Retention curves and the inference bill

Why inference changes retention maths#

In a conventional game, a player who stays longer is pure upside: serving them costs almost nothing. In an AI-native game, every minute played calls a model, so cost follows engagement. That flips some of live-ops' instincts. This post fits a standard retention model, builds LTV from it, and then adds the inference bill, for free-to-play and for premium. The two business models turn out to fail in completely different ways.

Code: research/retention_ltv.py. Spend, session length and playtime figures are assumptions, flagged as such.

Fitting the retention curve#

Day-$n$ retention in mobile and free-to-play games is commonly modelled as a power law (see Russell Ovans' Game Analytics 100: The Retention Curve):

We fitted it to a published "good" benchmark profile of D1/D7/D30 = 35%/15%/5% by nonlinear least squares, which gave $a=0.352,b=-0.490$. A log-space fit gives $a=0.377,b=-0.565$. Two parameters can't pass exactly through three points, and the fit predicts D7 13.6% and D30 6.7%. The benchmark's D30 is steeper than a pure power law, which is common: late churn often has a second, faster component.

Fitted curve (line) against the three benchmark points (dots).

Expected active days per install over a horizon $H$ is the area under the curve:

Free-to-play: the ratio that decides everything#

With average revenue per daily active user $\text{ARPDAU}$, LTV is $\text{ARPDAU}\cdot D(H)$. If each active day involves $\tau$ minutes of play at an inference cost of $\kappa$ per hour, inference per install is $(\tau /60)\kappa D(H)$. The share of revenue spent on inference is then

$D(H)$ cancels. In free-to-play, retention doesn't change the fraction of revenue you spend on inference. Better retention makes the business bigger, not healthier. The ratio is fixed by how long people play per day, what an hour costs and what a day earns. With $\tau =28$ minutes and $\text{ARPDAU}=\$0.12$ (both assumptions):

At $0.18 per player-hour, a perfectly reasonable figure for a dialogue-heavy game before optimisation (see our cost guide), inference eats 70% of revenue before platform fees, user acquisition or salaries. Free-to-play AI games need either very cheap inference (under ~$0.05/hour at typical ARPDAU) or monetisation that scales with AI usage itself.

Premium: your fans are your cost centre#

A premium game earns its net price $N$ once ($12.32 on a $20 game after fees), but inference scales with hours played $h$. Playtime is heavy-tailed, so we model it as lognormal with a median of 12 hours and log-sd 1.0 (an assumption: mean ≈ 19.8 hours). Per-player cost $C=\kappa h$ is lognormal too, so the fraction of buyers who cost more than a fraction $\alpha$ of their purchase is

and the top 5% of players account for $1-\Phi (z_{0.95}-{\sigma }_h)=1-\Phi (1.645-1)\approx 26\%$ of all inference spend, whatever $\kappa$ is.

At $0.18/hour, 4.1% of buyers (anyone past 68 hours) cost more in inference than the game earned from them. These are your biggest fans, the ones who stream it and write the reviews. You can't price them out. You can design for them:

• Taper model usage with mastery. Late-game players need fewer explanations. Route familiar interactions to small models.
• Move the long tail on-device. Heavy players are on good hardware more often than not; a local small model for routine turns cuts their marginal cost to nearly zero.
• Sell what scales. Expansions, new characters and seasons convert engagement into revenue again.

Limitations#

A single power law under-fits late retention, and real cohorts mix player types with different curves. ARPDAU isn't constant over a player's life (payers churn slower), which shifts $\varphi$ in favour of games that monetise long-term players. The premium playtime distribution is assumed, not measured. Replace it with your playtest or early-access telemetry. Treat every number here as an illustration of the maths, not a benchmark.

Reproduce it#

The full script is research/retention_ltv.py (Python 3.10+, numpy, scipy; fixed seeds, so you'll get the same numbers). This is the core of it:

(a, b), _ = curve_fit(lambda n, a, b: a * n ** b, [1, 7, 30], [0.35, 0.15, 0.05])
r = a * np.arange(1, 366) ** b
active_days_365 = 1 + r.sum()

phi = (MIN_PER_DAU / 60 * cost_per_hour) / ARPDAU      # F2P inference share: retention cancels

hours = 12 * np.exp(1.0 * rng.standard_normal(500_000)) # premium playtime (assumed)
p_over_net = np.mean(hours * cost_per_hour > NET)

Run it with python3 research/retention_ltv.py; results land in research/out/, and the site's tables and charts are rendered from that JSON at build time. Replace the D1/D7/D30 points with your own cohort and hours with your playtime distribution from early access. The rest follows.

Frequently asked questions#

Sources#

1. Russell Ovans, "Game Analytics 100: The Retention Curve". gameindustrylibrary.com/documents/game-analytics-100-the-retention-cur
2. Playio, "D1, D7, D30 retention benchmarks 2026". blog.playio.co/d1-d7-d30-retention-benchmarks-2026