Optimal Assignment of Durable Objects to Successive
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Optimal Assignment of Durable Objects to Successive Agents ? Francis Bloch † Nicolas Houy ‡ September 29, 2009 Abstract This paper analyzes the assignment of durable objects to successive generations of agents who live for two periods. The optimal assignment rule is stationary, favors old agents and is determined by a selectivity function which satisfies an iterative func- tional di?erential equation. More patient social planners are more selective, as are social planners facing distributions of types with higher probabilities for higher types. The paper also characterizes optimal assignment rules when monetary transfers are allowed and agents face a recovery cost, when agents' types are private information and when agents can invest to improve their type. JEL Classification Numbers: C78, D73, M51 Keywords: Dynamic Assignment, Durable Objects, Revenue Management, Dy- namic Mechanism Design, Overlapping Generations, Promotions and Intertemporal Assignments. ?We are grateful to Gabrielle Demange, Philippe Jehiel and seminar audiences at NUS (Singa- pore) and PSE (Paris) for their comments. Correspondence: and . †Department of Economics, Ecole Polytechnique, 91128 Palaiseau France, tel: , ‡Department of Economics, Ecole Polytechnique, 91128 Palaiseau France, tel: , nhouy@free.
- agent
- transfer rules
- vickrey-clarke-groves mechanisms
- optimal assignment
- assignment policy
- when monetary transfers
- all agents
OptimalAssignmentofDurableObjectstoSuccessive
Agents
∗
FrancisBloch
†
NicolasHouy
‡
September29,2009
Abstract
Thispaperanalyzestheassignmentofdurableobjectstosuccessivegenerationsof
agentswholivefortwoperiods.Theoptimalassignmentruleisstationary,favorsold
agentsandisdeterminedbyaselectivityfunctionwhichsatisfiesaniterativefunc-
tionaldi
ff
erentialequation.Morepatientsocialplannersaremoreselective,asare
socialplannersfacingdistributionsoftypeswithhigherprobabilitiesforhighertypes.
Thepaperalsocharacterizesoptimalassignmentruleswhenmonetarytransfersare
allowedandagentsfacearecoverycost,whenagents’typesareprivateinformation
andwhenagentscaninvesttoimprovetheirtype.
JELClassificationNumbers:C78,D73,M51
Keywords:
DynamicAssignment,DurableObjects,RevenueManagement,Dy-
namicMechanismDesign,OverlappingGenerations,PromotionsandIntertemporal
Assignments.
∗
WearegratefultoGabrielleDemange,PhilippeJehielandseminaraudiencesatNUS(Singa-
pore)andPSE(Paris)fortheircomments.Correspondence:
francis.bloch@polytechnique.edu
and
nhouy@free.fr
.
†
DepartmentofEconomics,EcolePolytechnique,91128PalaiseauFrance,tel:+331693330
45,francis.bloch@polytechnique.edu
‡
DepartmentofEconomics,EcolePolytechnique,91128PalaiseauFrance,tel:+331693330
15,nhouy@free.fr
1
1Introduction
Thispaperconsidersdurableobjectswhicharesuccessivelyreassignedtoagents.
Theprimeexampleofobjectswhichareregularlyreassignedarepositionsinor-
ganizationsandbureaucracies,likeexecutiveo
ffi
cesordiplomaticpostings.Other
examplesincludeo
ffi
ces,roomsindormitories,computerservers,largescalescien-
tificequipmentliketelescopesandparticleaccelerators,hotelroomsandrentalcars.
Inalltheseexamples,agentsaregiventemporarypropertyrightsoverthedurable
objectforagivenperiod,andcannotbeforcedtoreturntheobject.Thesetempo-
rarypropertyrightscreateaconstraintontheoptimalassignmentpolicychosenby
abenevolentsocialplanner,andourobjectiveinthispaperistocharacterizethe
e
ffi
cientdynamicassignmentofadurableobjecttosuccessivegenerationsofplayers
takingintoaccounttheseindividualrationalityconstraints.
Weconsiderenvironmentswhereoverlappinggenerationsofagentsenterthemar-
ketdeterministically,andagentsareassignedtheobjectfortheirentirelifetime.
Agentsdi
ff
erintheirvaluationsfortheobjects,ortheirqualificationsforthepo-
sitions.Agentscharacteristicsarethustwo-dimensionalandincludeatypewhich
determinesthevalueoftheassignment,andanagewhichdeterminesthelengthofthe
assignment.Ourobjectiveinthispaperistocharacterizeoptimalassignmentrules
inthistwo-dimensionalmodel,andconstructrevelationmechanismswhenagents’
typesareprivatelyknown.
Thebasictrade-o
ff
betweenageandvalueisbestunderstoodinatwo-period
overlappinggenerationsmodel.Whenassigningthegoodtoanoldoryoungagent,
thesocialplannermakesthefollowingcomputation.Assigningthegoodtotheold
agentforoneperiodhasapositiveoptionvalue,asthegoodcanbereassignedatthe
endoftheperiod;assigningthegoodtotheyoungagentfortwoperiodsprevents
theplannerfromreassigningthegoodimmediately.Hence,iftheoldandgood
agentswereofthesametype,itwouldalwaysbeoptimaltoassignthegoodtothe
oldagent
.
1
Thislineofreasoningshowsthat,foranytype
θ
oftheyoungagent,
theplannerwillprefertogivetotheanyoldagentoftypegreaterorequalto
φ
(
θ
),
where
φ
(
θ
)
<
θ
.
Themaincontributionofthepaperistocharacterizetheselectivityfunction,
φ
(
θ
),whichisadoptedintheoptimalassignmentpolicy.Thisfunctionisdefinedby
aniterativefunctionaldi
ff
erentialequation,whichissimilartotheequationsused
inphysicsandmathematicstostudydynamicalsystemswithstate-dependentde-
1
Theoptionvalueofgivingapositiontoanolderagentisawelldocumentedhistoricalfact.
Forexample,thehistoryofthepapacyrecordsanumberofelectionswherecardinalsvoluntarily
chosetheoldestcandidate.Oftentimes,these”transitionpopes”turnouttobethemostenergetic
ande
ff
ectivepopesoftheirtimes.SeeCollins(2009)andhisaccountofthereignoftwofamous
”transitionpopes”,JohnXXII(1316-1334)andJohnXXIII(1958-1965).
2
lay(Eder1984).Whilewecannotprovideanaanalyticalsolutiontothefunctional
di
ff
erentialequation,weproveexistenceanduniquenessoftheoptimalassignment
policyandshowthattheselectivityfunctionisincreasingandconvex.Weillustrate
theoptimalassignmentpoliciesdorunfiromandquadratictypedistributions,and
derivecomparativestaticspropertiesofthesolution.Undersomeregularitycon-
ditions,weshowthatwhenthediscountfactorincreases,orwhenthedistribution
oftypesisshiftedsothathighertypeshaveahigherprobability,thesocialplanner
becomesmoreselective,andassignstheobjecttotheyoungagentlessoften.
Inthesecondpartofthepaper,weconsiderdi
ff
erentextensionsofthemodel.
First,weanalyzetheoptimalassignmentpolicywhenmonetarytransfersareallowed
andagentscanbecompensatedwhentheyreturntheobject.Ifthereisnorecovery
cost,theoptimalassignmentpolicyisidenticaltothefirst-bestpolicywithoutindi-
vidualrationalityconstraints:theobjectisassignedateveryperiodtotheagentwith
thehighestvalue.Ifanoldagentwhocurrentlyholdstheobjecthasasmallervalue
thantheyoungagent,theyoungagentcaneasilytransfermoneyinreturnforthe
objectandtheindividualrationalityconstraintceasestobebinding.Withpositive
recoverycosts,theoptimalassignmentstrategybecomesmorecomplex,andinvolves
twodi
ff
erentselectivityfunctions,onewhichisusedatperiodswherenoagentholds
theobject,andonewhichisusedwhentheoldagentholdspropertyrightsoverthe
objectandneedstobecompensated.Wecharacterizetheoptimalassignmentpoli-
ciesassolutionstosystemsofdi
ff
erentialfunctionalequations,bothwithfixedand
proportionalrecoverycosts.Wealsoillustratethesecomplexassignmentstrategies
fortheuniformandquadraticdistributions.
Inasecondextensionofthemodel,weanalyzedirectrevelationmechanisms
whenthetypesoftheagentsareprivatelyknown.Giventhetimestructureofthe
assignmentrule,wecanbuilddi
ff
erentmodelsofrevelationmechanisms.Inthefirst
model,wesupposethatagentsareaskedtorevealtheirtypeswhentheyentersoci-
etyasyoungagents,whethertheobjectisreassignedornot.Inthesecondmodel,
weassumethatagentsareonlyaskedtorevealtheirtypes(andpayatransfer)at
periodswherethegoodisreassigned.Forbothmodels,weusestandardarguments
tocharacterizetransferrulesimplementingtheoptimalassignmentpolicy.Notsur-
prisingly,thesetransferrulesinvolveastepfunction,wheretransfersjumptoaflat
positivevaluewhentheagent’stypereachesthethresholdvalueforwhichthegood
isassignedtoher.
Finally,weinvestigatetheagents’incentivestoinvestinordertoimprovetheir
typesbetweenthetwoperiodsoftheirlives.Inthatmodel,ayoungagentwhodoes
notreceivethegoodinthefirstperiodhasanincentivetoinvestinhistypeboth
inordertoincreasetheprobabilityofreceivingtheobjectwhenold,andtoincrease
3
thevalueofthematch.Inthismodel,thesocialplanners’optimalassignmentpolicy
andtheagents’incentivestoinvestaredeterminedsimultaneously,asthesolutions
toasystemofdi
ff
erentialequation.Again,whileweareunabletosolvethesystem
analytically,weprovideanumericalillustrationofthesolutionsfortheuniformand
quadraticdistributions.
Axiomaticcharacterizationsofassignmentrulesfordurablegoodshaverecently
beenstudiedbyKurino(2008)andBlochandCantala(2008).Kurino(2008)con-
sidersadynamicextensionofAbdulkadirogluandSo¨nmez(1999)’sstudyofhouse
allocationwithexistingtenants–thefirstexampleofanassignmentproblemwith
individualrationalityconstraints–,andanalyzeswhethertherulesproposedinthe
staticpaperstillsatisfye
ffi
ciencyandincentivecompatibilityinthedynamiccontext.
BlochandCantala(2008)consideramodelwhereagentsareassignedtodi
ff
erent,
verticallyrelatedobjects,andcharacterizeMarkovianassignmentruleswhichsat-
isfymyopice
ffi
ciencyandfairness.Bycontrast,inthispaper,weconsiderasimpler
modelwhereagentsonlylivetwoperiodsandcanonlybeassignedonegood.Inthis
simplermodel,weareabletocharacterizedynamicallye
ffi
cientrules,andtodiscuss
theincentivepropertiesoftransferrules.
Thispaperisalsorelatedtotherapidlygrowingliteratureondynamicmecha-
nismdesign.ParkesandSingh(2003),AtheyandSegal(2007),Bergemannand
Valima¨ki(2006)andGershkovandMoldovanu(2008a,2008b,2008c)studydy-
namicassignmentproblems,whereagentsentersequentially,andparticipateina
Vickrey-Clarke-Grovesrevelationmechanismwhichdeterminestransfersandgood
allocations.TheyshowthatVickrey-Clarke-Grovesmechanismsandoptimalstop-
pingrulescanbecombinedtoobtaine
ffi
cientdynamicmechanisms.Inthesemod-
els,objectscanonlybeassignedonceatthetimeofentry.Someofthesestudies
(likeGershkovandMoldovanu(2008b,2008c))distinguishbetweenbenevolentand
revenue-maximizingplanners.Whenagents’typesareknown,theliteratureonyield
managementinmanagementscienceandoperationsresearch(seeTalluriandVan
Rysin(2004))providesanin-depthstudyofoptimalpricingstrategies.
Therestofthepaperisorganizedasfollows.WeintroducethemodelinSection
2.Weanalyzee
@
