These Companies Are Using The Math of War Against Massive Incumbents... And They're Winning

How a century-old war theory explains modern small-cap disruption

Polymath Investor. (2025). “Sopwith Camels Patrol the Coastline in Perfect Formation, 1918”. It was in this era of aerial combat that Frederick W. Lanchester, an English engineer, first published his Aircraft in Warfare: The Dawn of the Fourth Arm (1916), introducing the mathematical laws of attrition that would later be known as Lanchester’s Laws.

I’ve always been interested in World War I, so when something related to that period comes along, I pay attention. Last week, I stumbled upon Lanchester’s Laws — a mathematical framework for understanding attrition that still reverberates in the boardrooms of Corporate America. But who was Lanchester, and how do his laws apply to business strategy? That’s the topic of today’s article. Let’s dive in.

An Overlooked Genius at the Dawn of a New Era

In the mud-soaked trenches of World War I and in the skies over the Western Front, military commanders faced a new and terrifying form of industrial warfare.

The outcomes of battles, that were once decided by cavalry charges and tactical formations, now seemed lost in the chaotic noise of machine guns and artillery.

Yet, an English engineer named Frederick Lanchester¹ saw past the chaos to find an underlying mathematical order.

He proposed that the brutal arithmetic of attrition could be understood mathematically by breaking the complexity of combat into distinct, manageable scenarios.

Lanchester’s Laws would then describe two different worlds.

His Linear Law modelled ancient combat, where one soldier could only fight one other at a time. Here, attrition was a simple one-for-one exchange; a force twice as large was twice as strong.

But was that enough to describe what he was seeing on the battlefield? It was then that he came up with the Square Law.

His Square Law, described modern warfare, where multiple units could concentrate their fire on a single target. In this world, the fighting power of an army was proportional to the square of its numbers. A force twice as large was now four times as strong.

Lanchester's genius was in "deconstructing the problem" and recognizing that different battlefield conditions (terrain, weapons systems) required different mathematical models.

So how can these insights be applied in the world of business strategy?

Lanchester, F. W. (n.d.). Mathematics in warfare. In Encyclopaedia Britannica (pp. 2138–2139).

How Lanchester’s Laws Became Business Strategy

Just as trench commanders learned from Lanchester’s Laws that concentration of fire mattered, business strategists realized the same logic applied in markets: win first in narrow segments, then expand. Building on this insight, business thinkers over the years derived practical rules and strategies that came to be known as the Lanchester Patterns:

The laws explain why that narrow focus works, while the patterns show how companies put it into practice. These patterns were gradually developed and incorporated into business strategy over the following decades as you can see below:

As it happens, a handful of small-cap public companies in today’s markets are using these very strategies to outmaneuver massive incumbents. For investors, they could be especially compelling research candidates.

Below, I unpack the strategies of a few hand-picked challengers to see how they’re putting Lanchester’s playbook into action.

Let’s begin.

The Playbook in Action: Small Caps Waging Asymmetrical War

Each of the challengers I selected below concentrate its firepower on a narrow ‘beachhead’² segment and demonstrates at least two classic Lanchester patterns—whether through force concentration, local superiority, divide-the-enemy tactics, firepower scaling, a guerrilla-to-positional shift, or counter-positioning.