This is actually kind of entertaining to do and then play games at that size).

5.) We added more Help files, modified some others, and so forth and so on.

So, anyway, bottom line - the new "auto-play memory system" is now up and running. It will remember individual settings for each of the slots, as well as for Keno, for the duration of your session. Once you exit the casino, all settings are wiped.

I remember someone (I think it was Jasminebed) suggesting/requesting that the memory settings persist across sessions. However, my personal philosophy that "we do not use persistent cookies" finally trumped that decision (for better or for worse). Maybe I've done too much backpacking - leave nothing behind to indicate that you were even there.

Chris