Efficiency and traffic safety with pay for performance in road transportation

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Transportation Research Part B

journalhomepage:www.elsevier.com/locate/trb

Efficiency and traffic safety with pay for performance in road transportation

Harald Bergland

^a

, Pål Andreas Pedersen

^b,∗

aBusiness School, University of Tromsø – The Arctic University of Norway, Norway

bNord University Business School, Norway

a r t i c l e i n f o

Article history:

Received 18 March 2019 Revised 9 October 2019 Accepted 17 October 2019 Available online 1 November 2019 Keywords:

Road transportation Traffic safety Efficiency Game theory Principal-agent model Pay for performance

a b s t r a c t

Weproposeatheoreticalmodelinordertostudythebehaviorofaroadtransportcom- panyand adriver.The driveris supposed toface apay forperformance contract. The expectedprofitforthe company and the expectedutilityfor the driverdependonthe inputchosenbythemselvesandtheotheractor.Byanalyzingthepossibleinteractiongo- ingonbetweentheactorsinasimultaneousgameandthetwopossible leader-follower games,itisseenthattheeffortscouldbestrategiccomplements,independentorstrategic substitutes.Incaseswheretheeffortsarestrategiccomplements,andtheexpectedprofit forthecompanyandtheexpectedutilityforthedriverareincreasingintheotheractor’s effort,leader-followergamestriggerhigheraccidentrisksthanthesimultaneousgame.If effortsarestrategiccomplements,andtheexpectedprofitandutilityaredecreasinginthe otheractor’seffort,leader-followergamesproduceloweraccidentrisksthanwhentheac- torsmovesimultaneously.Presumingthatthetransportcompanyistheprincipalandthe driveristheagent,wededuceanoptimalpaycontract.Theoptimalcontractischaracter- izedbyapayforperformancecontractwherethedriver’sshareofnetrevenuebecomes higherthehigherinfluenceamarginalincreaseinherefforthasonthenetrevenue,the lowerinfluenceamarginalincreaseinherefforthasontheprobabilityofaccidents,and thelowerthelossthecompanyexperiencesifanaccidentoccurs.Whensuchacontractis used,thedriverfacesasituationwherethetransportcompany’sinterestsareperfectlyin- ternalized,meaningthatthecompanyalsomaximizesthesumoftheexpectedprofitand utility.However,sinceaccidentsalsomeancostsforothersapartfromthedriverandthe company,publicregulationisneededtoensureoverallwelfare.

© 2019TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense.

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1. Introduction

Companiesthat supply professional transport serviceson roads employ workerswho drive the carryingvehicles. The wagecontractsinroadtransportation,asinthelabormarketgenerally,mightvaryfromfixed paymentper timeunit toa totalflexiblewage,dependentonobservableandverifiablevariablescorrelatedwiththeworker’sperformance.Thetheoret- icalargumentsforinsistingon somekindofpayforperformance wagecontractsstemfromprincipal-agenttheory where theworkersbecome motivatedto behaveefficientlywhen their wagesare dependent onthe company’s economicresult.

∗ Corresponding author.

E-mail addresses: harald.bergland@uit.no (H. Bergland), pal.a.pedersen@nord.no (P.A. Pedersen).

https://doi.org/10.1016/j.trb.2019.10.005

0191-2615/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license.

( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

Flexiblewagepaymentisalsodesirableinordertoensurethathighskilledandqualifiedworkersfinditattractivetosign contractswiththecompany.Anotheraspectofwagepaymentoftenhighlightedintheprincipal-agentliteratureisthatthe wagepaymentcontractsshouldbedesignedinordertosharerisksamongthecompanyandtheworkers.(E.g.Alchianand Demsetz 1972; Stiglitz1974; Holmstrom1979; LazearandRosen 1981;Stiglitz 1987; JensenandMurphy, 1990;Bergland 1995;GrepperudandPedersen2006).Anessentialassumptionintheprincipal-agentmodelsisthatthecompanydoesnot directlyobservetheworkers’ effort,implyingthereisapervasivetendencyforshirkingamongtheworkers.Hence,an im- portant conclusionfromsuchmodels isthat adoptingsomekindofpayforperformance inwagecontractsisanefficient wayforthecompanytoensurethatworkershaveincentivestobeproductive.

Onthe other hand,researchersengaged intraffic safetyhave questionedintheoretical andempirical studies whether payment systems rewarding drivers’ production performance may influence the traffic safety outcomes negatively, see, for instance, Freyer et al. (1997), Johansson et al. (2010), Williamson and Friswell (2013), Soccolich et al. (2013), Mooren et al. (2014), Newnamand Goode (2015), Phillipset al. (2015), Thompson et al. (2015), Nævestad et al.(2015), NewnamandOxley (2016),Warmerdam etal.(2017)andNævestadetal.(2018).Oneofthe pointsmade intheseworks are that ifthe drivers face pay forperformance contracts, their level of effortwill be determined by their eagerness to makemoney.Forinstance,inordertoobtainhighwages,theywilldrivefasterandpaylessattention,implyingthattraffic accidentrisksincrease.

Nævestad etal.(2018)give a comprehensiveliterature overviewonresearch regarding trafficaccident risks inprofes- sional roadtransportation. In their discussion,the traffic accident risk factors fall into differentgroups. Theydistinguish betweenrisk factorsrelatedto thedrivers,the vehicles,the roads,theroadenvironment, theworkingconditionsandfa- tigue amongdrivers.Theirarticle focusesinparticularon developingmanagement strategiesforreducing traffic accident risksamongroadtransportoperators.Otherresearchershavediscusseddifferentregulatorytoolsthatmaydecreaseaccident risks,see,forinstance,BjørnskauandElvik(1992),PerssonandOdegaard(1995),Peirsonetal.(1998),Jara-Diazetal.(2000), Dickersonetal.(2000),JørgensenandPedersen(2002),Ryeng(2012),Bentham(2015),Haddaketal.(2016) andBergland andPedersen(2018).Researchershavealsobeeninterestedinthedrivers’perceptionsandreactionstotheriskstheyface– see,forinstance,Jørgensen(1993),JørgensenandPolak(1993)andVukinaandNesti´c(2015).Gametheoreticalapproaches to the possible strategic interactions going on between road users and the consequences for accident risks have also beenstudied– see,forinstance,Pedersen (2003),AnderssonandAuffhammer(2014),Elvik(2014),Arbisetal.(2016) and Bjørnskau(2018).

Eventhough thetrafficsafetyinterdisciplinaryresearch sofar hasrevealedmanypossiblerisk factorsandsafetychal- lengesinprofessionalroadtransportation,toourknowledgeno-onehasconductedaformaleconomicallyanalysisdescrib- inganddiscussingpayforperformance contractsbetweenatransportcompanyandadriver.Carryingoutsuchan analysis isinteresting forseveralreasons.Economictheory emphasizes thefavorableefficiencypropertiesofflexible paymentsys- temsto thedrivers,whiletrafficsafetyresearchersemphasizethe negativeimpact suchcontractsmighthaveonaccident risks.Hence,thereisa needtoclarifythereasoningbehindthesedifferentviews.The factthat payforperformance con- tractsseemtobe widelyusedinprofessionalroadtransportationisanotherreasonwhyeconomicmodelingisinteresting.

Thirdly,eventhoughmanyresearchershavebeenengagedinmodelingstrategicinteractionbetweenroadusers,application ofgametheorytoanalyzetherelationbetweenmanagementandworkerswithinthetransportcompaniesisnotthatoften conducted. Hence, withtheaim offillingthis gap,wewill analyzeboth the efficiencyandtraffic safetyconsequencesof practicingpayforperformance wagecontracts. Inparticular,we willtryto identifysituationswherepayforperformance leadsto relatively highaccident risks,andsee whetherthe opposite also could happen. Moreover, we are interesting to deduceanoptimalwagecontract,anddiscusswhether,andeventuallytowhatextent,suchacontractshouldbe basedon payforperformance.

Thefirstpartofouranalysisstudiestherelationbetweenthetransportcompanyandthedriverasa gamefora given wagecontract,whereeachoftheactorschooseseffortsaffectingbothproductivityandtrafficsafety.Thediscussioncontains thetraditionaldefinitionofstrategiccomplementsandsubstitutesintheindustrialorganizationliterature(seeforinstance Bulowetal.,1985).Weidentifysituationswhereahigherlevelofeffortfromoneoftheactorsinducesahighereffortfrom theother actorandtermsuch casesaseffortsbeingstrategiccomplements.Theoppositesituations,wherean increasein oneoftheactors’effortgivesalowereffortfromtheotheractor,aretermedcaseswheretheeffortsarestrategicsubstitutes.

Suchamodelingrepresentanextensionandrefinementoftheframeworkusedfordiscussingcontractsinthetransportation field. Moreover,we usethisgametheoreticalanalysisasan inputfordeducingan optimalwagecontractin aprincipal– agentmodel,wherethe transportcompany isthe principalandthe driveristhe agent.Unlikethe traditionalprincipal– agentmodels, where theprincipal only chooses thewage contract based onthe agent’sparticipation constraintand her behavioralreactionondifferentcontracts, theprincipalinourmodelissupposedtoadjusthereffortthataffectsboththe productionandtrafficaccidentrisks.

InSection2we proposeaconceptualmodelwhereatransportcompany mayaffectitsexpectedprofitandthedrivers expectedutilityby choosingeffortinproduction.The driver’seffortinproductionalsoinfluences thecompany’sexpected profitandherownexpectedutility.Section3containsadiscussiononthepossiblestrategicinteractionsgoingonbetween thecompany andthedriverfora givenwagecontract.Using a traditionalprincipal-agentframework,we deducetheop- timallinearwagecontractbetweenthecompanyandthedriverinSection 4.InSection 5wediscussthedriver’sandthe company’s behavior inan overall social welfare perspective. Finally, Section 6 offers a summary of the main theoretical conclusionsanddiscusseslimitationsandpossibleextensionsofouranalyses.

2. Themodel

Toanalyzebehavior intheroadtransport industry,we presentasimple modeldescribingthe company’s choiceofan inputinproductionandadriver’seffortatwork.Supposethat atypicaltransportcompanyfacesanexogenousproduction technologyandexogenouspricesforitstransportservices.Thecompany’sscaleofproductionisdeterminedbythechoiceof thenumberofvehiclesithas,i.e.duplicationofidenticalproductionunits.Tosimplify,weassumethatthecompanymakes useofonedriverforeachvehicle.Whentheseassumptionshold,wecanformulateanetrevenuefunction,R,dependenton theoperationalinputandeffortvalueschosenbythecompanyandby thedriver.Rmeasurestheincomefromproduction minusoperationalcoststhecompanyhasinthesameperiod,exceptthecostofhiringthedriverandthecoststhecompany suffersifanaccidentoccurs.¹ Supposetsymbolizestheeffortlevelchosenbythecompanyandetheeffortlevelchosenby thedriver,then R=R(t,e).Ifwetake intoaccount laborandaccident costsaswell, theexpectedprofitforthefirminthe period,

π

^,becomes

π

^=R

(

^t,e

)

^−w−p

(

^t,e

)

^{LC (1a)}

wherewisthewagepaidtothedriverintheperiod,pistheprobabilityforantrafficaccidentandL^Cisthelossexperienced bythetransportcompanyifanaccidentoccurs,forsimplicityassumedtobeunaffectedbytheparties’efforts.²L^Cmeasures thecostsofdamagetovehicleandcargoifanaccidentoccurs,andpossiblelossesduetodowntimeandassignmentsthat thecompanymayexperienceinthecaseofanaccident.Moreover,weonlyconsideronetypeofatrafficaccident,definedby anexogenousprobabilityfunction,p=p(t,e).Similartothenetrevenue,thismeansthattheaccidentrisk isdependenton theparties’effortlevels.Regardingtherevenueandprobabilityfunctions,thefollowingpropertiesaresupposedtohold:³

Rt>0, Re>0,Rtt<0,Ree<0,Ret>0 pt>0, pe>0,ptt>0,pee>0andpet>0 (1b) First,(1b)meansthat increasedindividual efforts implyhigherproductionvalue(R_t > 0andRe > 0),butsteadilyata lowerrate(Rtt <0andRee< 0).Moreover, itseemsreasonabletoassume thathighereffortfromthecompanyincreases themarginalproductivityfromthedriver’seffort⁴(Ret>0).Additionally,itisassumedthathigherproductioneffortsfrom thepartiesmakethetransportoperationsontheroadmoredangerous(p_t >0andp_e >0) andthattheseeffectsbecome strongerthehighertheeffortsareoriginally(ptt>0andpee>0).Itisalsoassumedthattheincreaseinaccidentprobability forhighereffortonthepartofthedrivergrowsasthecompanystepsupitsowneffort(pet >0).

Inourreasoning,wecould thinkoft asa variabledescribingtheoperationalintensitycontrolled bythecompany. For instance,thisvariablecouldbemeasuredbythetotalhoursofoperationperperiod,thenumberoftransportassignments, transportedvolumes,ordistance,oranindexthatbuildsonacombinationofallofthesemeasures.Moreover,weconsider easthedriver’sintensityinundertakingthetransporttask.Thedriver’sintensityrelates,forinstance,totrafficspeed,effort andtimeusedinloadingandunloadingthevehicleinoperation,thedriver’suseofresourcesinplanningandpreparingfor transportassignments,oranindexthatbuildsonacombinationofallofthesevariables.Itisoftenthecasethatthedriver doesnotexertasingledimensionaleffort.Shemaybeinvolvedinmanyrelatedactivitiesassociatedwithundertakingthe transport mission.The driver performs multi-tasking, wherethese tasks mightaffect both traffic safety andproductivity inamorecomplexpatternthan assumedin (1b).Effort inonetask,forinstancepreparingvehicleanduseofequipment (e.g.snow chains),mightreduceshortrunproductivityandatthesametimeincreaseroadsafety.Hence,consideringthis specifictaskisolated,the firstorderderivative withrespecttothedriver’s effortin (1b) aretheopposite. Specifyingonly oneactionvariable,interpretedasanindexofcombinedvariablesforeachoftheactors,representofcourseasimplification ofreality.However,we havechosen tokeepthemodelastractable andsimpleaspossible,inordertoconcentrateonthe strategicinterdependenciesbetweendriver andcompany.⁵ Now,letusassume that thedriver facesa linearpaycontract givenby

w=a+bR

(

t,e

)

⁽²⁾

whereais afixed wage,independentofthecompany’s revenueandb isthepaymentto thedriverper netrevenueunit obtainedintheactualperiod.Werestrictourselvestodiscussingcaseswhere0≤b≤1,whereb=0meansfixedpayment, andb=1 impliesaperfectlyflexible wagecontract. Inpracticeone oftenfinds thatdriver’s wageispartlydependent on productionperformance, 0< b< 1.In reality,thedriver’sperformance can be measuredby,forinstance, thenumberof

1In reality, the company might have both revenue and costs in inserting efforts in production. Suppose for instance that R = R ( e, t ) = F ( t, e ) −v ( t, e ), where F ( t, e ) and v ( t, e ) measure the company’s revenue and costs respectively. Without loss of generality, however, we have dropped to specify the costs, and are interpreting R ( e, t ) as a net revenue for the company.

2In general, accident losses are often presumed to be positively correlated with less risk attention and more production intensive individual behavior, see, for instance, Jørgensen and Pedersen (2002) and Pedersen (2003) .

3Here and throughout the text we use the conventional description A x= ^∂_∂^A_x in order to simplify our equations

4In ordinary production theory the inputs are often regarded as technical complements in production.

5In traffic safety research, it is sometimes considered that the driver assigns an effort to shortening the travel time and an effort f or securing the traffic safety simultaneously, see for instance in Risa (1994) and Jørgensen and Pedersen (2002) . In the latter work it is modeled a level of care, additional to the choice of speed among drivers, where an increased caretaking means lower accident risks. For further examples on multitask environments in principal – agent modeling, see also Holmstrom and Milgrom (1991) and Laffont and Martimort (2002) . We do not pursue this extended analysis here.

kilometresdriven,thenumberoftonscarried,thenumberoftonkilometresproduced,orothervariablesconcerning pro- duction.Forsimplicity,herewediscussawageconsistingofafixedelement,andanelementdependentonthecompany’s net revenue.This means thatwe suppose that the netrevenue stemming fromthe driver’s productionactivityis ‘highly correlated’withthedriver’sproduction.Moreover,thedriver’sexpectedutilityfunctionissupposedtobegivenby

U=w−g

(

^e

)

−p

(

t,e

)

^{LD (3a)}

whereg(e)isthedriver’sdisutilityfunctionofeffort,andL^D isthelossexperiencedbythedriverifanaccidentoccurs.The driver’slossmeasureseconomicfactorsaspossiblelossesinbonusesandoccupationandfeescoveredbythedrivershould anaccidenthappen.Inaddition,L^Dalsocontainsphysicalandmentalhealthconsequencesthatmightoccurintheeventof anaccident.Inaccordancewiththestandardprincipal-agentmodels,itissupposedthat thedisutility ofthedriver’seffort isconvexlyincreasingineffort,i.e.

ge>0andgee>0 (3b)

Tobeawareoftheparties’interrelationgivenbythemodelin(1)-(3),letusseehowtheexpectedprofitforthecompany isaffectedbymarginalchangesinthedriver’seffort.Byusing(1)and(2),anddifferentiatingtheexpectedprofitw.r.t.e,we obtain

π

e=

(

^1−b

)

^Re−peLC≥

(

^<

)

^0as

(

^1−b

)

^Re≥

(

^<

)

^{peLC (4)}

From (4)we seethatamoreintense productioneffortfromthedriverhastwodifferenteffectsontheexpectedprofit for thecompany. The company obtains an increase in thepart of the netrevenue that it keeps, (1−b)Re.However, the increasedprobability ofaccidentsthatfollowsfromahigheremeansanegativeimpactonexpectedprofits,peL^C.Whether thesumispositive,zero,ornegative,indicating whetherthecompanywouldprefer amoreintensedriver’s effortornot, isdependent onthesizeoftheseeffects.Moreover, wenote thatthe higherbis,thelower Re is,thehigherpeis,andthe higherL^C is,themorelikelyitbecomesthatthecompany’sexpectedprofitreducesaseissteppedup.

Letusnowdoasimilaranalysisonhowmarginalchangesintaffectthedriver’sutility,i.e.

Ut=bRt−ptL^D≥

(

<

)

^0asbRt≥

(

<

)

^{ptLD (5)}

From (5)weseethat itisalsoambiguouswhetherthedriver wouldprefer amoreorlessintense transportoperation fromthecompany.Ifthedriverobtainsanincreaseinhervariablewagethatmore(less)thancompensatesfortheincreased expectedlossesfollowingfromahighert,herexpectedutilitywillincrease(decrease).Itfollowsthatitbecomesmorelikely thatthedriverispositivelyaffectedbyanincreaseintforhighervaluesofbandRtandlowervaluesofpt andL^D.

In theway themodel is constructed,the company andthe driver havepartially the sameinterests, even though the companypreferslowwagestohighwages,whilethedriverhastheoppositepreference.Firstly,itfollowsthatbothparties prefer low accident risk to highaccident risk, and, secondly,when 0< b < 1, they are both interestedinhigh netrev- enue.⁶ Theseinterdependenciesbetweentheactors’economic interestsare thebackgroundforstudyingpossiblestrategic interactionbetweenthepartieswhentheyaresupposedtoactrationally.

3. Optimalbehaviorandstrategicinteractionsforagivenlinearpaycontract

An exogenous pay contract, defined by a andb in(2), might stem fromnegotiations betweenrepresentatives of the drivers’ labourunionandthe roadtransportation companies’ industryorganization. Inorder tostudythe transport com- pany’s andthedriver’s optimalbehaviorfora givenwage contract,we first lookintothe problemwherethe partiesare supposed to choose their efforts simultaneously, without knowing the other’s actions. Secondly, we look into the game wherethecompanychoosesits effortfirst,andthedriver,afterobservingtheother’seffort, adjustshereffort.Thirdly,we studythecasewherethedriveractsasa leader,choosinghereffortlevelfirst,andthecompany,afterbeingawareofher effort,choosesitsoperationintensity.Whichoftheproposedgamesismostrealisticdependsonthecontext.Thesimulta- neousonedescribesasituationwherethecompany’smanagementisunawareoftheactualchoicemadebythedriver,and, analogously,thedriverdoesnotknowthemanagement’sdecisionregardingthecompany’schoiceofinput.Ifthecompany’s decisioncomes first andthedriver knowsthisdecision, the company becomestheleader andthe driveris thefollower.

Ifthemanagersofthecompany waittoseethe driver’sbehaviorbeforechoosing thecompany’s effort,we havethecase wherethedriveristheleaderandthecompanyisthefollower.Afterwehavededucedthesolutionsofthesethreegames, wewillcomparethedifferentcasesanddiscusssimilaritiesanddifferencesintheoutcomes.

3.1. Simultaneousmoves

Usingthedefinitionsin(1)and(2),maximizingtheexpectedprofitforagivenwagecontractandagivenlevelofeffort chosenbythedriver,givesusthefollowing1.and2.orderconditions

π

t=

(

^1−b

)

^{Rt−ptLC=0and}

π

tt=

(

^1−b

)

^{Rtt−pttLC<0 (6)}

6If b = 0 , the driver would have no interest in doing more than necessary (to keep her job), and if b = 1 , there would be no incentive for the company’s management to do something that affects the net revenue.

whereit follows directly from the assumptions made in(1b) that the2.order condition holds.The optimallevel of t is characterizedbyasituationwherethecompany’sincreaseinitsshareoftherevenueforthelastunitinsertedintransport operations,(1−b)Rt,equalstheexpectedincreaseinitslossifanaccidentoccurs,ptL^C.

Analogously,usingthedefinitionsin(2)and(3),maximizing theexpectedutilityforagivenwagecontractandagiven leveloft,meansthatthe1.and2.orderconditionsforanoptimalecanbewrittenas

Ue=bRe−ge−peL^D=0andUee=bRee−gee−peeL^D<0 (7) whereitfollowsfromtheassumptions madein(1b) and(3b) thatthe2.orderconditionissatisfied.Theoptimaleisob- tainedwhenthedriver’spartoftheincreasednetrevenueforaunit effortlevel,bR_e−g_e,isequaltothe higherexpected lossfromaccidentsthesameeffortunitgives,peL^D.Thefirstexpressionsin((6)and(7)aretwoequalitiestogetherdefining theNash-equilibriuminthegamewherethepartiespicktheireffortlevelssimultaneously.Thesolutionofthesimultane- ousgameillustratesthattheactorsforagivenpayforperformancecontractarebothconcerned aboutthepositiveeffects higherefforts haveontheir portionof theincomeandthenegativeconsequences flowingfromexpectedhigheraccident risks.Inthefollowing,wedenotetheequilibriumvaluesinthesimultaneouscaseby(t^s,e^s).

Result1. Fora givenpay forperformance wagecontract, thetransport company andthe driverina simultaneous game injecteffortsinproductionuntiltheirindividualmarginalnetgainsthroughincreasedincomeareequaltotheirexperienced marginalincreaseinexpectedlossescausedbyahigheraccidentprobability.

3.2.Thecompanymovesfirst

We now assume that the transport company chooses its operational effort, t, before the driver chooses her level of effort. We study thiscase by backward induction, meaning that we first deducethe driver’s reaction to differentlevels oft. Implicitlythefirst equation in(7) definesthedriver’s response tothe company’s actualchoice ofeffort, i.e.e=e(t).

Differentiationofthefirstequationin(7)withregardtotthengivesus de

dt =−Uet

Uee =− bRet−petL^D

bRee−gee−peeL^D ≥

(

^<

)

^{0andUet=bRet−petLD≥}

(

<

)

^{0 (8)}

Itfollows from(8)that the driverwill react eitherby increasing or decreasingher level ofeffortwhen the company stepsup itseffort, dependingonwhetherhermarginal expectedutilitywithregard toher owneffortisincreasedorde- creasedasthecompany becomesmoreintensiveinits operations. Thereare twoeffects toconsider, drawinginopposite directions.First, when the company injects more effort, the driver’s effortbecomesmore valuable in creatinga positive wageincrease from the netrevenue, i.e.bRet is positive.However, an increasedeffort fromthe company alsomeans an increaseinthemarginalgrowthinprobabilityforaccidentswithregardstothedriver’seffort,implyingreducedexpected marginalutilityfrominjectingeffort,p_etL^D.Whethertheproductiontermortherisk termdominates,isambiguous.Ifthe driverincreases(decreases)her effortwhenthecompanystepsup itseffort,wefollowtheordinaryconceptingamethe- oryandsaythattheeffortsarestrategiccomplements(substitutes)seenfromthecompany’sperspective,see,forinstance, Bulowetal.(1985)andPedersen(2003).

Now,we consider the company’s optimal behavior, given the response from the driver. Using the expressions in (1), (2)and(8)implies that the 1.order condition for an optimaltnow must satisfy⁷

d

π

dt =

(

^1−b

)

^Rt−ptLC+

π

^ede_dt ⁼

(

^1−b

)

^Rt−ptLC+

(

^1−b

)

^Re−peLC

de

dt =0 (9)

Fromcomparingequationthefirstequationin(6)andtheequationin(9),weseethattheleadingcompany,inaddition tonotingthedirecteffectthasonits expectedprofit, isinterestedinhowthedriverreactsbychangingher levelofeas tis increasedordecreased,andwhetherthedriver’s responseincreasesordecreases thecompany’s expectedprofit. This ismeasured by theterm

π

^ede_dt^{.We havefourpossiblecases. First,ifthereaction fromthedriver ispositive formarginal}

changesin t,i.e. ^de_dt >0becauseUet > 0, and thecompany’s expectedprofitis growing ine,

π

^e > 0, thecompany will setahighertthaninthesimultaneouscase,andtheresponsefromthedriverisahighere.Second, ifthereactionisstill positiveandtheexpectedprofitisdecreasingine,

π

^e <0,thecompanywillpreferalowerintensityeffortfromthedriver, andtherefore,reducesitseffort.Third,ifthedriverreactsbyreducinghereffortaststepsup, ^de_dt <0becauseUet<0,and theprofitisincreasing ine,

π

e >0,thecompanywillseta lowertto inducehighereffortfromthedriver.Finally,when bothexpressions arenegative, i.e. ^de_dt <0becauseUet < 0and

π

^e < 0,thecompanychooses alow tinordertoforce the drivertochoosealowe.Itisalsoseenthatif^de_dt =0or

π

^e=0,orbothareequaltozero,bothactorschoosethesamelevel ofeffortasinthesimultaneouscase.Thefirstequation in(7)andtheequationin(9)aredefiningtheNash-equilibriumin thegamewherethecompanymakesitseffortfirstandthedriverobservesitseffortandsubsequentlydecidesherlevelof effort.Inthefollowing,wedenotetheseequilibriumvaluesby(t^L,e^F).

7Here we have dropped to present the 2.order condition in the text. However, it is seen that the sufficient condition is given by ^d_d²_t^π2 = (1 −b )R tt−

p ttL ^C+ 2[ (1 −b )R et−p etL ^C] ^de_dt+ [ (1 −b )R ee−p eeL ^C] [ ^de_dt] ²+ [ (1 −b )R e+ p eL ^C] ^d_d²_t2^e< 0 , meaning that the assumptions already made are not sufficient to se-

cure that the 2.order condition is satisfied. However, in the following analysis, we presume that this condition holds.

Result2. Inthecasewherethetransportcompanydecidesfirst,itwillchooseahighereffortthaninthesimultaneouscase when

π

^e > 0andUet >0andwhen

π

^e <0andUet < 0aresatisfied.When

π

^e < 0andUet >0andwhen

π

^e >0and Uet <0hold,thecompanywillchoosealowerlevel.Thedriver,beingthefollower,willchooseahighereffortthaninthe simultaneous casewhen

π

e > 0andU_et > 0andwhen

π

e >0 andU_et < 0are satisfied.If

π

e <0 andU_et > 0andif

π

^e <0andUet <0hold,thedriverwillinjectalowerlevel.

3.3. Thedrivermovesfirst

Now the company chooses its level ofeffortaccordingto the first equation in(6). From thisequation the company’s optimalvalueoftisdependentone,meaningthatthisequationimplicitlydefinesthecompany’sresponsefunctiont=t(e).

Differentiatingthefirstequationin(6)withregardtoe,givesus dt

de =−

π

^et

π

^{tt =−}

⁽

1−b

)

^Ret−petL^C

(

^1−b

)

^{Rtt−pttLC ≥}

(

^<

)

^0as

π

^et=

(

^1−b

)

^{Ret−petLC≥}

(

^<

)

^{0 (10)}

Theslope ofthe reactionfunctionin(10)isdependentonthesignof

π

et.Ifthemarginalprofitintisincreasing (de- creasing) in e,the efforts become strategic complements (substitutes) seen fromthe perspective of the driver. Now,the driverwillmaximize herutility,takingintoaccountthereactionofthecompany.Using(2),(3)and(10),thengivesusthe following1.ordercondition⁸

dU

de =bRe−ge−peL^D+Ut

dt

de =bRe−ge−peL^D+

bRt−ptL^D

dt

de =0 (11)

Ifwe compare(11)andthe firstequation in(7),itisseen thatthe driver,inadditionto evaluatingthedirect effecte hasonherexpectedutility,mustalsobeawareoftheindirecteffectehasonthecompany’seffort,andwhetheramarginal increaseinthecompany’seffortincreasesorreducesherexpectedutility.ThisindirecteffectismeasuredbythetermU_tde^dt. Inthecasewherethiseffectispositive,eitherwhenboth Ut and ^dt_de are positiveornegative, thedriverastheleaderwill chooseahigherlevelofeffort.When ^dt_de>0,thiswillencouragethecompanytoincreaseitseffort,whichisfavourablefor driverasUt>0.InthecasewheretheslopeofthereactionfunctionisnegativeatthesametimeasUt<0,theincreased effortforthedriverwillreducethecompany’seffort,whichincreasestheexpectedutility.Ifthefactorshaveoppositesigns, the driverwill choosea lower effortthan inthe simultaneous case. When theslope ofthe reactionfunction is negative andUt > 0,thelower effortmeansthat thecompanyincreasesits effortcomparedwiththesimultaneous case,whichis favorablefor the driver. Finally,when the slope ofthe reaction function is positive andUt < 0,the driver’s lower effort reducesthecompany’seffort, increasingthedriver’sutility.IfUt=0or _de^dt =0,orboth areequaltozero,thesolutionwill bethesameasinthesimultaneouscase.Thefirstequation in(6)andtheequation in(11) definetheNash-equilibriumin thegamewherethedrivermovesfirst,andthecompany observeshereffortandchooses itsinput level.Inthefollowing, wedenotetheseequilibriumvaluesby(t^F,e^L).

Result3. Inthecasewherethe driveristhe leader, shewillchoose a highereffortthaninthe simultaneouscasewhen U_t >0and

π

et>0orwhenU_t <0and

π

et <0aresatisfied.IfU_t>0and

π

et <0orifU_t <0and

π

et>0hold,shewill choosealowerlevel.Thetransportcompany,beingthefollower,willchooseahighereffortthaninthesimultaneouscase whenUt >0and

π

^et >0orwhenUt >0and

π

^et <0aresatisfied.IfUt <0and

π

^et >0orif Ut< 0and

π

^et <0hold, thecompanywillmakealowerlevelofeffort.

3.4. Comparingthedifferentgamesandoutcomes

FromtheanalysesitisseenthatthesignsofU_t,

π

e,U_et and

π

et,crucialfortheactualchoicesofeandtinthedifferent cases,canbe eitherpositive,zeroornegative,andgenerallyallcombinationsofsignsare possible.Fromtheequationsin (4),(5),(8)and(10)itfollowsthatthesignsofthesefunctionsisconditionalonwhetherthemarginaleffectsontheactors’

portionofthenetrevenue– beingpositive– ortheeffectsontheexpectedaccidentlosses– beingnegative– dominateor not.Inordertocompareanddiscusstheconsequencesfortrafficsafetyfromthedifferentgames,wehaveinTables1and 2summarizedtheresultsfoundintheanalysesofthethreeproposedgames.InTable1wehavepresentedthecasewhere the companyis theleader andthe driveristhe follower.The linesin Table1 illustratewhetherthe company’s expected profit increases,is zero, orreduces as the driversteps up her effort, while the columns tellus whetherthe efforts are strategic complements, independent or strategic substitutes in the eyes of the company. All together nine combinations becomepossibleandwecomparetheoutcomesfromthesituationswherethecompanyistheleaderwiththesimultaneous case. Inaddition tothe effortlevels,we havealso calculatedthe accident probabilitiesforthe differentcases,wherethe followingdenotationsareused: p^S=p(t^S,e^S),p^LF=p(t^L,e^F) andp^FL=p(t^F,e^L).Forthecasewherethedriveristheleaderand thecompanyisthefollower,Table2presentsthedifferentsituations.Herethelinesillustratewhethertheexpectedutility

8The second order condition for this problem is given by ^d_d²_e^U2= b R ee−g ee−p eeL ^D+ 2(b R et−p etL ^D)^dt_de+ (b R tt−p ttL ^D)(^dt_de)²+ (b R t−p tL ^D)^d_d²_e^t2< 0 . As com- mented on when considering the case where the company moves first, also in this case extra assumptions are needed in order to secure that the 2.order condition is satisfied. In the following discussions, we presume it holds.

Table 1

Comparing the case where the company is the leader with the simultaneous case.

U et > 0, ^de_dt > 0

strategic complements U et= 0, ^de_dt= 0

strategic independent U et < 0, ^de_dt < 0 strategic substitutes πe > 0 t ^L> t ^S, e ^F> e ^S, p ^S< p ^LF t ^L= t ^S, e ^F= e ^S, p ^S= p ^LF t ^L< t ^S, e ^F> e ^S, p ^Sp ^LF πe= 0 t ^L= t ^S, e ^F= e ^S, p ^S= p ^LF t ^L= t ^S, e ^F= e ^S, p ^S= p ^LF t ^L= t ^S, e ^F= e ^S, p ^S= p ^LF πe< 0 t ^L< t ^S, e ^F< e ^S, p ^S> p ^LF t ^L= t ^S, e ^F= e ^S, p ^S= p ^LF t ^L> t ^S, e ^F< e ^S, p ^Sp ^LF

Table 2

Comparing the case where the driver is the leader with the simultaneous case.

πet> 0, _de^dt> 0

strategic complements πet= 0, ^dt_de= 0

strategic independent πet< 0, _de^dt< 0 strategic substitutes U t> 0 t ^F> t ^S, e ^L> e ^S, p ^S< p ^FL t ^F= t ^S, e ^L= e ^S, p ^S= p ^FL t ^F> t ^S, e ^L< e ^S, p ^Sp ^FL U t= 0 t ^F= t ^S, e ^L= e ^S, p ^S= p ^FL t ^F= t ^S, e ^L= e ^S, p ^S= p ^FL t ^F= t ^S, e ^L= e ^S, p ^S= p ^FL U t < 0 t ^F< t ^S, e ^L< e ^S, p ^S> p ^FL t ^F= t ^S, e ^L= e ^S, p ^S= p ^FL t ^F< t ^S, e ^L> e ^S, p ^Sp ^FL

forthedriverisincreasing,constantordecreasingasthecompanyincreasesitseffort, whilethecolumnsexpresswhether theeffortsarestrategiccomplements,independentorstrategicsubstitutesseenfromtheperspectiveofthedriver.

Firstwe shouldnote thatthesecond lineinTable1,definedby

π

^e=0,impliesthat

π

^et=0,meaningthat thissecond linecorresponds to thesecond columninTable 2.Also thesecond line inTable 2corresponds to the second columnin Table1becauseUt=0impliesUet=0.Altogether,thismeansthatifUt=0,

π

^e=0,orboththesearezero,thenallcasesgive thesameefforts,i.e.t^L=t^S=t^Fande^L=e^S=e^F,andconsequentlythesameaccidentrisk,i.e.p^S=p^LF=p^FL.Furthermorewe seefromTable1thatinthecaseofstrategiccomplements,illustratedinthefirstcolumn,thesimultaneousgamepossibly givesboth lower andhigheraccident risk than thecase wherethe companyis theleader. Thisdependson whetherthe expectedmarginalprofitwithregardtothedriver’seffortispositiveornegative.If

π

e>0,thecaseofstrategiccomplements meansthatbothactorschoosehighefforts,resultinginrelativelyhighaccidentrisk,whiletheoppositeistrueif

π

^e<0.In thecaseofstrategicsubstitutes,illustratedinthethirdcolumninTable1,theactorsmoveindifferentdirectionsregarding theeffortstheymakecomparedwiththesimultaneouscase,meaningthattheeffectonaccidentriskisinconclusive.

Thesameobservationsareseen inTable2wherewecomparethecasewherethedriveristheleaderwiththesimul- taneouscase.Fromthefirstcolumn,wheretheeffortsarestrategiccomplementsseenfromthedriver,itfollowsthatboth actorshavehighereffortsintheleader-followercasethaninthesimultaneouscaseifU_t>0andlowereffortsifU_t<0.This meansthattheleader-followermodelcouldboth increaseordecreaseriskscomparedwiththesimultaneouscase, depen- dentonwhetherthedriverprefershighereffortfromthecompanyornot.Ontheotherhand,whentheeffortsarestrategic substitutes,seeninthethirdcolumninTable2,itisinconclusivewhetherthesimultaneouscaseortheleader-followercase givesthehighestaccidentrisk ornot.Thisisbecausetheactors’efforts, comparedwiththesimultaneouscase, moveina differentdirection.

Moreover,we summarizesome ofthecasesfromTables 1and2 bydrawing conceptualfigures, consistingofthe two reactionfunctionsandthe iso-profitandindifference curvesfortheleadingcompany andtheleadingdriver respectively, seeFigs.1−6below.Inall diagrams,curvesrepresentingthedriverarecoloredred,andcurvesrepresentingthe company arecolored blue.The intersectionlabelledS representsthesimultaneous move solution,thepoint oftangencylabelled C representsthesolutionwhenthecompanyisthefirstmover, andthepointoftangencylabelledDrepresentsthesolution whenthedriveris thefirstmover. Inthespecialcases whereallthemarginal effectson theactors’partsofthe revenue followingforamarginalincreaseineffortsdominatetheeffectsonexpectedaccidentlosses– i.e.thesignsofUt,

π

^e,Uetand

π

etareallpositive– it isseen fromTables 1and2that theleader-follower solutionsmean higher accidentrisks thanin the simultaneouscase.Insuch particularcases,wheretheefforts arestrategiccomplements,andthedriverandthecompany haveinterests inhigheffortbytheopponent,theleader-followergamesarenotpreferableseenfromatrafficsafetypoint ofview,seeFig.1.However,ifUtand

π

^{earenegativeandUet and}

π

^{etarepositive,suchstrategiccomplementcasesinduce}

theactorstomakelessintensiveeffortsintheleader-followercasesthanthesimultaneouscase.Asthemarginaleffectson expectedlossesdominatetheeffectsontheportions ofnetrevenue keptbytheactors,theyprefer lowlevelsofeffortby theopponent.Theleader thenchoosesalowlevelofeffortinordertostimulatethefollowertodothesame,resultingin relativelylowaccidentprobabilities,seeFig.2.

Inthe casewherethe actors’effects onaccident risks dominatetheeffects on their sharesof thenet revenuefor all functions– i.e. thesignofUt,

π

^{e,Uet and}

π

^{et areallnegative– wehavecaseswheretheeffortsare strategic}substitutes.

Thentheleadersinthetwodifferentgameswouldchoosearelativelyhigheffortinordertoforcethefollowertochoose alow effort,andthe consequenceforaccidentrisk isinconclusive comparedwiththesimultaneous case,see Fig.3.IfUt

and

π

^{e arepositive andUet and}

π

^{et arenegative, westill havecaseswhere theefforts arestrategic} substitutes.Now the leaderwouldchoosearelativelyloweffortlevelinordertostimulatethefollowertochoosearelativelyhighlevelofeffort, implyingthatitisambiguouswhethertheaccidentrisksbecomelowerorhigherthaninthesimultaneouscase,seeFig.4. However,bycombiningtheinformationinTables1and2itispossibletoidentifytwocaseswhentheeffortsarestrategic complementswhere the accident risks fora specific leader-follower game become low compared withthe other games.

Fig. 1. An illustration of the case where U et> 0, πet> 0, U t> 0 and πe> 0.

Fig. 2. An illustration of the case where U et> 0, πet> 0, U t< 0 and πe< 0.

GiventhatUet>0and

π

^et>0,and

π

^e<0andUt>0hold,itisseenfromTables1and2thatt^L<t^S<t^F ande^F<e^S<e^L. Thismeansthatwenowhavep^LF<p^S<p^FL,implyingthatthegamewherethecompanyismovingfirst,inducesthelowest accidentriskcomparedwiththeothergames.Inthisgamethecompanyhasaninterestinreducingtheeffortofthedriver, and,astheeffortsare strategiccomplements,thecompanyastheleader choosesarelativelylowinput,resultinginalow effortfromthe driver. Thisgivesa relatively low accident risk. Forthe driver,it isthe opposite. Herutility is increasing in thecompany’s effort inthiscase, and ifshebecame theleader, shewouldchoose a higheffort. Since the efforts are strategiccomplements,thecompany wouldanswer bychoosinga highlevelofeffort, resultinginrelativelyhighaccident risks,seealsoFig.5.Theother casewherewefindanunambiguousrankingofaccident probabilitiesissimilar. Giventhat Uet >0and

π

^et >0,and

π

^e >0andUt<0hold,therankingsofeffortsfollowingfromTables1and2becomet^F <t^S<t^L ande^L <e^S <e^F.Theserankingsgivetheoppositeconclusion,i.e.p^FL<p^S<p^LF,implyingthatinthiscaseitispreferable, fromatrafficsafetyapproach,thatthedrivershouldbetheleaderandthecompanyshouldbethefollower.Nowthedriver,

Fig. 3. An illustration of the case where U et < 0, πet < 0, U t < 0 and πe < 0.

Fig. 4. An illustration of the case where U et < 0, πet < 0, U t > 0 and πe > 0.

bychoosing arelatively low effortfirst,induces thecompany tofollowup by inputtinga relativelylow effort, altogether implyingarelativelylowaccidentprobability.Ifthecompanychoosesfirst,itschoiceisarelativelyhigheffortinorderto promptthedrivertochoosearelativelyhigheffort,meaningarelativelyhighaccidentprobability.Thiscaseisillustratedin Fig.6.⁹

9By combining the information in Tables 1 and 2 , it follows that when U t> 0, U et^0, πe^{0 and} πet< 0 are satisfied, the rankings t ^L< t ^s< t ^F and

e ^L< e ^s< e ^Fhold, (see Fig. 4 ), and when U t < 0, U et< 0, π^e< 0 and π^et< 0 hold, the rankings t ^F < t ^s< t ^L and e ^F< e ^s< e ^Lhold, (see Fig. 3 ). Even

though in these two cases, additionally to the cases identified above, we are able to rank the effort s, it is not possible to rank the accident risks following from these two cases.

Fig. 5. An illustration of the case where U et> 0, πet> 0, U t> 0 and πe< 0.

Fig. 6. An illustration of the case where U et> 0, πet> 0, U t< 0 and πe> 0.

Result4.

a) Inthecaseswherethecompanyistheleaderandtheeffortsarestrategiccomplements,thesimultaneousgamegives higheraccidentrisk thantheleader-follower caseif

π

^e <0.When

π

^e >0andweare inthe strategiccomplement case,thesimultaneousgamemeansloweraccidentriskthanwhenthecompanyistheleader.

b) Incaseswherethedriveristheleaderandtheeffortsarestrategiccomplements,thesimultaneousgamegiveshigher accident risk thanthe leader-followergameifUt < 0.WhenUt > 0andtheeffortsare strategiccomplements,the simultaneousgamegivesloweraccidentrisksthantheleader-followergame.

c) Ifthe efforts are strategiccomplements,seen both fromthedriver andthecompany, and

π

^e <0 andUt > 0,the lowest accident riskswill be realized ifthe companyis the leader.When

π

^e > 0andUt < 0 andthe efforts still are strategic complementsforboth parties, the driverasthe leader will give the lowestaccident risk. Inall cases