The Lanchester Model

Business Warfare

During the 2nd world war in 1916, an engineer by the name of Frederick William Lanchester wrote Aviation in Warfare: The Dawn of the Fourth Arm. Lanchester was very bright and made several contributions to automobiles and aircraft. He turned his attention to the scientific study of aircraft in battle and launched the Lanchester Strategy. The fourth arm referred to the fourth weapon, the airplane. This strategy was revised by Bernard Koopman of Columbia University and used successfully by the Allies during the 2nd World War.

What is especially significant is the path that the model took both in the US and in Japan. In the US, it developed into logistical based Operations Research (OR). In 1952, Dr. W. Edwards Deming introduced OR and the Lanchester Strategy into Japan . In 1962, Nobuo Taoka, founder of the Management Statistics Research Society in Tokyo, restructured the Lanchester Strategy into a model for capturing market share in business operations. Japan has been beating US companies in market share competition since then.

The Model

Lanchester looked at two scenarios, single combat (hand-to-hand) and theater campaigns (armies). The first scenario involves a simple linear relationship between the number of troops and the loss rate. The second scenario involves a square relationship. The second is more appropriate for the business model, so we will look at it more closely.

The basic approach is to write a differential equation that equates the loss rate of troops with certain two factors; (1) the size or number of opposing troops and (2) the effectiveness of the killing power of each troop. Fundamental rules and laws arise out of this theoretical study. Then the rules and laws are applied to business.

However, we cannot use the size of companies as a surrogate for troops. We must use market share. When market share is used for troop strength, then the relevant "effectiveness factor" becomes the marketing mix; product, place, promotion, and price.

Simple Mathematical Derivation

We start by noting that for large forces fighting each other it is a matter of probability whether a troop on one side hits a troop on the other. This is quite different than hand-to-hand combat where there is a certainty that an effective kill weapon will terminate an enemy. Under this assumption, we have the rate of casualties of army x is proportional to the number of troops of army y times the effectiveness of its weapon ky. The same goes for army y's rate of casualties.


For parity, we would have that the fractional loss rates of casualty are equal:


Now by substitution we derive:


This simple expression states that size is more important than the affectivity of weapons. For instance, if you are shooting down four times as many troops as the enemy, your size must still be more than half their number in order to win.

3 to 1 Rule

Of the many relationships that have been developed during the years, none is more arcane than the 3:1 rule. This rule states that in order to vanquish an enemy that has established a defensive position, you must have 3 times as many troops. This means, of course, that Generalismo Santa Anna had more than 3 times as many soldiers as Davy Crocket, Jim Bowie, and the other defenders of the Alamo. It is difficult to find authoritative sources justifying the 3:1 Rule quantitatively in any detail . A good discussion of it can be found at the website:

We will see its use as we turn to the business applications of Lanchester's Strategy.

Marketing Strategy

It would be extremely difficult to apply Lanchester's model to business by taking the employees of a company as "troops." In particular, the effectiveness of each of the numerous roles in a company would be difficult to quantify, let alone quantify them in a differential equation. Furthermore, employees are terminated in stepwise fashion depending on the company's success. Therefore, we use the market itself as a surrogate for the troops. In other words, when a company captures market share, it "kills" the competition proportionately. We use market share, measured in dollars, to represent troops.

The two kinds of combat that I mentioned above suggest strategies that favor the weak or the strong competitors.

Weak Competitors

Since hand-to-hand combat does not give any "extra" weight to numbers of troops, it would behoove companies with lower market share to segment the market in order to compete one-to-one with an entrenched foe. In this case, the effectiveness of the weapons is elevated to the same weight as the number of troops in importance.

But what are the weapons? The weapons are the 4 Ps - product, price, place (distribution), and promotion. In this way, a weaker competitor can isolate factors of the marketing strategy, for instance, product, price, and promotion, and then compete head-to-head on distribution strategy. This could take the form of forming joint ventures or alliances with companies that have a particularly efficient channel. Each market segment will suggest different ways in which to isolate and compete.

Strong Competitors

If it is in the interest of weak competitors to segment, it is in the interest of competitors with larger market share to resist segmentation. This is because of the square law in that it weights numbers of troops as the square of the ratio of the strength of the sides. By keeping the competition between groups covering a large market, it forces the weaker competitors to raise the efficiency of their strategy (weapons) to the square of the ration of the strength (market share) of the two sides.

Market Share Targets

It is important to know when to fight and when to run. B.O. Koopman's paper in 1943 outlined several plateaus that establish a certain market dominance. His equations were further investigated by Dr. Taikobo Onoda in the 1960s, who described five kinds of markets:

  • Monopoly: The leading company has more than 73.9% of the market share
  • Premium: The market leader has more than 41.7% of the market, and has at least 1.7 times the market share of the next largest company.
  • Duopoly: The combined market share for the market leader and second ranking company is greater than 73.9%, and the first company is within 1.7 times the share of the second.
  • Oligopoly: The combined market share of the first three companies is greater than 73.9% and the combined market share of the second and third companies is greater than that of the market leader.
  • Polyopoly: The market leader has less than 26.1% of the market and each company is within 1.7 times the market share of its nearest rival. In this case, the market is unstable with a strong possibility of abrupt shifts in the company rankings.

These combinations of market shares arise from Koopman's and Onoda's equations and the reliance on the 3:1 law. For instance, the figure of merit of 1.7 comes about because it is the square root of 3. If a competitor is entrenched, the rule states that there is a 3 to 1 advantage. Therefore, the effectiveness of weapons becomes the square root of the ratio or 1.7.


The analysis of market share and market targets is paramount in the game of business. Market share provides the capital resources with which to affect replenishments on the field of battle. It is clear that profitability increases linearly with market share in any given industry.

We see in the Lanchester Strategy many of the same strategies that have been presented by Michael Porter in his Competitive Strategy and Competitive Advantage. Lanchester's strategy and the application of it to business by Taoka and Onoda show us an effective quantification of market strategy. Most importantly, it gives us a means to know what battles can be won and what type and amount of resources are necessary to win them.

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Last modified: August 13, 2005