The Mathematics of Leadership: Team Size and Catastrophe Theory

Why do so many start-ups that have achieved product-market fit still fail soon afterwards? And why do so many scale-ups break during growth? The mathematics of Catastrophe Theory provides the answers to these questions.

So, you decided to start a business.

After some good early progress on your own, you managed to secure a little funding, which you used to hire some great people. Those people built a product and it started to get exciting market traction. So, you hired some more people to make the product even better, and to market it more widely. Things continued to go well and you went around this loop a few more times, hiring more people to further expand and accelerate outcomes. Everything was going just great, and you even found time to enter the local CEO Of The Year awards.

But now, suddenly and apparently without warning, there's a collapse in productivity and morale. Infighting and frustration are on the rise, customer satisfaction is falling and user growth is stuttering (and, to add insult to injury, you didn't even win that CEO award). What just happened?

Catastrophe Theory is the answer; where a previously continuous output suddenly undergoes an abrupt and discontinuous change in response to only a small change in the inputs. In this case, the output is the rate of desired business outcomes achieved. What about the input?

Although there are many inputs to a business, it's the change in the number of employees in your business that most often triggers the effects predicted by Catastrophe Theory. There are several points during a company's lifetime where adding more employees causes a sudden collapse in productivity, which is, of course, the opposite of what we intended when we added more people. (By the way, note that Chaos Theory, which you have probably also heard about, is a different concept from Catastrophe Theory and one which we'll cover in the context of business, in a future article).

So prevalent are the effects of Catastrophe Theory in organizations that it's arguably the most common reason that businesses fail, after misunderstandings about what product-market fit means.

We previously discussed the value of applying a mathematical mindset to the never-ending task of organizational planning and development. Let's see if we can bring such a mindset to the present problem - of businesses experiencing sudden, major difficulties as employee numbers rise - and let's see if, by doing that, we can gain some insight into how best to navigate such difficulties, and get back to productive growth.

We'll start by visualizing what we expect to happen as we add employees to the business. The reason we add them, of course, is to increase the rate of desired outcomes, and perhaps also to increase the quality of those outcomes. You can think about outcomes here as whatever is appropriate to the context of your own business. For example, product updates released, customers added, markets opened, contracts closed, etc., are all desirable outcomes, which we'd like to increase as a result of adding people to our teams.

Ideally, we'd experience the dotted line below; adding more people would disproportionately increase outcomes. But, even if we can't have that, we'd at least expect to experience the solid line - outcomes rising in line with company size.

But that almost never happens.

Instead, most businesses will experience the following phenomenon: other things being equal, a company typically grows pretty smoothly up until about 40-50 people. Then, as we continue to add employees, outcomes suddenly level-off, so that we experience effectively no marginal benefit from the marginal increase in team size. With the right response, it's possible to return to growth again, as depicted by the solid line in the figure below.

But the majority of businesses that hit this plateau never progress beyond it. They may well keep hiring, but doing so in isolation from other remedies only compounds the original problem that led to the plateau, such that the organization will eventually take one of the trajectories illustrated by the dotted lines, and head into failure.

For those businesses that do progress past this first inflection point, they are destined to meet others in future. In a moment, we'll examine the causes of these recurring inflection points. But, for now, it's useful to note that, while the proximate causes of the inflection points are different in each case, they nevertheless all share the same underlying, systemic cause.

Predicting the date of your next Productivity Catastrophe

The pattern illustrated above continues beyond those inflection points shown, reoccurring at approximately 400 and 800 employees, and so on. In fact, so regular is the pattern, that we can express it as a simple formula:

where N is the set of numbers of employees at which the inflection points will occur, and i = 1, 2, 3... is the sequence of inflection points.

For example, the formula predicts that your first inflection point (i = 1) will occur at approximately 50 employees, the next (i = 2) at 100, then 200, and so on.

Of course, this is a somewhat simplified model and would be more correctly expressed as a probability density function, with a concentration around each of the points 50, 100, 200, etc., recognizing that inflection points occur within a region centred upon those numbers rather than occurring exactly at those numbers. For example, this diagram illustrates the probability of an inflection point occurring, centred on the case where n = 50.

I also haven't enhanced this equation for very large values of i. But these points don't greatly alter the principle and pattern, so we'll ignore them for now, in the interests of clarity. The key takeaways, therefore, are that inflection points will occur regularly with employee growth and that the point at which your business will experience its next inflection point is relatively predictable within a certain range.

to clear an inflection point, you have to be prepared to change almost everything about how your organization works

The Systemic Cause of Organizational Inflection Points

So, what causes these inflection points? To answer this question, it's important to look beyond the proximate causes. Otherwise, we thrash around, treating symptoms, without getting out of the productivity plateau in which we have found ourselves. For example, when your organization hits an inflection point, you may experience any number of symptoms, such as more bugs, a slow-down in bug fixing rates, less roadmap progress, falling user growth, higher employee attrition, falling staff-survey scores, more infighting amongst teams, and so on. Attempting to address each of these issues symptomatically won't get the business out of the inflection point.

No, to clear an inflection point, you must be prepared to change almost everything about how your organization fundamentally works. That's because the inflection point was caused by almost everything to do with how your organization currently works.

Specifically, your org consists of three major components, the intersection of which determines the organization's performance. What happens during an inflection point is that at least one - and usually all - of these components scales-out, such that the intersection of all three components no longer works at the current team size. Structures and processes that used to work very well now, suddenly, work against the organization.

There are some important properties associated with this effect.

• Productivity hits a tipping point, where it suddenly drops drastically, rather than experiencing a gradual decline. Therefore, it's common to hear people say that the issues arrived "without warning".
• When one or more of these components scale-out, the effect is not neutral to the organization but is, instead, actively harmful. In other words, productivity at this point often actually decreases in absolute terms, not just per-capita, even though there are more people in the business overall.
• We tend to prolong this effect by "doubling-down" on what used to work, just at the point where we should be changing everything. It's hard to accept that what used to work very well is now, all of a sudden, working against us. It is the refusal to accept this reality that causes most businesses that enter inflection points never to leave them.

Let's illustrate with a typical example. Imagine that you have a small product development team of 20 people, with most roadmap and architecture decisions being made by the product manager and two engineering team leaders, together and consensually. They know and trust each other, and so this works really efficiently, and roadmap progress is fast. Consensus-based decision making often works very well on a small scale. Further, at this scale, the founder-CEO knows precisely where to go to input her guidance on product direction and to be briefed on progress.

Time passes, and the team grows. As it does, we occasionally add another team lead to manage the greater team size. So, for example, when we have 30 engineers, we now have four team leaders and two product managers. It's a little more difficult to make decisions and it's harder for the founder-CEO to interact with the larger group of leaders, but it still works well enough. And so we go on, slowly adding, incrementally, to the existing leadership model.

Then one day we find ourselves with eight team leaders and three product managers, to manage fifty engineers. Consensual decision-making scales-out very quickly beyond a certain point. Now, it's suddenly very difficult to achieve consensus - we have 11 people in the meeting - so decision latency increases significantly. Further, because everyone is in charge, in reality, no-one is in charge - no-one feels a sense of true ownership responsibility. So nobody wants to take the difficult or risky decisions, and decision quality drops accordingly.

This, in turn, hits the productivity of all of the engineers, who depend on this leadership team. Technical debt rises, market-alignment falters and frustration levels rise. The founder-CEO, now having no single point of interaction, begins to struggle to understand status, or starts getting sucked into making detailed product engineering decisions that she's too far away from to make well. But she still has to make them because she's the only person who is really in charge...

And this isn't even the whole issue, because the same problem is almost certainly breaking out, simultaneously, right across the company, both inside and between departments, and for the same reasons.

In this example, we didn't change out our ownership-structures or processes, nor did we invest in the leadership capabilities of our people until what used to work well suddenly started working against our goals. It's not hard to imagine - or perhaps recall - many other examples similar to this one, in cause and effect.

It's straightforward-enough to conclude that fixing the problems described in this example will require changes to how we structure and operate the business. But, how do we best decide what those changes should be? What are we optimizing for? And what about other inflection points, including those that occur at much larger levels of scale than the one illustrated here? How do we diagnose and respond to those? And is it possible to create and early-warning system that an inflection point is beginning, before it does too much damage?

In the next article, we'll tackle these questions, and develop a general model for navigating organizational inflection points, and living with Catastrophe Theory.

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The Mathematics of Great Leadership, Part I: Markov Organizations

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