The U.S. Army Three-to-One Rule

Various Three-to-one rules of thumbs have existed in the U.S. Army and in writings possibly as early as the American Civil War (1861-1865). These are fine as “rules of thumb” as long as one does not take them seriously and understands what they really mean. But, unfortunately, we have now seen something that is a loose rule of thumb turned into a codified and quantified rule. This is annoyingly overstating its importance and as given in U.S. Army manuals, is patently false.

The U.S. Army has apparently codified the “three-to-one rule” in its documentation and has given it a value. In the 2014 edition of FM 6-0, paragraph 9-103, it states that “For example, historically, defenders have over a 50 percent probability of defeating an attacking force approximately three times their equivalent strength.” This statement, on the surface, simply is incorrect. For example, the following table from my book War by Numbers is drawn from a series of 116 division-level engagements in France in 1944 against the Germans (see War by Numbers, page 10) They show the following relationship between force ratio and outcome:

European Theater of Operations (ETO) Data, 1944


Force Ratio………………..Result…………………Percent Failure…Number of cases

0.55 to 1.01-to-1.00………Attack Fails…………………..100%……………….5

1.15 to 1.88-to-1.00………Attack usually succeeds……21%………………..48

1.95 to 2.56-to-1.00………Attack usually succeeds……10%………………..21

2.71-to-1.00 and higher…Attacker Advances…………….0%……………….. 42


Now these engagements are from fighting between the U.S., UK and Germany in France and Germany in 1944. These are engagements between forces of roughly equal competence. As can be seen, based upon 42 division-level engagements, in all cases of attacks at three-to-one (more specifically 2.71-to-1 and greater), the attacker advanced. Meaning in all cases of attacks at three-to-one, the attacker won. This directly contradicts the statement in FM 6-0, and contradicts it based upon historical data.

This is supplemented by the following two tables on the next page of War by Numbers. The first table shows the German performance when attacking Soviet units in 1943.

Germans attacking Soviets (Battles of Kharkov and Kursk), 1943


Force Ratio………………..Result………………….Percent Failure…Number of cases

0.63 to 1.06-to-1.00………Attack usually succeeds……..20%……………………..5

1.18 to 1.87-to-1.00………Attack usually succeeds……….6%……………………17

1.91-to-1.00 and higher…Attacker Advances……………….0%……………………21


The next table shows the Soviet performance when attacking German units in 1943:

Soviets attacking Germans (Battles of Kharkov and Kursk), 1943


Force Ratio………………Result…………………..Percent Failure…Number of cases

0.40 to 1.05-to-1…………Attack usually fails…………70%……………………10

1.20 to 1.65-to-1.00…….Attack often fails…………….50%……………………11

1.91 to 2.89-to-1.00…….Attack sometimes fails…….44%……………………..9


These charts are from the fighting around Kharkov in February, March and August of 1943 and the fighting during the Battle of Kursk in July 1943. It is 73 engagements between the German and Soviet armies.

Now, there is a clear performance difference between the German and the Soviet armies at this time. This is discussed in considerable depth in War by Numbers and will not be addressed here. But, what it amounts to is that the German Army has an advantage in the casualty exchange and that advantage also shows up in the outcomes of the battles, as show above. If they attacked at two-to-one odds are greater, they would win. The Soviets attacking at the same odds would win only 56 percent of the time. Clearly, at the division-level, in a unit to unit comparison, the Germans were two or three times better than their Soviet opponents.

Still, even in the worse case, which is the Soviets attacking the Germans, we do not get to the claim made in FM 6-0, which is the defender won 50% of the time when attacked at three-to-one. In fact, the Soviets managed to win 50% of the time when attacking at 1.20 to 1.65-to-1. Something is clearly wrong with the statement in FM 6-0.

Now, at the time I wrote War by Numbers, I was not aware of this sentence planted in FM 6-0 and so therefore did not feel a need to respond to the “three-to-one rule.” It is a rule of thumb, not completely without value, that had been discussed before (see Dupuy, Understanding War, pages 31-37). I thought this issue was properly understood in the U.S. analytical and defense community, therefore I did not feel a need to address it further. It turns out that I do. So, I will take a moment to tap into our databases and properly address this using all the resources at my disposal. This will be in subsequent blog posts.

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Christopher A. Lawrence
Christopher A. Lawrence

Christopher A. Lawrence is a professional historian and military analyst. He is the Executive Director and President of The Dupuy Institute, an organization dedicated to scholarly research and objective analysis of historical data related to armed conflict and the resolution of armed conflict. The Dupuy Institute provides independent, historically-based analyses of lessons learned from modern military experience.
Mr. Lawrence was the program manager for the Ardennes Campaign Simulation Data Base, the Kursk Data Base, the Modern Insurgency Spread Sheets and for a number of other smaller combat data bases. He has participated in casualty estimation studies (including estimates for Bosnia and Iraq) and studies of air campaign modeling, enemy prisoner of war capture rates, medium weight armor, urban warfare, situational awareness, counterinsurgency and other subjects for the U.S. Army, the Defense Department, the Joint Staff and the U.S. Air Force. He has also directed a number of studies related to the military impact of banning antipersonnel mines for the Joint Staff, Los Alamos National Laboratories and the Vietnam Veterans of American Foundation.
His published works include papers and monographs for the Congressional Office of Technology Assessment and the Vietnam Veterans of American Foundation, in addition to over 40 articles written for limited-distribution newsletters and over 60 analytical reports prepared for the Defense Department. He is the author of Kursk: The Battle of Prokhorovka (Aberdeen Books, Sheridan, CO., 2015), America’s Modern Wars: Understanding Iraq, Afghanistan and Vietnam (Casemate Publishers, Philadelphia & Oxford, 2015), War by Numbers: Understanding Conventional Combat (Potomac Books, Lincoln, NE., 2017) , The Battle of Prokhorovka (Stackpole Books, Guilford, CT., 2019), The Battle for Kyiv (Frontline Books, Yorkshire, UK, 2023), Aces at Kursk (Air World, Yorkshire, UK, 2024), Hunting Falcon: The Story of WWI German Ace Hans-Joachim Buddecke (Air World, Yorkshire, UK, 2024) and The Siege of Mariupol (Frontline Books, Yorkshire, UK, 2024).
Mr. Lawrence lives in northern Virginia, near Washington, D.C., with his wife and son.

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  1. Lawrence has an excellent assessment and demonstrates the pitfalls of bad habits using numbers in wartime analyzes in a way unintended, to produce consequences that could in the future prove fatal. A fine analysis an discussion, with well planned follow ups. Look forward to future blogs and where the author takes the reader.

  2. >”This is discussed in considerable depth in War by Numbers and will not be addressed here”<

    I just read through the Kursk chapter in War by numbers. I did not find it particularly convincing, to put it mildly. A few points regarding the overall analysis:

    1.) A misinterpretation of the factors that defined the disparity in casualty infliction potential between both belligerents, the firepower levels, inventory combat ratings and or economic factors/outlays involved (especially per capita investment). This would have a severe impact on the evaluations of all post war/cold war engagements.
    There was a pronounced difference in the development levels between the Soviet Union and Germany. Of significance here are patent registrations, per capita GDPs, HDIs, QoM and R&D indices. The US/UK and Germany were compartively developed, nations such as the Soviet Union or Japan would rank lower at that time.

    2.) An increase in barrels does not necessarily result in a proportional increase of firepower which would translate into greater casualty infliction. Additionally, the material density of all combatants was similar, while the numerical factor is completely ignored, i.e. meeting the demands of a larger army, which involves the distribution of equipment over more troops (though this also depends on doctrines/organization).
    This can be illustrated by inspecting their focus, for instance, the Soviet Army being a tank heavy Army.
    We can take the example of AFVs, with their shock effect and supporting elements (since it is such a popular topic and rarely observed from a different perspective, which also continues to plague historiography). The Soviet Army made more tanks because:
    a.) It had a greater focus on tanks.
    b.) It was larger.
    c.) they suffered greater losses.
    and hence => c.) allocated a greater share from their resource base to tank production.
    The weight/per capita ratio however does not differ greatly (here, by combat weight, for the Kursk operation), resulting in a difference of ~1.54 tons per soldier invested in favour of the Axis, or 85% of German levels, expressed in volume.
    “Ad valorem” the difference increases even further (~2.31 net worth, for 1943 figures). This is a considerable investment differential per soldier, which presses casualties down.
    Also, the avg. consumption of ground munitions of larger caliber guns > 75mm (annual, 1943 figures) was over 15 tons per Division and about 4 tons for a Soviet Division (a normalization does not change the overall picture).
    From the 4th to 14th July the Wehrmacht consumed 49,662 t of munitions (110 trainloads), 4,966 daily (7,691 peak), or on average 200 tons for a Division (of the 37 Divisions involved) during the opening phase of Zitadelle, for a total of 232,621 t expended during July.
    This does not factor in the quality of the respective material, or communications/coordination devices/sound ranging equipment etc., though the preferred choice on the EF were probably telephone lines.
    The Soviets had to compensate via sacrifice. Everything else would be a misunderstanding of the nature of WW2 Eastern Front battles.

    3.) No differentiation between concept and sophistication (e.g. the field artillery journal gives interesting insights into the cutting edge of modernity of the 20s and 30s, the adoption of 105 and 150mm concepts, or see the “Westervelt board”, a review of the interwar years
    A good summary is also found here:

    4.) The tactical performance, CEVs to SEVs, i.e. a correlation between investment and training. I only know of few examples in modern warfare which would involve such lopsides rates in a protracted conflict.
    As Dr. Zetterling already asserted, there was not much which would indicate that the Red Army managed to bridge the qualitative gap and whatever advantage they gained during this period, was offset by the learning effects experienced by their opponent’s armed forces (it goes both ways). In a prolonged war of attrition however, the quality of personnel will suffer, personnel will be replaced or shifted, substituted by different classes, often less educated or even illiterate (also moved to the war industry, which will result in a deterioration, e.g. midwar, the Soviets returned to the production of simpler projectile variants , due to an ever increasing absence of skilled labour). Notable is also the increase in accidents experienced during flight training in the air forces.
    In fact, the tactical performance (expressed in odds, relative to the ratio of Ng/Ns) seemed to have been lower, indicating that the RKKA of 1941 must have been qualitatively superior to the standing Army of 1943/44 (with a slight upswing after the stabilization post 1942), unless the forces in the later phases were progressively worse armed. A theory easily refutable, considering that nations will undergo a natural arms race.
    Hence, I fundamentally disagree with most of the conclusions formulated at the end of this chapter and the implication that the Soviets were technologically superior to the United States, the United Kingdom and Germany is just absurd.

    • ….the implication that the Soviets were technologically superior to the United States, the United Kingdom and Germany is just absurd.

      Really? Where did I imply that? You may have inferred something that I did not mean to imply.

      • ETO/MTO:
        “The technology and quality of all weapons were similar among all three armies”.

        “although one could certainly make the argument that the Soviets were the technologically superior force”

        Average military expenditure per soldier for the major combatants in 1943 (in 1905 British Sterling), compare with casualty infliction potential:

        US 591
        UK 422
        USSR 169
        Germany 525
        Japan 197

        In the case of the US Army the additional investment did not yield any increase in effectiveness, we can expect diminishing returns dictated by capabilities, asymptotically. The UK figure appears too low, this is divided by their entire forces, but since only a condensed force was deployed to the continent it would be closer to US values and also to per capita GDPs (Germany 88% of UK, US 98% of UK).

        • This also answers the question whether a 55,000$ salary is justified or if this individual shows any advantage over an officer that is paid 50,000$. No, not necessarily.
          Superior to personnel with a 20,000$ payment? Hell yeah.

          I hope this puts a dent into the mythical Wehrmacht.

          • Hello Stiltzkin! To obey strange axishistory forum’s requirement of posting to get an opportunity to contact you, I ask here: do you have an access to Askey’s volume IIB of Operation Barbarossa? I’m really interested in ‘SDE calculation’ part.

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