I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favour to men of skill; but time and chance happeneth to them all.

Ecclesiastes 9:11

Theory of aces
A New Year present for the untalented

How achievement depends on skill

During the “Manhattan project” (the making of nuclear bomb), physicist Enrico Fermi asked General Leslie Groves, the head of the project, what is the definition of a “great” general. Groves replied that any general who had won five battles in a row might safely be called great. Fermi then asked how many generals are great. Groves said about three out of every hundred. Fermi conjectured that if the chance of winning one battle is 1/2 then the chance of winning five battles in a row is (1/2)5 = 1/32. “So you are right, General, about three out of every hundred. Mathematical probability, not genius.”

Balm for the untelanted, or unbridled truth?

We can't verify the numbers given by Groves, as we are not familiar with an existing database of generals' victories and losses. Fortunately there is a plenty of data on WWI fighter pilot aces. Similarly to a great general, an ace is a fighter pilot who achieved five or more victories. Can the latter be explained by simple probability, like the former? At first glance this does not appear to be so, as some aces scored way too many victories. For example, the probability to achieve by pure chance Manfred von Richthofen’s 80 victories is (1/2)80 = 10-24 . One is tempted to conclude that high-scoring aces had outstanding skills.

A more careful analysis proves this conclusion wrong. During WWI British Empire Air Forces fully credited their pilots for moral victories, such as forcing the enemy aircraft to land within enemy lines, driving it down “out of control”, or driving it down in damaged condition (See ref. [1], p.6). It is not that unlikely to achieve five moral victories if you can have five moral defeats in between. In addition British Air Force fully credited their pilots for shared victories (Ref. [1], p.8). That is if e.g. three British airplanes shot one German airplane – all three were credited with a victory. The French did not count moral victories, but allowed for shared ones (Ref. [2], p.6). The Americans were either under French or British command and had the corresponding rules applied to them. In contrast, the Germans had ideal scoring system (Ref. [3], p.6-7). They did not count moral victories (see, however, Appendix A). The opponent aircraft had to be either destroyed or forced to lend on German territory and its crew taken prisoners. They did not allow shared victories as well (see, however, Appendix A). We, therefore, will use German war records in our analysis.

Ref. [4] contains the list of all German WWI fighter pilots, with all of their victories and casualties. The total number of credited victories is 6759. The number of casualties, however, is a lot smaller. They amount to 618 KIA (killed in action), 52 WIA/DOW (wounded in action and later died of wounds), 140 POW (prisoner of war), and 431 WIA (wounded in action and survived). According to the official German scoring system, for a pilot to be credited with a victory his opponent should be killed or taken prisoner. Let us compute the number of defeats suffered by the Germans using their own scoring system for victories. Obviously, KIA, WIA/DOW, and POW should be counted as defeats. These add up to 810. This is by a factor of 8.3 less than the number of credited victories. We are not supposed to include WIA in defeats if we wish to follow the German scoring system. However, even if we count all of the WIA as defeats we get 1,241 defeats, which is still by a factor of 5.4 less than the number of credited victories. We don’t know for sure why the number of victories exceeds the number of casualties by such a large factor, but can suggest several possible reasons:

Fortunately, we don’t need to know the exact reason to compare German fighter pilots between themselves. Let us, given the statistics of defeats and victories, compute the probability to get Richthofen’s score. German pilots were credited with 6759 victories (this number probably includes moral and shared victories). Germans also recorded 810 defeats. The total number of engagements was probably not 6759 + 810, but 6759 + 810 + X. Here X is the unknown number of moral defeats. As long as moral defeat does not affect the ability of a pilot to participate in further battles we don’t need to know X. We will call a “recorded engagement” an engagement which resulted in either credited victory or in a defeat. The rate of defeat in recorded engagements is r = 810/(810+6759) = 0.107. The probability of 80 victories in a row is p = (1-r)80 = 1.17*10-4. The probability that at least one of 2894 German fighter pilots will achieve 80 or more victories is 1 - (1-p)2894 = 0.29. Richthofen’s score is thus within the reach of chance.

A more sofisticated analysis [5] of the complete distribution of pilots by the number of victories shows that the assumption that everyone has identical skill can not be correct. We nead to introduce some variation in pilots' skill to explain the data. However, one can show that Manfred von Richthofen, most likely had a skill in the top quarter of the active WWI German fighter pilots, and was no more special than that.

How fame depends on achievement

An easily assessable index to fame is the number of web pages, as found using Google, that mention the person in question. One thus can compare achievement of aces, measured as the number of victories, with their fame, measured as the number of Google hits. A statistical analysis of such data for 392 German WWI aces shows that fame grows exponentially with achievement (see [6]). To be precise, there is a stong correlation, 0.72, between achievement and logarithm of fame. The correlation could have been stronger would not some aces get additional fame for reasons other than their performance in WWI.

Exponential growth of fame with achievement leads to its unfair distribution. With 80 confirmed victories Manfred von Richthofen is the top-scoring ace of the WWI. With 4,720 Google hits he is also the most famous. The total amount of opponent aircraft destroyed by German aces in WWI is 5050. At the same time there are 17,674 Google hits for all of the German aces. This means that Manfred von Richthofen accumulated 4,720/17,674 = 27% of fame, while being personally responsible for shooting down only 80/5050 = 1,6% of opponent aircraft. On the opposite side 60 lowest scoring aces (with 5 victories each) together shot down 300 aircraft, or 300/5050 = 5.9% of all aircraft destroyed. However, together they got only 463 Google hits, or 463/17,674 = 2.6% of fame. And all this happens when the achievement itself is mostly due to chance.

Mikhail Simkin
December 31, 2008

Discuss this article in reverent forum


  1. C. Shores, N. Franks, and R. Guest, Above the trenches: a complete record of the fighter aces and the units of the British Empire air forces 1915-1920 (Grub Street, London, 1990)
  2. N. L.R. Franks and F.W. Bailey, Over the front: a complete record of the fighter aces and the units of the United States and French air services 1914-1918 (Grub Street, London, 1992)
  3. N. L.R. Franks, F.W. Bailey, and R. Guest, Above the Lines: The Aces and Fighter Units of the German Air Service, Naval Air Service and Flanders Marine Corps, 1914 – 1918 (Grub Street, London, 1993)
  4. N. L.R. Franks, F.W. Bailey, and R. Duiven, The Jasta Pilots: detailed listings and histories, Aug 1916 – Nov 1918 (Grub Street, London, 1996)
  5. M.V. Simkin and V.P. Roychowdhury, Theory of aces: high score by skill or luck?, http://arxiv.org/abs/physics/0607109 , Journal of Mathematical Sociology, 32, 129 (2008)
  6. M.V. Simkin and V.P. Roychowdhury, Theory of Aces: Fame by chance or merit?, http://arxiv.org/abs/cond-mat/0310049 , Journal of Mathematical Sociology, 30, 33 (2006)

Appendix A

German scoring system was ideal in theory. In practice, however, military historians “have found a number of ‘victories’ where, say, three Allied aircraft have been claimed and credited when there is absolutely no doubt that only one or two of those Allied planes were lost” (Ref. [3], p.7). This means that in reality some moral or shared victories were counted by the Germans. Not allowing shared victories brought another problem. It happened that there were two or more claims for one destroyed opponent aircraft. Military historians had found that “In some of these cases rank or being a higher scoring ace helped win the decision over a more lowly pilot” (Ref. [3], p.7). Several such cases are documented in Ref. [4]: Vizefeldwebel (Sergeant-Major) Boldt claimed a victory, but it was also claimed by and awarded to Ltn von Schönebeck (Ref. [4], p.108); Vizefeldwebel (Sergeant-Major) Hegeler claimed a victory, but it was also claimed by and awarded to Ltn d R Müller (Ref. [4], p.157).

Appendix B

Unfortunately, it is not enough to paste a name into Google search window and record the number of search results, because many aces have namesakes. For this reason search results were filtered to contain words related to war and aviation. The complete list of used filter words is: flying, pilot, ace, flieger, Jasta, Fokker, and WWI. In many cases thismethod did not remove all of the namesakes andGoogle search results had to be visually inspected to complete the filtering. The data used in this article were collected around May 2003. Today’s numbers of Google hits are different.

Highest scoring German WWI aces

Manfred von Richthofen left (80 victories, 4,720 Google hits). Killed in action in 1918. Ernst Udet center (62 victories, 1,390 Google hits). In 1935 became Colonel General of the Luftwaffe. Shot himself in 1941. Erich Löwenhardt right (54 victories, 73 Google hits). Killed in action in 1918.

Some aces got additional fame for other reasons

Johann Baur left (6 victories, 276 Google hits) mostly known for being Hitler's personal pilot. Spent 10 years in Gulag, where lost a leg due to a gangrene. Died in 1993. Hermann Göring center (22 victories, 1,790 Google hits) is mostly known for being Commander-in-Chief of the Luftwaffe during WWII. Poisoned himself with cyanide to avoid hanging in 1946 in Nürenberg. Wilhelm Frankl right (20 victories, 71 Google hits) is also known for his name being removed from the list of aces by National Socialists after they discovered that he was of Jewish origing. Killed in action in 1917.

Aces in eyeglasses

Some pilots managed to become aces despite wearing eyeglasses. Walter Kypke left (9 victories, 12 Google hits). Died in 1924 in flying accident. Otto Kissenberth center (20 Victories, 63 Google hits). Died in 1919 in a climbing accident in Alps. Kurt Wintgens right (19 victories, 54 Google hits). Killed in action in 1916.

Sometimes fame is even less deserved

During the discussion of this research on The Aerodrome forum von Lemming wrote:

This one seems to be more along the lines of what you are after:

From Google:

Searched the web for "mikhail simkin". Results 1 - 10 of about 352. Search took 0.19 seconds

Mikhail Simkin replied:

352 Google hits for "Mikhail Simkin". According to The Theory of Aces to achieve such level of fame an average WWI ace had to down over forty opponent aircraft.

What can I come up with to justify such number of Google hits in my case?

A private pilot license; 203.5 hrs total flight time; Cessna 152/172, Piper Cherokee.

Not a single combat mission. Shame. Nothing, but shame.

Some press about this research

Theory of aces is the sixths essay in the Ecclesiastes 9:11 series. To be posted of our future releases subscribe to our newsletter.