Theory of aces: high score by skill or luck?

Transcription

1 Theory of aces: high score by skill or luck? M.V. Simki ad V.P. Roychowdhury Departmet of Electrical Egieerig, Uiversity of Califoria, Los Ageles, CA We studied the distributio of WWI fighter pilots by the umber of victories they were credited with alog with casualty reports. Usig the maximum etropy method we obtaied the uderlyig distributio of pilots by their skill. We fid that the variace of this skill distributio is ot very large, ad that the top aces achieved their victory scores mostly by luck. For example, the ace of aces, Mafred vo Richthofe, most likely had a skill i the top quarter of the active WWI Germa fighter pilots, ad was o more special tha that. Whe combied with our recet study [], showig that fame grows expoetially with victory scores, these results (derived from real data) show that both outstadig achievemet records ad resultig fame are mostly due to chace. Durig the Mahatta project (the makig of uclear bomb), physicist Erico Fermi asked Geeral Leslie Groves, the head of the project, what is the defiitio of a great geeral []. Groves replied that ay geeral who had wo five battles i a row might safely be called great. Fermi the asked how may geerals are great. Groves said about three out of every hudred. Fermi cojectured that if the chace of wiig oe battle is /2 the the chace of wiig five battles i a row is 2 5 = 32. So you are right, Geeral, about three out of every hudred. Mathematical probability, ot geius. Similarly to a great geeral, a ace is a fighter pilot who achieved five or more victories. Ca the latter be explaied by simple probability, like the former? At first glace this does ot appear to be so, as some aces scored way too may victories. For example, the probability to achieve by pure 8 24 chace Mafred vo Richthofe s 8 victories is 2. Oe is tempted to coclude that highscorig aces had outstadig skills. A more careful aalysis proves this coclusio wrog. Durig WWI British Empire Air Forces fully credited their pilots for moral victories (Ref. [2], p.6). It is ot that ulikely to achieve five moral victories if you ca have five moral defeats i betwee. I additio British Air Force fully credited their pilots for shared victories (Ref. [2], p.8). That is if e.g. three British airplaes shot oe Germa airplae all three were credited with a victory. The Frech did ot cout moral victories, but allowed for shared oes (Ref. [3], p.6). The Americas were either uder Frech or British commad ad had the correspodig rules applied to them. I cotrast, the Germas had ideal scorig system (Ref. [4], p.6-7). They did ot cout moral victories. The oppoet aircraft had to be either destroyed or forced to led o Germa territory ad its crew take prisoers. They did ot allow shared victories as well 2. This was i theory. I practice, however, military historias have foud a umber of For example, forcig the eemy aircraft to lad withi eemy lies, drivig it dow out of cotrol, or drivig it dow i damaged coditio (Ref. [2], p.6). 2 This brought aother problem. It happeed that there were two or more claims for oe destroyed oppoet aircraft. Military historias had foud that I some of these cases rak or beig a higher scorig ace helped wi the decisio over a more lowly pilot (Ref. [4], p.7). Several such cases are documeted i Ref. [5]: Vizefeldwebel (Sergeat-Major) Boldt claimed a victory, but it was also claimed by ad awarded to Lt vo Schöebeck (Ref. [5], p.8); Vizefeldwebel (Sergeat-Major) Hegeler claimed a victory, but it was also claimed by ad awarded to Lt d R Müller (Ref. [5], p.57). This pheomeo, if widespread, ca aloe geerate aces through the cumulative advatage

2 victories where, say, three Allied aircraft have bee claimed ad credited whe there is absolutely o doubt that oly oe or two of those Allied plaes were lost (Ref. [4], p.7). This meas that i reality some moral or shared victories were couted by the Germas. Ref. [5] cotais the list of all Germa WWI fighter pilots, with all of their victories ad casualties. The total umber of credited victories is The umber of casualties, however, is a lot smaller 4. They amout to 68 KIA (killed i actio), 52 WIA/DOW (wouded i actio ad later died of wouds), 4 POW (prisoer of war), ad 43 WIA (wouded i actio ad survived). Accordig to the official Germa scorig system, for a pilot to be credited with a victory his oppoet should be killed or take prisoer. Let us compute the umber of defeats suffered by the Germas usig their ow scorig system for victories. Obviously, KIA, WIA/DOW, ad POW should be couted as defeats. These add up to 8. This is by a factor of 8.3 less tha the umber of credited victories. We are ot supposed to iclude WIA i defeats if we wish to follow the Germa scorig system. However, eve if we cout all of the WIA as defeats we get,24 defeats, which is still by a factor of 5.4 less tha the umber of credited victories. We do t kow for sure why the umber of victories exceeds the umber of casualties by such a large factor, but ca suggest several possible reasos: Moral ad shared victories. Aces flew fighter-plaes, while their oppoets ofte were less well armed aircraft. Germa Air Force fought mostly defesive war behid their frot lies [4]. So, if a Germa aircraft was shot dow, it could lad o their territory. I cotrast, whe Allied aircraft was shot dow, it had to lad o the eemy territory ad its pilot was take prisoer. The Germas were better. Fortuately, we do t eed to kow the exact reaso to compare Germa fighter pilots betwee themselves. Let us, give the statistics of defeats ad victories, compute the probability to get Richthofe s score. Germa pilots were credited with 6759 victories (this umber probably icludes moral ad shared victories). Germas also recorded 8 defeats. The total umber of egagemets was probably ot , but X. Here X is the ukow umber of moral defeats. As log as moral defeat does ot affect the ability of a pilot to participate i further battles we do t eed to kow X. We will call a recorded egagemet a egagemet which resulted i either credited 8 victory or i a defeat. The rate of defeat i recorded egagemets is r =. 7. The probability of 8 victories i a row is ( r ).7. The probability that at least oe of Germa fighter pilots will achieve 8 or more victories is (.7 ). 29. Richthofe s score is thus withi the reach of chace. We ca also compute the probability distributio of the victory scores, assumig that everyoe fights util he gets killed. The probability to wi fights ad lose the ext is: 2894 mechaism. However, we have o evidece that this practice was widespread, ad will igore its effect i this study. 3 This umber is the sum of 55 victories credited to aces ad 79 victories credited to o-ace pilots. The first umber is accurate (i the sese that o ew error was itroduced i this study), as aces victory scores are available i electroic format (for example o this website: The secod umber is a result of the had-cout usig the listig i Ref. [5], so some error was most likely itroduced. 4 The casualties, which are listed o pages of Ref. [5], were maually couted. 2

3 P ( ) ( r) r = () Figure shows the result of Eq.() (with r =. 7 ) compared with the actual distributio of the victory scores (which are give i Table ). While the agreemet is ot perfect, it is clear that chace ca accout for most of the variace i the umbers of victories. Apart from ot leadig to a quatitative agreemet with the data, the above simple aalysis assumes that fighter pilots always fight util they get killed. I reality may of them did ot eve look for a fight. There were over eight hudred Germa fighter-pilots who did ot score a sigle victory ad also were ever wouded or shot dow. Also may pilots with just few victories survived the war. I may of such cases they joied the Air Force shortly before the ed of the war. umber of victories probability.... Figure. The distributio of Germa WWI fighter-pilots by the umber of victories, computed usig the data i Table, is show by rhombs. The lie is the predicted distributio, computed usig Eq.() with r =. 7. 3

4 Table. Distributio of pilots by umber of victories. umber of credited victories total umber of pilots defeated i the ext fight umber of credited victories total umber of pilots udefeated udefeated defeated i the ext fight cotiued A better way to address the problem is to study the probability of defeat as a fuctio of the umber of previous victories. Table 2 shows the statistics of casualties (KIA + WIA/DOW +POW) as a fuctio of the umber of previous victories. For example, 434 pilots were defeated before they achieved a sigle victory. At the same time 32 pilots achieved oe or more victory. This makes the 434 rate of defeat i the first fight Similarly 3 people were defeated after they achieved 4 victories, while 392 pilots achieved 5 or more victories (ad became aces). This makes the 3 rate of defeat i the fifth fight The rate of defeat computed this way usig the data of Table 2 is show i Fig.2. The rate drops strogly for the first few fights, but appears ot to chage after about te fights. The reductio i rate of defeat ca be explaied assumig that pilots have differet iate rate of defeat, which depeds o their skill. The uskilled oes get killed at a higher rate, ad, as we progress to higher umber of fights, the average skill icreases. Variace i iate skill is oe possible explaatio. Aother explaatio is that the pilots simply get more experieced. A hard-core idealist ca attribute all the decrease of the defeat rate with the icrease of the fight umber to learig. I reality both factors play role ad their relative cotributios are impossible to determie. I the followig aalysis we assume that a defeat rate is iate to a pilot ad does ot chage with the umber of fights he participates i. 4

5 Table 2. Numbers of defeated ad wiig pilots as fuctios of the fight umber. fight umber of umber of fight umber of umber of umber defeats victories umber defeats victories cotiued

6 defeat rate fight umber Figure 2. The actual defeat rate as a fuctio of fight umber, determied usig the data of Table 2, is show by rhombs. The lie represets the same rate, theoretically computed usig the distributio of the rate of defeat i the pool of pilots give i Fig. 3. The error bars are of the size of oe stadard deviatio, which correspods to 68% cofidece iterval. If the distributio of the iate defeat rates is p ( r) the the average rate of defeat i first fight is r = r p ( r )dr. The probability distributio of defeat rate of pilots survivig the first fight is p ( r) = ( ( r ) p( r ) dr. The rate of defeat i the secod fight is r r p ( r) 2 = dr = I geeral, the probability distributio of defeat rates of pilots, survivig fights, is r ( dr ( dr. p ( r) = ( ( r ) p( r ) dr, (2) ad the rate of defeat i th fight is: 6

7 r r ( ( dr = = dr ( dr ( dr. (3) Obviously, r, give by Eq.(3), mootoically decreases with. Whe the miimum defeat rate i the distributio p ( r) is greater tha zero, r approaches this rate at some value of ad the decreases o further. Fig. 2 suggests that this miimum defeat rate is aroud 3%. Oe ca use Eq.(3) to compute the defeat rates for trial distributios, p ( r), ad search for the distributio which best fits the data of Fig. 2. However, a better way to fid p ( r) is to use the method of maximum likelihood [6]. Let us cosider a udefeated pilot with victories. We assume that he was selected at radom from a pool where defeat rates are distributed accordig to p ( r). The probability that a radomly selected pilot, who participates i battles, wis all of them is: ( ) = drp( r)( r) P. Similarly, the probability that a radomly selected pilot wis battles ad loses the ext is: P d = ( ) drp( r)( r) r = P( ) P( + ). Now the probability to get the whole set of data i Table is give by the likelihood fuctio: f = ( ) N ( ( )) ( ) d P P ( ) where N ( ) ad N d ( ) fid the distributio ( r) d N ( ) are the umbers of udefeated ad defeated pilots with victories. We should p which maximize this fuctio. I computatios it is more coveiet to work with the logarithm of the likelihood fuctio: = [ + ( )] d d ( f ) N( ) l( P( ) ) N ( ) l P ( ) l. (4) The distributio, p ( r), obtaied by maximizig f (see the Appedix) is show i Fig. 3 by rhombs. It looks irregular. The maximum likelihood estimatio we just performed assumed that all possible distributios, p ( r), are, a priori, equally probable. The Maximum Etropy Priciple [7] provides a more reasoable way of assigig a priori probabilities to distributios. As we are iferrig a probability distributio the the relevat etropy is the iformatio (Shao) etropy [8]: = ( r) l( p( r) ) s drp (5) A priori probability of a give probability distributio, p ( r), is e s [7]. The combied probability of realizig a particular distributio ad that this distributio produces the observed data is ~ e s f. 7

8 This is the quatity which should be maximized, or, alteratively, its logarithm, l ( f ) + s, which is more coveiet. The result of this maximizatio is show i Fig. 3 by a lie. The defeat rate as a fuctio of fight umber, computed usig this distributio ad Eq.(3) is show i Fig.2 by a lie maximum lilelihood maximum likelihood + maximum etropy probability deafeat rate Figure 3. Distributio of the rate of defeat computed usig maximum likelihood method (rhombs). Solid lie is the distributio computed by combiig the maximum likelihood ad the maximum etropy method. Now we ca use p ( r) to do Bayesia iferece [6] for itrisic defeat rate of ay give pilot (icludig those who were ever defeated 5 ). We will use p ( r) as a prior distributio of defeat rate ad will make a estimate of pilot s defeat rate based o this prior distributio ad actual umber of fights he wo ad lost. For example, if we do t kow how may fights a pilot had wo, the all we ca say is that the probability distributio of his defeat rate is p ( r). If we kow that he wo fights, the the probability distributio of his defeat rate is the probability distributio of defeat rates of pilots, who wo fights, which is give by Eq.(2). The same iferece for defeat rate for pilots, who wo fights ad were defeated i the ext, ca be obtaied similarly: p d ( r) = r r ( ( r ) p( r ) dr (6) 5 Similar approach was previously used to estimate the true dropped calls rates whe o dropped calls happeed durig the test [9]. 8

9 The iferece for five represetative pilots, computed usig Eqs. (2) ad (6) is give i Fig. 4. I particular, Mafred vo Richthofe most likely had the itrisic defeat rate of 2.5%. Accordig to the distributio of itrisic defeat rates show i Fig. 3 about 27% of pilots have the defeat rate of 2.5% or lower. This meas that MvR is most likely merely i top 27% accordig to his skill. Note that we completely eglected the effects of learig i our aalysis. It is clear that at least part of the variatio i perceived ability is due to the fact that pilots are gettig more experieced as they participate i more ad more fights. Thus takig ito accout the effects of learig will make the variace i iate ability oly less. Cosequetly, our estimate of the uiqueess of MvR is a upper boud Mafred vo Richtofe Erst Udet Max Immelma Werer Juk Kurt Wissema probability.3.2. % % 2% 3% 4% 5% 6% 7% 8% 9% % defeat rate Figure 4. Bayesia iferece for the defeat rate of five aces, computed usig the distributio, show i Fig. 3, ad Eqs.(2) ad (6). Udefeated aces (umber of victories i brackets): Erst Udet (62), Werer Juck (5). Defeated aces: Mafred vo Richthofe (8), Max Immelma (5), Kurt Wissema (5). I our previous paper o the theory of aces [] we foud a strog correlatio betwee the logarithm of fame (measured i umbers of Google hits) ad the achievemet (umber of victories), suggestig that fame grows expoetially with achievemet. I other words fame gives icreasig retur o achievemet, but still is determied by this achievemet. This would be acceptable if 9

10 achievemet was proportioal to skill. However, ow we have show that the differece i the umber of victories is mostly due to chace. This meas that the fame i the ed is due to chace. There were a couple of papers ([], [2]) which speculated, usig argumets aki to the oe by Fermi i the begiig of the article, that people ca be perceived as havig extraordiary ability whe i reality they are simply lucky. However, this paper is the first oe, which argues it usig real data. Refereces. See e.g. W.E. Demig, Out of the crisis (MIT, Cambridge, 986). 2. C. Shores, N. Fraks, ad R. Guest, Above the treches: a complete record of the fighter aces ad the uits of the British Empire air forces (Grub Street, Lodo, 99) 3. N. L.R. Fraks ad F.W. Bailey, Over the frot: a complete record of the fighter aces ad the uits of the Uited States ad Frech air services (Grub Street, Lodo, 992) 4. N. L.R. Fraks, F.W. Bailey, ad R. Guest, Above the Lies: The Aces ad Fighter Uits of the Germa Air Service, Naval Air Service ad Fladers Marie Corps, (Grub Street, Lodo, 993) 5. N. L.R. Fraks, F.W. Bailey, ad R. Duive, The Jasta Pilots: detailed listigs ad histories, Aug 96 Nov 98 (Grub Street, Lodo, 996) 6. W. Feller, A itroductio to probability theory ad its applicatios (Joh Wiley, New York, 957) 7. Naraya, R. ad Nityaada, R., Maximum Etropy Image Restoratio i Astroomy, A. Rev. Astro. Astrophys. 24, 27-7, M.V. Simki ad J. Oless, What is the true dropped calls rate whe i the test it was foud to be zero?, M.V. Simki ad V.P. Roychowdhury, Theory of Aces: Fame by chace or merit?, Joural of Mathematical Sociology, 3, 33 (26). K.W. Deutsch ad W.G. Madow, A ote o the appearace of wisdom i large bureaucratic orgaizatios, Behavioral Sciece, 6, 72 (96). 2. S. Turer ad D.E. Chubi, Aother appraisal of Ortega, the Coles, ad Sciece policy: Ecclesiastes hypothesis, Social Sciece Iformatio, 5, 657 (976). Appedix (The maximizatio algorithm) The method used is as follows. The defeat rates are discretized: r k =.5 +. k, k =,,99. The probability also is discretized i uits of.. These uits of probability are iitially distributed at radom over the defeat rates. The we use the followig maximizatio algorithm. Start with k =. Reduce the umber of probability uits at k = (if there is ay) by oe. Tetatively l f i each case. Stick with the move which move this probability uit to k =,,99 ad compute ( ) maximizes l ( f ). Proceed to k = ad repeat the procedure ad so o. After k = 99 go back to k = ad repeat the whole cycle. Stop whe o further move icreases l ( f ). The whole ru was repeated 5 times each time startig with a differet radom probability distributio. These rus eded up i 33 distict local maximums. The result of the most successful ru is show i Fig. 3. Note, that all of the rus produced very similar probability distributios. The best ru resulted i the likelihood, exceedig that of the worst ru by the factor of oly.2. The maximum etropy case was treated exactly the same way. The oly differece was that l ( f ) was replaced with ( f ) l( s) l +. Oe hudred rus eded up i the same maximum.