Budget allocation under uncertainty

Day 4 Module 10 110 minutes (45 briefing, 55 lab, 10 debrief) 14:15 briefing · 15:00 lab · 15:55 debrief

Turn response curves into a budget: write a constrained SLSQP allocator, compare it with PyMC-Marketing’s optimizer, add minimum and maximum spend per channel, make the allocation robust to posterior uncertainty, and credit channels for the long-run value of the customers they bring.

Open the Python lab in Colab Python

The decision. Recommend next quarter’s weekly spend per channel within the agreed bounds, with expected incremental sales and a 94% interval, and show how the recommendation changes under a risk-averse or a CLV-weighted objective.

Day: Day 4 · Where the next dollar goes (14:15–16:05)
Time: 110 minutes (45 briefing, 55 lab, 10 debrief)
Languages: Python. The optimizer works directly on PyMC-Marketing’s posterior draws, and SciPy’s SLSQP, which you call yourself, is the algorithm the library uses.
Labs: Python: labs/python/10-budget-allocation.ipynb
Data: Synthetic weekly MMM (mmm_weekly.csv, seed 2028), Synthetic lift tests (mmm_lift_tests.csv)

What you will be able to do

Derive steady-state response curves and marginal ROAS per channel from the posterior.
Write a constrained allocator with scipy.optimize.minimize (SLSQP) and check it against the allocation that is optimal under the true parameters.
Run PyMC-Marketing’s budget optimizer with minimum and maximum bounds and reconcile it with your allocator.
Evaluate allocations over posterior draws and choose one by a stated risk rule (mean versus 10% quantile of response).
Add a CLV-weighted term so channels that acquire more valuable customers get credit beyond short-run sales.

You leave with:

Your own allocate() function with bounds and a budget constraint
A comparison of your allocation, PyMC-Marketing’s and the true optimum
Three allocations (mean, risk-averse, CLV-weighted) with expected sales and 94% intervals

Before you start

Module 9 · Calibrating and validating an MMM and its lab.
A Google account for Colab. The default Python runtime (CPU) is enough.
Or the workshop’s Docker image, which has every package installed.

Where we are. Module 8’s marketing mix model said what each channel’s dollars returned, and Module 9 calibrated it with lift tests so its response curves can be trusted further. Neither says what to do next quarter. A planner has a fixed weekly budget, bounds the business has agreed for each channel, and a model whose curves are uncertain. This module turns response curves into a budget: which split of the money maximizes expected sales, how sure we are that it beats the current plan, and how the answer changes when we care about the bad case or about the long-run value of the customers each channel brings.

The lab

Python lab · `10-budget-allocation`

Open in Colab

Step	Time
Part A · Response curves
Exercise 1 · Steady-state response curve	7 min
Exercise 2 · Marginal ROAS	5 min
Part B · Allocate
Exercise 3 · An SLSQP allocator	10 min
Exercise 4 · PyMC-Marketing’s optimizer, in its units	6 min
Part C · Uncertainty and the long run
Exercise 5 · Risk of an allocation	7 min
Exercise 6 · A CLV-weighted objective	5 min
Decision · Next quarter’s weekly budget	—
Python subtotal: 6 exercises	40 min

Stretch (optional) · Tail risk and a contract. For anyone who finishes early.

6 exercises, 40 minutes between them, in a 55-minute lab slot. Each exercise has a stub to complete, a folded solution and a checkpoint.

1. The planning problem

The planner’s facts, as finance would supply them (from the synthetic truth file):

a weekly media budget of $72,000, the mean weekly total over the last 52 weeks;
bounds for each channel: between half and twice its mean weekly spend over those 52 weeks;
the current plan, the recent mix scaled to the budget.

Channel	Current plan ($/week)	Lower bound	Upper bound
TV	31,581	15,716	62,864
search	20,562	10,233	40,930
social	12,229	6,086	24,343
display	7,628	3,796	15,185

The model is Module 8’s specification refitted with one lift test per channel added to its likelihood (the calibration of Module 9), so that this lab stands on its own. The question this module answers is how to split a budget, not how large it should be. Section 7 comes back to the second question.

Notation for this module

Symbol	Meaning
s_c	planned weekly spend on channel c
r_c(s)	steady-state weekly sales response of channel c to weekly spend s
\beta_c, \lambda_c	the fitted saturation parameters (Module 8), per posterior draw
x_c, y_{\text{scale}}	the channel’s largest weekly spend and the largest weekly sales in the data (the model’s scaling)
B	the weekly budget
\ell_c, u_c	lower and upper bounds on s_c
m_c(s)	marginal ROAS: dr_c/ds, the return on the next dollar
\mu	the Lagrange multiplier of the budget: the marginal ROAS shared at the optimum
q_{0.1}, CVaR	the 10% quantile of a plan’s weekly sales, and the mean of the draws below it
m	gross margin rate (30%)
n_c, \text{CLV}_c	new customers acquired per dollar on channel c, and a new customer’s future margin

2. Response curves and marginal ROAS

Objective for this section: derive steady-state response curves and marginal ROAS per channel from the posterior.

Hold weekly spend on channel c constant at s for the quarter. Normalized adstock (Module 8) averages recent weeks with weights that sum to 1, so the adstock of a constant spend is that spend, and after a few weeks the channel adds the same sales every week. That steady-state response is Module 8’s saturation curve scaled back to dollars:

r_c(s) = \beta_c \, y_{\text{scale}} \cdot \frac{1 - e^{-\lambda_c s / x_c}}{1 + e^{-\lambda_c s / x_c}} . \tag{1}

\beta_c y_{\text{scale}} is the most the channel can add in a week; \lambda_c / x_c says how fast spend gets there. Each posterior draw of (\beta_c, \lambda_c) gives one curve, so the posterior gives a band of curves, and the band’s width at the spend levels between the bounds is the model’s uncertainty about what a plan will return.

The curve is concave: each extra dollar adds less. On the true curve, doubling TV from its current $31,581 a week raises its weekly sales from $53,020 to $97,539, an increase of 84% for an increase in spend of 100% (FULL run).

The quantity a budget decision needs is the marginal ROAS,

m_c(s) = \frac{d r_c}{d s} \approx \frac{r_c(s + \varepsilon) - r_c(s - \varepsilon)}{2\varepsilon} , \tag{2}

the return on the next dollar (the lab uses a central difference with \varepsilon = one dollar; its error shrinks with \varepsilon^2). Module 8 reported the average ROAS, r_c(s)/s, which includes the steep first dollars. On a concave curve that starts at zero, the marginal ROAS is below the average at every spend level. At the current plan, on the true curves (FULL run):

Channel	Average ROAS	Marginal ROAS
TV	1.68	1.58
search	3.02	2.00
social	1.43	1.30
display	0.58	0.36

A dollar moved from one channel to another loses that channel’s marginal return and gains the other’s. Moving a dollar from display to search today gains about $2.00 − 0.36 = $1.64 of weekly sales. Only marginal ROAS says whether a move pays.

3. The allocation problem

Objective for this section: write a constrained allocator with scipy.optimize.minimize (SLSQP) and check it against the allocation that is optimal under the true parameters.

The problem is

\max_{s_1, \dots, s_C} \; \sum_c r_c(s_c) \quad \text{subject to} \quad \sum_c s_c = B, \qquad \ell_c \le s_c \le u_c . \tag{3}

Its solution has a simple shape. Attach a multiplier \mu to the budget constraint. At the optimum,

\begin{aligned} m_c(s_c^*) &= \mu && \text{if } \ell_c < s_c^* < u_c, \\[4pt] m_c(\ell_c) &\le \mu && \text{if } s_c^* = \ell_c, \\[4pt] m_c(u_c) &\ge \mu && \text{if } s_c^* = u_c . \end{aligned} \tag{4}

Every channel strictly inside its bounds earns the same marginal ROAS; otherwise moving a dollar from the lower one to the higher one would add sales. A channel at its lower bound returns less than \mu even there, so it would get less if the bound allowed. Because every r_c is concave, the problem has a single optimum, and a local method finds it.

SLSQP (sequential least squares quadratic programming) handles exactly this: a smooth objective, bounds, and an equality constraint. It replaces the problem near the current point with a quadratic model and linearized constraints, solves that, and repeats. The lab’s allocate(total, bounds, curves) calls scipy.optimize.minimize(..., method="SLSQP", bounds=..., constraints=[{"type": "eq", "fun": lambda s: s.sum() - total}]) on the negative of total sales. On curves that are not concave, SLSQP can stop at a local optimum: start it from several points.

FULL run, weekly spend:

Plan	TV	search	social	display	True weekly sales from media
current	31,581	20,562	12,229	7,628	136,901
your `allocate`, model curves	36,831	25,287	6,086	3,796	143,069
your `allocate`, true curves	36,878	25,240	6,086	3,796	143,069
answer key (exact optimum)	36,882	25,236	6,086	3,796	143,069

The best plan puts social and display at their lower bounds and splits the rest between TV and search where their marginal ROAS is equal (about 1.53 dollars of sales per dollar on the true curves). Allocating on the model’s posterior-mean curves gives a plan worth the same true sales as the exact optimum, $6,168 a week (4.5%) more than the current plan. Search, the channel with the highest average ROAS, does not get the most money: TV’s curve is less saturated at these spend levels, so its next dollar earns as much.

Check yourself. At some plan, search’s marginal ROAS is 2.0 and social’s is 1.3, and both are inside their bounds. What is the gain from moving $1,000 a week from social to search, roughly? Why does the gain shrink as you keep moving money?

Answer

About 1{,}000 \times (2.0 - 1.3) = \$700 of weekly sales, for a small move. As money moves, search climbs its curve (its marginal ROAS falls) and social moves down its curve (its marginal ROAS rises), so each further dollar gains less. The moves stop paying when the two marginal ROAS are equal (Equation 4), or when social hits its lower bound first, which is what happens here.

4. PyMC-Marketing’s optimizer, in its units

Objective for this section: run PyMC-Marketing’s budget optimizer with minimum and maximum bounds and reconcile it with your allocator.

PyMC-Marketing plans a budget for a future window of dates: mmm.budget_optimizer(start, end) builds an optimizer for the 13 weeks after the data end, and allocate_budget(total_budget=..., budget_bounds=...) runs it, with SLSQP by default. Two things differ from your allocator, and both matter.

Units. In pymc-marketing 1.2.0, total_budget is a budget per period (per week here): the optimizer repeats it for each of the 13 weeks (read in the installed source). Pass a quarterly total and it recommends spending the quarter’s budget every week, 13 times too much.
Time. The optimizer scores whole weeks, not the steady state: its window includes 8 weeks of history before the quarter (their spend still carries in) and 8 weeks after it (so the carry-over of the last weeks’ spend counts).

FULL run: the library’s plan is TV $36,510, search $25,608, social $6,086, display $3,796. It differs from your model-curve plan by 0.4% of the budget, on TV and search only. Two implementations of the same idea, with different handling of time, agree to a fraction of a percent; that is the reconciliation the checkpoint asks for.

5. Uncertainty: the bad case

Objective for this section: evaluate allocations over posterior draws and choose one by a stated risk rule (mean versus 10% quantile of response).

A plan’s sales are uncertain because the model’s parameters are. For each posterior draw, the lab computes the plan’s incremental weekly sales (the optimizer’s window total minus the same total with zero spend, divided by 13 weeks). Draw j of two plans comes from the same parameter draw, so plans can be compared draw by draw, as Module 5 compared segments.

Two summaries of the bad case:

\begin{aligned} q_{0.1} &= \text{the 10\% quantile of the draws}, \\[4pt] \text{CVaR}_{0.1} &= \text{mean of the draws at or below } q_{0.1} . \end{aligned} \tag{5}

The 10% quantile says how bad the bad case is; CVaR (conditional value at risk) also says how bad the tail is beyond it. A risk-averse plan maximizes q_{0.1} instead of the mean. FULL run, weekly incremental sales from media:

Plan	Mean	10% quantile	CVaR
current	121,462	105,910	102,673
mean-optimal (PyMC-Marketing)	129,081	113,546	110,026
risk-averse (best 10% quantile)	129,025	113,732	110,154

Two lessons.

Even is not safe. The current plan spreads money more evenly, yet its whole distribution, tail included, sits lower: it keeps money on channels with low marginal returns.
The risk-averse plan moves little: $1,385 of the weekly budget, from search to TV. That raises the 10% quantile by $186 a week and costs $56 of expected sales. At the mean-optimal plan TV and search share one marginal ROAS, so a small shift costs almost nothing on average, and it trims exposure to search, the less certain curve. Most of that uncertainty is in search’s overall level, which every allowed plan carries, so no reallocation within the bounds removes it. A lift test that narrows search’s curve would (Module 9).

Note the levels: the model’s expected sales for a plan are about 10% below the true ones (129,081 against 143,062 for the mean-optimal plan), because its curves are biased low in level. Its estimate of the gain over the current plan, $7,620 a week (94% HDI $3,739 to $10,848), contains the true gain of $6,161. For a decision between plans the gain is what matters, and differences between plans cancel much of the shared error.

6. The long run: a CLV-weighted objective

Objective for this section: add a CLV-weighted term so channels that acquire more valuable customers get credit beyond short-run sales.

Short-run sales miss what new customers buy later. Module 5 priced that: the margin a new customer brings after the first purchase, by acquisition channel. Adding it to the objective,

V(s) = m \sum_c r_c(s_c) + \sum_c s_c \, n_c \, \text{CLV}_c , \tag{6}

where n_c is the number of new customers a dollar on channel c acquires and \text{CLV}_c a new customer’s future margin, excluding the first purchase, which the MMM’s short-run sales already contain. The long-run term is linear in spend, so it adds a constant n_c \text{CLV}_c to each channel’s marginal value, m\, m_c(s) + n_c \text{CLV}_c, and the optimum moves money toward channels where that is largest until diminishing short-run returns or a bound stop it.

The lab reads n_c and \text{CLV}_c from the synthetic world’s truth file, and says so. Media channels (where money is spent) and acquisition channels (how the CRM records a new customer’s first visit: search, social, referral) are different lists, and the synthetic world links them: TV viewers mostly arrive by word of mouth (recorded as referral) or by searching for the brand, so a TV-acquired customer is valued as that mix. Cost per new customer: TV $150, search $150, social $100, display $300. Future margin per new customer: TV $58, search $41, social $27, display $34. Resulting long-run margin per dollar of spend: TV 0.388, search 0.275, social 0.271, display 0.113. In practice the cost per new customer comes from attribution in the CRM and the CLV from Module 5’s model, both with uncertainty.

FULL run: the CLV-weighted plan moves $5,982 a week from search to TV: TV $42,492, search $19,626, social $6,086, display $3,796. TV gains although a new customer costs as much there as on search, because its customers arrive mostly by referral, the most loyal acquisition channel. The short-run curves are nearly flat around the optimum, so a modest difference in long-run value per dollar moves thousands of dollars a week. On short-run sales alone the plan gives up a little (true 141,348 against 143,062 a week) and still beats the current plan with posterior probability 1.00. On the true curves the long-run optimum is TV $40,336, search $21,782: the model’s curves put $2,156 a week (3% of the budget) more on TV.

Check yourself. With m = 0.30 and long-run margin per dollar of 0.388 for TV and 0.275 for search, at what difference in short-run marginal ROAS do TV and search have equal marginal value under Equation 6? Which way does that move money compared with the short-run optimum?

Answer

Equal marginal value means 0.30\, m_{\text{TV}} + 0.388 = 0.30\, m_{\text{search}} + 0.275, so m_{\text{search}} - m_{\text{TV}} = 0.113 / 0.30 \approx 0.38. At the short-run optimum the two marginal ROAS are equal; now TV’s must be about 0.38 lower than search’s, so money moves to TV until its curve has flattened that much more. The size of the move depends on the 0.388 and 0.275, which a real business would have to measure.

7. The decision

Decision. The rule, stated before the numbers: recommend the mean-optimal plan unless its 10% quantile is below the current plan’s; then recommend the risk-averse plan if its 10% quantile beats the current plan’s, and otherwise keep the current plan. Use the CLV-weighted plan only if the cost per new customer and the CLV by channel are measured (Module 5’s model and an acquisition count per channel). Here they come from the synthetic world’s truth file, not from a model you fitted, so the CLV-weighted plan is shown as the long-run alternative, not recommended.

FULL run, weekly budget $72,000 for 13 weeks:

The 10% quantile of the mean-optimal plan ($113,546) is above the current plan’s ($105,910), so the rule picks the mean-optimal plan: TV $36,510, search $25,608, social $6,086, display $3,796 a week.
Expected incremental sales $129,081 a week (94% HDI $105,344 to $157,483); a gain over the current plan of $7,620 a week (94% HDI $3,739 to $10,848); posterior probability that it beats the current plan 1.00.
The risk-averse plan moves $1,385 a week from search to TV for a 10% quantile $186 higher; the rule keeps the mean-optimal plan because its own 10% quantile already beats the current plan’s. The CLV-weighted plan (TV $42,492, search $19,626) shows how far long-run value could move the split once acquisition costs and CLVs are measured.

The truth confirms the choice: the plan’s true weekly sales from media are $143,062, against $143,069 at the true optimum and $136,901 for the current plan. What would change it: a lift test that narrows search’s response curve (the widest band), measured acquisition costs and CLVs for the CLV-weighted plan, or new bounds.

One more question the allocation does not answer. At the optimum the shared marginal ROAS is about 1.53 on the true curves: the last dollar returns $1.53 of sales, which at a 30% margin is $0.46 of margin. On short-run sales alone, the last dollars of this budget lose money, which matches Module 8’s finding that no channel pays back at a break-even ROAS of 3.33. Adding the truth file’s long-run value does not close the gap: the last dollar returns about 0.30 \times 1.53 + 0.39 \approx \$0.85 of margin on TV and 0.30 \times 1.53 + 0.28 \approx \$0.73 on search. Whether to spend less in total is a decision about the size of the budget, not its split, and with real data it needs measured acquisition costs and CLVs.

Summary

With normalized adstock, a constant weekly spend has a steady-state weekly response equal to the saturation curve in dollars (Equation 1).
Budget decisions need the marginal ROAS (Equation 2), which is below the average ROAS on every concave curve.
At the optimal split of a fixed budget, every channel inside its bounds has the same marginal ROAS (Equation 3, Equation 4). Here social and display sat at their lower bounds and TV and search shared the rest.
SLSQP finds the optimum of this concave problem. Your allocator, PyMC-Marketing’s (whose total_budget is per period) and the exact optimum agreed to within 0.4% of the budget.
Evaluate plans draw by draw. The mean-optimal plan beat the current one with probability 1.00 and had the better bad case too (Equation 5); a risk-averse objective moved it little ($1,385 a week).
A CLV-weighted objective (Equation 6) moves money toward channels that bring valuable customers; how far depends on acquisition costs and CLVs that a real business must measure.

Equations and where you implement them

Equation	What it is	In Lab 10
Equation 1	steady-state response curve	Exercise 1: `response_curve(spend, beta, lam, x_scale, y_scale)`; Checkpoint 1 checks it against the true curves, broadcasting over draws, and zero response at zero spend
Equation 2	marginal ROAS	Exercise 2: `marginal_roas(spend, curve_fn)`; Checkpoint 2 compares with the exact slope and checks it falls with spend
Equation 3, Equation 4	the allocation problem and its optimum	Exercise 3: `allocate(total, bounds, curves)` with SLSQP; Checkpoint 3 checks the budget, the bounds, sales within 0.5% of the true optimum and equal marginal ROAS inside the bounds
The library’s optimizer	budget per period, with carry-in and carry-over	Exercise 4: `pymc_allocation(opt, weekly_total, bounds)`; Checkpoint 4 checks the weekly total and reconciles the plan with yours within 5% of the budget
Equation 5	10% quantile and CVaR	Exercise 5: `allocation_risk(draws, q)`; Checkpoint 5 checks it on the draws 1 to 10 and on the real plans; the risk-averse plan is provided
Equation 6	long-run value of a plan	Exercise 6: `long_run_value(plan, curves, margin, new_customers_per_dollar, clv)`; Checkpoint 6 checks it with the CLVs set to zero and with them, then allocates on it

Gotchas

Average ROAS is not marginal ROAS. Allocating by Module 8’s average ROAS would pour money into search past the point where its next dollar pays.
total_budget is per period in pymc-marketing 1.2.0’s allocate_budget; a quarterly total is spent every week.
Bounds decide a lot. Two of four channels ended at their lower bounds; the bounds, not the model, set their spend. Say so when you present the plan.
SLSQP is local. These curves are concave, so one start suffices; with other curve shapes, start from several points.
Compare plans draw by draw. The level of every plan carries the model’s shared error; the difference between plans carries much less of it.
“Even” is not “safe”. Spreading money across low-return channels lowers the whole distribution, tail included.
A CLV-weighted plan is only as good as its acquisition costs and CLVs. Here they come from the truth file; a real plan needs them measured, with their uncertainty.
Allocation is not budget size. The optimal split of a budget can still be a budget whose last dollars lose money.

Next steps

Allocation decides how much to spend on each channel. It treats every customer the same: a dollar of email or retention spend is worth the same whoever receives it. But an offer that makes one customer buy can be wasted on another who would have bought anyway, and Module 5’s retention break-even was provisional for exactly that reason. Module 11 asks whom to target: it estimates, customer by customer, how much an email offer changes behaviour, evaluates targeting rules on held-out data, and sends the offer only where the expected incremental margin exceeds its cost.