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1、TheoryDec.DOI10.1007/s11238-011-9281-4InternalslackeningscoringmethodsMarcoSlikker·PeterBorm·RenévandenBrink©TheAuthor(s)2011.ThisarticleispublishedwithopenaccessatSpringerlink.comAbstractWedealwiththerankingproblemofthenodesinadirectedgraph.Thebilateralrelationshipsspecifiedbyadirectedgraph
2、mayreflecttheoutcomesofasportcompetition,themutualreferencestructurebetweenwebsites,oragroupprefer-encestructureoveralternatives.Weintroduceaclassofscoringmethodsfordirectedgraphs,indexedbyasinglenonnegativeparameterα.Thisparameterreflectstheinter-nalslackeningofanodewithinanunderlyingiterati
3、veprocess.Theclassofso-calledinternalslackeningscoringmethods,denotedbyλα,consistsofthelimitsofthesepro-cesses.Itisseenthatλ0extendstheinvariantscoringmethod,whileλ∞extendsthefairbetsscoringmethod.Methodλ1correspondswiththeexistingλ-scoringmethodofBormetal.(AnnOperRes109(1):61–75,2002)andca
4、nbeseenasacompromisebetweenλ0andλ∞.Inparticular,anexplicitproportionalityrelationbetweenλαandλ1isderived.Moreover,theinternalslackeningscoringmethodsareappliedtothesettingofsocialchoicesituationswheretheygiverisetoaclassofsocialchoicecor-respondencesthatrefineboththeTopcyclecorrespondenceand
5、theUncoveredsetcorrespondence.M.Slikker(B)SchoolofIndustrialEngineering,EindhovenUniversityofTechnology,P.O.Box513,5600MB,Eindhoven,TheNetherlandse-mail:M.Slikker@tue.nlP.BormCenterandDepartmentofEconometricsandOR,TilburgUniversity,P.O.Box90153,5000LE,Tilburg,TheNetherlandsR.vandenBrinkDepa
6、rtmentofEconometrics,VUUniversityandTinbergenInstitute,DeBoelelaan1105,1081HV,Amsterdam,TheNetherlands123M.Slikkeretal.KeywordsDigraphs·Scoringmethods·Invariantmethod·Fairbetsmethod·SocialchoicecorrespondencesJELClassificationD711IntroductionRecently,bytheapplicationofscoringandrankingmethod
7、sinsearchenginesontheWorldWideWeb(suchasPageRankintheGooglesearchengine),scoring,andrank-ingmethodsfordirectedgraphsregainedattentionintheliterature.ForanextensivesurveyofvariousscoringmethodswerefertoLaslier(1997).Kendall(1955)studiedamethodfromWei(1952
