Attrition Models at Sea

Share this article:

The Lanchester Square Law of 1914 is, by design, limited in scope. The author did not concern himself with victory or defeat, advance or retreat, or control over a given area of land; attrition rate (casualties inflicted and suffered over time by the contending sides) is its only output, and any deeper insights are left to the practitioner’s best judgement.

Some have questioned its validity for exactly this reason, arguing that combat is much more complex than a mere exchange of fire and an ensuing butcher’s bill. And it is a valid point; but as James G. Taylor reminds us, we should not be too ready to blame a model for our incorrect implementation of it, or for not validating satisfactorily in a scenario it was never designed to describe. In its original form, Lanchester’s Square Law is an attrition model and nothing else.

This could help explain why the first proposals of a similar set of equations were originally offered by Navy men – J.V. Chase, Bradley A. Fiske, and Ambroise Baudry. At sea, there is a predominance of attrition over manoeuvre: as Wayne P. Hughes points out in his classic reference work Fleet Tactics and Coastal Combat, “Forces at sea are not broken by encirclement; they are broken by destruction”. An attrition model may never take us beyond a mere casualty count in a land battle, but in the realm of naval action it might be just good enough.

There are other factors that make the sea an ideal testing environment for the practitioner of Operational Research. Hughes continues:

“The potential to effect this concentration [of force] is greater at sea than on land. At sea there is no high ground, no river barrier, no concealment in forests that requires what is often used as a rule of thumb on land, a 3:1 preponderance of force to attack a prepared position […]. Sun, wind, and sea state all affect naval tactics, but not to the extent that terrain affects ground combat.”

He concludes that the main objective of naval tactics throughout history has been the successful attack, and that is achieved by concentration of firepower – Lanchester would have agreed. Although today we still refer to the Square Law as Lanchester’s creation and by his name, it can be argued that the models created by Chase, Fiske and Baudry not only predate it, but are also more fortunate in their application from their conception, being better adjusted to the reality of war at sea.


As presented in 1902, Lieutenant J.V. Chase’s own approach to the Square Law made the following assumptions, very similar to what Lanchester would postulate a decade later:

  • When averaged out, shell fire from a fleet can be considered a continuous stream, rather than a series of individual projectiles.
  • The two opposing fleets (let us call them Blue and Red) participating in the action are homogeneous: all ships in the Blue fleet are identical, and so are all ships in the Red fleet.
  • The fighting power of a given ship is constant: it is unaffected by range, target aspect, spotting effectiveness, or morale.
  • All ships in the fleet are able to fire at all ships in the opposing fleet, and no shots are wasted on a target that is already out of action.

The variables that describe the encounter are:

  • The fleets engaged each have a number of ships, which we will refer to as A and B respectively.
  • Ships have a “fighting power”, which we define as the number of accurate shots each of them can fire in a given unit of time. All ships in a fleet are assumed to have the same fighting power. We refer to the fighting power of Blue’s ships as α, and that of Red’s ships as β.

Finally, and as the main deviation from Lanchester’s formulation,

  • Ships are also defined by their “staying power”: the number of shots they can withstand before being taken out of action. Again, all ships in a fleet are expected to be identical in this regard. We write the staying power of Blue’s ships as a, and that of Red’s as b.

It is worth mentioning that ships are, in this model, not necessarily taken out of action one by one: one intact ship is equal in every way to two ships that are halfway out of action. For this, we will refer to the results of Chase’s equations as the “equivalent” number of surviving ships, which represents the remaining combined combat strength of the fleet rather than an exact number of vessels.

The mathematical formulation, as presented by Wayne P. Hughes in The value of warship attributes in missile combat (1992) would look like this:

\frac{dB(t)}{dt} = \frac{\alpha A(t)}{b}

\frac{dA(t)}{dt} = \frac{\beta B(t)}{a}

Where A and B are the number of ships in the Blue and Red fleets, α and β their fighting power, and a and b their staying power.

The state equations for a given time, obtained by integration:

\alpha a[A(0)^{2} - A(t)^{2}] = \beta b[B(0)^{2} - B(t)^{2}]

A “square law” much in the spirit of Lanchester’s, and an identical attrition process being described. The only new factor, from a mathematical perspective, is the notion of “staying power”.

This difference might seem small at first (a mere denominator) but it is invaluable in practice: the model lets the practitioner explore the relative values of offence and defence, which in this formulation affect the end result equally, all else being the same. Chase himself, in a 1921 retrospective on his work, reflected:

“Having certain definite quantities of the various materials the question of ship design [is], …in the simplest form: “Shall we construct from these materials one ship or two ships?”…if we decide to build one ship instead of two, this single ship must be twice as strong offensively and twice as strong defensively as one of the two ships.”

History has taught us that fighting ships are susceptible to catastrophic damage; a torpedo strike, a bomb exploding below decks, or even a really unfortunate shell hit can take a well-armoured vessel out of action. This should make it abundantly clear that a ship twice the displacement of another is not necessarily twice as strong defensively. A design trend towards smaller vessels would seem mathematically reasonable.

As for the value of the concentration of firepower, let us run the experiment Chase proposed in the same retrospective we mentioned earlier.


Say we have two opposing fleets, both identical in every aspect.

Each has eight ships of comparable power; for the sake of our example, let us say all of them can fire on average 0.2 accurate shots per time increment, and all of them can take, say, twelve shots before they are disabled.

Of course, two such fleets (with identical combat strength and staying power) would eventually annihilate each other if they attacked at the same time and neither retreated.

Instead, let fleet Blue manoeuvre in such a way that one ship of fleet Red is masked and unable to fire. The contest is now temporarily eight ships versus seven – an apparently small imbalance. We apply the equations and plot the strength of both fleets as a function of time:

Phase 1 plot

Predictably, the larger fleet wins this first phase of the engagement. We might find it more surprising that it does so by a comfortable margin, with the equivalent of 3.87 ships still in the fight – almost one half of the fleet’s original strength. These could now engage the lone remaining hostile ship, as shown below:

Phase 2 plot

The last ship of fleet Red is eliminated with minimum damage to Blue – now at the equivalent of 3.74 battle-ready ships.

And so, a minor alteration in the initial conditions of the action grants fleet Blue a decisive victory, preserving almost half of its original force.


Chase’s model is a testament to the importance of concentration: the side engaging the fewest enemies while bringing the most guns to bear can expect to win, and manoeuvre should aim to secure this advantage. This much should be evident from our earlier example.

It also makes a compelling argument for the desirability of smaller ships. With offensive power and staying power having equal weight, the latter simply cannot be relied upon; the effects of enemy fire are difficult to quantify, and the possibility of catastrophic damage poses a great risk to any fleet concentrating its resources.

However, and perhaps more importantly, it shows how the shortcomings of attrition models are less conspicuous in the setting of naval warfare than they are on land; context matters.


The Python implementations of the models used in this article for plotting simulated combat results can be found in the author’s GitHub page.

Further reading

  1. Wayne P. Hughes – Fleet Tactics and Coastal Combat
  2. Wayne P. Hughes –  The Value of Warship Attributes in Missile Combat
  3. James G. Taylor – Lanchester-type Models of Warfare, Vol. I
Share this article:

Lanchester’s Laws of Combat

Share this article:
Early combat models

Just over a hundred years ago now, US Navy lieutenant (later rear admiral) J.V. Chase developed a differential equation to model engagements between two homogeneous fleets. About a decade later, British engineer Frederick W. Lanchester arrived independently to an almost identical equation, with examples covering air and land combat as well. A Russian contemporary to both, M. Osipov, also reached similar conclusions in a paper published in the tsarist journal “Military Digest” (Voennyi Sbornik) in 1915.

Of the three, history has been kinder to Lanchester: Chase’s work was not declassified until 1972, too late for him to enjoy the public recognition he rightly deserved; and Osipov, although sometimes mentioned alongside his British counterpart, is still a relative stranger to us.

In practical applications, the simple combat models described by Lanchester and his contemporaries are mostly surpassed today – partly because of the more complex and dynamic nature of the modern battlefield, but mostly due to the power of calculation brought by the computer age, which makes much more sophisticated simulations possible.

And yet, Lanchester’s Laws, as we know them today, never quite seem to lose relevance. Many combat models introduced in the last few decades are directly evolved from them (COMAN, Bonder-Farrell), or use them at the local level to describe smaller engagements within a broader framework.

Lanchester’s Linear Law

Lanchester began by postulating a simple yet reasonable condition of ancient combat (close formations of men using swords or spears) in what he called the “Linear Law”: a combatant armed with a melee weapon can only engage one target at a time, within arm’s reach. What this means is that the casualties that a force could sustain or cause at any given instant have a practical upper limit, proportional to the number of its combatants that are in direct physical contact with the enemy. At most, this upper limit would be the number of active units in the smaller force, but further limitations could be imposed by terrain, or formation frontage. We will now examine an example of the latter:

In the situation above, and given equal fighting skill for both sides, the Blue army and the Red army would cause the same exact number of casualties (five) per unit of time and would go on like that until the smaller army (in this case, Red) were wiped out.

From the Blue army, 12 units survive the encounter.

Now, let us imagine the Red army is technically superior to the Blue army: due to a greater fighting skill, or better equipment, or a combination of both, any Red soldier can eliminate two enemies in the time it takes a Blue soldier to eliminate just one. In that case, the plot of the forces of both sides as a function of time would look like this:

Red wins this time around, with eight survivors.

We can make the following observations from these two engagements:

  • The attrition rate of both armies is constant throughout the battle (except by the end, when fewer soldiers remained than would fit the original frontage)
  • A larger army does not confer a greater killing power – only a greater staying power, as the larger army can soak up more casualties before being eliminated.
  • An army that is outnumbered two to one can compensate by being exactly twice as lethal. In other words, lethality and numbers have the same weight in deciding the outcome of a battle.
  • Frontage does not really alter the outcome of the battle, only its duration.

According to Lanchester, exactly the same rules could be applied to a modern scenario of unaimed fire, in which both sides fired at an enemy they could not see, and merely swept a constant target area: in such a case, the larger side would have greater firepower, but this would be compensated by a lower density of enemy targets; the outnumbered side would have fewer guns, but their shots would find their mark more often. The result would be a constant attrition rate for both armies, exactly as in the melee combat examples above.

Lanchester’s Square Law

In modern combat using long range weapons and aimed fire, Lanchester continued, the restriction established by the Linear Law is lifted: now, any combatant can engage any target in range, and in turn receive fire from multiple enemies. Thus, attrition rate is no longer limited by frontage – every element can participate in every stage of the battle.

In these conditions, if an army had 20 fighting elements, and each were expected to eliminate one enemy unit on average every two “time steps” of a given encounter (whatever we actually mean by a “time step”, be it a minute or a day, depends on the scope of our simulation) then the army would be expected to eliminate 10 enemies on average per time step. In a general case, an army with a number of elements A, all of which participate in the encounter in one time step, will always cause a number of casualties to an army B of A times α, where α is the expected number of kills per element per unit of time on average. This translates into the following equations:

\frac{dB}{dt} = -\alpha A

\frac{dA}{dt} = -\beta B

Where A and B are the number of soldiers in each army, and α and ß their average expected number of kills per soldier per unit of time. Today, we refer to α and ß as Lanchester attrition-rate coefficients.

Let us apply these equations to our prior example of a Blue army numbering 42 soldiers, and a Red army with a strength of 30. As before, we make both armies exactly as lethal – both will be expected, for this example, to neutralise one enemy every five time steps (1/5 = 0.2 kills per time step). Substituting these values in the previous equations, we obtain:

\frac{dA}{dt} = -0.2 \times 42

\frac{dB}{dt} = -0.2 \times 30

Which, plotted as a function of time, gives the following result:

In our Linear Law example, with the same numbers, the Blue army won with only 12 surviving units. This time around, applying the Square Law, things are considerably more lopsided, as 29 elements of Blue survive unscathed.

Intuitively we can appreciate how, in the conditions established by Lanchester’s Square Law, superiority in numbers has a greater effect in the outcome of an encounter than the lethality of each individual unit – every casualty inflicted not only reduces the number of enemies that can fire back, but also increases the concentration of fire on each target in the next time pulse. A tacit agreement with Nathan Bedford Forrest’s military maxim “get there first with the most men”.

Analytically, we may divide the equations for the casualty rates, obtaining:

\frac{dA}{dB} = \frac{\beta B}{\alpha A}

Which we can rewrite as:

\alpha A \times dA = \beta B \times dB

to relate the instantaneous losses of both armies, and then integrate to find that:

\alpha(A_0^2 - A^2) = \beta(B_0^2 - B^2)

Which shows that the fighting strength of an army is proportional to the square of its size – hence “Square Law”.

Validation of Lanchester’s Square Law:

As a combat model, Lanchester’s Square Law has some considerable limitations, among others:

  • It assumes perfect intelligence, as both sides know the exact location of the enemy at all times.
  • It assumes perfect fire control, as both sides can concentrate fire in the most efficient way.
  • It assumes the conditions of the battlefield are constant, as the efficiency coefficients α and ß are unaffected by changes in terrain, manoeuvres, morale, supply, etc.
  • It is a deterministic model, as the same input will always yield the same result.
  • It is a homogeneous model, in that all elements of a combat unit are assumed to have the same fighting characteristics.

Many authors have proposed revisions and expansions of Lanchester’s work to correct some of these shortcomings, or to account for a variety of factors such as reinforcement and withdrawals, operational losses, command and control, guerrilla warfare and insurgency, the suppressive effects of weapons, close air support, etc. A detailed list of these developments up to the late 1970s can be found in James G. Taylor’s Lanchester-Type Models of Warfare, vol. I, 1980.

This said, we must look at the Square Law not as a combat model, but as an attrition model. In other words, we cannot expect it to paint a detailed view of the evolving battlefield, but rather to describe the mathematical process by which casualties are dealt and sustained, once all other factors are known.

And in that regard, it does surprisingly well. Although other studies exist that illustrate the matter clearly, J.H. Engel’s 1954 paper on the battle of Iwo Jima shows a remarkable fit between theoretical vs. actual attrition rates using Lanchester’s Square Law:


On the subject of Lanchester models of warfare, Professor James G. Taylor said in 1980:

We should, perhaps, be more amazed that such simple models yield intuitively appealing results than be critical because of the factors omitted from them. As is usually the case with simple analytical models, they may be too abstract to solve any specific real operational problem. They can, however, illustrate a general principle such as concentration, clearly delineate modelling issues, warn about potential difficulties, and serve as a basis for communication among analysts

Whether Lanchester models continue being practical is a different debate entirely, but one cannot deny that they continue being relevant, both in the field of Operational Research and outside it. The civilian study of military history, in particular, strikes us as a doctrine with no need to solve specific operational problems, but which could do with some insights into the dynamics of combat – a historian would not do far too wrong revisiting some past conflicts from an analytical perspective.


The Python implementations of the models used in this article for plotting simulated combat results can be found in the author’s GitHub page.

Further reading

  1. Lanchester Equations and Scoring Systems (RAND)
  2. Ronald L. Johnson – Lanchester’s Square Law in Theory and Practice
  3. James G. Taylor – Lanchester-type Models of Warfare, Vol. I


Share this article: