Daily Brief: MECE & Hilbert Spaces

Dissecting the wave function of capital markets.

TLDR; MECE frameworks are orthonormal vector spaces, and capital markets are quantum wave functions. Interesting? Read on.

To understand a business engine, you need to know how to measure its inputs and outputs. Innovation in business and discovery in science always follows good data collection. Without data, we are string theorists.

That's where the MECE principle comes in—Mutually Exclusive (ME), Collectively Exhaustive (CE)—pronounced “mee-see” (like Meeseeks, for those that went to college in the 20-teens). The term is most commonly used in the business consulting world.

Imagine trying to decode a complex system without clear boundaries—categories overlap, gaps emerge, and clarity dissolves into a money laundering scheme. MECE is your structured approach to avoid this chaos, ensuring every piece of data has a home, with no overlaps and no omissions.

Set Theory from my Favorite Bot

I first encountered MECE in a case interview tutorial, where I was instructed to make my consulting framework “as MECE as possible” — a friend of mine caught my live reaction here, in this video.

Exasperated, I asked for my favorite LLM to break it down for me in terms I could understand (Llama3, if you’re wondering):

Let's break down MECE using basic set theory concepts.

Mutually Exclusive (ME)

In set theory, two sets are mutually exclusive if they have no elements in common. In other words, their intersection is the empty set.

• Example: Consider two sets, A and B:

  • A = {1,2,3}

  • B = {4,5,6}

Here, A∩B=∅ (the intersection of A and B is empty), meaning A and B are mutually exclusive.

Collectively Exhaustive (CE)

Sets are collectively exhaustive if their union covers the entire universal set, meaning every element in the universal set is included in at least one of the sets.

• Example: Consider the universal set U = {1,2,3,4,5,6} and two sets A and B as before:

  • A = {1,2,3}

  • B = {4,5,6}

The union of A and B is A∪B = {1,2,3,4,5,6}, which equals the universal set U. This means A and B are collectively exhaustive.

MECE in Set Theory

When sets are both mutually exclusive and collectively exhaustive, they cover all possible elements without any overlap or omission. Together, they form a MECE structure in set theory.

This means every element in the universal set U belongs to exactly one of the sets A or B, with no overlap (mutually exclusive) and no element left out (collectively exhaustive). This is the essence of MECE: a clear, organized division of a whole into distinct, non-overlapping parts that fully cover all possibilities.

That’s pretty good! Bertrand would be proud.

But I promised you physical insight! So physical insight I shall give (at least in abbreviated form).

The Wave Function of Capital Markets

There’s arguably a better way to think about MECE framework in the context of splitting up revenue, cost structures or a customer base — and that’s as a set of orthonormal basis vectors that span some Hilbert space (I know, the quantum fanatics are already foaming at the mouth).

Then, we can treat our market as some Wave Function of Capital Markets that exists out there in the universe, and our job as a business owner is simply to measure its features along a set of basis vectors. Easy enough?

Let’s flesh this out a little in practice.

Suppose I want to sell shoes — there are a bunch of MECE framework I could use to describe my customer base. For example,

• Foot Type,

• Purpose/Usage,

• Buying Motivation,

• Brand Loyalty,

• Sustainability Preference,

• Shoe Size,

• Lifestyle,

• Frequency of Use,

just to name a few. Each one of these will absolutely span (i.e. be collectively exhaustive) — and I can even think of some ME categories within them:

• Foot type = {wide, medium, narrow},

• Lifestyle = {Couch potato, active, Eliud Kipchoge},

• Shoe Size = {smaller than mine, greater than mine, my size¹}.

and so on.

But already you can see that there’s some aspects of enumerating MECE frameworks that we are missing with simple set theory. Many of these categories, while independently MECE, are correlated with other categories.

If you are TOMS, sustainability preference and brand loyalty will have strong correlations — and Lifestyle and Frequency of Use will as well — as will Foot Type and Shoe Size.

But MECE is all about creating non-overlapping sets — did something break? It seems we have a real pickle.

Nothing to see here - Quantum Mechanics and Algebra to the rescue!

We have essentially identified two aspects of set theory alone that miss out on some key features of MECE:

1. Different MECE frameworks are not necessarily independent from each other.

2. Information about some MECE frameworks could tell us about other MECE frameworks that are possibly more optimal for our business model.

This is precisely what the vector algebra used in quantum mechanics (known as Hilbert spaces) does for a living. Let’s define our client demographics for our shoe business as some wave function vector, ∣ψ⟩:

\(|\psi \rangle = |\text{client demographics}\rangle\)

Then we can define a bunch of MECE frameworks by enumerating bunch of MECE framework vectors, that are mutually exclusive (orthogonal) and collectively exhaustive (unit normalized):

\(\langle a_i | a_j \rangle = \delta_{ij}\)

Using this orthonormal basis, we can start extracting information from our client base, ∣ψ⟩:

\( |\psi \rangle = \sum_i |a_i \rangle \langle a_i|\psi \rangle\)

where the value assigned to each bucket is just the inner product with the bucket item, ∣ai⟩. This is just like quantum mechanics!

More importantly, when using Hilbert spaces, we can start exploring how to transform between different MECE frameworks by projecting one basis element onto a new MECE basis element:

\( |a_i \rangle = \sum_j |b_j \rangle \langle b_j|a_i \rangle\)

where ⟨bj​∣ai​⟩ tells us about the correlation between categories in different frameworks — like shoe size vs. shoe width.

Example of Shoe Market Hilbert Space

Let’s give this a try with some numbers and a simple example. Consider a customer base with the following demographic features:

1. shoe size = { less than 40 == 10, 40 to 45 == 15, greater than 45 == 20}

2. shoe width = { wide == 25, medium == 17, narrow == 8}

Now the number of clients with shoes will be the same regardless of which observable feature we measure. Thus, we find:

\(\sum_\text{size} | \langle \psi | \text{size}\rangle |^2 = \sum_\text{width} | \langle \psi | \text{width}\rangle |^2 = 1\)

which is to say both Hilbert spaces are complete (i.e. collectively exhaustive).

However, now we can see that depending on which feature we measure within each MECE framework, we will get different expected values. For example,

\(\langle \psi | \text{wide} | \psi \rangle = 25 \quad \langle \psi | 40 \leq \text{size}\leq 45| \psi \rangle = 15 \quad \langle \psi | \text{size} > 45| \psi \rangle = 20 \quad \)

and so on.

We can also start building unitary transformations that map between the size Hilbert Space and the width Hilbert Space:

\(U \equiv \sum_{\text{size, width}}|\text{size}\rangle \langle \text{size}|\text{width}\rangle \langle \text{width}| = \left(\begin{matrix} 0.71 & 0.7 & 0.07\\ 0.71 & -0.7 & -0.07\\ 0.0 & -0.1 & 0.98 \end{matrix}\right)\)

where

\(U: \mathcal{H}_{\text{width}} \rightarrow \mathcal{H}_{\text{size}} \)

As one can see, there is strong mixing between medium and wide with shoes above size 40, and the narrow shoes are strongly correlated with small feet.

And there you have it… MECE from Hilbert spaces.

In closing, as promised…

MECE frameworks are orthonormal vector spaces, and capital markets are wave functions.

But before you start imagining management consultants wielding quantum state vectors in boardrooms, let’s bring it back to reality.

In practice, MECE doesn’t magically emerge from mathematical elegance. Remember, businesses are human-level machines for ordering capital — so they don’t themselves have the capacity for self-reference.

They must instead be maintained by the physics intrinsic to the boardroom system — namely, professional services, armed with armadas of consultants, accountants, and spreadsheets, painstakingly identify these Hilbert space “buckets” through detailed analysis, workshops, and many cups of coffee.

The structure of this super economy we will discuss tomorrow.

1. European size 42, if you’re wondering. Does this mean my feet hold the answer to life, the universe, and everything?