The problem
Almost every acquisition model in use treats organic installs as a fixed share of paid installs. Spend doubles, organics double. It is a convenient assumption and it is wrong.
Organic installs do not come from paid spend. They come from people playing the game: word of mouth, store ranking, streams, friends joining friends. All of those scale with the size of the active player base, not with the media plan. And the size of the active player base is set by retention.
So the fixed share assumption breaks the chain at exactly the point where the interesting behaviour lives. Retention is treated as a revenue input. It is also a volume input.
The model
Kadar makes one change. Organic installs on a given day are proportional to the number of active players on that day.
Where k is the daily organic installs produced per active player.
Active players on day t are whatever survives from every cohort installed before it. Given a retention curve r(d), the active base is the convolution of installs with retention.
That is the whole structural change. Everything below follows from it.
The consequence nobody notices
Once installs produce players and players produce installs, acquisition is a branching process. It has a reproduction number, in the same sense an epidemic does.
Follow one install. It produces some number of active days over its life. Call that L, the area under the retention curve.
Each of those active days produces k organic installs. So one install produces, on average, R further installs directly.
Those installs do the same thing, and so on. Total installs generated by one paid install are the sum of a geometric series.
That factor, 1 / (1 − R), is the amplification. It is the number every studio is missing when it divides revenue by paid installs.
Where it breaks
The series only converges below one. At R = 1 every install replaces itself exactly, the sum diverges, and the model predicts infinite installs from a single download. Above one it is worse than useless.
This is not a numerical inconvenience to be clipped. It is the reason the naive parameterisation is dangerous.
The trap
The obvious thing to do is let the user set k, the organic installs per active player. It looks like the natural parameter. It is the worst possible choice.
Because R = k · L, and L is the area under the retention curve, the same k means completely different things on different games. A weak retainer has a small L, so a given k leaves R comfortably low. A strong retainer has an L several times larger, so the same k pushes R past one and the model diverges.
One input, two games, one safe answer and one explosion. Nobody using the tool would see it coming, because k reads like a property of the marketing, not of the retention curve.
The fix
Do not ask for k. Ask for the organic share of installs, which is a number every studio already has on a dashboard, and derive k internally from the fitted retention curve.
Organic share is bounded between zero and one by construction, so the divergence condition becomes unreachable by accident. It is observable, so it can be validated against reality. And it is retention invariant: 30 per cent organic means the same thing on any game, while k does not.
The rate the engine actually uses is then a consequence of the fitted curve rather than a guess the user has to make.
What it means in practice
Your paid installs generated organic installs. Those organic installs are a return on your ad spend. They are not a separate free channel that happened to grow at the same time.
No attribution tool can see this, and it is not a flaw in the tools. Attribution answers the question of where a given install came from. This is a question about what a given install caused. Those are different questions and only one of them has a tracking link.
The practical consequences are unglamorous and large. Your true cost per install is lower than your reported cost per install by the amplification factor. Improving retention raises volume as well as revenue per user, so retention work competes with acquisition work for budget on equal terms. And a game that looks marginally unprofitable on paid attribution may be comfortably profitable once the loop is counted.
All of which is arithmetic, once the edge from the player base back into installs is drawn.
What is not on this page
No fitted parameters, no genre curves, no calibration values. Those are the product. The model above is complete and anyone is welcome to implement it.