Media saturation: comparing the functions before optimising the budget.
Directly verifiable evidence
Primary sources to check the key claims.
- Official Google Meridian documentation (opens in a new tab)
Response curves, saturation and use for allocation.
Key distinctions
Three conditions before calculating.
Observed range
A curve is informed by the variation genuinely present in the data. Recommending twice the historical maximum often amounts to letting the function choose in place of the evidence.
Interpretable parameters
The half-saturation level, the slope around the inflection point and the asymptote must be translated into spend, exposures and expected response.
Marginal, not average
Average ROI summarises the past. Allocation requires the local derivative of the curve at the candidate spend level, with its uncertainty.
Method
Choosing and using a diminishing-returns function
- 01Profile the variation
Document minimum, median, maximum, number of periods close to each level and correlation with the other channels.
- 02Compare the families
Fit several functions under comparable priors or bounds; examine validation, parameters, monotonicity and extrapolation.
- 03Compute the derivative
Publish the response to the next budget tranche and not only the total attributed contribution; the decision must compare tranches of the same size.
- 04Bound the optimisation
Limit the recommendation to an envelope close to the history, or require a test when the curves no longer agree.
Four functions, four extrapolations
The curves are normalised over the observed range. Their marginal return diverges strongly when the candidate budget exceeds the history.
Michaelis-Menten · diminishing marginal return
Doubling the spend adds only €86k with Michaelis-Menten
The maximum response is parameterised at €500k and the half-saturation at €50k. Spend moves from €40k to €80k.
Response at €40k = 500 × 40 ÷ (50 + 40) = €222k. At €80k = 500 × 80 ÷ 130 = €308k. The marginal gain is €86k, not €222k.
The doubling is acceptable only if €86k of response covers the cost and stays robust to the other functions. Otherwise the tranche must be tested or reallocated.
Acceptance conditions
What must be true to act.
- 01
The candidate spend stays within a covered or explicitly tested range.
- 02
Several families of functions are compared with the same data.
- 03
The marginal return and its interval are published at the candidate point.
- 04
The optimisation is stressed against the curves that remain plausible.
Limits
What this analysis does not prove.
- Adstock and saturation are hard to separate when spend varies little.
- A saturation in spend can reflect changes in media price or inventory quality.
- The asymptote is rarely well identified without significant variation or external information.
References
Works cited.
- Hill, A. V. (1910), “The Possible Effects of the Aggregation of the Molecules of Haemoglobin”.
- Michaelis, L. & Menten, M. L. (1913), “Die Kinetik der Invertinwirkung”.
- Jin, Y. et al. (2017), “Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects”.