How to calculate the RTP on a Megaways slot?

I recently learned about Megaways-style slots. I don't plan on coding one (unless a client asks for it), but they seem so complex that I started wondering how hard it would be to calculate the RTP. Usually I do a cycle test, summing the payout for every possible combination of symbols/stops, and for a 5-reel slot with 32 stops per reel, that's 5^32 = 33,544,432 combinations. But a Megaways slot would seem to have a gazillion more combinations than that.

Here's my understanding about how Megaways works:

(1) There are six reels instead of 5.

(2) On each spin, each reel is randomized as to how many symbols are visible in the viewport, 2 to 7. I presume the game picks a random number from 2-7, and then serves up that particular reel style for that column. I also presume that, for example, the 3-symbol in column 1 could be different from the 3-symbol reel in other columns.

(3) An article I read somewhere said that the arrangement of symbols on the reels is also randomized each spin (but not the symbol frequency). I don't know whether that's true, but I don't think it matters, shouldn't affect the RTP.

(4) The paylines are simple: If you have the same symbol *anywhere* in the first 3+ reels, you win, even if the symbols don't touch. So, if you got a 7-stop reel for each of the 6 reels, there would be 117,649 potential paylines.

(5) And article I read somewhere said that the *paytable* also changes each spin, but I'm strongly skeptical.

(6) When there's a winning combo, after the win is shown, the winning symbols are erased and new symbols cascade down for a new possible win. For the purposes of the discussion below, I'm ignoring that feature.

Here's an example slot, Monopoly.

I figure there are a whopping 46,656 sets of reels (e.g., 2-2-2-2-2-2, 2-2-2-2-2-3, 2-2-2-2-2-4, etc.), and them, um, the number of combos gets fuzzy for me. With 32 stops per reel, regardless of how many are actually visible in the viewport, is it 32 stops ^ 6 reels * 46,656 sets = 1,073,741,824 • 46,656 = 50,081,568,620,544? That can't be right.

Then I'm thinking, when we do a cycle test for a regular slot, we don't care about how many stops are visible in the viewport, so would it be as simple as 32^6 = 1,073,741,824 combos? But with the screwy viewport and paylines, and the fact that on each spin every payline counts (up to 117,649 of them), it seems that there's got to be more to it than that. Certainly if every reel has 7 visible stops, wins are more likely than if each reel has only 2 visible stops. But for putting that into a formula or method, the math is over my head.

This is not even the most complicated version. Other variants do the random reel sizes and include a reel along the top that cascades from right to left.

Consider the complicated nature a blessing. IGT has a patent on random ways of winning. You will get sued into oblivion if you make one for commercial use. I cannot find the original BTG (now part of Evolution) patent.

Okay, wow, thanks. But that raises even more questions. Like, did IGT buy the patent from BTG? And there are lots of Megaways slots on the web that are branded BTG (like the Monopoly game I linked to), so, what, did IGT grant BTG the right to keep using Megaways for games it already developed? I also know that Robert Muir's Slot Designer software says that it can create Megaways-style slots, so it seems that others may be developing them. I'd love to know more.

Incidentally, I did just figure that while my computer takes 2 seconds to run through the 33M combos in a normal slot, it would take over a month to run through every combo on every set of reels for the Megaways type slot I described, making me think that there has to be some shortcut one could use in the analysis.

Quote: MichaelBluejay

Incidentally, I did just figure that while my computer takes 2 seconds to run through the 33M combos in a normal slot, it would take over a month to run through every combo on every set of reels for the Megaways type slot I described, making me think that there has to be some shortcut one could use in the analysis.

Or, you could upgrade your computer. If you could get that time down to about an hour that would seem reasonable to me.

Do you know anyone working at Lawrence Livermore National Laboratories? My best friend used to work there but he has since moved on. They have a computer with 11 million CPUs. I am guessing that would take less than an hour.

Quote: MichaelBluejay

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Incidentally, I did just figure that while my computer takes 2 seconds to run through the 33M combos in a normal slot, it would take over a month to run through every combo on every set of reels for the Megaways type slot I described, making me think that there has to be some shortcut one could use in the analysis.
link to original post

One way to do that is to replace combinations with probabilities, where possible.

I did something like that for a baccarat analyzer. For an 8 deck shoe, 416 cards, your possible 6-card combinations are 416x415x414x413x412x411 for a total of 5 quadrillion combinations.

But being it's baccarat and only 6 cards at most will be involved in a hand, and for the most common bets we only care about the value 0-9 of each card, rather than counting all the combinations I can calculate the probability of each combination, leaving only 10^6 probabilities to analyze. If my needs are rank-aware like for pair bets it's 13^6, and if it's the kind of thing where I need suits too it's 52^6, but even that is only 20 billion which is manageable on a good desktop, unlike 5 quadrillion.

So it may help you to determine how many non-redundant combinations (in terms of outcome) there are on the machine and solve for the probabilities of those, assign an integer label to each probability, and then do your CA on the probabilities rather than the reel positions themselves.

Quote: DRich

Quote: MichaelBluejay

Incidentally, I did just figure that while my computer takes 2 seconds to run through the 33M combos in a normal slot, it would take over a month to run through every combo on every set of reels for the Megaways type slot I described, making me think that there has to be some shortcut one could use in the analysis.

Or, you could upgrade your computer. If you could get that time down to about an hour that would seem reasonable to me.

Do you know anyone working at Lawrence Livermore National Laboratories? My best friend used to work there but he has since moved on. They have a computer with 11 million CPUs. I am guessing that would take less than an hour.
link to original post

That seems a bit dramatic.
Is the software compatible with one of the IaaS providers, so you can farm out the heavy calculations to "the cloud"?

Quote: DRich

Or, you could upgrade your computer. link to original post

I thought about it, but a newer computer is only about twice as fast as mine, so that would knock it down from 34 days to 17.

Quote: AutomaticMonkey

One way to do that is to replace combinations with probabilities, where possible.

Oh, right! That's what I (well, we all) used to do with 3-reel slots back in the day. But then the Wizard and others said that for five reels with wilds it was too cumbersome and error-prone to do it by hand, so they just ran a cycle test. So now I thinking a hybrid: Run the probabilities, but not manually with a spreadsheet, automate the process with some code.

Quote: Dieter

Is the software compatible with one of the IaaS providers, so you can farm out the heavy calculations to "the cloud"?

I've never done IaaS, but that makes me think I could buy several computers myself and give each of them a slice of the project.

Again, all this is theoretical, I'm just interested in how it's done, I have no intention of actually doing it.

Would still like to hear how BTG et al are publishing Megaways games if IGT holds the patent.

Nik Robinson, founder of BTG, has a patent for an "Apparatus for playing a game". Filed in 2020. Currently assigned to BTG. Not cited in the IGT patent though. Weirdly, a patent to alter payouts dynamically is (if only "slotenthusiast' knew about that one, he could have used it in his scam courses!) TESS says that the "megaways" and "megapays" trademarks were registered in 2019.

The IGT patent was in house. Redick is a still a mathematician at IGT. DeBrabander (not a math guy) is at LNW now.

Internet lists a phone number for BTG. I tried calling it to ask them about this, but it is 8pm in australia time. No one answered and it did not go to a recorded message, so I am not entirely sure this is the right number.

edit: replaced number with a link in case it aint the real numba. remember to check international rates before calling any far off numbers. don't want a huge bill for a nonsense call.

@itsmejeff, thank you for your detective work.

BetMGM says Megaways started in 2016 with BTG's Bonanza, and suggests that Megaways slots at other providers is courtesy of BTG licensing the idea. (Well, now Evolution, which BetMGM says acquired BTG in 2021.)

Casino Life says that BTG's Bonanza launched in 2012, not the 2016 that BetMGM said. And btw, that Casino Life article is the first time I've seen the trademark symbol (™) in a url.