by John Schuler
Manager of Equipment Programs
Semiconductor Equipment and Materials International


The conventional wisdom has it that the Japanese economic juggernaut has been powered by low interest rates and workers fastidious attention to detail. But, a key element of Japan’s success is also the development and application of Lanchester Strategy.

Based on the equations of combat developed by the early automobile and aeronautical pioneer F. W. Lanchester (1868-1946) and published in his 1916 book Aircraft in Warfare [1], the Lanchester Strategy gives precise values for characterizing market structure and segmenting markets. The strategy also offers valuable insights into the defense and attack of market share positions. Several million books on these aggressive “military based” sales and marketing strategies have been distributed in Japan. Lanchester Strategy has enabled the Japanese to develop a powerful methodology for market penetration of its products in world markets that is unrivaled in the West.

F. W. Lanchester (1868 – 1946 )

The British automobile and aeronautical engineer F. W. Lanchester, is a well known contributor to a number of different fields of knowledge:

Automobile Engineering

Lanchester built the first car in Britain in 1895 and was responsible for many of the key inventions of the automobile industry, such as disc brakes, power steering, fuel injection, four wheel drive, and dynamic balancing of engines. The Lanchester cars from 1901 to 1931 were the epitome of quality, reliability, and advanced technical features [2].

Aeronautical Engineering

In the 1880s Lanchester conducted a number of experiments on gliders and small powered models. Using the methods of dimensional analysis (which he invented), he formulated the theory of flight. Lanchester published two books in 1907 on flight and aerodynamics. The general equations of flight (circulation theory of lift) subsequently became known as the Lanchester-Prandtl equations [3]. In the 1930s, Lanchester also published a number of papers on aircraft engines, stability of flight, counter-rotating propellers, and rocket-assisted flight. For his achievements in aeronautics, Lanchester was awarded the Daniel Guggenheim Gold Medal in 1931.

Operations Research

Lanchester’s extensive writings on military affairs and the conduct of warfare became the foundation of the science of Operations Research. For his work, Lanchester is honored by an annual prize in his name by the Operations Research Society of America [4]. Many of the concepts Lanchester introduced in his 1916 publication Aircraft in Warfare were used during WW II. These concepts were described in Foundations of Operations Research published in 1951 by Morse and Kimball [5].

Military Strategy

Two chapters of Lanchester’s Aircraft in Warfare deal with the power of concentration of military force and the Lanchester linear and squared laws of combat. These chapters form the basis of military strategy and are an important part of the course content at military colleges [6]. Lanchester’s equations of combat are widely used in calculating military force depositions, in particular to ensure a three to one advantage by the attacking force [7].

Marketing Strategy

F. W. Lanchester is well known to Japanese readers because of the many books published in Japan on Lanchester strategy of sales and marketing. These books typically begin with a brief historical perspective of Lanchester and his book Aircraft in Warfare . For example, New Lanchester Strategy Volume 1 by Shinichi Yano [8] has two chapters on this subject. The book titles almost always invoke Lanchester’s name such as New Lanchester Strategy or Lanchester Strategy to beat the Competition . As more of these Japanese books are translated and published in English, the contribution of F. W. Lanchester to sales and marketing strategy will be acknowledged.

Lanchester’s 1916 publication Aircraft in Warfare

A series of articles on the use of aircraft in warfare was written by Lanchester and published in 1914 in the British journal Engineering . These articles, with some additions, were published in 1916 as Aircraft in Warfare – the Dawn of the Fourth Arm published by Constable and Co. England. Lanchester regarded aircraft as a new military force, following infantry, cavalry and artillery, hence the sub-title “the dawn of the fourth arm.” In his book, Lanchester describes the various aspects of the use of aircraft in warfare, tactics and strategy, armament, attack on troops, other aircraft, ground positions, shipping, and submarines. The key ideas introduced by Lanchester in his book are the linear and N-squared laws of combat and the power of concentration of military force. These laws are explained in detail below.

Lanchester’s Linear Law of Combat

In his discussion of early battles, Lanchester wrote: “In olden times, when weapon directly answered weapon, the act of defense was positive and direct, the blow of sword or battle-ax was parried by sword and shield. Under the old conditions, it was not possible by any strategic plan or tactical maneuver to bring other than approximately equal numbers of combatants into the actual firing line; one man would ordinarily end himself opposed by one man. Even were a general to concentrate twice the number of men on any given portion of the field to that of the enemy, the number of men actually wielding their weapons at any given time, was, roughly speaking, the same on both sides.” [Aircraft in Warfare , Chapter V, section 20].

This consideration leads to Lanchester’s linear law of individual combat:

Ao – A = E ( Bo – B )

where: Ao = initial number of army A troops
A = final number of army A troops
E = exchange rate of weapon efficiency (performance of weapons, skill of user)
Bo = initial number of army B troops
B = final number of army B troops

An alternative formulation is:

Combat strength = E ( Number of troops )

The large number of martial arts establishments that have sprung up for teaching self defense is an example of the use of Lanchester’s linear law of individual combat. If you don’t want to be mugged, have your bag stolen etc., then you need to improve your weapon efficiency E. This is accomplished by training, repetitive exercises, and simulation of combat situations.

The Principle of Concentration of Military Force

Weapons used in modern warfare give rise to concentration of fire and effect from several independent sources on one point of the enemy. In addition, each combatant of each side is able to cause damage to any number of the opposing side. As Lanchester puts it: “…the root of all strategy is that of concentration ; the concentration of the main strength of force, whether naval or military, at one point in the field of operations,” [Aircraft in Warfare , Chapter V, section 19]. “With modern long-range weapons, the concentration of superior numbers gives an immediate superiority in the active combatant ranks, and the numerically inferior force finds itself under a far heavier fire, man for man, than it is able to return,” [Aircraft in Warfare , Chapter V, section 20].

Lanchester’s N-Squared Law of Combat

Applying the principle of concentration to the case of modern warfare, the attrition rate of the combatants is proportional to the number of troops of the opposing side. The solution to this first order differential equation leads to the result that fighting strength of a force is proportional to the square of its numerical strength multiplied by the fighting values of its individual units. Thus Lanchester’s N-Squared law may be stated as:

Ao*2 – A *2= E ( Bo*2 – B*2 )

Where the symbols are as defined above.

An alternative formulation is:

Combat Strength = E ( Number of troops squared )

Application of this equation to combat situations leads to two important results. Firstly, a superior force should always choose to fight under the N-Squared law, because the end result will leave the victor with a greater number of survivors than when fighting under the Linear law. Secondly, for an inferior force, there is an opportunity for winning by using a tactical device to split the larger opposing force and annihilate each section of the enemy in sequence. There are many examples in history, myth, and fiction of the success of an inferior force in combat with a seemingly overwhelming opposition. The example described by Lanchester is the battle of Trafalgar in 1805. Admiral Lord Nelson, with 27 ships at his command, defeated an opposing force of 33 ships. Nelson divided his force into two columns, attacking the center and rear of the enemy ships, preventing the advance column from engaging in battle while the rear half of the opposing forces were defeated.

The Deming Connection

During the reconstruction of Japan following the Second World War, quality expert and statistician, W. E. Deming made a number of presentations to the Japan Union of Scientists and Engineers. It appears that Deming also sent a copy of the book by Morse and Kimball Methods of Operations Research to his contacts in Japan. Chapter three titled “Strategical Kinematics” discussed Lanchester’s concept of concentration and the two laws of combat and gave numerous examples of the use of Lanchester strategy in the Second World War. For the Japanese in the 1950s, faced with rebuilding a shattered economy and striving to enter world trade markets, the possibility of applying Lanchester’s concepts to sales and marketing was, no doubt, an interesting concept.

Outline of the Lanchester Strategy of Sales and Marketing

The Japanese adoption of Lanchester’s laws for sales and marketing proceeded on two fronts. On one hand, Lanchester’s laws, idea of concentration and military terminology were absorbed and adapted into the Japanese marketing language. The strategies provide unique insights into competition between companies and rules for selecting a strategy depending upon if the company was attacking a new market or defending an existing market position. For small companies or new product introduction, the choice is one-on-one combat using concentration of effort to secure a market beachhead. Well established companies with existing markets need to battle under N-Square law conditions, where the survival rate is greater. Books such as New Lanchester Strategy, Sales and Marketing Strategy for the Weak Volume 2, by Shinichi Yano [9] have the following “military” chapter headings:

Local Battles
Single Combat
Close Combat
One-Point Concentration
Diversionary Operations

Each chapter illustrates a particular tactic for market share capture by the small company.

The second thrust of the Japanese work on Lanchester sales and marketing strategy was to take the basic Lanchester theory and innovate powerful new concepts of market segmentation, market structure, and market position. While the concepts may be familiar in the West, the analysis gives precise market share values and key ratios that are of great practical value when considering a marketing strategy.

Maximum Target (73.9%). A company with 73.9% or greater occupies the monopoly position.

Equilibrium Target (41.7%). A company with 41.7% or greater will control the market.

Minimum Target (26.1%). If no company has more than 26.1%, the market tends to be unstable.

Applying these market share percentages allows the following five market structures to be distinguished, which provides a valuable insight to a company considering entry into a new market or a new product introduction.

1. Monopoly Market; the leading company has more than 73.9% of the market share

2. Premium Market; 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.

3. Duopoly Market; 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.

4. Oligopoly Market; 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.

5. Polyopoly Market; 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.

A additional key concept developed by the Japanese is that of the “shooting range” distance which allows a company to estimate which rivals can have market share wrested away and which companies are too well established and should be avoided. For the case of one-on-one market share conflict, the rival company needs to be within three times the market share of the attacking company. For the case of multiple market participants, (under Lanchester’s N-Squared law) the critical multiplier is 1.7 times the market share of the company under attack.

Publications in English on Lanchester strategy

Three journal articles articles have been published and two books have been translated from Japanese and published in English.

Campbell and Roberts, writing in the 1986 Strategic Management Journal [10], give an overview of the development in Japan of the Lanchester Strategy. The correlation between return on investment (ROI) and market share is investigated using data from the PIMS company data base.

A second article by Campbell in the 1987 International Studies of Management and Organization [11], compares the PIMS data for North American companies with the data from Yano Research Institute in Japan in terms of market share and ROI.

The third article appeared in Upside Magazine of March 1996 by Dr. Ted Lewis [12] and discusses the market share position of software vendors. Making use of the Japanese approach to market share segmentation, Lewis proposed a model of competition in the software industry in relation to opportunities for mergers and acquisitions.

Two books by Yano have recently been translated and published in California. New Lanchester Strategy Volume 1 [8] and New Lanchester Strategy – Sales and Marketing Strategy for the Weak Volume 2 [9].
It is reported by the publisher that further works by Yano and works by Taoka, Onoda and Takeda are also in translation for publication later in 1996 and in 1997 [13].


This paper serves as an overview of the historical foundation for the strategies in F. W. Lanchester’s work, and in provides insights into the Japanese focus on market share. It is hoped that recognition of Lanchester’s influence on marketing strategy will be recognized and that others will take up the task of applying these theories in the search for competitive advantage.

As Dr. Campbell of the Manchester School of Business succinctly puts it: “Given the emphasis which the Japanese place on this aspect of corporate strategy, it is unfortunate that Lanchester’s followers in the west are largely confined to the narrow worlds of mathematics and operations research” [14].

A question remains, however, why is no mention made of the Japanese development of Lanchester sales and marketing strategy in the many books on Japanese marketing strategies? In particular, why are these strategies not addressed in books on competing with the Japanese?

Perhaps the answer lies in the observation by Ruth Benedict, who, writing in her 1946 book The Chrysanthemum and the Sword ; states: “A Japanese who writes about Japan passes over really crucial things which are as familiar to him and as invisible as the air he breathes” [15].


[1] Lanchester, F. W. (1916) Aircraft in Warfare – the Dawn of the Fourth Arm , Constable and Co., London, United Kingdom.

Lanchester, F. W. (New Edition 1995) Aircraft in Warfare – the Dawn of the Fourth Arm , Lanchester Press ,Forestville, CA, ISBN 1-57321-002-1

[2] Clark, C. S. (1995) The Lanchester Legacy 1895 – 1931 Volume 1, Coventry University, City of Coventry, United Kingdom. ISBN 0-905-949-30-7

[3] Fletcher, J. (1996) The Lanchester Legacy – A Celebration of Genius Volume 3, Coventry University, City of Coventry, United Kingdom. ISBN 0-905-949-47-1

[4] Operations Society of America, Lanchester Prize Committee, 901 Elkridge Landing Road, Suite 400, Linthicum, MD 21090.

[5] Morse, M. M., Kimball, G. E. (1951) Methods of Operations Research , MIT and John Wiley & Sons New York.

[6] American Military University, 9104-P Manassas Park, Virginia 22111, Tel: 703-330-5398, Fax: 703-330-5109,

[7] Taylor, J. G. (1983) Lanchester Models of Warfare , (two volumes) Kentron Inc., Arlington, VA.

[8] Yano, S. (1995) New Lanchester Strategy , Vol. 1, Lanchester Press, Forestville, CA., ISBN 1-57321-000-5.

[9] Yano, S. (1996) New Lanchester Strategy – Sales and Marketing Strategy for the Weak , Vol. 2 Lanchester Press, Forestville, CA., ISBN 1-57321-004-8.

[10] Campbell, N. C. G., Roberts, K. J. (1986) Lanchester Market Structures: a Japanese Approach to the Analysis of Business Competition , Strategic Management Journal, Vol. 7, pp 189-200. John Wiley & Sons.

[11] Campbell, N. C. G. (1987) Market – Share Patterns and Market Leadership in Japan , Int. Studies of Mgt. & Org. , Vol XVIII, No 1, pp 50-66. M. E. Sharpe, Inc.

[12] Lewis, T. (1996) Surviving in the Software Economy , Upside Magazine, March 1996, pp 67-78.

[13] Lanchester Press, P.O. Box 341, Forestville, CA 95436.

[14] Campbell, N. C. G. (1986) A British Military Theory Finds Favor Among Japan’s Businesses . Financial Times London, October 5th 1986.

[15] Benedict, R. (1946) The Chrysanthemum and the Sword, Patterns of Japanese Culture , p7 Charles Tuttle and Co.
Copyright (c) Lanchester Press, 1996. All rights reserved, including the right of reproduction in whole or in part by any means.