Tag land power

Dupuy’s Verities: Combat Power =/= Firepower

A U.S. 11th Marines 75mm pack howitzer and crew on Guadalcanal, September or October, 1942. The lean condition of the crewmembers indicate that they haven’t been getting enough nutrition during this period. [Wikipedia]

The ninth of Trevor Dupuy’s Timeless Verities of Combat is:

Superior Combat Power Always Wins.

From Understanding War (1987):

Military history demonstrates that whenever an outnumbered force was successful, its combat power was greater than that of the loser. All other things being equal, God has always been on the side of the heaviest battalions and always will be.

In recent years two or three surveys of modern historical experience have led to the finding that relative strength is not a conclusive factor in battle outcome. As we have seen, a superficial analysis of historical combat could support this conclusion. There are a number of examples of battles won by the side with inferior numbers. In many battles, outnumbered attackers were successful.

These examples are not meaningful, however, until the comparison includes the circumstances of the battles and opposing forces. If one take into consideration surprise (when present), relative combat effectiveness of the opponents, terrain features, and the advantage of defensive posture, the result may be different. When all of the circumstances are quantified and applied to the numbers of troops and weapons, the side with the greater combat power on the battlefield is always seen to prevail.

The concept of combat power is foundational to Dupuy’s theory of combat. He did not originate it; the notion that battle encompasses something more than just “physics-based” aspects likely originated with British theorist J.F.C. Fuller during World War I and migrated into U.S. Army thinking via post-war doctrinal revision. Dupuy refined and sharpened the Army’s vague conceptualization of it in the first iterations of his Quantified Judgement Model (QJM) developed in the 1970s.

Dupuy initially defined his idea of combat power in formal terms, as an equation in the QJM:

P = (S x V x CEV)

When:

P = Combat Power
S = Force Strength
V = Environmental and Operational Variable Factors
CEV = Combat Effectiveness Value

Essentially, combat power is the product of:

  • force strength as measured in his models through the Theoretical/Operational Lethality Index (TLI/OLI), a firepower scoring method for comparing the lethality of weapons relative to each other;
  • the intangible environmental and operational variables that affect each circumstance of combat; and
  • the intangible human behavioral (or moral) factors that determine the fighting quality of a combat force.

Dupuy’s theory of combat power and its functional realization in his models have two virtues. First, unlike most existing combat models, it incorporates the effects of those intangible factors unique to each engagement or battle that influence combat outcomes, but are not readily measured in physical terms. As Dupuy argued, combat consists of more than duels between weapons systems. A list of those factors can be found below.

Second, the analytical research in real-world combat data done by him and his colleagues allowed him to begin establishing the specific nature combat processes and their interaction that are only abstracted in other combat theories and models. Those factors and processes for which he had developed a quantification hypothesis are denoted by an asterisk below.

Dupuy’s Verities: The Inefficiency of Combat

The “Mud March” of the Union Army of the Potomac, January 1863.

The twelfth of Trevor Dupuy’s Timeless Verities of Combat is:

Combat activities are always slower, less productive, and less efficient than anticipated.

From Understanding War (1987):

This is the phenomenon that Clausewitz called “friction in war.” Friction is largely due to the disruptive, suppressive, and dispersal effects of firepower upon an aggregation of people. This pace of actual combat operations will be much slower than the progress of field tests and training exercises, even highly realistic ones. Tests and exercises are not truly realistic portrayals of combat, because they lack the element of fear in a lethal environment, present only in real combat. Allowances must be made in planning and execution for the effects of friction, including mistakes, breakdowns, and confusion.

While Clausewitz asserted that the effects of friction on the battlefield could not be measured because they were largely due to chance, Dupuy believed that its influence could, in fact, be gauged and quantified. He identified at least two distinct combat phenomena he thought reflected measurable effects of friction: the differences in casualty rates between large and small sized forces, and diminishing returns from adding extra combat power beyond a certain point in battle. He also believed much more research would be necessary to fully understand and account for this.

Dupuy was skeptical of the accuracy of combat models that failed to account for this interaction between operational and human factors on the battlefield. He was particularly doubtful about approaches that started by calculating the outcomes of combat between individual small-sized units or weapons platforms based on the Lanchester equations or “physics-based” estimates, then used these as inputs for brigade and division-level-battles, the results of which in turn were used as the basis for determining the consequences of theater-level campaigns. He thought that such models, known as “bottom up,” hierarchical, or aggregated concepts (and the prevailing approach to campaign combat modeling in the U.S.), would be incapable of accurately capturing and simulating the effects of friction.

Dupuy’s Verities: The Effects of Firepower in Combat

A German artillery barrage falling on Allied trenches, probably during the Second Battle of Ypres in 1915, during the First World War. [Wikimedia]

The eleventh of Trevor Dupuy’s Timeless Verities of Combat is:

Firepower kills, disrupts, suppresses, and causes dispersion.

From Understanding War (1987):

It is doubtful if any of the people who are today writing on the effect of technology on warfare would consciously disagree with this statement. Yet, many of them tend to ignore the impact of firepower on dispersion, and as a consequence they have come to believe that the more lethal the firepower, the more deaths, disruption, and suppression it will cause. In fact, as weapons have become more lethal intrinsically, their casualty-causing capability has either declined or remained about the same because of greater dispersion of targets. Personnel and tank loss rates of the 1973 Arab-Israeli War, for example, were quite similar to those of intensive battles of World War II and the casualty rates in both of these wars were less than in World War I. (p. 7)

Research and analysis of real-world historical combat data by Dupuy and TDI has identified at least four distinct combat effects of firepower: infliction of casualties (lethality), disruption, suppression, and dispersion. All of them were found to be heavily influenced—if not determined—by moral (human) factors.

Again, I have written extensively on this blog about Dupuy’s theory about the historical relationship between weapon lethality, dispersion on the battlefield, and historical decline in average daily combat casualty rates. TDI President Chris Lawrence has done further work on the subject as well.

TDI Friday Read: Lethality, Dispersion, And Mass On Future Battlefields

Human Factors In Warfare: Dispersion

Human Factors In Warfare: Suppression

There appears to be a fundamental difference in interpretation of the combat effects of firepower between Dupuy’s emphasis on the primacy of human factors and Defense Department models that account only for the “physics-based” casualty-inflicting capabilities of weapons systems. While U.S. Army combat doctrine accounts for the interaction of firepower and human behavior on the battlefield, it has no clear method for assessing or even fully identifying the effects of such factors on combat outcomes.

Toward An American Approach To Proxy Warfare

U.S.-supported Philippine guerilla fighters led the resistance against the Japanese occupation of Luzon during World War II. [Warfare History Network]

U.S. Army Major Amos Fox has recently published the first two of a set of three articles examining nature of proxy warfare in the early 21st century and suggests some ideas for how the U.S. might better conduct it.

In “In Pursuit of a General Theory of Proxy Warfare,” published in February 2019 by the The Institute of Land Warfare at the Association of the U.S. Army, and “Time, Power, and Principal-Agent Problems: Why the U.S. Army is Ill-Suited for Proxy Warfare Hotspots,” published in the March-April 2019 edition of Military Review, Fox argues,

Proxy environments dominate modern war… It is not just a Russian, Iranian or American approach to war, but one in which many nations and polities engage. However, the U.S. Army lacks a paradigm for proxy warfare, which disrupts its ability to understand the environment or develop useful tactics, operations and strategies for those environments.

His examination of the basic elements of proxy warfare leads him to conclude that “it is dominated by a principal actor dynamic, power relationships and the tyranny of time.” From this premise, Fox outlines two basic models of proxy warfare: exploitative and transactional.

The exploitative model…is characterized by a proxy force being completely dependent on its principal for survival… [It] is usually the result of a stronger actor looking for a tool—a proxy force—to pursue an objective. As a result, the proxy is only as useful to the principal as its ability to make progress toward the principal’s ends. Once the principal’s ends have been achieved or the proxy is unable to maintain momentum toward the principal’s ends, then the principal discontinues the relationship or distances itself from the proxy.

The transactional model is…more often like a business deal. An exchange of services and goods that benefits all parties—defeat of a mutual threat, training of the agent’s force, foreign military sales and finance—is at the heart of the transactional model. However, this model is a paradox because the proxy is the powerbroker in the relationship. In many cases, the proxy government is independent but looking for assistance in defeating an adversary; it is not interested in political or military subjugation by the principal. Moreover, the proxy possesses the power in the relationship because its association with the principal is wholly transactional…the clock starts ticking on the duration of the bond as soon as the first combined shot is fired. As a result, as the common goal is gradually achieved, the agent’s interest in the principal recedes at a comparable rate.

With this concept in hand, Fox makes that case that

[T]he U.S. Army is ill-suited for warfare in the proxy environment because it mismanages the fixed time and the finite power it possesses over a proxy force in pursuit of waning mutual interests. Fundamentally, the salient features of proxy environments—available time, power over a proxy force, and mutual interests—are fleeting due to the fact that proxy relationships are transactional in nature; they are marriages of convenience in which a given force works through another in pursuit of provisionally aligned political or military ends… In order to better position itself to succeed in the proxy environment, the U.S. Army must clearly understand the background and components of proxy warfare.

These two articles provide an excellent basis for a wider discussion for thinking about and shaping not just a more coherent U.S. Army doctrine, but a common policy/strategic/operational framework for understanding and successfully operating in the proxy warfare environments that will only loom larger in 21st century international affairs. It will be interesting to see how Fox’s third article rounds out his discussion.

TDI Friday Read: Tank Combat at Kursk

Today’s edition of TDI Friday Read is a roundup of posts by TDI President Christopher Lawrence exploring the details of tank combat between German and Soviet forces at the Battle of Kursk in 1943. The prevailing historical interpretation of Kursk is of the Soviets using their material and manpower superiority to blunt and then overwhelm the German offensive. This view is often buttressed by looking at the  ratio of the numbers of tanks destroyed in combat. Chris takes a deeper look at the data, the differences in the ways “destroyed” tanks were counted and reported, and the differing philosophies between the German and Soviet armies regarding damaged tank recovery and repair. This yields a much more nuanced perspective on the character of tank combat at Kursk that does not necessarily align with the prevailing historical interpretations. Historians often discount detailed observational data on combat as irrelevant or too difficult to collect and interpret. We at TDI believe that with history, the devil is always in the details.

Armor Exchange Ratios at Kursk

Armor Exchange Ratios at Kursk, 5 and 6 July 1943

Soviet Tank Repairs at Kursk (part 1 of 2)

Soviet Tank Repairs at Kursk (part 2 of 2)

German Damaged versus Destroyed Tanks at Kursk

Soviet Damaged versus Destroyed Tanks at Kursk

Comparative Tank Exchange Ratios at Kursk

Has The Army Given Up On Counterinsurgency Research, Again?

Mind-the-Gap

[In light of the U.S. Army’s recent publication of a history of it’s involvement in Iraq from 2003 to 2011, it may be relevant to re-post this piece from from 29 June 2016.]

As Chris Lawrence mentioned yesterday, retired Brigadier General John Hanley’s review of America’s Modern Wars in the current edition of Military Review concluded by pointing out the importance of a solid empirical basis for staff planning support for reliable military decision-making. This notion seems so obvious as to be a truism, but in reality, the U.S. Army has demonstrated no serious interest in remedying the weaknesses or gaps in the base of knowledge underpinning its basic concepts and doctrine.

In 2012, Major James A. Zanella published a monograph for the School of Advanced Military Studies of the U.S. Army Command and General Staff College (graduates of which are known informally as “Jedi Knights”), which examined problems the Army has had with estimating force requirements, particularly in recent stability and counterinsurgency efforts.

Historically, the United States military has had difficulty articulating and justifying force requirements to civilian decision makers. Since at least 1975, governmental officials and civilian analysts have consistently criticized the military for inadequate planning and execution. Most recently, the wars in Afghanistan and Iraq reinvigorated the debate over the proper identification of force requirements…Because Army planners have failed numerous times to provide force estimates acceptable to the President, the question arises, why are the planning methods inadequate and why have they not been improved?[1]

Zanella surveyed the various available Army planning tools and methodologies for determining force requirements, but found them all either inappropriate or only marginally applicable, or unsupported by any real-world data. He concluded

Considering the limitations of Army force planning methods, it is fair to conclude that Army force estimates have failed to persuade civilian decision-makers because the advice is not supported by a consistent valid method for estimating the force requirements… What is clear is that the current methods have utility when dealing with military situations that mirror the conditions represented by each model. In the contemporary military operating environment, the doctrinal models no longer fit.[2]

Zanella did identify the existence of recent, relevant empirical studies on manpower and counterinsurgency. He noted that “the existing doctrine on force requirements does not benefit from recent research” but suggested optimistically that it could provide “the Army with new tools to reinvigorate the discussion of troops-to-task calculations.”[3] Even before Zanella published his monograph, however, the Defense Department began removing any detailed reference or discussion about force requirements in counterinsurgency from Army and Joint doctrinal publications.

As Zanella discussed, there is a body of recent empirical research on manpower and counterinsurgency that contains a variety of valid and useful insights, but as I recently discussed, it does not yet offer definitive conclusions. Much more research and analysis is needed before the conclusions can be counted on as a valid and justifiably reliable basis for life and death decision-making. Yet, the last of these government sponsored studies was completed in 2010. Neither the Army nor any other organization in the U.S. government has funded any follow-on work on this subject and none appears forthcoming. This boom-or-bust pattern is nothing new, but the failure to do anything about it is becoming less and less understandable.

NOTES

[1] Major James A. Zanella, “Combat Power Analysis is Combat Power Density” (Ft. Leavenworth, KS: School of Advanced Military Studies, U.S. Army Command and General Staff College, 2012), pp. 1-2.

[2] Ibid, 50.

[3] Ibid, 47.

U.S. Army Releases New Iraq War History

On Thursday, the U.S. Army released a long-awaited history of its operational combat experience in Iraq from 2003 to 2011. The study, titled The U.S. Army in the Iraq War – Volume 1: Invasion – Insurgency – Civil War, 2003-2006 and The U.S. Army in the Iraq War – Volume 2: Surge and Withdrawal, 2007-2011, was published under the auspices of the U.S. Army War College’s Strategic Studies Institute.

This reflects its unconventional origins. Under normal circumstances, such work would be undertaken by either the U.S. Army Combat Studies Institute (CSI), which is charged with writing quick-turnaround “instant histories,” or the U.S. Army Center of Military History (CMH), which writes more deeply researched “official history,” years or decades after the fact.[1] Instead, these volumes were directly commissioned by then-Chief of the Staff of the Army, General Raymond Odierno, who created an Iraq Study Group in 2013 to research and write them. According to Odierno, his intent was “to capture key lessons, insights, and innovations from our more than 8 years of conflict in that country.[I]t was time to conduct an initial examination of the Army’s experiences in the post-9/11 wars, to determine their implications for our future operations, strategy, doctrine, force structure, and institutions.”

CSI had already started writing contemporary histories of the conflict, publishing On Point: The United States Army in Operation IRAQI FREEDOM (2004) and On Point II: Transition to the New Campaign (2008), which covered the period from 2003 to January 2005. A projected third volume was advertised, but never published.

Although the Iraq Study Group completed its work in June 2016 and the first volume of the history was scheduled for publication that October, its release was delayed due to concerns within the Army historical community regarding the its perspective and controversial conclusions. After external reviewers deemed the study fair and recommended its publication, claims were lodged after its existence was made public last autumn that the Army was suppressing it to avoid embarrassment. Making clear that the study was not an official history publication, current Army Chief of Staff General Mark Milley added his own forward to Odierno’s, and publicly released the two volumes yesterday.

NOTES

[1] For a discussion of the roles and mission of CSI and CMH with regard to history, see W. Shane Story, “Transformation or Troop Strength? Early Accounts of the Invasion of Iraq” Army History, Winter 2006; Richard W. Stewart, “‘Instant’ History and History: A Hierarchy of Needs” Army History, Winter 2006; Jeffrey J. Clarke, “The Care and Feeding of Contemporary History,” Army History, Winter 2006; and Gregory Fontenot, “The U.S. Army and Contemporary Military History,” Army History, Spring 2008.

 

Active Defense, Forward Defense, and A2/AD in Eastern Europe

The current military and anti-access/area denial situation in Eastern Europe. [Map and overlay derived from situation map by Thomas C. Thielen (@noclador) https://twitter.com/noclador/status/1079999716333703168; and Ian Williams, “The Russia – NATO A2AD Environment,” Missile Threat, Center for Strategic and International Studies, published January 3, 2017, last modified November 29, 2018, https://missilethreat.csis.org/russia-nato-a2ad-environment/]

In an article published by West Point’s Modern War Institute last month, “The US Army is Wrong on Future War,” Nathan Jennings, Amos Fox and Adam Taliaferro laid out a detailed argument that current and near-future political, strategic, and operational realities augur against the Army’s current doctrinal conceptualization for Multi-Domain Operations (MDO).

[T]he US Army is mistakenly structuring for offensive clashes of mass and scale reminiscent of 1944 while competitors like Russia and China have adapted to twenty-first-century reality. This new paradigm—which favors fait accompli acquisitions, projection from sovereign sanctuary, and indirect proxy wars—combines incremental military actions with weaponized political, informational, and economic agendas under the protection of nuclear-fires complexes to advance territorial influence…

These factors suggest, cumulatively, that the advantage in military confrontation between great powers has decisively shifted to those that combine strategic offense with tactical defense.

As a consequence, the authors suggested that “the US Army should recognize the evolved character of modern warfare and embrace strategies that establish forward positions of advantage in contested areas like Eastern Europe and the South China Sea. This means reorganizing its current maneuver-centric structure into a fires-dominant force with robust capacity to defend in depth.”

Forward Defense, Active Defense, and AirLand Battle

To illustrate their thinking, Jennings, Fox, and Taliaferro invoked a specific historical example:

This strategic realignment should begin with adopting an approach more reminiscent of the US Army’s Active Defense doctrine of the 1970s than the vaunted AirLand Battle concept of the 1980s. While many distain (sic) Active Defense for running counter to institutional culture, it clearly recognized the primacy of the combined-arms defense in depth with supporting joint fires in the nuclear era. The concept’s elevation of the sciences of terrain and weaponry at scale—rather than today’s cult of the offense—is better suited to the current strategic environment. More importantly, this methodology would enable stated political aims to prevent adversary aggression rather than to invade their home territory.

In the article’s comments, many pushed back against reviving Active Defense thinking, which has apparently become indelibly tarred with the derisive criticism that led to its replacement by AirLand Battle in the 1980s. As the authors gently noted, much of this resistance stemmed from the perceptions of Army critics that Active Defense was passive and defensively-oriented, overly focused on firepower, and suspicions that it derived from operations research analysts reducing warfare and combat to a mathematical “battle calculus.”

While AirLand Battle has been justly lauded for enabling U.S. military success against Iraq in 1990-91 and 2003 (a third-rank, non-nuclear power it should be noted), it always elided the fundamental question of whether conventional deep strikes and operational maneuver into the territory of the Soviet Union’s Eastern European Warsaw Pact allies—and potentially the Soviet Union itself—would have triggered a nuclear response. The criticism of Active Defense similarly overlooked the basic political problem that led to the doctrine in the first place, namely, the need to provide a credible conventional forward defense of West Germany. Keeping the Germans actively integrated into NATO depended upon assurances that a Soviet invasion could be resisted effectively without resorting to nuclear weapons. Indeed, the political cohesion of the NATO alliance itself rested on the contradiction between the credibility of U.S. assurances that it would defend Western Europe with nuclear weapons if necessary and the fears of alliance members that losing a battle for West Germany would make that necessity a reality.

Forward Defense in Eastern Europe

A cursory look at the current military situation in Eastern Europe along with Russia’s increasingly robust anti-access/area denial (A2/AD) capabilities (see map) should clearly illustrate the logic behind a doctrine of forward defense. U.S. and NATO troops based in Western Europe would have to run a gauntlet of well protected long-range fires systems just to get into battle in Ukraine or the Baltics. Attempting operational maneuver at the end of lengthy and exposed logistical supply lines would seem to be dauntingly challenging. The U.S. 2nd U.S. Cavalry ABCT Stryker Brigade Combat Team based in southwest Germany appears very much “lone and lonely.” It should also illustrate the difficulties in attacking the Russian A2/AD complex; an act, which Jennings, Fox, and Taliaferro remind, that would actively court a nuclear response.

In this light, Active Defense—or better—a MDO doctrine of forward defense oriented on “a fires-dominant force with robust capacity to defend in depth,” intended to “enable stated political aims to prevent adversary aggression rather than to invade their home territory,” does not really seem foolishly retrograde after all.

TDI Friday Read: Multi-Domain Battle/Operations Doctrine

With the December 2018 update of the U.S. Army’s Multi-Domain Operations (MDO) concept, this seems like a good time to review the evolution of doctrinal thinking about it. We will start with the event that sparked the Army’s thinking about the subject: the 2014 rocket artillery barrage fired from Russian territory that devastated Ukrainian Army forces near the village of Zelenopillya. From there we will look at the evolution of Army thinking beginning with the initial draft of an operating concept for Multi-Domain Battle (MDB) in 2017. To conclude, we will re-up two articles expressing misgivings over the manner with which these doctrinal concepts are being developed, and the direction they are taking.

The Russian Artillery Strike That Spooked The U.S. Army

Army And Marine Corps Join Forces To Define Multi-Domain Battle Concept

Army/Marine Multi-Domain Battle White Paper Available

What Would An Army Optimized For Multi-Domain Battle Look Like?

Sketching Out Multi-Domain Battle Operational Doctrine

U.S. Army Updates Draft Multi-Domain Battle Operating Concept

U.S. Army Multi-Domain Operations Concept Continues Evolving

U.S. Army Doctrine and Future Warfare

 

Force Ratios in Conventional Combat

American soldiers of the 117th Infantry Regiment, Tennessee National Guard, part of the 30th Infantry Division, move past a destroyed American M5A1 “Stuart” tank on their march to recapture the town of St. Vith during the Battle of the Bulge, January 1945. [Wikipedia]
[This piece was originally posted on 16 May 2017.]

This post is a partial response to questions from one of our readers (Stilzkin). On the subject of force ratios in conventional combat….I know of no detailed discussion on the phenomenon published to date. It was clearly addressed by Clausewitz. For example:

At Leuthen Frederick the Great, with about 30,000 men, defeated 80,000 Austrians; at Rossbach he defeated 50,000 allies with 25,000 men. These however are the only examples of victories over an opponent two or even nearly three times as strong. Charles XII at the battle of Narva is not in the same category. The Russian at that time could hardly be considered as Europeans; moreover, we know too little about the main features of that battle. Bonaparte commanded 120,000 men at Dresden against 220,000—not quite half. At Kolin, Frederick the Great’s 30,000 men could not defeat 50,000 Austrians; similarly, victory eluded Bonaparte at the desperate battle of Leipzig, though with his 160,000 men against 280,000, his opponent was far from being twice as strong.

These examples may show that in modern Europe even the most talented general will find it very difficult to defeat an opponent twice his strength. When we observe that the skill of the greatest commanders may be counterbalanced by a two-to-one ratio in the fighting forces, we cannot doubt that superiority in numbers (it does not have to more than double) will suffice to assure victory, however adverse the other circumstances.

and:

If we thus strip the engagement of all the variables arising from its purpose and circumstance, and disregard the fighting value of the troops involved (which is a given quantity), we are left with the bare concept of the engagement, a shapeless battle in which the only distinguishing factors is the number of troops on either side.

These numbers, therefore, will determine victory. It is, of course, evident from the mass of abstractions I have made to reach this point that superiority of numbers in a given engagement is only one of the factors that determines victory. Superior numbers, far from contributing everything, or even a substantial part, to victory, may actually be contributing very little, depending on the circumstances.

But superiority varies in degree. It can be two to one, or three or four to one, and so on; it can obviously reach the point where it is overwhelming.

In this sense superiority of numbers admittedly is the most important factor in the outcome of an engagement, as long as it is great enough to counterbalance all other contributing circumstance. It thus follows that as many troops as possible should be brought into the engagement at the decisive point.

And, in relation to making a combat model:

Numerical superiority was a material factor. It was chosen from all elements that make up victory because, by using combinations of time and space, it could be fitted into a mathematical system of laws. It was thought that all other factors could be ignored if they were assumed to be equal on both sides and thus cancelled one another out. That might have been acceptable as a temporary device for the study of the characteristics of this single factor; but to make the device permanent, to accept superiority of numbers as the one and only rule, and to reduce the whole secret of the art of war to a formula of numerical superiority at a certain time and a certain place was an oversimplification that would not have stood up for a moment against the realities of life.

Force ratios were discussed in various versions of FM 105-5 Maneuver Control, but as far as I can tell, this was not material analytically developed. It was a set of rules, pulled together by a group of anonymous writers for the sake of being able to adjudicate wargames.

The only detailed quantification of force ratios was provided in Numbers, Predictions and War by Trevor Dupuy. Again, these were modeling constructs, not something that was analytically developed (although there was significant background research done and the model was validated multiple times). He then discusses the subject in his book Understanding War, which I consider the most significant book of the 90+ that he wrote or co-authored.

The only analytically based discussion of force ratios that I am aware of (or at least can think of at this moment) is my discussion in my upcoming book War by Numbers: Understanding Conventional Combat. It is the second chapter of the book: https://dupuyinstitute.dreamhosters.com/2016/02/17/war-by-numbers-iii/

In this book, I assembled the force ratios required to win a battle based upon a large number of cases from World War II division-level combat. For example (page 18 of the manuscript):

I did this for the ETO, for the battles of Kharkov and Kursk (Eastern Front 1943, divided by when the Germans are attacking and when the Soviets are attacking) and for PTO (Manila and Okinawa 1945).

There is more than can be done on this, and we do have the data assembled to do this, but as always, I have not gotten around to it. This is why I am already considering a War by Numbers II, as I am already thinking about all the subjects I did not cover in sufficient depth in my first book.