Shut up and take my money... but use it wisely! On the optimisation of budget allocation (...and other internet memes.) At the end of 2019 and beginning of 2020, many of our clients had planning sessions for the marketing budget of 2020. Unfortunately for everybody, the pandemic disruption threw those plans away and they needed to react swiftly to the new conditions: most likely a budget cut. What is the best way to use those leftovers and make the market aware of my product? I will try my best to use the most internet memes (and other drawings of my own) to give an idea of what we may do to help our clients to answer this question under a lot of assumptions.

1. On the optimisation of media investment…
and other internet memes.
Horacio González Duhart
01/07/2020 1

2. Some context
• We had a plan.
• Suddenly… a pandemic.
• A budget cut is enforced.
• What should we do?
2

3. Marketing Mix Model
• We now call this: TMROI (Total
Marketing Return On
Investment). See the video
here.
• It is an econometric model:
given some inputs what do we
expect as an output
𝑆𝑎𝑙𝑒𝑠 = 𝑓(𝐼𝑛𝑝𝑢𝑡𝑠)
3

4. What are the inputs?
• Clearly, sales of anything may
depend on a lot of things:
price, seasonal pattern, other
economic variables…
• One, an important one, is the
marketing effort.
This is not a meme
4

5. Marketing efforts
• There are different ways to
reach an audience and make
them aware of a product: TV,
out of home displays, radio,
cinema, digital, etc.
• During this pandemic, it is
clear that digital channels
have an advantage over out of
home displays.
5

6. Adstocks
• When someone is exposed to
an ad, that person may recall
the ad for a long period even if
the person is no longer
exposed to the ad.
• The “long” term effect in the
memories of users is what we
call adstocks.
6

7. Threshold
• Sometimes one needs a
considerable amount of
investment in order to use a
channel. We call this
threshold.
• Different channels have
different investment
thresholds
7

8. Saturation
• There is also a chance of
overspending on a specific
channel.
• There’s evidence that media
investment has diminishing
returns.
8

9. Saturation curves
• Taking into account all of these
effects, brings into play the
saturation curves.
• We model the effect of media
investment into sales via an S-
shaped functional form like
the one of the right.
This is not a meme
9

10. 10

11. Additive model
• One way of modelling sales is
to add the media contribution,
given by the saturation curves
of the media channels, to the
contribution of the other
inputs:
𝑆𝑎𝑙𝑒𝑠 = 𝑂𝑡ℎ𝑒𝑟 𝑠𝑡𝑢𝑓𝑓 +
𝑐ℎ ∈ 𝑐ℎ𝑎𝑛𝑛𝑒𝑙𝑠
𝑆𝑐ℎ(𝑆𝑝𝑒𝑛𝑑 𝑖𝑛 𝑐ℎ)
Contribution of media
investment.
Obviously not a meme!
11

12. The optimisation
problem
• What we want it maximise the
contribution of media
investment given a budget
• To simplify notation, let’s
denote the spend on the 𝑘-th
channel by 𝑥 𝑘, the 𝑘-th
saturation curve by 𝑆 𝑘, and
the total marketing budget by
𝐵.
• Let us assume that there are 𝑛
channels. Then,
mathematically, our problem
looks like this:
𝑚𝑎𝑥
𝑘=1
𝑛
𝑆 𝑘(𝑥 𝑘)
𝑠. 𝑡.
𝑘=1
𝑛
𝑥 𝑘 = 𝐵
𝑥 𝑘 ≥ 0
Ok, wait… where are the memes?
12

13. The Lagrange multiplier
• The standard way of dealing with this is to
define a new objective function that includes
the restriction:
𝐿 𝑥, 𝜆 =
𝑘=1
𝑛
𝑆 𝑘(𝑥 𝑘) − 𝜆
𝑘=1
𝑛
𝑥 𝑘 − 𝐵
• Differentiate for x and lambda, equate to 0,
and solve hoping the solution is positive. If it
isn’t, the solution must be in the boundary…
but it is a rather big boundary.
• Moreover, se solution we do find might be a
minimum instead of a maximum, so the
maximum must be in the boundary, but then
again…
𝑚𝑎𝑥
𝑘=1
𝑛
𝑆 𝑘(𝑥 𝑘)
𝑠. 𝑡.
𝑘=1
𝑛
𝑥 𝑘 = 𝐵
𝑥 𝑘 ≥ 0
13

14. What is happening?
• Let’s start by seeing a simple
case: 2 channels, TV and
digital.
• If the budget is 600, what is
best?
1. All in TV.
2. 50/50
3. All in Digital
Option 1:
Around 700
1000
1400
Option 2:
Around 200
Option 3:
Around 500
Option 1:
Around 900
Option 2:
Around 800
Option 3:
Around 1000
Option 1:
Around 1000
Option 2:
Around 1400
Option 3:
Around 1100
14

15. In summary
• From these 9 cases we see
where are the optimal
solutions:
Budget: 600 1000 1400
All TV X
50/50 X
All Digital X
• Imagine we do this exact same
procedure but for more
budgets and for more possible
scenarios (10/90, 20/80, etc.)
Then, we would find this:
15

16. The 3D world
• Here’s how the total contribution
looks in 3D. (It’s an interactive plot,
have a look here).
• The x axis shows the investment on
TV.
• The y axis shows the investment on
Digital.
• The z axis is the contribution of
both channels.
• Note the lines at the bottom. These
lines are the set of possible
allocations giving as a result the
same contribution.
e.g: x = 1000, y=0 and x=900, y=200
16

17. Contour plots
• When we plot all these lines
having the same values of
contribution in a single 2D map,
we call it contour plot.
• The budget restriction in this
map is a line with slope -1.
• The optimisation problem is
basically finding the the contour
farthest from the origin that
touches the budget line.
• The discontinuities in the
optimal path cause difficulties
for classic optimisation
problems
17

18. Multi-channel
• The 2 channels case can be
solved numerically by
evaluating a grid and
observing the maximum.
• Can we do the same thing
with many channels?
Budget: 600 1000 1400
All TV X
50/50 X
All Digital X
Size of grid: N
Number of
channels: n
Steps:
1. Take the a value of the grid for the first channel, then for the second,
so on until the n-1-th channel. The n-th channel is then fixed because
they all add to the budget.
2. Evaluate…. And repeat for all possible values of the grid and channels.
This implies we need to do 𝑁 𝑛−1 evaluations. Exponential time!
18

19. Bundling
• Instead of doing that, consider only
the first two channels. We already
know how to solve this problem.
• Consider those channels as if they
were one: a bundle channel and
then find the optimum allocation of
this bundled channel and the third.
Those are two channels again! We
can solve that.
• Continue doing this until we have all
channels bundled.
• We only need to do 𝑁(𝑛 − 1)
evaluations. Polynomial time!
(actually linear)
19

20. Bundling: Visually
1. Take the optimal path for a set of increasing budgets
Part 1
Part2
2. Consider each budget as spend and find the
contribution
Part 2
3. Bundling is done
Part 3
20

21. Back to reality
• This optimization method is
not only applied in a regular*
TMROI study but also in our
new Agile Budget Allocation
offer.
* There is nothing regular about a regular
TMROI study
21

22. 1. FMCG: Food & Bev, Home
Care, Personal Care
2. Auto
3. Travel & Hospitality
4. Retail
5. Financial Services
6. Tech
7. Telecom
8. Insurance
9. Pharma
CrossMedia
Norms
Industry
Curves
Output:
• Revenue
• ROI
• LT vs. ST Impact
Client MMM Data
Agile Budget Allocation – media allocation for a particular brand / market
Analytics ROI
DB Norms By media channel
Short-term (direct)
Budget Allocation
Simulator
Spend by channel
Simulations
• X% budget cut
• What if going dark
• Evaluate proposed
media mix
• Optimize new reduced
budget
BrandZ Norms
By media
channel
Equity impact
Brand
Contribution
Inputs
• Category, Market
• Market share
• Brand size, Volume/Revenue
• Current - Annual media budget
• Current - Media mix/allocation
• Scenario - Budget cuts
22

23. An example
• The offer is under
development but we have
tested the tool against some
projects.
• It has some areas of
opportunity but the
optimisation algorithm is
ready.
23

24. Thank you!
(Btw… Leo was in this presentation 3 times! All different characters.) 24

25. Questions, comments,
suggestions… memes?
• If you can communicate it with
a meme… just do it.
25