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Hill saturation

Additional investment often delivers less incremental effect than the initial spend. The Hill function models this relationship with a half-saturation point and a curve shape that describes how quickly returns increase and then level off.

6 min read · Updated September 7, 2026

k: half-saturation0×1×2×3×0%50%100%Spend, as a multiple of the average weekShare of the maximum effects = 0.8, concaves = 2.5, S-shaped
Two channels with the same half-saturation point k, set at the average weekly spend. The concave one (s = 0.8) starts steep and flattens early. The S-shaped one (s = 2.5) needs a threshold before it works, then saturates hard. Both reach half their maximum at k.

What it is

Saturation is diminishing returns: each additional euro on a channel earns less than the one before, because the audience is finite, frequency piles up, and the cheapest inventory goes first. The Hill function is the standard form for it:

saturation(x) = x^s / (k^s + x^s)

It runs from 0 to 1. Two parameters shape it. The half-saturation point k is the spend at which the channel delivers half of its maximum effect. The shape s controls the steepness: below 1 the curve is concave from the start; above 1 it is S-shaped, flat at low spend, steep in the middle, flat again at the top.

Why it matters for a media plan

Saturation is the whole reason a plan cannot be made from average returns. A channel at three times its half-saturation point still shows a good average return, because the first euros earned it, while the next euro earns almost nothing. The average describes the past; the slope decides the next euro.

It is also why budget moves. If channel A is past its knee and channel B is before it, the same euro moved from A to B earns more, even if A’s average return is higher. That is the arithmetic behind every reallocation the Media Planner proposes.

How it works

In a media mix model, saturation is applied to the adstocked spend, so what saturates is the accumulated effect, not one week’s outlay. The channel’s contribution is then the coefficient times the saturated value.

Spend should be normalised before fitting, usually divided by its mean. Then k is read in multiples of the average week: k = 1 means the channel is half-used at its typical spend, k = 2 means there is a lot of room, k = 0.5 means it is well past the knee. Normalised priors also transfer between brands, which matters when a plan starts from benchmarks.

A useful set of default priors on the normalised scale: k centred near 1 with a wide spread, and s centred near 2. Channels with a known threshold, television for a small brand, out-of-home, can start with a higher s. Channels that respond from the first euro, search, retargeting, can start below 1.

The response curve a planner reads is this function scaled by the channel’s coefficient and expressed in revenue. Its slope at the current spend is the marginal return.

How Kuwalyst uses it

Every channel on a data-backed plan carries its fitted k and s on the model card, and the planner reads the response curve at the committed spend to state where each channel sits: before the knee, at it, or past it. A recommendation to move budget names the source channel’s position on its curve, so the reader can see the arithmetic and not just the conclusion.

For benchmark-led plans, the half-saturation point comes from the registry, by channel type and budget band, tagged as a benchmark. The plan lists it among the assumptions to test first, because saturation is the parameter most specific to a brand.

Pitfalls

Extrapolation: the fit only knows the spend range it saw. A curve that says a channel could take three times its highest historical week is a guess about a place the data never went. Scenarios that leave the observed range should say so, and the planner caps them.

Confounding with adstock: a long adstock tail and a low k can explain the same weeks as a short tail and a high k. Strong decay priors keep saturation honest.

Threshold effects on small data: an S-shape needs weeks at low spend to locate the threshold. Without them, s is the prior.

Saturation is not fatigue: Hill describes diminishing returns at a point in time. Creative wearing out over months is a different effect and needs a different term.

FAQ

What is a typical half-saturation point?

It is set relative to the channel's own spend. With spend normalised to its mean, a prior centred at 1.0 says the channel is halfway to saturation at its average week, which is a reasonable starting point for most paid channels. Search with a small brand tends to saturate earlier; awareness channels later.

What does the shape parameter mean in practice?

Below 1, the curve is concave: every euro does a little less than the one before, from the first euro. Above 1, it is S-shaped: a threshold where small budgets do almost nothing, a steep middle, then a plateau. Television and out-of-home often show a threshold. Search and retargeting are usually concave. A default prior centred around 2 works for most channels until the data says otherwise.

Is Hill the only option?

No. Logistic saturation, with a single parameter, and Michaelis-Menten, which is Hill with the shape fixed at 1, are also common. Hill is preferred because the two parameters map to two things a planner can reason about: where the channel is half-used, and whether it has a threshold.