Attributing credit to coauthors in academic publishing: the 1n rule, parallelization, and team bonuses

Attributing credit to coauthors in academic publishing The 1/n rule, parallelization, and team bonuses ARTICLE IN PRESS JID EOR [m5G; January 20, 2017;20 1 ] European Journal of Operational Research 0[.]

ContentslistsavailableatScienceDirect

European Journal of Operational Research

journalhomepage:www.elsevier.com/locate/ejor

Interfaces with Other Disciplines

Attributing credit to coauthors in academic publishing: The 1/n rule,

parallelization, and team bonuses R

Louis de Mesnard

Univ de Bourgogne Franche-Comté, CREGO (EA 7317), 2 Bd Gabriel, Dijon 210 0 0, France

a r t i c l e i n f o

Article history:

Received 17 June 2015

Accepted 4 January 2017

Available online xxx

Keywords:

OR in scientometrics

Academic publishing

1/n rule

Parallelization

Productivity gains

a b s t r a c t

Universities lookingtorecruit orto rankresearchershave toattribute creditscorestotheiracademic publications.Whiletheycoulduseindexes,thereremainsthedifficultyofcoauthoredpapers.Itisunfair

tocountann-authoredpaperasonepaperforeachcoauthor,i.e.,asnpapersaddedtothetotal:thisis

“feedingthemultitude” Sharingthecreditamongcoauthorsbypercentagesorbysimplydividingbyn (“1/nrule”)isfairerbutsomewhatharsh.Accordingly,weproposetotakeintoaccounttheproductivity gainsofparallelizationbyintroducingaparallelizationbonusthatmultipliesthecreditallocatedtoeach coauthor

Itmightbeanideaforcoauthors toindicatehowtheyorganized theirwork inproducingthe pa-per.However,theymightsystematicallybiastheiranswers.Fortunately,thenumberofparalleltasksis boundedbythenumberofcoauthorsbecauseofspecializationandthecreditisboundedbyalimiting Paretomaximum.Thus,thecreditisgivenby( N+2) /3nforNparalleltasks.Astheremaybe,atmost,

asmanyparalleltasksasco-authors,creditallocatedtoeachcoauthorisgivenby( n+2)/3n,thatvaries between2/3ofasingle-authoredpaperfortwocoauthorsand1/3whenthenumberofcoauthorsisvery large.Thisisthe“maximumparallelizationcredit” rulethatweproposetoapply

Thisnewapproachisfeasible.Itcanbeappliedtopastand presentpapersregardlessofthe agree-mentofpublishinghouses.Itisfairanditrewardsgenuinecooperationinacademicpublishing

© 2017 The Authors Published by Elsevier B.V ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1 Introduction

Inmostcountriesscholarsareevaluatedbytheiracademic

out-put Most scholars coauthor their papers This is standard

prac-ticeanddesirableinmanyfieldsofscience.Multi-authorshipisa

feature of scientific research which is not commonplace in other

domains such asart (Galenson, 2007; Nabout et al., 2015; Sahu

& Panda, 2013 ) Multi-authorship is a growing practice (Wuchty,

Jones, & Uzzi, 2007 )1 and is oftenperceived as a signof quality

research.ThisexcerptisfromNarin and Hamilton (1996 p.296):2

Countsofcoauthorships,andespeciallyinternational

coauthor-ships, are an indicator of quality,and that scientists who

co-operatewiththeircolleaguesinotherinstitutionsandoverseas

aremorelikelytobedoingqualityresearchthanthosethatare

relativelyisolated

R The author thanks the anonymous reviewers and the editor of the journal who

greatly helped to improve this paper

E-mail address: louis.de-mesnard@u-bourgogne.fr

1 See also the answer of Brandão (2007)

2 On international coauthorship, see Narin, Stevens, and Whitlaw (1991)

Laband (1987) reports that the acceptance rateof coauthored papersishigherandsuch papersare citedmore.Hsu and Huang (2011) find a correlation between collaboration and higher im-pact Harsanyi (1993) discusses multi-authorship as a source

of prestige Moreover, it might be argued that multi-authored manuscripts allow economies of scale For example, Durden and Perri (2003) think that coauthorship,3 increases total and per capitaproductivityineconomics Moreover, itmaybe more diffi-cultto get publishedwhen thepaper issingle-authored: Gordon (1980) reportsthattheleadingjournalAstronomy and Space Science

accepted only 63% of single-authored papers but 81% of multi-authoredpapersand100%ofpaperswithmorethansixcoauthors (althoughitshould besaid that veryfew papershavemore than three coauthors).4 The rate of multi-authorship may be highly variableacrossdisciplines(Wang, Wu, Pan, Ma, & Rousseau, 2005 )

3 And membership of the American Economic Association

4 The number of coauthors may be very large, up to 5154 coauthors in the recent Aad et al.’s ( Aad & ATLAS Collaboration, CMS Collaboration, 2015 ) paper, the world record

An anonymous referee suggested to us that multi-author papers are more likely to

be accepted possibly because multiple authors tend to criticize each other and dis- http://dx.doi.org/10.1016/j.ejor.2017.01.009

0377-2217/© 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Pleasecitethisarticleas:L.de Mesnard,Attributingcredittocoauthorsinacademicpublishing:The1/nrule,parallelization,andteam

However,forZi-Lin (2009) ,internationallycoauthoredpapers“[do]

nothavemoreepistemicauthority”.Coauthorshipisnotapanacea

It may slow down the reviewing process For example, Hartley

(2005) showsthat, inpsychologyjournals, single-authoredpapers

arereviewedfasterthanmultiple-authoredones.Coauthoringdoes

notprevent bad papers(de Mesnard, 2010 ) orsimply unreadable

papers(Remus, 1977 )frombeingproduced.Throughoutthispaper

we work on the assumption that coauthors are rational: if they

cooperate, it is because they derive a benefit,either in terms of

quality or in terms of time and effort Therefore, cooperation is

simplyameans toobtain a paperofagivenlevel ofqualityata

lowercost.5

In the context ofacademic evaluation, andwhen no

informa-tionabouteachauthor’scontributionisavailable,amulti-authored

paperisgenerallycountedtodayasone paperforeach coauthor,

which means that the n-coauthored paper is counted for n

pa-pers This is unfair: counting each paper coauthored by many

researchers as one paper each leads to an obvious bias For

example,considerthree papers andthree scholars.Scholarb has

produced “Beta” and during that time, scholars a1 and a2 have

coauthored papers “Alpha1” and “Alpha2” If an unsophisticated

count is made we have three papers: “Beta” appears in b’s

cur-riculum vitae, which is fair enough But “Alpha1” and “Alpha2”

appear in the curricula of both a1 and a2 Therefore, if n

re-searchersproduce Ppaperstogether,theyseemtohaveproduced

nPpapers, a phenomenon wemight call feeding the multitude,to

use a biblical metaphor, with no disrespect intended Moreover,

coauthorsa1anda2ontheonehandandbontheotherhandare

nottreatedequally Thisiswhysome scholarsmaygo toofarby

forming what we term publication club to artificially boost their

output.Apublicationclub6isarathersmallgroupofscholarswho

mutuallyagree,nottocollaborateonwritinga paperincommon,

but to cosign each other’s papers, even if they have not really

been involved in writing them This obviously results in more

coauthorship.7

All authors want to be publishedin the leadingjournals, but

thismay turn into a farce, in the words of McDonald and Kam

(2007) Evenifthe academicevaluationsystemignores

coauthor-ship, powerful incentives for coauthorship will be created,

lead-ingimmediatelytotheideaof“publicationclub” Agoodstrategy

foranyscholarwantingto increasehis scoreistojoin a

publica-tion club This raises the difficult questionof whether

coauthor-shipisnecessary,asinapublicationteam,orisartificialandeven

completely fake as in publication clubs Publication teams are a

growingphenomenon,whichisa goodthing(Rey-Rocha, Garzon-

Garcia, & Martin-Sempere, 2006 ),butthedevelopmentof

publica-tionclubscouldproveveryharmful

Although itis impossibletoprevent the formationof

publica-tionclubs, or to eradicate them, there remains the possibility of

reducingtheir impactby “punishing” multi-authorship.Imaginea

paperwrittenby asingleauthor.Itisobviouslyattributedtohim

in full Now, imagine a paper written by two coauthors Should

wecount the paperforzeroto eachcoauthor becauseneither of

cuss their publication, so that a kind of internal evaluation improves the manuscript

before submission much more effectively than a single author can do

5 Cooperating may generate coordination costs: if coordination costs are higher

than the gross benefit derived from cooperation, the paper will not be written in

cooperation This topic will not be examined in this paper

6 The term “club” is not here used in Buchanan’s sense ( Buchanan, 1965 ) but in

the sense of “brotherhood,” “fraternity” or “union.” Wang et al (2005) draw a dis-

tinction between collaboration in the same institution, and regional, national, and

international collaboration The term “team” indicates that genuine collaborative

work has been done by coauthors On the identification of research teams, see also

( Calero, Buter, Valdes, & Noyons, 2006 ) See also Hou, Kretschmer, and Liu (2008)

7 This is plainly unethical On some types of misconduct in publishing, see List,

Bailey, Euzent, and Martin (2001)

themisabletoproducethefullpaperinafinitetime?Thiswould

beplainlyunfair.Shouldweattributethefullpapertoeach coau-thor? Forexample, should we count one paper for the coauthor specializedintheoryandonepaperforthecoauthorspecializedin econometrics,althoughneitherofthemhasproducedthefull pa-peralone?Thisleadstocountingtwopapersintotalalthoughonly onehasactuallybeenproduced.Orshouldwedividethemeritby two,countingonly1/2paperforeachcoauthor(andonepaperfor thetotal)?Thislastsolutionisthe“1/n” rulewherenisherethe numberofcoauthors

Evenifthismight seem fair,each ofthe two coauthors could rightlyarguethathe/shehasdonehis/herfulljobandcontributed decisivelytothepaper— thefirstoneinthinkingaboutand writ-ingthetheoryandthesecondoneinconductingthetests.Iseach author’smeritlessthanforasingleauthoredpaper?Thus,we pro-pose to reward coauthors who are more efficient,that is, to re-ward justified specializationin a teamby means of a paralleliza-tion bonusthat takesinto account the parallelization induced by collaborationamongspecialists

However,asthedegreeofparallelizationcannotbedetermined exogenously discipline by discipline,we could propose that each teamofcoauthors indicates howthe laborwasorganizedto pro-duce the paper Unfortunately, coauthors may systematically bias theiranswersinordertoincreasetheparallelizationbonusand re-ceivealargerallocationforeachcoauthor.Nevertheless,wewillbe abletodemonstratethatalimitationmechanismoperates—a limit-ingParetomaximumenablingustodefineaparallelizationbonus For the first time, to our knowledge, we will look inside the

“machine” by“liftingthehood”,thatis,bytakingintoaccountthe organizationoftasksnecessarytowriteapaper.Thisapproachwill concernacademicdisciplinesinwhichsingle-authorshipisa cred-iblepossibility,butalsootherdisciplinesinwhichteamsmayrun

to dozens and even thousands of coauthors as in medicine and wheresubtlerulesamongsignatoriesallowspecialiststoinferwho hasdonewhat

2 Sharing the credit: contribution weights and 1/nrule

Even if the connection between collaboration and productiv-ityisunclear (Fox, 1983 ), itmaybe arguedthat coauthorship di-videstheeffort:it isundoubtedlyunreasonable tocount asingle paper produced by a team of n coauthors, as n papers, one for each of the n coauthors A paper produced by a team of n > 1 scholars cannot be worth n times a paper produced by a single scholar.8

We never claim that the number of publications is a sign of productivityorperformance forresearchers9butthe formationof publication clubs should be prevented.This iswhy themerit at-tributed to the coauthor i ofa givenacademic paper p must be sharedamongthecoauthors.Ingeneralterms,foranacademic pa-per p, thecreditsystemfora nonemptysetp ofcoauthors isa vectorAp ofdimensionnwhere

n=card

p



∈N\0 such that

acreditA p i isattributedtoeachcoauthori∈pbythefollowing mapping:

f: p→[0,1]n: {i}i∈ p→Ap

Forexample,acreditofA p i=.45means thatcoauthor iofpaper

piscredited of45%ofafull paper.Inwhat follows,we omitthe

8 In some disciplines where the list of coauthors follows subtle rules (“first au- thor” / “first authors”, last author, other authors, etc.), all coauthors are not equal

We consider here in the set of n coauthors only the real coauthors, that is, in prac- tice, the “first coauthors” See the discussion below in Section 2.1

9 The quality of the research is important: in scoring systems based on the qual- ity of the journals in which papers are published, the quality of the output is eval- uated by the journal’s ranking

Pleasecitethisarticleas:L.deMesnard,Attributingcredittocoauthors inacademicpublishing:The1/nrule,parallelization,andteam

paper.Weidentifythreemainmethods;(i)itispossibletoidentify

coauthors’ relative contributions or(ii) the coauthors self-declare

theirrespectivecontributions,(iii)noinformationatallisavailable

andthecreditissharedamongcoauthorsequally

2.1 Identifying the relative input by authors: Is this a solution?

One simple wayto shorten discussion abouthow to attribute

merittoeachcoauthorwouldbetoidentifytherelative

contribu-tionmadebyeachcoauthor.Thisisdifficult.Inmedicine,therank

of coauthors is indicated by the list of coauthors: generally, the

names are not sortedby alphabetical orderbut thefirst nameis

thepersonwho hasdone thegreatestshareofthe work(aPh.D

student, a PostDoc, etc.) Authorship may even be honorary,

par-ticularly inmedicine(O’Brien, Baerlocher, Newton, Gautam, & No-

ble, 2009 ).10 Galam (2010) and Prathap (2011) pay particular

at-tention to the rank of authors Egghe, Rousseau, and Hooydonk

(20 0 0) examinetheroleofdifferentindexestodeterminethe

re-spectivecontributionofeachauthorandtheyshowthattheresult

depends largely on the index chosen On the contrary, in

math-ematics orin the social sciencesorhumanities, papers need not

be written by large teams (one, two, or three contributors

suf-fice)andthelistofcoauthors maybeinalphabeticalorder:being

thefirst-namedauthormeansnothing.Insuchinstances,andalso

toevaluate theroleofsuccessiveauthorswhenbeingfirstmeans

something, one might conduct sophisticated analyses of

publica-tionnetworks11 toidentifytheleader(Yoshikane, Nozawa, & Tsuji,

2006 ).Thisisunrealisticinthecontextofageneralscoringsystem

whichmustevaluatehundredsofscholarsatonce.Moreover,there

isnocertaintythatfocusingononedisciplinealonewouldbe

suf-ficientbecauseofinterdisciplinary collaboration,whichwould

in-volve examining thousands ofscholars atthe same time Onthe

otherhand,forKatz and Martin (1997) ,whorefertoSubramanyam

(1983) ,“ ifthe levelof workingtogether ofa numberof

scien-tistswasbelowthisminimumthreshold,theywouldneverappear

ascoauthorsofapublication.”

Anothermethodconsistsinconsideringthatthecorresponding

authoristheleaderasinRoyle, Coles, Williams, and Evans (2007)

However,whencontributionsareeven,choosingthecorresponding

authorasleadermightgenerateabias;indicatingacorresponding

authorismandatoryandbeingthecorrespondingauthorofateam

ofequalsmightmeanverylittle.Bycontrast,Krapf (2015)

consid-erstheageofthecoauthors

2.2 The Utopian solution: self-declared authoring weights

Authors might alsostate their respectivecontributions in

per-centage terms: fora given paper,coauthor a has contributed for

a percentageofw a ,coauthorb forw b ,etc.Then, percentagesare

usedtoascertainthemeritsofeachauthorandwehavean

objec-tive indication ofwhat the respective contribution ofeach

coau-thoriis,intheformofaweightw iwithn

i=1w i=1:

A i=w i foranyi∈

For example,ifthe weights of thethree coauthors 1, 2 and3 of

paper p are respectively 45%, 25%, and 30%, the credit is Ap=



.45 .25 .3

, which means that the three coauthors are

re-spectivelycredited of45%,25%,and30%ofafull paper.Thetotal

10 In physics, the old practice of the laboratory director to add his name to the

end of the of the list of coauthors is now considered unethical Katz and Martin

(1997) underline that the list of coauthors, that is, the multiple signatures on a

paper, may not coincide completely with the list of scholars who have actually col-

laborated on this paper

11 On networks of coauthorship, see also Cardillo, Scellato, and Latora (2006)

creditreceivedbyanypaperisequalto1:theruleisneutralwith respecttoeachpaperbecausen

i=1A i=1 However,sucharuleposestwoproblems.(i)Itisfeasibleonly

ifall journalsofall publishinghousesdecide atthesametime to compelauthorstodoso.(ii)Itisvalidforfuturearticlesonly,not forpreviouslypublishedpapers.Inshort,theprocedureis unsatis-factoryandUtopian

Moreover,if aset ofcoauthors is ableto indicate percentages other than 1/n, it is because all the coauthors really worked on thetopic.However,thisopensupthewayforagame:ifthegame

iscooperative,theresultwouldbeaNash (1951) equilibrium(i.e., fortwocoauthors,abilateralmonopolywhichsharesoutthe com-mongain equallywheninformationabouttherespectiveeffortis public12),whichleadstothenextsubsection.Actually,treatingthe questionoftheself-declarationoftheweightsw iwouldrequire in-vokinggametheoryinafullpaper.Thatwouldbeanotherstory

2.3 Sharing the credit without information: the “1/n” rule

Intheabsenceofanyinformationabouttherespective contri-butions,i.e.,ifnopercentagesareindicated,theauthorsshouldbe considered tobe equals: thismustbe found fairif we apply the Bernoulli-Laplace principle In other words, because we have no information aboutthe proportions, we should attribute an equal fractionofthetotaltoeachone.Intherestofthepaper,we con-sideronlythecasewhereallw iareequal

Wemakean assumption:asaneditorhasnomeansof detect-ingfakecoauthors,weconsiderasgiventhelistofcoauthors,i.e.,

nisgivenbytheteam.Therefore,theprincipleissimple.Ifapaper hasbeencoauthoredbynauthors,eachcoauthoriscredited equally

dependingonthenumberofcoauthorsn:

Eq (1) means that each coauthor receives 1/n ofthe pointsthat thispaperwouldhavebeenallocatedhaditbeenwrittenbya sin-gleauthor.Forexample,ifa paperhasbeenauthoredbyjustone authoritcountsasonepaperforthisauthor;ifithasbeen coau-thoredby twoauthorsitcountsasone-half foreachcoauthor;by threeauthors one-third, etc Hence thename,the “1/n rule” Be-tweenzeroand1/n,i.e., [0,1/n[,anyruleisunfair:thewhole pa-periscountedforlessthanonepaper;1/ncorrespondstothe sim-pledivisionamongallcoauthors andunitycorresponds tothe al-lotmentofthewholepapertoeachcoauthor.Weseethatwehave

amarginofmaneuveronthesegment[1/n,1].The1/nruleisalso neutralwithrespecttoeachpaperbecausen

i=1A i=n×1

n=1 However, unlike other systems discussed for personal impact factors by Galam (2010) ; Hirsch (2009) , and Prathap (2011) , in whichthe rankwithin thelist ofauthorsis meaningful,the “1/n

rule” treats all authors in the same way, independently of their respectivecontributions:underthe“1/n” formula,eachscholaras coauthoristreatedequally

The“1/n” ruleiscertainlythesimplestrulefortaking coauthor-shipinto account.Schreiber (20 08a, 20 08b, 2010 ) proposesmuch the same thing in the context of the h index Such a ranking is computedforeveryscholar,withoutconsideringanyothercriteria, suchascitations,theh-index,andsoon;nofractionaloradjusted accountismade.13

Ontheone hand,multi-authorshipmaybe one wayof allow-ing some researchers topublish who are incapable ofpublishing alone;on theother hand,coauthorship could bea genuineasset, speedinguppublicationbyallowingarealdivisionoflaboramong

12 This leads on to the theory of contests; see Tullock (1980)

13 Obviously, more sophisticated rules could be chosen, such as 1/ n αwhere α >

1, instead of the 1/ n rule, meaning the benefit of large coauthorship decreases very rapidly However, justifying them would be difficult

bytwoormorescholars,itiseitherbecausetheproblemthatthe

paperpurportstosolve istoo bigto behandled bya single

per-son,eveniftheoutputisonlyasinglepaper,orbecausethepaper

needsdifferentskillsthatarerarelyfoundinjustoneperson

Thus,itcouldbepreferableorevennecessaryinsomeacademic

disciplinesto associatemultipleauthorsto produceonepaper.In

such cases,a multi-authoredpaper isworth morethan a

single-authoredpaper In a nutshell, the 1/n rule is tooharsh and

dis-couragesmulti-authorshipevenwhenitcannotbecharacterizedas

apublicationclub.Therefore,the1/nruleshouldbecorrected.This

iswhyweproposetotakeintoaccounttheparallelizationoftasks

inducedbycoauthoring,andtheproductivitygainsthatisimplies,

andtorewardit

3 Beyond the 1/nrule: parallelization of tasks

3.1 Idea of tasks

Writing anacademicpaperinvolvescompletingacertain

num-ber of tasks What are the tasks in writing an academic paper?

Theyincludedoingsome preliminarythinkingabouttheproblem,

compilingthebibliographybyreadingabstractsorperusingalarge

numberofpapers,readingafew papersthoroughly, situatingthe

contributionmadebythepaperwithrespecttootherpapers,

de-signingthe model,makingnumerical simulations onthe basis of

themodelorperformingthestatisticalandeconometric

computa-tions,writinguptheresults,writingtheintroductionandthe

con-clusion,etc

If we consider two tasks i and j, we have here two extreme

cases

Definition 1. (i) A taskj is independentof a taski if the results

ofj do not depend on the results of i.The tasks of a couple i,

j thatareindependentofeachothercanbeperformedinparallel

(ii)Tasksthatarenotindependentmustbeperformedserially,one

aftertheother

Toperformtasksinparallel,multi-authorshipisobviously

nec-essary.Tasksthat canbe performedin parallelrequireeitherthe

sameskills (e.g., performing clinical testsin two different

hospi-tals)ordifferentskills(e.g.,onecoauthorspecializedintheoryand

one coauthorspecialized in testing).14 ByRicardo ’s (1817) theory

ofcomparativeadvantage,we assume thatcoauthors practicethe

horizontaldivisionoflaborandspecializeintasksthatrequire

dif-ferentiatedskills Eveniftwo authorsare capableof workingon

twoparalleltasks becausetheyare multi-skilled,itisrationalfor

themtoinvesttheirenergyinthetaskswhichtheyaremore

pro-ductive.Inthat sense,coauthorsarenotsubstitutable.15This

divi-sionoflabor isanintelligent wayofworking(providedthateach

scholarunderstandswhattheothersaredoing)

On theother hand,inthecontext ofacademicpublishing and

followingthephilosophyofthe1/nrule,we considerthatseveral

peopleworkingonthesametaskdonotaddmorethanonetask

14 Specialized coauthors are able to produce a paper that could be impossible to

produce alone Actually, it could take each one a very long time, perhaps an infinite

time, to obtain the skills of the other coauthor As underlined by Krapf (2015) , the

paper is of better quality, and the complementarity between coauthors’ inputs is

maximum, if the difference in age is of ten years or so

Moreover, working in parallel obviously produces better papers because there is

interaction between specialized coauthors For example, if Theory and Testing can

be performed simultaneously, the coauthors will interact and improve what they

are doing The Theory will be better and the Testing will be more appropriate to

the Theory for accepting or rejecting its assertions

15 Not to be confused with the different idea of substitution of inputs in Krapf

(2015) On the other hand, a specialized author cannot be efficient on a paper which

requires different unfamiliar skills

3.2 Productivity gains generated by parallelization

Twoideasmustbeintroducedatthispoint:theideasofcycles andproductivitygain

We donot usethe idea ofcalendar time becausethe time to write apaperisnobetteran indicatorthan thenumberofpages

to determine its importance in a ranking We all know papers that have beenwritten rapidlybut that are excellent andpapers that took years and that are poor.16 Moreover, in the academic world, the idea of calendar time does not make sense because academictimeisall fitsandstarts Scholarsperformmany activi-tiesina sameweek ormonth.Theyteach,receivestudents,meet colleagues,runtheirdepartmentoruniversity,conductresearch.17

Therefore,thecalendartimeusedtowriteapapercannotbe mea-sured,asunderlinedbyKrapf (2015) ,thatis,“apaperisa paper”, whateverthetime spentandthenumberofpages.Ifweconsider thedifferenttasksnecessarytowriteanacademicpaper,itwould

begoodtobeabletodefinethetimenecessarytoperformthe dif-ferenttasksbuttherelativetime ofthetasksthathavebeen nec-essarytowritethepaperisdifficulttomeasureobjectively.Evenit mightmakesensetosaythatforonepaper,theoryrepresentsthe main partofthe job,orthat foranother paper,performing trials represents themain effortcompared to thinkingabout theory,it

isoftenimpossibletoreallyquantifyonetaskwithrespectto an-other.So,wedonotdeterminethetotalnumberofhoursanddays

oflaborthatapaperandthedifferenttasksrequiredbyadoptinga qualitativeapproach withrespecttothesetasks.We attributethe sametemporalimportancetoeachtask

Definition 2. A cycle isthe moment duringwhich a task is per-formed Any task is completed during only one cycle but many taskscanbeconductedinthesamecycle

It ensuesfrom specializationthat a coauthor can work on only one parallel task in a same cyclebecauseifanauthorcanworkon twoormoreparalleltasksinthesamecycle,thenthosetaskshave beeninadequatelydefinedandoverlaponeanother.Consequently

whereNdenotesthenumberofparalleltasks.Therefore,nandN

cannotbeconsideredasindependent

Taskandcyclewouldbeone-to-oneconcepts ifparallelization wasimpossiblebutwithparallelization—acrucialideathatwewill develop below— itwillbepossible toconducttwo ormoretasks

inthesamecycle

Definition 3. The productivity gain istheratioofthenumberof cyclesbeforeintroducingparallelismandthenumberofcycles af-terintroducingparallelism:

ηP

(2)

16 Moreover, we differ here from Krapf ’s (2015) approach which uses what he terms “human capital” to evaluate the comparative input of two coauthors by a CES function (which implies that both authors’ input is perfectly substitutable) Human capital is measured as a discounted function of the number of papers published Following McDowell (1982) for academic economists, this function depreciates at the rate 13.18 but the function itself is U-shaped Here, we consider the effort for completing a paper or a task and not the personal input of each author

17 Scholars also wait for answers from journal editors Obviously, the time spent writing a paper may be short compared to the time spent waiting for the answers

of the journal to which the paper has been submitted, and so it might be argued that the time is not of importance in academic publishing However, there is a big difference: in the first case, scholars work, in the second case, they wait Even if

“waiting is harder to bear than fire,” according to the Arab proverb, when a scholar waits, he can do other things: thinking, writing another paper, teaching, correcting exams, etc We would all prefer to produce a paper in collaboration with a colleague

in three months than alone in six months, if it is possible, even if we have to wait one year for the answer from the journal: this would enable us to write a second paper in collaboration, or half a paper alone

Pleasecitethisarticleas:L.deMesnard,Attributingcredittocoauthors inacademicpublishing:The1/nrule,parallelization,andteam

Table 1

Matrix of tasks Legend: Pr, Preliminary task; Th, Theory; T,

Test/econometrics; F, Final task

where ∈[1,∞[,ηandηPbeingrespectivelythenumberofcycles

before andafterparallelization Byconstruction, η isalso the

to-talnumberoftasks.Whenwe havenoparallelization,ηP=η and

s=1

Example 1. Considerapaperwithfourtasksorganizedasfollows:

- one task for thinking about the paper and preparing the job

(preliminarytask,denotedPr),

- onetaskforwritingtheory(theorytask,denotedTh),

- one task for testing and/or doing econometrics

(tests/econometricstask,denotedT),

- onetaskforsynthesisandfinalwriting:(finaltask,denotedF)

We are able to determinewhich tasks are dependent,that is,

which task must be completed before or afterwhich other task,

andwhichtasksareindependent,i.e.,canbeperformedinparallel

InTable 1 ,anumber“1” incell i,j indicatesthattaskishouldbe

placedaftertaskj,i.e.,inthecyclethatfollowsthatofj,anumber

“−1” indicatesthattaskishouldbeplacedbeforetaskj,i.e.,inthe

cyclethatisbeforethatofj,andazeroindicatesthattasksiandj

canbeperformedinparallel,i.e.,inthesamecycle

Wehavefourcases:

(i) If thepaper is single-authored, theauthor doesthe whole

jobandperformsthefourtasksaloneforallfourcycles.See

theGanttdiagram(Wilson, 2003 )inFig 2 ,upperdiagram

(ii) Ifwehavetwoormorecoauthors,butthetasksare

depen-dent,thesuccessionoftasksisthefollowing:Pr,Th,T,Fand

four cyclesare used Inthiscase, a single authorcould be

sufficienttodothejob.SeetheGanttdiagraminFig 2 ,

up-perdiagram

(iii)Ifwehavetwo coauthorsandthetasksThandTare

inde-pendent,it isbetter forthecoauthors tospecialize: oneof

themshouldspecializeintheory(taskTh)andtheotherone

in testing/econometrics (task T) Therefore, the team

per-forms Prcollaborativelyin thefirst cycle;then it performs

ThandTinasecondcycle;finally,theteamperformsF

col-laborativelyinathirdcycle.AsPrandFcouldbeperformed

byonlyasingleauthor,wesharethecreditbetweenthetwo

coauthorsasabovebutweattributetaskThtoonecoauthor

andtaskTtotheotherone.Insteadofcompletingthepaper

infourcycles, itisnow completedinthree cycles:this

re-flectsthebenefitofspecialization/collaboration.Thenumber

ofcyclesgoesfromη=4toηP=3.So,theproductivitygain

sgeneratedbyparallelism isequalto4/3.Thiscorresponds

tothefunctionaldiagramofFig 1 andtothelowerdiagram

ofFig 2

(iv) If we have three or more coauthors (i.e., n > N) and the

tasksThandTareindependent,weconsideronlytwo

coau-thors would be sufficient: the 1/n rule will correct that

Again,weproceedasincase(iii)

3.3 Productivity gain in practice

WedenotebyN∈N\ {0}thenumberofparalleltasks.The

num-berofparalleltaskscannotbehigherthanthenumberoftasks.η

beingequaltothetotalnumberoftasks,wehave

Proposition 1. When the paper is composed of some preliminary se-rial tasks, then of N parallel tasks, and finally of some final serial tasks

as in Fig 1 , 18 we have:

s( η, N)= η

where N∈N\ {0}andη∈N\ {0}and ∈[1,∞[.

Proof. We have initially η tasks over η cycles Among the η

tasks, we have η− N serial tasks that take η− N cycles and

N parallel tasks that take N cycles before parallelization and 1 cycle after parallelization There remain ( η− N)+1 cycles after parallelization 

Whenwehavenoparallelism,thatis,acompletelyserialpaper,

s=1,whichimplieshereN=1.SeeFig 2 ,upperdiagram.Thisis theminimumproductivitygain,asprovedbelow

Proposition 2. The productivity gain is larger or equal to unity, whateverη∈N\ {0}and N∈N\ {0}, N≤η,are: ≥ 1.

Proof. From the minimum of reached for =1, which corre-spondstoN=1, isan increasingfunctionofNbecause∂/∂N=

η ( η −N+1 )2 ≥ 0 isalsoadecreasingfunctionofη because∂ /∂ η=

1−s

η −N+1≤ 0 Asη∈N\0,the minimumof is reachedforη → ∞ andlimη→∞ =1 

Remark. N=ηisaveryspecialcasewherewehaveηcompletely independenttasks,whichmeansthatthecoauthorswork indepen-dently,asiftherewereN=ηindependentpapers.PosingN=ηin (4) gives =η

3.4 Productivity gain and multiple phases

Wehave implicitlyassumedthat we invariablyhadN parallel tasks,whilewecouldhavetwoparalleltasks(e.g.,theoryandtest)

atacertainmomentofthejobandthreeparalleltasks(e.g.,theory, andtwo typesoftests) atanothermoment.This iswhywe pro-posetogeneralizetheaboveapproachbyconvenientlydividingthe efforttoproducethepaperintophases.Wewillbeabletohandle morecomplicatedsituationsanditwillbeusefultodemonstratea fundamentaltheorem

Definition 4. One passes from one phase to another when the numberofparalleltasksvaries,whichincludesthecasewhere se-rialandparalleltasksalternate

Then,wecountineachphasehowmanyparallelorserialtasks are performed: this determines N k ≤ ηk, the numberof parallel tasksineach phase k,ηk beingthenumber ofcyclesofphase k, knowingthatN k=1meansthatthejobisperformedserially dur-ing phasek Ineach phase k,we proceed asbeforeto determine theproductivitygain k attachedtoit.Then allteamproductivity gainsareaggregated

Proposition 3. If P denotes the number of phases,

s( η, N, P)= η

where N∈N\ {0,1},η∈N\ {0}, P∈N\ {0}and ∈[1,∞[.

Proof. Wehaveinitiallyη tasksoverηcycles.Amongtheη tasks,

wehaveη− N serialtasksthat takeη− N cycles, Nparallel tasks

18 We say that the paper has only one “phase” In the next section, we examine the case of multiple phases to demonstrate a fundamental theorem, which interest- ingly allows us to handle resubmissions

Fig 1 Paper with two parallel tasks: functional diagram

Fig 2 Paper with four tasks and two parallel tasks: Gantt diagram Legend: Pr =

Preliminary task; Th = Theory; T = Test/econometrics, etc.; F = Final task Parallel

phases are shown in gray

thattake N cyclesbefore parallelization,andP cyclesafter

paral-lelization.Thereremain( η− N)+P cyclesafterparallelization 

Lemma 1. For a given number of cyclesη and a given number of

parallel tasks N, (η,N,P)is decreasing with the number of phases P.

Proof.Theproofisobviousfrom(5) 

We can alsohandleheterogeneous phases, wherethe number

ofcycleschangesfromonephasetoanother

Proposition 4. The total productivity gain is the harmonic mean of

the productivity gains of phases, weighted by the number of tasks:

s( { ηk},{s k}, P)= η

P

k=1η k

s k

(6)

where ηk is the number of cycles in phase k withη=P

i=kηk and where η∈N\ {0}, ηk∈N\ {0}, P∈N\ {0}, k(ηk, N k) ∈ [1, ∞[ and

s({ηk}, s k},P)∈[1,∞[.

Proof. From(4) , k= η k

η k −N k+1.Thus,

P

k=1

ηk

s k =

P

k=1

ηk − N k+1=η− N+P

 However,wehavetodiscussthenumberofcoauthors:itcannot

be lower than the maximum numberof tasks to be parallelized, thatis,

We exclude the casewhere the number ofcoauthors could vary among the phases to follow the number of parallel tasks: if n

coauthorshavesignedthepaper,ncoauthorsarecountedforeach phaseeveniffewercoauthorsmightbenecessaryinsomephases

Inthis, thereasoningissimilartothat ofthe homogeneoustasks case

Example 2. Apaperwithtwo resubmissions:we havethree het-erogeneous phases, asshownby Fig 3 Wehave 1=2, 2=4/3 and 3=5/3.Intotal,wehaveaproductivitygainof =5/3

4 Parallelization bonus and credit

4.1 The reward

We now propose to reward parallelization because it corre-sponds to an intelligent and efficientway ofusing the resources thatcoauthorshaveattheirdisposalinordertoproducebetter pa-pers Wewillcorrecteachcoauthori’scontribution(measuredby 1/n)bytheproductivitygain consideredasaparallelization bonus

toincreasewhatisattributedtoeachcoauthorofamulti-authored manuscript.Indoingso,werewardtruemulti-authorshipbecause

itcorrespondstoarealadvantagewhenitiscomparedtoasimple additionofncoauthorswhoworkseriallyonthesamepaper,and

itisthe signthatcoauthors havebetterskills (e.g.,theyare each abletoworkontheoryandtesting).However,theauthorswillonly

berewarded,not punished:theparallelizationbonusmustnot be lowerthanunity,i.e., ≥ 1

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Fig 3 Paper with resubmission, three heterogeneous phases: Gantt diagram Legend: Pr = preliminary task; Th1 = theory 1; T1 = test/econometrics 1; T2 =

test/econometrics 2; T3 = test, econometrics 3; F1 = synthesis, final writing 1; A1 = analysis of reports; Th2 = theory 2; T4 = test/econometrics 4; F2 = synthesis, fi- nal writing 2; A2 = analysis of reports; Th3 = theory 3; T5 = test/econometrics 5; T6 = test/econometrics 6; F3 = synthesis, final writing 3 Parallel phases are shown in gray

Now,we canproducethenewcreditrule.Tostayon thecase

ofthe1/nrule(1) ,wedefine

withrespectto , beinggivenby(2), (4), (5) ,or(6) 19

Example 3. If we return to the data of Example 1 , each

coau-thorhascontributedtohalfoftheproductivitygainof (2)=4/3,

which is a bonus If we have no information about each

coau-thor’srespectivecontribution,it willbe dividedby n=2,that is,

2/3, we find A i=2/3 for i=1,2 The paper is counted for 2/3

to each coauthor instead of 1 under the usual system and 1/2

by the 1/n rule The minimum number of coauthors is two In

19 We have simplified the notations here Actually it should be: s ({ η k }, { N k }, P ) and

A i ({ η k }, { N k }, P )

Example 2 where =5/3,wehavenecessarilyatleastfour coau-thorsandtheallocationforeachcoauthorisA i=5/12

The total allocation attributed to a paper is greater than or equalto1:thenewruleA( )isnotneutralbutisfavorabletothe teamwithrespecttoeachpaperbecause

n

i=1

A i=s≥ 1 Thisshould be compared tothe situationwhere norule atall is applied:becauseeachcoauthorisawarded100%ofthepaper,the wholepaperisawardedn,whichistoomuch

Example 4. In Example 1 , A i=2/3for i=1,2 and2

i=1A i=4/3 obviously

Remark. Whenn=1, N=1, =1andthusA i=1:theunique au-thorreceivesthewholeaward,asexpected

Fig. 4 Paper with only N parallel tasks Functional diagram

4.2 A maximum limiting case

We have positedthat N ≤ n because of specialization.When

we have more coauthors than parallel tasks, the 1/n rule comes

to“punish” themby dividing thecredit betweenthem then − N

coauthorscanbeconsideredsurplustorequirements.Cheating

oc-curshere:theteam maydeclarea higherparallelizationthan

re-allyoccurred Fortunately,wewillshow nowthat thereisalimit

tocheating

Is afullparallelization possible?Certainlynot:we wouldhave

Ncompletely independent papers(see Fig 4 ) Similarly,one

pre-liminarytaskandNparalleltasks(seeFig 5 )isanimpossible

con-figuration:writingthepaperwouldneverend.Nparalleltasksand

one final task is also impossible (see Fig 6 ): writing the paper

would never begin Therefore, we should have at least one

pre-liminarytask,Ntasks thatcan be performedinparallel,andone

finaltask,asdescribedbyFig 7 Wewillshownowthatthisisthe

limiting caseinafundamentaltheorem

Definition 5. L(N)istheparallelizationbonusthatcorrespondsto

theparticularcaseofonephase,withonepreliminarytask,Ntasks

performedinparallel,andonefinaltask

Theorem 1. If we assume that each team tries to maximize its bonus,

the particular case of one preliminary task, N tasks performed in

par-allel, and one final task, is a Pareto maximum Hence the name

limit-ing case.

WeareinthecaseofFigs 7 or1

Proof.FromLemma 1 , (η,N,P)isdecreasingwithP.So, (η,N,1)

isthemaximumofthe (η,N,P).Moreover,we haveη− N serial

tasks.As (η,N,P)isincreasingwhenη− N decreases,20the

max-imumof (η,N,1)isfoundforη− N minimum,thatisη− N=2,

whichcorrespondsto L(N) Thus, L(N)isthe maximum(it is

im-possibleto go beyond it) It isa Pareto equilibrium inthe sense

20 If we write η − N = x, s ( η , N, P ) = x+ N

x+1 and d

x+N

x+1

dx = ( x1+1 −N )2 < 0 for N > 1

Fig. 5 Paper with one preliminary task and N parallel tasks Functional diagram

Fig. 6 Paper with N parallel tasks and one final task Functional diagram

that iftheteamattempts todeviate fromthismaximumby self-declaring someother formof organizationoflabor (where possi-ble),thiswillpenalizeatleastoneteammember 

InTheorem 1 wehaveonlyone phaseinthelimitingcase.So,

itensuesfrom(4) that:

s L(N)=N+2

Pleasecitethisarticleas:L.deMesnard,Attributingcredittocoauthors inacademicpublishing:The1/nrule,parallelization,andteam

Fig. 7 Paper with one preliminary task, N parallel tasks, and one final task Functional diagram

Weseethat L(N) →

N→∞∞andthat L(N) issimplylinear.To L(N) corresponds

A L

i= N+2

When thenumberofparalleltasksismaximumandequaltothe

numberofcoauthors,i.e.,N=n ,themaximummaximorumcredit

is

A L imax=n+2

It varies between 2/3 for two coauthors and 1/3 for an infinite

numberofcoauthors (from(9) ,whichis adecreasingfunction as

dA Lmax

i

dn =− 2

3n2 <0).Fig 8 illustratesthecreditA Lmax

i andcompares

ittothe1/nrule

5 Conclusion

Universities thatare recruiting facultyorthat mustrank their

researchersforpromotingthem,havetoattributethefairmeritto

each scholar.Itis nota questionofindexes (suchasthe h-index,

etc.)butofsharingthecreditbetweencoauthorsofmulti-authored

papers.Amulti-authoredpaperisgenerallycountedtodayasone

paper for each coauthor when no information about each

au-thor’scontributionisavailable, meaningthatann-coauthored

pa-percountsfornpapers.Thisistoogenerousandwetermit

“feed-ing the multitude” A contrasting approach consists in allocating

themeritsbyapplyingthe“1/nrule” whennoinformationis

avail-able,21 thereby dividing the credit given to a complete paper by

the number ofcoauthors This is too harsh.Even ifthis

discour-ages what we term publication club,the credit allocated to each

21 Or by applying self-declared percentages of coauthors’ contributions, but this

could be Utopian

coauthortends to zeroin large teams(commonly found insome experimentaldisciplines).22

Thisis whywe propose a completely newapproach.We take intoaccount theproductivitygainsofparallelizationby introduc-ingaparallelizationbonusthatrewardscooperation—i.e.,the pos-sibilityofparallelwork—bymultiplyingthecreditallocatedtoeach coauthor

Correcting the1/n lawby introducingthe idea ofteambonus encouragesscholars tocooperate whereit isnecessary and justi-fied.Nevertheless, italso encouragescoauthors tobe involved in manypaperswhere theyalways perform thesame type oftasks However,itdependsonwhichscientificdomainisconsidered.The criticismmustberejectedindisciplineswhereextensiveresearch

is necessarily conductedby large teams The criticism should be accepted inall domains where theory predominates or where it

iscommonpracticeto publishalone(wehaveall hearditsaidof scholars that “they have to prove their worth”) The situation is morecontrasted whenthe approach can be partially experimen-tal or when a rangeof skills may be useful forwriting a paper Thus,thedegreeofparallelizationmayvaryfromonedisciplineto anotherbutastheintra-disciplinevariabilitymaybeverylarge it variesalsofromonepapertoanother.Therefore,thetruedegreeof parallelizationcannotbedeterminedexogenously,whichwouldbe artificial.So,in orderto determinethe parallelization bonus,one couldproposethateachteamofcoauthorsindicates,whenthe pa-perissubmittedtoajournal,howthelaborwasorganizedto pro-ducethepaper.However,thisimpliesthattheycollectandprovide

alotofinformationaboutthewaythey workedandisvalidonly forfuturepapers

Yet,evenifthelaboratorynotebook(whenitexists)couldhelp

toreveal the truth,the coauthors maycheat by exaggeratingthe

22 For example, for the 5154 coauthors of the Aad and ATLAS Collabora- tion, CMS Collaboration (2015) paper, the 1/ n rule allocates 1/5154 ࣃ 0.02% of a complete paper to each coauthor, a ridiculously low reward

Fig. 8 Limiting credit A Lmax

i and 1/ n for some values of n = N n is an integer

degreeofparallelism andaddingparallel tasks.Obviously,degree

of parallelism and team organization are linked So, after

show-ingthat we cannot havemore parallel tasksthan co-authors

be-causeofspecialization,thatis,N ≤ n(nbeingthenumberof

coau-thors and N the number of parallel tasks),23 we show that the

limitingcase (which has one preliminary task, N tasks that can

be performed in parallel, and one final task), is a Pareto

maxi-mumwitha credittoeachcoauthori ofA L

i=(N+2)/3nofa pa-perwrittenbyasingleauthor.So,evenifcoauthorsexaggerateby

declaring too manyparallel tasks, they are limitedby this

max-imum.In short, whateverthe true organization of the tasks and

theirparallelization,andevenifcoauthorstrytocheat,wecannot

givemoreto eachcoauthor Obviously,itcouldbe difficultto

ap-plythe(N+2)/3nruleforpast papersorwithoutthe agreement

ofpublishinghouses

Fortunately,themaximummaximorumcreditisreachedwhen

thenumber of parallel tasks isitself maximum, that is, equalto

thenumberofcoauthors,i.e.,N=n.So,weattributetoeach

coau-thoriacreditofA Lmax

i =(n+2)/3nofapaperwrittenbyasingle author.When this credit is attributed, coauthors donot wantto

cheataboutparallelization.This is2/3 ofa paperto each oftwo

coauthors,5/9toeachofthreecoauthors,etc.,upto1/3toeachof

averylargenumberofcoauthors:thisismuchmorefavorableto

coauthorsthan1/2,1/3ofapaper,etc.,uptozero,distributedby

the1/nrule.This“maximumparallelizationcredit” rulecanbe

ap-pliedtofuturepapersaswellaspast papers24orwithout any

agree-ment of publishing houses,evenwhencoauthorsdonot,orcannot,

23 For example, a team of three coauthors may not claim four parallel tasks

24 Most bibliometric indicators (impact factor, h-index ( Hirsch, 2005 ), etc.) have

been applied ex post , on the papers published before they were devised

communicatehowtheywereorganized,orwhenthejournaldoes not wish to require such information We only need to know the number of coauthors.For example,forthe 5154 coauthors of Aad and ATLAS Collaboration, CMS Collaboration (2015) ,weareableto credit each coauthorwith one-third ofa paperwritten by a sin-gleauthor.Thismightseemtoogenerousbutitisafairrewardas demonstratedabove.Itislargelyabove the0.02%allocatedbythe 1/nrulebutitisclearlybelowtheunfaircredit ofone fullpaper percoauthor

Wehopethatthepublicationofthepresentarticlewould gen-eratea progressivechangeinattitudes tothe necessityof under-standinghow each coauthored paper is produced in order to go beyondthesubtle butunclearandshifting rulesthat we havein some disciplines Itcould largelyhelp interdisciplinarityprogress Thisnewapproach isfeasible,andfairanditcredits genuine co-operationinacademicpublishing.Itcould giveriseto anewway

of comparing scholars, especially forrecruitment and promotion

“Lifting the hood” and looking inside the “machine” to see how thejob ofwritingpapers isdone,is new.It couldopen the door

tofuturedevelopmentsandcreatea newbranchofbibliometrics

Wehopethatthisnewrulewillencourageresearchesonhow aca-demicpapersareproducedbytakingintoaccounttheorganization

ofthetasksthatarenecessarytowriteapaper

References

Aad, G , & ATLAS CollaborationCMS Collaboration (2015) Combined measurement of the Higgs boson mass in pp collisions at s = √ 7 and 8 tev with the ATLAS and CMS experiments Physical Review Letters, 114 , 191803 Published 14 May, 2015 Brandão, A A (2007) Small versus big teamwork: E-letter response to Stefan Wuchty et al Science Online: 02 august 2007

Buchanan, J M (1965) An economic theory of clubs Economica, New Series, 32 (125), 1–14

Pleasecitethisarticleas:L.deMesnard,Attributingcredittocoauthors inacademicpublishing:The1/nrule,parallelization,andteam