Fun With Numbers: The Anime Sequel Probability Equation (Alpha)

Ever wonder what anime get sequels and what don’t? The simple answer to this fairly simple question is “sales, plus a few mitigating lesser factors”. We know this, beyond the obvious intuition, because over the years, we see those with robust sales totals get continued much more often than those with lackluster ones:

FWN-Sequel-B-SalesA more interesting question is perhaps this; which anime this year have the best chances of getting continued? After some delving into the subject, I can finally answer this question to a respectable degree of confidence. Based on data from 615 shows airing from from 2005-2012, the first season of an anime which sells x units per volume has P odds of getting a sequel, meaning either a second season or a movie.

P(x, t, L)=.01+θ(x-2250)*(.7-1650/(x+L*750))*e^(-((t-1)*θ(t-1))/.7)

Where t is the time in years passed since the first season aired, and L=1 corresponds to the series having been licensed. θ is the step function, 0 when the number inside the brackets is less than 0, and 1 when the number inside is greater than 0.

Before going into detail on this equation, it’s worth noting what this equation does and does not account for:

Things this equation does not specifically account for:

1. Which studio is making the show
2. Source Material Status (whether the Manga/LN is still running or not)
3. Shows that rely primarily on other income streams (e.g. toy sales and TV ratings)

^These aren’t not accounted for, but factor into the percentages in the rest of the equation. It’s blind to, but not ignorant of, these factors.

Things this equation does account for:

1. Split Cours (These were a huge factor in overestimating the true odds of a sequel happening for non-profitable shows. )
2. Licenses (A license is essentially worth approximately 750 units per volume on the probability curve. Matters a bunch for shows between 2000 and 5000, not so much for shows over 10000.)

That out of the way, let’s take a look at where each term of the equation comes from. Skip this part if you’re not big on math!

θ(x-2250)*(.7-1650/(x+L*750)) accounts for both Japanese sales totals and non-Japanese licenses (see reference 3 for why that matters). 2250 is approximately the area where an anime becomes break-even with a really good license (which is worth about 750 average sales per volume in gross profit, hence the coefficient for L). I didn’t pull those numbers out of a hat; they’re the result of a scaled fit to the data below. Green is the hyperbolic fit I ended up using, red is a logarithmic fit which overestimates the odds of a high-seller’s continuation:


e^(-((t-1)*θ(t-1))/.7) is the product of an integral of a time-integrated probability distribution. Essentially, the odds of an anime getting a sequel 1 year or later decline as time passes. 1 year after the initial airdate is as early as “true” sequels that weren’t already in production start being aired, which accounts for the t-1 being in there as opposed to t. The fit below is integrated from whatever t is to infinity, because the model accounts for the chances that a show may get a sequel at any *future* point in time, while assuming that one hasn’t already been made:


.01 is a factor to account for the fact that low-selling shows do get sequels on occasion, mainly due to loss-leader effects that are near impossible to account for because manga sales aren’t as cleanly available and gains are really tough to dissect for shows not named Blue Exorcist.

Resume reading here if you skipped the last part!

So, what does this all mean? For one thing, it makes it pretty easy to calculate the odds that a given show will get a sequel, answering the burning question with some accuracy even before we get any “official” news.

Granted, this is a less straightforward method of measuring a show’s success than just straight-up sales, but it’s a different perspective on what makes success, and something that’s more tangible in that it emphasizes how much more important sales are for series on the margin. For example, a series that sells only 2500 units per volume has only about a 19% chance of getting a sequel. Add 1000 disks per volume to that average, and that figure shoots up to 31%. By contrast, adding 1000 DPV to a series at 10,000 only bumps the odds up from 55% to 56%, a tiny swing. Incidentally, the series with the highest sequel odds at present in 2013 is, unsurprisingly, Attack on Titan (at 67%). The ones with the biggest swing potential, where one additional customer could have made a big impact on the odds of a sequel, are Uchoten Kazoku and Devil Survivor 2, both around 0.028%. That changes a bit, but not much, as time passes:

FWN-Sequel-B-PlotShortI’m still working on an input-output calculator, figuring out both how to code and where to host it. It’s coming. For now, that bit of user interface is as far as I plan to go with the data. Most of the remaining points to be addressed are possible to handle, but only provided I’m willing to spend time going back through my sample and individually adding values for upwards of 600 datapoints.

Plus, there are interesting issues I’ve yet to really address; the large scale effectiveness of anime as a 12-episode manga commercial, the kinda-resurgance of Weekly Shonen Sunday, and the burgeoning OAD market. That last one’s probably coming first, courtesy of Yozakura Quartet’s astounding DVD-bundled volume sales of almost 17000. That’s huge, enough by my calculations to pay for *two* super-glitzy max-budget episodes. From a disk with one total episode. More to the point, it’s probably enough (combined with prior OAD sales and an achievable 1000+ average sales total from the Fall TV series) to make the anime a break-even proposition. Coming from an otherwise statistically unimpressive series, that’s all kinds of fascinating stat geek fuel. And then you get to Seitokai Yakuindomo’s case…

Yep, definitely expect something about OADs in this space in the near future.*





4. Sheet where most of the non-graphical math was done:

*Read: before 2014

7 thoughts on “Fun With Numbers: The Anime Sequel Probability Equation (Alpha)

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  6. I keep finding myself referencing this article again and again. Sequel chances are probably the most important result of sales chasing, so I hope you’ll develop this model more in the future!

    • Accuracy does demand I revise this eventually – some of the high-end series that hadn’t had sequels at the time this was written (SAO, Mushishi, P4), so the odds there are going to bump up. Plus, I’ve seen mounting evidence that borderline print-source adaptations don’t often get sequels because of the boosts they produce; Emma is the only solid bad sales+boost example, and Kore Zombie is a borderline one. Other series like Maken-Ki didn’t get visible boosts and sold poorly but got sequels anyway.

      All of that will eventually be incorporated into the next model whenever I get to it, though I just finished a major project, so I’m probably going to take a little break from stuff involving 50+ spreadsheets for a bit and focus on less-intensive stuff, namely OP/ED singles-disk sales relations, the list of directors helming 2 or more 10k+ sales hits, and the average performance of genres other than ecchi pre and post blu-ray.

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