Ths artcle s ublshed n Aled Economcs Letters htt://www.tandfonlne.com/do/full/0.080/350485.05.0435 Chess layers fame versus ther mert M.V. Smkn and V.P. Roychowdhury Deartment of Electrcal Engneerng, Unversty of Calforna, Los Angeles, CA 90095-594 We nvestgate a ool of nternatonal chess ttle holders born between 90 and 943. Usng Elo ratngs we comute for every layer hs exected score n a game wth a randomly selected layer from the ool. We use ths fgure as layer s mert. We measure layers fame as the number of Google hts. The correlaton between fame and mert s 0.38. At the same tme the correlaton between the logarthm of fame and mert s 0.6. Ths suggests that fame grows exonentally wth mert. Earler we reorted [], [3] that the fame of WWI fghter-lot aces (measured as the number of Google hts) grows exonentally wth achevement (measured as the number of vctores). For most other rofessons a unversally acceted measure of achevement does not exst. However, we hyotheszed that the exonental relaton between achevement and fame holds for all rofessons. We used ths hyothess to estmate the achevement of obel Prze wnners n Physcs based on ther fame [], [3]. In artcular we found that Paul Drac, who s a hundred tmes less famous than Ensten, contrbuted to hyscs only two tmes less. Afterward Claes and De Ceuster [4] used our method to estmate the achevement of obel Prze wnners n Economcs. However one may argue that ths aroach s on a shaky foundaton snce there s only a sngle actual observaton of exonental relatonsh between achevement and fame. In the resent artcle we reort a second actual observaton of such a relatonsh: the case of chess-layers. Elo ratng [5] (see the Aendx) s a number comuted based on chess layers erformance and s used to measure ther strength. Fgure shows the number of Google hts versus Elo Ratng for all 37 nternatonal ttle holders (nternatonal masters and grand masters) born between 90 and 943. Ths samle ncludes 7 consecutve world chamons: Euwe, Botvnnk, Smyslov, Tal, Petrosan, Sassky, and Fscher. Elo ratngs of the chess layers ncluded n Fgure range between 5 and 780. The number of Google hts ranges between 67 and,60,000. The correlaton coeffcent between Elo ratng and the number of Google hts s 0.40 ( r = 0. 6 ). The best lnear ft between Elo ratng (E) and fame (F) F ( E) A E B = () has A = 358.73 and B = -86703. ote, that ths gves negatve values of fame for Elo ratngs below 404 (see Fgure (a)). The correlaton coeffcent between Elo ratng and the logarthm of the number of Google hts s 0.6 ( r = 0. 37 ). The corresondng exonental ft ( E) = C ( β E) F ex () has β. 9 C = 6.938 0 and = 0. 03 The r for the exonental ft s more than two tmes bgger than the corresondng value for the lnear ft. In addton exonental ft does not roduce negatve fame anomaly. Ths suggests that fame grows exonentally wth Elo ratng rather than lnearly.
400000 0,000,000 00000 000000,000,000 y = 7E-09e 0.03x 800000 00,000 Google hts 600000 y = 358.73x - 86703 0,000 400000 00000 Google hts,000 0 00-00000 00 400 600 800 Elo ratng (a) 0,00,400,600,800 Elo ratng (b) Fgure. Elo ratng of 37 nternatonal chess ttles holders born between 90 and 943 versus ther fame, measured as the number of Google hts. Plotted usng lnear (a) and logarthmc (b) scale for fame. For every chess layer we comuted the dfferences between the logarthm of fame comuted usng Eq.() and the logarthm of the actual fame. Fgure shows the hstogram of these errors. The bn 0 ncludes the errors between -0.5 and 0.5, bn the errors between 0.5 and.5 and so on. As you can see the dstrbuton of errors s close to normal. Ths means that the data has no unnatural outlers, whch stand out from the general attern. ote that 303 of 37of the observatons fall nto the three central bns. Ths means that n 8% of the cases the fame estmated usng Elo ratng s between 4.5 tmes less and 4.5 tmes more than the actual fame. Furthermore we analyzed the average error as a functon of Elo ratng. See Fgure 3(a). We used 00 Elo onts wde bns for averagng. The data ont at the Elo ratng value of 350 s the root mean square (RMS) error of logarthm of fame of all chess layers wth Elo ratng of more or equal to 300 and less than 400. And so on. As one can see the heteroskedastcty though resent s moderate. The hghest RMS error (at 350 Elo onts) s.7. It s about two tmes bgger that the lowest RMS error (at 650 Elo onts) whch s 0.83. In contrast, f we use lnear ft, that s Eq.(), and comute the corresondng errors heteroskedastcty s enormous. The dfference between the hghest and lowest RMS error s 40 tmes. See Fgure 3(b). Ths s one more argument n favor of usng exonental ft. number of observatons 000 00 0-6 -4-0 4 6 ln(redcted fame) - ln(actual fame) actual data normal dstrbuton Fgure. The hstogram of the dfferences between the logarthm of fame comuted based on Elo ratng usng Eq.() and actual fame. The bn 0 ncludes the errors between -0.5 and 0.5, bn the errors between 0.5 and.5 and so on. The lne s a ft usng normal dstrbuton.
RMS error of logarthm of fame (a).8.6.4. 0.8 0.6 0.4 0. 0 00 300 400 500 600 700 800 Elo ratng 000000 Google hts 0000000 000000 00000 0000 y = 07.0e 8.6795x RMS error of fame 00000 000 00 (b) 0000 00 300 400 500 600 700 800 Elo ratng Fgure 3. The average (usng 00 Elo onts wde bns) fame error as a functon of Elo ratng: (a) the root mean square (RMS) error of logarthm of fame comuted usng Eq.(); (b) the RMS error of fame comuted usng Eq.(). Elo ratngs seak only to secalsts. However usng them one can comute exected outcome of the match between any two layers (see the Aendx). We decded to comute exected score n a game wth a randomly selected layer for all 37 nternatonal chess ttles holders. By randomly selected layer we mean a layer randomly selected from the very same ool of 37 layers. We can use ths exected score as a tangble measure of mert. The mert, M, of the layer s thus gven by the followng equaton M = j= ( S > S ) j (3) Here ( S > S j ) s comuted usng Eq. (A), and s the number of layers n the ool (n our case = 37). Mert of dfferent chess layers n our ool, comuted usng Eq.(3), ranges between 0.9 and 0.85. 0 0. 0.3 0.5 0.7 0.9 mert Fgure 4. Fame (number of Google hts) of 37 nternatonal chess ttles holders versus ther mert (exected score n a game wth a randomly selected layer). The straght lne s a ft usng Eq.(4) wth C = 07 and β = 8. 68. Fgure 4 shows ths exected score versus the number of Google hts. The correlaton coeffcent between mert and fame s 0.38 ( r = 0. 4 ). The correlaton between mert and the logarthm of fame s 0.6 ( r = 0. 37 ). The above numbers are qute close to the correlaton coeffcents between Elo ratng and fame. Ths s not surrsng snce the correlaton coeffcent between mert, comuted usng Eq.(3), and Elo ratng s 0.999. Smlarly to growng exonentally wth Elo ratng, fame grows exonentally wth mert (M): ( M ) = C ( β M ) F ex (4) The best exonental ft has C = 07.0and β = 8. 6795. Exonental growth of fame wth achevement leads to ts unfar dstrbuton. For examle Mkhal Botvnnk has a mert fgure of 0.80, whch s only 6% below the mert fgure of
Robert Fscher, whch s 0.85. However Botvnnk s fame measures 73,000 Google hts, whch s 7 tmes less than Fsher s fame of,60,000. At the bottom of the lst s a chess layer wth a mert of 0.9. Ths s 4.5 tmes less than Fshers mert. However hs fame fgure of 76 s 7 thousand tmes less than Fshers fame. robablty densty 0 0. 0.0 0 0. 0.4 0.6 0.8 mert Fgure 5. Dstrbuton of mert (exected score n a game wth a randomly selected layer) for 37 nternatonal chess ttle holders. robablty.e-03.e-04.e-05.e-06.e-07.e-08.e-09 fame (n Googfle hts).e+0.e+03.e+04.e+05.e+06.e+07 Fgure 6. Dstrbuton of fame (number of Google hts) for 37 nternatonal chess ttle holders. In Refs [], [3] we reorted a smlar observaton n the case of fghter lot aces and roosed a model whch exlans exonental growth of fame wth mert. ote, however, n the case of fghter lot aces the correlaton coeffcent between the number of vctores and fame was 0.48, and the correlaton between the number of vctores and logarthm of fame was 0.7. The correlaton s less n the case of chess layers. Ths could be because Elo ratngs are only estmates of layer s actual strength, or because our measure of mert s not erfect. Fgure 5 shows the dstrbuton of mert for our ool of chess-layers, whle Fgure 6 shows the dstrbuton of fame. As we can see the dstrbuton of fame s far more sread than the dstrbuton of mert and requres a logarthmc scale to lot. Ths s not surrsng snce fame grows exonentally wth mert. The dstrbuton of mert of chess layers looks somethng lke a Gaussan. In contrast, the dstrbuton of the mert of fghter-lot aces (measured as the number of vctores) looks close to exonental (see Fgure 3 of Ref.[] and Fgure of Ref. [6]). Ths dfference s because we are lookng at two dfferent thngs. The Elo ratngs and comuted from them mert fgures deend only on skll, whle the numbers of aces vctores deend also on chance. The dfference between chess layers and lots s that whle a chess layer can easly lay another game next day after hs defeat, ths s an mossble thng for a lot. At least accordng to the offcal olces, a lot s granted a vctory f hs oonent s ether klled or taken rsoner (see Ref. [6]). So a lot can fght untl hs frst defeat. To comare chess layers wth fghter-lots we decded to comute the dstrbuton of the number of games before frst defeat for each of chess-layers n our ool. There s a comlcaton ntroduced by draws, whch are not recorded n the case of lots. To elmnate ths comlcaton we wll nterret exected average score, M, as the robablty of vctory. The robablty to acheve n vctores before frst defeat for a layer s gven by the equaton: n ( n) ( M ) ( M ) = (5)
number of vctores before the frst defeat robablty 0. 0.0 0.00 0.000 0.0000 0 5 0 5 0 5 30 Fgure 7. Probablty dstrbuton of the number of games before frst defeat for our ool of chess layers comuted usng Eq.(6). We obtan the overall robablty dstrbuton by averagng Eq.(5) over all layers: ( n) = ( n) = (6) The result of alyng Eq.(6) to our samle of chess layers s shown n Fgure 7. One can notce a remarkable resemblance between ths fgure and Fg. () of Ref. [6]. The only dfference s that the robablty decays faster wth the number of vctores n the case of chess layers. The reason s aarently that we comuted t assumng that nternatonal ttle holders lay between each other, whle fghter lot aces do not always fght wth aces. If we nclude n our ool of chess layers aart from nternatonal masters and grand masters also natonal masters and exerts then the mert fgure of the to layers wll ncrease and, accordng to Eq.(5), ther robablty of achevng hgher number of vctores wll be hgher. Perhas, n that case the smlarty wth Fgure of Ref. [6] wll become erfect. References. M.V. Smkn and V.P. Roychowdhury, Theory of Aces: Fame by chance or mert? Journal of Mathematcal Socology, 30, 33 4 (006); htt://arxv.org/abs/cond-mat/030049. M.V. Smkn and V.P. Roychowdhury, Von Rchthofen, Ensten and the AGA: Estmatng achevement from fame Sgnfcance, 8, 6 (0); htt://arxv.org/abs/0906.3558 3. M.V. Smkn and V.P. Roychowdhury A mathematcal theory of fame, Journal of Statstcal Physcs: Volume 5, Issue (03), Page 39-38; htt://arxv.org/abs/30.706 4. Claes, A.G. P. and De Ceuster M. J. K. (03) Estmatng the economcs obel Prze laureates' achevement from ther fame, Aled Economcs Letters, 0, 884-888; htt://dx.do.org/0.080/350485.0.758836 5. Elo, A.E. (978). The ratng of chesslayers, ast and resent. ew York: Arco ublshng. 6. M.V. Smkn and V.P. Roychowdhury, Theory of Aces: hgh score by skll or luck? Journal of Mathematcal Socology, 3, 9-4 (008); htt://arxv.org/abs/hyscs/060709 Ths work was suorted n art by the IH grant o. R0 GM05033-0.
Aendx In Elo s model [5] every chess layer has certan average strength S and hs actually demonstrated strength vares from game to game accordng to a Gaussan dstrbuton. Elo assumed that whle average strength vares from layer to layer, the strength varance s the same for all layers and s equal to 00 Elo onts. So the robablty densty of layer s strength s ( S) ( S ) S ex 00 π 00 = (A) The layer who demonstrated a hgher actual strength n a gven game wns that game. Elementary calculaton gves that the robablty that the layer of average strength S wns over the layer of average strength A S s B + D 400 ( S > S ) = ex( t ) A B π dt (A) where D = S A S B. Interretaton of Eq. (A) as an exected score elmnates the comlcatons caused by the draws. In Elo s model average layer strength s not constant over lfetme but changes tycally ncreasng at the start of layer s career and declnng at ts end. Elo comuted average strengths of chess layers (Elo ratngs) by alyng certan teratve rocedure to the results of the games. Chess layer s Elo ratngs roved to be good redctors of the outcome of matches. Elo ratngs can be used to comare layers who never layed wth each other and even those who belong to dfferent generatons. The table n chater 9.4 of [5] contans Elo ratngs of all nternatonal ttle holders as of //978. Accordng to Chater 6 of [5] chess layer s ratng eaks n md-thrtes. So we selected those chess layers born n 943 or earler. The lower bound on brth year was 90 to ensure that selected chess-layers belong to more or less the same eoch. The table n chater 9.4 of [5] has two ratng columns: Best 5-yr average and Ratng as of //78. For some of the layers both fgures are gven and for other only one. In the case when two fgures were gven we took the hgher of the two fgures.