Retention is an acquisition channel, and the maths says so.

The problem

Most acquisition models treat organics as a fixed share of paid installs. Spend doubles, organics double. It's convenient and it isn't how it works.

Organic installs don't come from your spend. They come from people playing the game: word of mouth, store ranking, streams, friends joining friends. All of that scales with how many people are playing, not with what you put into the media plan. And how many people are playing is set by retention.

So the fixed share assumption cuts the chain in the one place where something interesting is happening. Retention isn't only a revenue input. It's a volume input.

The model

Kadar changes one thing. Organic installs on a day are proportional to how many people are playing that day.

organic(t) = k · active(t)

Where k is the daily organic installs produced per active player.

Active players on day t are whatever is left of every cohort that installed before it. Given a retention curve r(d), that's the convolution of installs with retention.

active(t) = Σ installs(s) · r(t − s)

That's the whole change. Everything below follows from it.

Figure 1. The loop. The only unconventional edge is the one from the player base back into installs.

What that does to the arithmetic

Once installs make players and players make installs, you have a loop. It behaves like any other branching process, which means it has a reproduction number, the same idea epidemiologists use.

Follow one install. It produces some number of active days over its life. Call that L, the area under the retention curve.

L = Σ r(d), d = 1 … ∞

Each of those active days produces k organic installs. So one install produces, on average, R further installs directly.

R = k · L

Those installs do the same thing, and so on. Total installs generated by one paid install are the sum of a geometric series.

1 + R + R² + R³ + … = 1 / (1 − R), for R < 1

That factor, 1 / (1 − R), is the amplification. It's the number you're missing every time you divide revenue by paid installs.

Where it breaks

The series only converges below one. At R = 1 every install replaces itself exactly, the sum runs away, and the model says one download produces infinite installs. Above one it's worse than useless.

You can't clamp that and move on. It's the reason the obvious way to set this up will hurt you, which is the next section.

Figure 2. Total installs per paid install. The interesting range and the unusable range are separated by very little.

Why we don't ask you for k

The obvious move is to let you set k, the organic installs per active player. It looks like the natural dial. It's the one that will catch you out.

Because R = k · L, and L is the area under your retention curve, the same k means very different things on two different games. A game that retains badly has a small L, so that k leaves R comfortably low. A game that retains well has an L several times bigger, so the same k pushes R past one and the whole thing runs away.

Same number typed in, two games, one sensible answer and one that falls over. And you wouldn't see it coming, because k reads like something about your marketing when it's really something about your retention curve.

Weak retainer

small L, R stays low

Strong retainer

large L, same k sends R past 1

Figure 3. The shaded area is L. Identical k, different curve, different R. Illustrative shapes, no fitted values.

What we ask for instead

So we don't ask for k. We ask for your organic share of installs, which is already on your dashboard, and work k out from the fitted retention curve.

share = R, so k = share / L

Three things that buys you. It sits between zero and one by construction, so you can't wander into the runaway case by accident. You can check it against what actually happened. And it means the same thing on any game: 30 per cent organic is 30 per cent organic, where k isn't.

The rate the engine uses then falls out of your curve, instead of being a guess you have to make.

What this means for your spend

Your paid installs produced organic installs. Those are a return on your ad spend, not a free channel that happened to grow at the same time.

No attribution tool sees this, and that isn't a criticism of them. Attribution tells you where an install came from. This is about what an install caused. Different question, and only one of them has a tracking link on the end of it.

Three things follow, none of them exciting and all of them worth money. Your true cost per install is lower than the one you're reporting, by the amplification factor. Retention work buys volume as well as revenue per user, so it's competing with acquisition for the same budget on the same terms. And a game that looks marginally unprofitable on paid attribution can be comfortably profitable once you count the loop.

What isn't here

No fitted parameters, no genre curves, no calibration values. That's the product. The model above is complete, so go and build it if you want to.

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