Reference-Dependent Stochastic User Equilibrium with Endogenous Reference Points

Transcription

1 EJTIR Issue 3(2), 203 pp ISSN: Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Polo Delle Site, Fncesco Filippi nd Cludi Cstldi Deptment of Civil Achitectul nd Envionmentl Engineeing, Univesity of Rome L Spienz, nd Resech Cente fo Tnspot nd Logistics, Univesity of Rome L Spienz We conside the ppliction of efeence-dependent consume choice theoy to tffic ssignment on tnspottion netwoks. Route choice is modelled bsed on ndom utility mximistion with systemtic utility embodying loss vesion fo the tvel time nd money expenditue ttibutes. Stochstic use equilibium models found in the litetue hve consideed exogenously given efeence points. The ppe poposes model whee efeence points e detemined consistently with the equilibium flows nd tvel times. The efeencedependent stochstic use equilibium (RDSUE) is defined s the condition whee (i) no use cn impove he utility by uniltelly chnging pth, (ii) ech use hs s efeence point the cuent tvel time nd the money expenditue of one of the vilble pths, nd (iii) if ech use updtes the efeence point to he cuent pth the obseved pth flows do not chnge. These conditions e fomlly equivlent to multi-clss stochstic equilibium whee ech clss is ssocited with pth nd hs s efeence point the cuent stte on the pth, nd the numbe of uses in ech clss equls the cuent flow on the pth. The RDSUE is fomulted s fixed point poblem in the pth flows. Existence of RDSUE is gunteed unde usul ssumptions. A heuistic lgoithm bsed on the method of successive veges is poposed to solve the poblem. The model is illustted by two numeicl exmples, one eltes to two-link netwok nd nothe to the Nguyen-Dupuit netwok. A efeence-dependent oute choice model clibted on stted pefeence dt is used. The second exmple seves lso to demonstte the lgoithm. The impct on the equilibium of diffeent ssumptions on the degee of loss vesion with espect to the tvel time ttibute e investigted. Keywods: Refeence-dependent theoy, Loss vesion, Stochstic use equilibium, Endogenous efeence point.. Intoduction Refeence-dependent theoy (Tvesky nd Khnemn, 99) poposes fundmentl chnge of pdigm in choice theoy becuse it ssumes tht cies of utility e not sttes but gins nd losses eltive to efeence point. Loss vesion is centl to the theoy: losses e vlued moe Vi Eudossin 8, Rome 0084, Itly, T: , F: , E: polo.dellesite@uniom.it

2 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points hevily thn gins. The theoy consides iskless choices with the utilities of ltentives being chcteized by ttibutes whose outcomes e cetin. The theoy llows fo both constnt nd diminishing sensitivity of the utilities with espect to chnges in the ttibutes. The expeimentl vlidtion of efeence-dependent theoy in the cse of the choice of the oute on od netwok hs been the subect of esech. Diffeent uthos (De Boge nd Fosgeu, 2008; Hess et l., 2008; Delle Site nd Filippi, 20) hve fomulted efeence-dependent oute choice models following the fmewok poposed by Tvesky nd Khnemn (99) nd estimted ndom utility vesions of these models bsed on multinomil logit ssumptions fo the stochstic tems. Evidence of symmeticl esponse with espect to gins nd losses in both tvel time nd money ttibutes of the oute ltentives is found. A common finding in Hess et l. (2008) nd in Delle Site nd Filippi (20) is tht uses e moe loss vese in the money ttibute thn in the time ttibute. A ntul subsequent step consists in the use of efeence-dependent oute choice models in netwok nlysis. A vint of the clssicl stochstic use equilibium (SUE) tht ssumes tht the ndom utilities of oute ltentives e efeence-dependent hs been poposed by Delle Site nd Filippi (20). They conside efeence-dependent oute choice model with constnt sensitivity of the utilities. They develop the equilibium model on the bsis of the ssumption tht the efeence points e the sttus quo. They ssume tht thee is multiplicity of efeence points fo uses of the sme oigin-destintion pi becuse in the sttus quo uses choose diffeent outes, ech hving distinct tvel time nd money expenditue. This gives ise to n equilibium poblem with multiple use clsses, with ech clss hving distinct efeence point detemined by the tvel time nd money expenditue in the sttus quo. The ppe by Delle Site nd Filippi (20) dels lso with the efeence-dependent vlution of time chnges ove the netwok. It is shown tht it is possible to ttibute to ech clss of uses distinct vlution of time svings nd time losses. Both time svings nd losses cn be vlues ccoding to compensting o n equivlent mesue. The sttus quo ssumption is commonly dopted in efeence-dependent models. Thee is limited esech into which efeence points should be employed in tnspottion pplictions, with existing litetue elying pe-dominntly on using cuent tip conditions s efeence (De Boge nd Fosgeu, 2008; Hess et l., 2008; Delle Site nd Filippi, 20). Thee e, howeve, othe possibilities. Chin nd Knetsch (2006) nd Knetsch (2007) conside the issue of which stte the individuls egd s the nom fo udging thei stisfction, guing tht thei legitimte expecttions, o wht they feel is deseving o ight, might be such nom. Fo od uses they suggest the fee-flowing tffic conditions s nom. Recently, Stthopoulos nd Hess (202) hve consideed efeencing occuing ginst diffeent plusible ncho points including cuent, idel nd cceptble tvel conditions. They estimte discete choice models nd find tht the best dt fit is chieved with model whee the espondent-epoted idel vlue is ssumed s efeence point. Thei investigtion eltes, howeve, to the int-modl choices fo bus nd il uses, not to oute choice. Also, they find evidence of symmetic pefeences only fo the monety cost ttibute, not fo tvel time. The sttus quo ssumption implies tht the efeence points e the cuent tvel conditions. In n equilibium setting, the sttus quo ssumption ises the issue whethe the equilibium is mintined when the efeence points e updted to the new sttus quo. This popety of efeence-dependent equilibium is efeed to s eflexivity in the economics litetue on tde (Muno nd Sugden, 2003; Muno, 2009). In tnspottion netwo Delle Site nd Filippi (20) hve shown tht the equilibium is mintined when the efeence points e updted to the new sttus quo if the dditionl ssumption is mde tht the stochstic tems of the oute choice model do not chnge with the updting. Thus the equilibium model in Delle Site nd Filippi (20) tets the efeence points s exogenous.

3 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Howeve, othe ssumptions on the dynmics of the stochstic tems cn be consideed to tke into ccount chnges in unobseved ttibutes of the outes o int-pesonl pefeence vition. If the stochstic tems e edwn, the eltive convenience of the ltentives my chnge when the efeence point is updted. A new equilibium would need to be computed on the bsis of the updted systemtic utilities. The efeence points would be futhe dusted nd new equilibium computed. Theefoe, the inteest of the obseve is in modelling the conditions whee thee is consistency between flows nd tvel times (this consistency identifies the conventionl equilibium ove congested netwok) nd, t the sme time, consistency between cuent choices nd efeence points. These extended consistency conditions epesent the efeence-dependent equilibium with endogenous efeence points which is the subect of the pesent ppe. The ppoch to equilibium is simil to the one found in Xu et l. (20) who hve consideed use equilibium with endogenous efeence points but in pospect-theoy bsed 2 nd deteministic choice setting. The contibutions of the ppe e the following: to define the equilibium conditions ove netwok in setting of ndom utility efeence-dependent oute choice unde the sttus quo ssumption with endogenous efeence points; to povide the mthemticl fomultion of these efeence-dependent equilibium conditions; to povide solution lgoithm tht computes flows, tvel times nd efeence points t equilibium; to illustte the equilibium model with numeicl exmples pplying to netwoks of diffeent size. The ppe is ognised s follows. Section 2 povides the definition of the equilibium conditions nd pesents the mthemticl model nd the solution lgoithm. Section 3 pesents some numeicl esults elting to two illusttive netwoks. Section 4 povides few concluding emks. 2. Netwok equilibium 2. Netwok epesenttion nd ssumptions Let G = ( N, A) be stongly connected od tnspottion netwo with N nd A being the sets of nodes nd links, espectively. Let be the link index. Oigins (O) nd destintions (D) constitute subset of N. Let R be the set of OD pis nd the OD pi index. Let K be the set of simple pths of OD pi, nd k the pth index. k Fo ech pth k K, F, denotes the coesponding pth flow. We denote by z the flow on link A. The link flows e obtined fom the pth flows by: z = R k K k, k, δ F A () k, whee δ is the element of the link-pth incidence mtix whose vlue is if pth k includes link, is 0 othewise. 2 Pospect theoy (Khnemn nd Tvesky, 979) diffes fom efeence-dependent theoy becuse it ssumes, in ddition to loss vesion, tht outcomes of the ttibutes e uncetin thus clling fo the need to model isk.

4 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points The demnd flow of the OD pi is denoted by q = k K F k, R q. We hve the demnd constints: (2) The fesible pth flows e ll the non-negtive stisfying the demnd constints (2). Theefoe, the set of fesible pth flows is non empty, compct nd convex. k Let T, denote the tvel time on pth k of OD pi. Let denote the tvel time on link. The link tvel times e continuous functions of the link flows: t. The pth tvel times = t z, A e obtined fom the link tvel times by the stndd link-dditive model: T = ( z, A) k K, R k, δ t (3) A 2.2 Refeence-dependent oute choice The uses of n OD pi peceive utility on ech pth. This pth utility is ndom vible given by the sum of systemtic, i.e. deteministic, component nd stochstic tem. The stochstic tems summise fctos tht e unobseved by the modelle. The stochstic tems e intepeted s individul specific thus ccounting fo both inte-individul nd int-individul vibility of tstes. The individul-specific stochstic tems my chnge coss epeted choices. A efeence-dependent model is dopted fo the pth systemtic utility ccoding to the following hypotheses. The pth systemtic utility (i) depends on two ttibutes: expenditue in tvel time T nd expenditue in money M; (ii) (iii) depends on gins G nd losses L in the two ttibutes defined eltive to efeence point, nd inceses with gins nd deceses with losses; is line in gins nd losses (constnt sensitivity) nd steepe fo losses thn fo gins (loss vesion). The uses of ech OD pi e gouped into clsses, with ech clss denoted by nd identified by efeence point in tems of pth tvel time nd money spent. Let J be the set of clsses of OD pi. The pth utilities hve the dditive fom: U = V k, + ε k, k F, t ( ) V GT LT GM LM whee: k U, = β k, GT = mx = mx GT = mx = mx + β LT LT ( T T,0) k, ( T T,0) ( M M,0) ( M M,0) + β GM GM + β is the pth peceived utility, LM LM k K, J, R (4) k V, ε k, is the pth systemtic utility, is the stochstic tem,

5 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points β GT, β GM e the gin coefficients, β LT, β LM e the loss coefficients, k, GT, GM k, e the gin, espectively, in tvel time nd in money, k, k, LT, LM k M, T, M e the loss, espectively, in tvel time nd in money, is the money spent on the pth; e the efeence point fo, espectively, the tvel time nd the money spent. Hypothesis (ii) implies tht the systemtic utility is decesing in ech ttibute, i.e. the gin coefficients e positive nd the loss coefficients e negtive. Hypothesis (iii) implies loss vesion, i.e. β LT > β nd GT β LM > β : in bsolute vlues, the loss coefficient is lge thn the GM gin coefficient fo ech ttibute. The systemtic utility in eqns (4) hs two tems fo ech ttibute: gin tem nd loss tem. If thee is gin in the ttibute the gin tem is positive nd the loss tem is zeo. Convesely, if thee is loss the loss tem is positive nd the gin tem is zeo. The single-ttibute pt of the systemtic utility is piecewise line in the ttibute with kink in the efeence point. Thus the function is eveywhee continuous in the ttibute but non diffeentible in the efeence point. If the bsolute vlues of the gin nd loss coefficients wee equl, the function would be symmetic bout the efeence point. Fo diffeent coefficients the function is symmetic with slope steepe in losses thn in gins if the coefficients stisfy loss vesion. This is shown in Figue whee the ttibute, i.e. the expenditue in tvel time o in money, is denoted by X. A constnt dditive tem my be included in the systemtic utility in eqns (4), e.g. to epesent othe time-independent pth ttibutes; howeve, without loss of genelity it is left out becuse it does not ffect the developments below. utility gin efeence point loss X loss vesion Figue. Single-ttibute systemtic utility Uses of clss of OD pi who choose pth k e those who peceive this pth to mximise thei utility. The choice pobbilities e defined s: P = P k, ( ) m, U U m k K k K, J, R (5)

6 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points k We ssume tht the stochstic tems ε hve non-degenete oint pobbility density function tht is continuous, stictly positive, nd independent of the pth systemtic utility. We ssume tht the choice pobbilities e single-vlued nd continuous in the pth systemtic utilities: P k, ( V, k K ) k, k, = P k K J, R, (6) The hypotheses e sufficiently genel to dmit nge of behvioul ssumptions though the fom of the oint distibution fo the stochstic tems, thus encompssing vious dditive models, including, but not esticting to, multinomil logit. In the cse of multinomil logit the pobbility function tkes n symmetic S shpe with kink in the efeence point due to the loss vesion ssumption (this is illustted gphiclly in Suzuki et l., 200). Let k f, denote the flow on pth k of clss of OD pi. The choice model is expessed in tems of these clss-specific pth flows s: f k, = q P k,, k K, J, R (7) whee q denotes the numbe of uses of clss of OD pi, with J q = q R (8) 2.3 Refeence-dependent stochstic use equilibium with endogenous efeence points In stochstic setting ech clss of uses is ssigned to ll vilble pths. The efeencedependent stochstic use equilibium (RDSUE) with endogenous efeence points is defined s the condition whee: no use cn impove he efeence-dependent utility by uniltelly chnging pth, ech use hs s efeence point the cuent tvel time nd the money expenditue of one of the vilble pths,, if ech use updtes the efeence point to he cuent pth the obseved pth flows do not chnge. This condition is obtined when: ech use chooses the pth with the mximum utility, ech use clss is ssocited with pth nd the efeence point of the clss is the tvel time of the cuent stte nd the money expenditue on tht pth, the numbe of uses in ech clss equls the cuent flow on the coesponding pth. In fct, the flow of pth of n OD pi is given by the union of the following two sets of uses: the uses who hve s efeence tht pth nd choose it, nd the uses who choose the pth while hving s efeence othe pths. The numbe of uses in the second set equls the numbe of uses who hve s efeence the pth while choosing othe pths. Theefoe, if ech use updtes the efeence point to the cuent pth, o, in othe wods, if ech use shifts clss, the pth flows do not chnge. It is notewothy tht in the RDSUE hee the popety tht the pth flows do not chnge if ech use updtes the efeence point to he cuent pth holds unde ny ssumption on the dynmics of the stochstic tems. In Delle Site nd Filippi (20) it hs been poved tht the popety holds in efeence-dependent stochstic equilibium with exogenous efeence points

7 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points if the stochstic tems e unchnged with the updting. The RDSUE hee is defined s the equilibium whee the popety holds fo ny chnge of the stochstic tems. Mthemticlly, the equilibium conditions e defined by the following: J q K R = F, K, R (0) whee: F Eqns (9) imply tht the efeence points in tems of tvel time nd money expenditue of ech clss tht ppe in the systemtic utilities of eqns (4) e given by the following: T M, = k K = T, K, R (2) = M, K, R (3) A RDSUE is solution to the fixed point poblem in the pth flows = f F, :,, F f =,..., K R (4) k k K, k K, R (9) () F K, K = q F =, R (5) with f F, = F 0, P K, ( F, K, R), R k K, K, R (6) whee K denotes the cdinlity of the set K. The dependence of the pobbilities on the pth flows which ppes in eqns (6) is obtined by chining the expessions (4) of the systemtic utilities in the pth tvel times, the expessions (3) of the pth tvel times in the link tvel times, the link tvel times in the link flows, nd the expessions () of the link flows in the pth flows. A solution, not necessily unique, to the fixed point poblem (4), (5) nd (6) detemines k uniquely the link flows z, the link tvel times t, the pth tvel times T,, s well s the clssspecific pth flows k f,. The stuctue of the poblem is s follows: eqns (4) nd (5) e system of non-line equtions, the inequlities in (6) estict the set of fesible pth flows to the non negtive othnt. Eqns (4) e fom the equilibium conditions, eqns (5) fom the demnd constints. The equilibium conditions (4) e witten, fo ech OD pi, only fo K pths becuse only K conditions e independent, s the condition fo one pth is line combintion of the othe K. By dding, fo ech OD pi, the demnd constint, well defined system of equtions, i.e. system with s mny equtions s unknowns, is obtined. It is convenient to e-wite in compct fom the fixed point poblem using vecto nottion (vectos in bold): ( ) 0 F = Ψ F F (7)

8 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points whee F is the K vecto of pth flows, nd Ψ the R K vecto mpping R epesenting the functions t the ight-hnd sides of eqns (4) nd (5) Existence nd uniqueness popeties In the light of the Bouwe s fixed point theoem, solution to RDSUE exists since the fesible set is non empty, compct nd convex (hving tken into ccount the demnd nd the nonnegtivity constints) nd ll the functions composed to fom the fixed point fomultion e continuous. We hve not estblished uniqueness of the RDSUE solution, which, howeve, cnnot be excluded Reduction to conventionl SUE The RDSUE collpses to conventionl SUE when the bsolute vlues of the loss nd gin coefficients e equl fo both the tvel time nd the money expenditue ttibutes, i.e. β LT = β GT nd β LM = β. In fct, due to the model dditivity, when these conditions occu GM choice pobbilities e not ffected by efeence points: = k K J, R P P, (8) The RDSUE fixed point poblem (4), (5) nd (6) educes then to the conventionl SUE fixed point poblem: F,, = q P K, R (9) which cn be e-witten in compct fom: F = q ~ o P (20) whee q ~ is the R K pth-bsed expnded vesion of the demnd vecto q = [ q ],...,q T, R P is the K vecto mpping of pobbilities, nd o denotes the Hdmd, i.e. R componentwise, poduct ( x o y is the vecto whose i-component is xi y ). The non-negtivity i constints become edundnt becuse the pobbilities e non-negtive. It is possible to pove tht in this cse, unde n dditionl ssumption on the monotonicity of the link time-flow functions, the solution is unique (see, mong the othes, Cscett, 2009) Deling with the oute ovelpping poblem It is possible in RDSUE to tke into ccount the poblem of ovelpping outes. To this im, it is sufficient to conside suitble oute choice models, othe thn the simple multinomil logit model which suffes fom the limittion of the independence of ielevnt ltentives popety. As esult, in tffic ssignment with multinomil logit oute choice the ovelpping pts of outes tht she common links e oveloded. To tckle the oute ovelpping poblem, diffeent fomultions of the oute choice model hve been poposed. These cn be gouped into two clsses ccoding to the level t which the simility of outes is tken into ccount. A eview of the diffeent models in the context of equilibium poblems is found in Chen et l. (202). The fist clss consides the simility t the level of the systemtic pt of the oute utility nd keeps fo the stochstic tems the independently nd identiclly Gumbel distibuted ssumption of the multinomil logit. This clss includes the c-logit poposed by Cscett et l. (996) nd the

9 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points pth-size logit poposed by Ben-Akiv nd Bielie (999). These models hve the dvntge tht the oute choice pobbilities hve the simple closed-fom expession of the multinomil logit. The second clss consides the simility t the level of the stochstic pt of the oute utility nd, theefoe, depts fom multinomil logit ssumptions becuse coeltion mong ltentives is ccounted fo. This clss includes the pobit model, which hs been specilized to tckle the ovelpping outes poblems by Yi et l. (997), nd genelised exteme vlue (GEV) models. The ltte include the coss-nested logit (Vovsh nd Bekho, 998) nd the pied-combintoil logit (Pshke nd Bekho, 998). The oute choice pobbilities need to be computed numeiclly in the cse of pobit, hve closed-fom expessions in the cses of coss-nested logit nd pied-combintoil logit. All the models educe the pobbilities of choice of the ovelpping outes. Fo ppliction in the cse of RDSUE, it needs to be consideed tht the systemtic utility is efeence-dependent. The c- logit nd pth-size logit models cn be dpted by dding to the efeence-dependent systemtic utility of eqns (4) the efeence-independent tem tht ccounts fo ovelpping. The pobit, coss-nested logit nd pied-combintoil logit cn be dpted by simply tking s systemtic utilities the efeence-dependent expessions of eqns (4) Compison with the model with exogenous efeence points The RDSUE with endogenous efeence points hee is fomulted s fixed point poblem in the K eqns nd K unknowns. In contst, the efeence-dependent pth flows with R R equilibium model with exogenous efeence points in Delle Site nd Filippi (20) is fomulted s fixed point poblem in the clss-specific pth flows nd hs, consequently, highe numbe of eqns nd unknowns. Anothe diffeence with the model in Delle Site nd Filippi (20) eltes to the policy pth independence popety. The RDSUE with endogenous efeence points hee is independent of the initil netwok conditions nd, theefoe, of the initil efeence points. This implies tht the RDSUE does not chnge if the inteventions on the netwok e implemented stepwise o simultneously (policy pth independence). In contst, the equilibium with exogenous efeence points in Delle Site nd Filippi (20) depends on the initil netwok conditions nd efeence points nd, theefoe, diffeent equilibium is obtined depending on how the inteventions e phsed (policy pth dependence) Solution lgoithm To solve the poblem we use heuistic ppoch bsed on the method of successive veges (MSA). The fomultion of RDSUE s fixed point in the pth flows suggests pth flow-bsed MSA. A pth-bsed lgoithm is in ny cse the only vible option since the pth systemtic utilities of eqns (4) e not dditive in the constituent links. The min steps e, in compct nottion, s follows. Step. Initilistion. Set link flows equl to zeo. Compute link nd pth tvel times. Set the efeence point of ech OD pi equl to the tvel time of n bity pth (e.g. the pth with minimum time in fee-flow conditions). Set itetion counte: t=. Compute the pobbilities P t whee the efeence point is set s bove. Set initil pth flows: F = q ~ o P t t

10 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Step 2. Convegence check. F Ψ < If ( ) δ t F t then stop nd povide outputs, othewise go to step 3. Step 3. Updting of pth flows. The symbol x = mx x i t + t t t Incement itetion counte: t=t+. Go to step 2. Set pth flows: F = F + [ Ψ( F ) F ] i t x denotes the infinity, o mximum, nom of the vecto x with components x i, i.e.. The convegence tolence δ cn be tken equl to unity, which mens diffeence in flows less thn vehicle pe unit of time, becuse highe ccucy is pcticlly ielevnt. The lgoithm genetes sequence of fesible pth flows, i.e. stisfying both the demnd nd non-negtivity constints. At ech itetion the solution F t+ is the vege of the fist t solutions Ψ, hence the nme successive veges. As in the MSA fo conventionl SUE, the ( ) F t convegence cn be slow becuse of the decesing flow coection which depends on the fcto /t. Enumetion of pths is equied, theefoe in lge netwoks the pth set my need to be educed selectively (eviews of selection citei e in Bekho nd Toledo, 2005, nd in Cscett, 2009). When the bsolute vlues of the loss nd gin coefficients e equl, the lgoithm educes to the conventionl pth flow-bsed MSA fo SUE. It is possible to pove tht the MSA lgoithm conveges to SUE if pobbilities e logit (Powell nd Sheffi, 982). Conditions of convegence of MSA to SUE fo othe ndom utility models e investigted in Cntell nd Velonà (200). 3. Illusttive exmples 3. Route choice model We use multinomil logit oute choice model estimted on the bsis of dt fom stted pefeence suvey which took plce in Rome in This model is found in Delle Site nd Filippi (20). Tble shows the esults of the estimtion. The hypothesis of loss vesion is suppoted by dt becuse, in bsolute vlue, the loss coefficient is highe thn the gin coefficient fo both tvel time nd money.

11 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Tble. Estimtion esults fo the oute choice model Coefficient t sttistic t sttistic fo diffeence in bsolute vlues time gin (minutes) time loss money gin (EUR) money loss numbe of obsevtions: 068 finl log likelihood ho-squed pmetes estimted 4 ho-squed dusted Two-link netwok In the fist exmple, we conside two-link netwok (Figue 2) epesenting town cente oute nd bypss oute. 2 Figue 2. Two-link netwok We ssume totl demnd of 200 veh/h. Fo supply, Bueu of Public Rods (BPR) time-flow functions deived empiiclly fo simil outes e used. The functions (in hous) e t = [ + ( z /800) ] fo the town cente oute, nd t = [ ( z /230) ] fo the bypss oute. The RDSUE fixed point poblem educes to the non-line eqution in the town cente oute flow F : F ( F, 200 F ) + ( 200 F ) P ( F,200 F ) = F P (2) 2 which in the cse of multinomil logit oute choice model specilizes to: F ( 200 F ) expv ( F ) expv2 ( F ) = F + (22) 2 2 expv ( F ) + expv (200 F ) expv ( F ) + expv (200 F ) Tble 2 povides the esults in the cse whee toll of EUR is chged on the bypss. Tble 3 povides the clss-specific pth flows. Of the 858 veh/h which e found on the town cente oute, 64 veh/h hve s efeence point the cuent stte on the oute, while the emining 27 veh/h e those hving s efeence point the cuent stte on the bypss oute. At the sme time, thee e 27 veh/h which hve s efeence point the cuent stte on the town cente oute nd choose the bypss oute. Theefoe, if the 27 veh/h hving s efeence point one oute nd choosing the othe updte thei efeence point to thei cuent oute, the totl flow on ech oute does not chnge. 2 2

12 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Tble 2. RDSUE esults with toll on the bypss time town cente oute bypss oute expenditue time flow (veh/h) time (minutes) flow (veh/h) (h) (minutes) Tble 3. RDSUE esults with toll on the bypss clss-specific nd totl pth flows (veh/h) Refeence pth chosen pth town cente oute bypss oute totl town cente oute bypss oute totl The equilibium tht is obtined with the RDSUE model is independent of both the initil netwok conditions nd the policy pth. This is diffeent fom the esults tht hs been obtined by Delle Site nd Filippi (20) fo the sme two-link netwok nd policy. The equilibium tht they hve found ccoding to the model with exogenous efeence points is dependent on the sequence of inteventions, i.e. the equilibium obtined with single-stge policy (bypss opening nd chging simultneously) is diffeent fom the equilibium obtined with twostge policy (bypss opened nd equilibium set, chge implemented in lte time). Anothe diffeence lies in the equilibium flows. In the RDSUE with endogenous efeence points hee the flow on the bypss (342 veh/h) is highe thn the flows obtined in Delle Site nd Filippi (20) with single-stge policy (32 veh/h) nd two-stge policy (333 veh/h). The eson lies in the efeence points. In the model with endogenous efeence points only pt of the uses found on the bypss hve efeence point tht implies the loss of the EUR chge (these e the uses hving s efeence the town cente oute, the othes hving s efeence the bypss do not suffe this loss), while in the model with exogenous efeence points ll the uses found on the bypss hve efeence point implying the loss of the EUR chge Sensitivity to loss vesion in time ttibute We investigted in the cse of bsence of tolls the sensitivity of the equilibium to the ssumption on the degee of loss vesion with espect to the tvel time ttibute. The degee of loss vesion is defined s the bsolute vlue of the tio between the loss coefficient nd the gin coefficient: β /. LT β GT In the oute choice model estimted the degee of loss vesion is.6. This vlue is low when comped with the esults obtined by Hess et l. (2008). They found fo the two demnd segments consideed degee of loss vesion in the fee-flow time ttibute of.49 nd This compison suggests tht it is meningful to exploe the sensitivity to the degee of loss vesion. We exploed the nge fom to 3 (only β LT is chnged while β GT is left unchnged). The cse of degee equl to is tht whee demnd exhibits no loss vesion. The RDSUE collpses then to conventionl SUE nd the solution is obtined by solving the eqution: F ( F,200 ) = 200 P F (23) which in the cse of multinomil oute choice model specilizes to: expv ( F ) F = 200 (24) 2 expv ( F ) + expv (200 F )

13 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points 2 whee in the systemtic utilities V nd V the lowe index is omitted becuse without loss vesion systemtic utilities e efeence independent. Tble 4 shows the esults of the sensitivity nlysis. As the degee of loss vesion inceses, the flow on the town cente oute, which is the oute whee tvel time is highe, deceses. The vition in the flow vlues is not lge (mximum of 30 veh/h out of totl flow of 200 veh/h). The diffeence between the flow on the town cente oute nd the flow on the bypss oute inceses s the loss vesion inceses. The Tble lso shows tht the totl time spent on the netwok deceses with loss vesion. Tble 4. RDSUE esults with no toll: sensitivity to loss vesion in the time ttibute time degee of loss town cente oute bypss oute expenditue vesion flow (veh/h) time (minutes) flow (veh/h) time (minutes) (h) Sensitivity to dispesion pmete In multinomil logit the dispesion pmete divides the systemtic utilities nd is diectly popotionl to the vince of the stochstic tems. The numeicl esults shown so f hve implicitly ssumed in the oute choice model dispesion pmete equl to unity. The esults of sensitivity nlysis of RDSUE to the dispesion pmete in the cse of the two-link netwok whee no toll is chged e shown in Tble 5. The dispesion pmete vies in the intevl between 0.25 nd.75. As the dispesion pmete inceses, the diffeence between the flow on the town cente oute nd the flow on the bypss deceses. This is consistent with the intuition tht with highe dispesion pmete the OD flow is less concentted nd tends towds unifom distibution coss outes. Tble 5. RDSUE esults with no toll: sensitivity to dispesion pmete time dispesion town cente oute bypss oute expenditue pmete flow (veh/h) time (minutes) flow (veh/h) time (minutes) (h) Nguyen-Dupuis netwok In the second exmple, the Nguyen-Dupuis netwok (Nguyen nd Dupuis, 984) is used. The netwo which includes 3 nodes, 9 diected links nd 4 OD pis, is shown in Figue 3. The link-pth incidence eltionship is shown in Tble 6.

14 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Figue 3. Nguyen-Dupuis netwok Tble 6. Link-pth incidence eltionship fo the exmple netwok OD pi pth link sequence OD pi pth link sequence (,2) 2-8- (,3) (4,2) (4,3) Thee is totl of 25 pths. The OD demnd flows e q,2=660, q,3=495, q 4,2=42.5, q 4,3=495 (s 0 4 ssumed in Xu et l., 20). The following BPR time-flow functions e used: t = t [ z / c, ( ) ] whee the fee-flow tvel time t 0 nd the cpcity c e given, fo ech lin in Tble 7 (the vlues in the Tble e fom Xu et l., 20). At the oute choice level, tvel time is the only ttibute consideed in the utilities. The RDSUE is found using the MSA lgoithm of section 2.4.

15 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Tble 7. Link chcteistics of the exmple netwok link fee-flow fee-flow cpcity link tvel time tvel time cpcity Tble 8 shows the RDSUE pth flows togethe with the clss-specific flows fo OD pi (-3). The pth flows e found in the bottom line. The cells in the cente of the tble povide the clssspecific flows, ech clss being identified by the pi constituted by the efeence pth nd by the chosen pth. The lst column on the ight povides the flows hving ech given efeence pth. Tble 8 seves to illustte the popety of RDSUE: if the uses of the OD pi updte the efeence point to the pth chosen, the pth flows do not chnge. In fct, the mtix of the clssspecific flows emins unchnged with updting. This is consequence of the popety tht the sum of the elements on ow equls the sum of the elements on the coesponding column. The sum of the elements on the ow equls the flow hving s efeence given pth, while the sum of the elements on the coesponding column equls the flow using tht pth. These two flows e equl by definition of RDSUE. As n exmple, the sum of the elements on ow nd on column, which efe to pth 9, e both equl to 29.. Thee cn be smll diffeences between the ow sum nd the column sum fo othe pths due to the ppoximte convegence of the computtions. In ll cses, these devitions e less thn unity becuse the lgoithm used convegence tolence δ =. Figue 4 shows the convegence of the MSA lgoithm. Initilly, the infinity nom chnges nonmonotoniclly s the numbe of itetions inceses, then it deceses monotoniclly nd with decesing te. The numbe of itetions equied fo convegence is 323 nd the computtion time is ound 00 milliseconds with n Intel i5-760 pocesso (2.80 GHz, 8 GB RAM). Tests hve been cied out to ssess the impct on the RDSUE computed by the lgoithm of chnge in the initil vlues of the pth flows. The esults pesented bove e obtined with initil flows computed in the initilistion step on the bsis of efeence pth equl to the fist pth of the OD pi ccoding to the ode in the link-pth incidence eltionship of Tble 6. The lgoithm ws e-un by consideing s efeence pth in the initilistion step the pth with minimum time in fee-flow conditions nd the pth with mximum time in fee-flow conditions. The lgoithm esulted to be obust, s no chnge in the RDSUE pth flows hs been found. Tble 8. RDSUE clss-specific nd totl pth flows fo OD pi (,3) chosen pth totl efeence pth totl

16 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points infinity nom numbe of itetions Figue 4. Convegence of the lgoithm 3.3. Sensitivity to loss vesion in time ttibute The esults of the sensitivity nlysis with espect to the loss vesion in the time ttibute e shown in Tble 9. The degee of loss vesion in the time ttibute is denoted by γ = β LT / β GT. The Tble shows the RDSUE pth nd link flows. In the Tble, thee vlues fo the degee of loss vesion e consideed: the cse of equl bsolute vlues of the gin nd loss coefficient ( γ = ) which is the conventionl SUE, i.e. no loss vesion; the bse cse with the estimtion vlues γ =.6; nd cse with mked loss vesion ( γ = 3). The ltte vlue is ustified by litetue s explined in the two-link netwok exmple. Agin, s in the two-link netwok exmple, only β LT is chnged while β GT is left unchnged. It is possible to detect the following ptten when the distibution of pth flows fo given OD pi is consideed. If stndd devition of pth flows is computed fo ech OD pi, the stndd devition inceses with the degee of loss vesion. This mens tht, s the degee of loss vesion inceses, the OD flow is distibuted coss pths with highe vibility. This ptten is shown in Figue 5, whee the stndd devition of pth flows is computed fte hving nomlised pth flows s pecentge vlues of the coesponding OD flows. The sme ptten hd been found in the two-link netwok exmple (Tble 4). Anothe ptten which cn be detected eltes to the totl tvel time spent on the netwok. This quntity deceses fo ech OD pi s the degee of loss vesion inceses. This is shown in Figue 6. The sme ptten hd been found in the two-link netwok exmple (Tble 4).

17 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points Tble 9. Sensitivity of RDSUE pth nd link flows to loss vesion in the time ttibute pth flows link flows pth γ = γ =. 6 γ = 3 link γ = γ =. 6 γ = 3 OD pi (,2) OD pi (,3) OD pi (4,2) OD pi (4,3) OD pi(,2) OD pi (4,2) stndd devition pth flows 2 0 OD pi (4,3) OD pi (,3) 8 6,2,4,6,8 2 2,2 2,4 2,6 2,8 3 degee of loss vesion in time Figue 5. Vition of stndd devition of pth flows with degee of loss vesion in the time ttibute

18 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points OD pi (,2) OD pi (4,3) tvel time spent OD pi (,3) OD pi (4,2) ,2,4,6,8 2 2,2 2,4 2,6 2,8 3 degee of loss vesion in time Figue 6. Vition of totl tvel time spent on the netwok with degee of loss vesion in the time ttibute Sensitivity to dispesion pmete The esults shown so f hve implicitly ssumed dispesion pmete of the multinomil logit oute choice model equl to unity. When the dispesion pmete is chnged, the OD flow tends to be distibuted coss pths with lowe vibility s the dispesion pmete inceses. This ptten cn be detected in Figue 7 which shows the vition of the stndd devition of nomlised pth flows with the dispesion pmete by OD pi. The esult is consistent with intuition, bsed on the mening of the dispesion pmete, nd is simil to wht hs been found in the two-link netwok exmple (Tble 5). The totl tvel time spent on the netwok hs n incesing tend s the dispesion pmete inceses (Figue 8), similly to the cse of the two-link netwok (Tble 5). 8 6 OD pi(,2) 4 stndd devition pth flows 2 0 OD pi (4,3) OD pi (4,2) 8 OD pi (,3) 6 4 0,25 0,5 0,75,25,5,75 dispesion pmete Figue 7. Vition of stndd devition of pth flows with dispesion pmete

19 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points OD pi (4,3) OD pi(,2) OD pi (,3) tvel time spent OD pi (4,2) ,25 0,5 0,75,25,5,75 dispesion pmete Figue 8. Vition of totl tvel time spent on the netwok with dispesion pmete Sensitivity to size of pth choice set The esults shown so f hve consideed fo ech OD pi the full set of pths. It is inteesting to exmine the impct of the size of the pth choice set on RDSUE. To this im, we vied the mximum numbe of pths llowed fo ech OD pi. The pths in ech OD pi hve been odeed ccoding to incesing fee-flow tvel time. The fist pths in the odeed set hve been consideed fo ech RDSUE computtion. As mesue of the devition of the equilibium solution with espect to the equilibium tht is obtined with the full pth choice set we consideed the oot men sque eo (RMSE) of the link flows defined by: RMSE = A ( z z*) A 2 (25) whee z * denotes the flow on link in the RDSUE with the full pth choice set. Figue 9 shows tht the RMSE deceses, t n incesing te, with the mximum numbe of pths pe OD pi. This is n encouging esult becuse, in cse it wee confimed in netwoks of lge size, it suggests tht the RDSUE my be pefomed without need fo complete pth enumetion. A simil esult hs been found by Bekho nd Toledo (2005) in the conventionl SUE model.

20 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points RMSE mximum numbe of pths pe OD Figue 9. Vition of RMSE of link flows with the mximum numbe of pths pe OD 4. Conclusion In efeence-dependent models of oute choice the efeence point of tvele needs to be detemined in tems of tvel time nd money expenditue ttibutes. When the oute choice model is pplied to n equilibium setting, the efeence point cn be exogenously given o teted s endogenous. The efeence point cn be egded s exogenous when it is ssumed to be independent of the cuent tvel conditions ove the netwok. This occus if the idel tvel conditions, typiclly the fee-flow tvel time nd money expenditue on the shotest oute, e consideed s efeence point. In contst, the efeence point needs to be teted s endogenous when uses e ssumed to tke s efeence the sttus quo, i.e. the cuent choices. In this cse, the efeence point of ech use is given by the cuently chosen pth with the elted tvel time nd money expenditue ttibutes. The ppe hs fomulted efeence-dependent stochstic use equilibium model, in shot RDSUE, with endogenously detemined efeence points. Equilibium conditions e defined whee efeence points e detemined consistently with cuent flows nd tvel times. It is consideed tht uses of ech OD pi e subdivided into clsses with ech clss hving s efeence one of the pths of the OD, nd ttendnt tvel time nd money expenditue. The equilibium is defined s the condition whee the updting of the efeence pth to the cuently used pth leves unchnged the esulting pth flows. This is obtined by setting the numbe of uses of pth equl to the numbe of uses hving tht pth s efeence. The equilibium cn be chcteized fom eithe mico-level o mco-level pespective. The mico level eltes to the individul use ccoding to the usul intepettion of the stochstic tems s individul specific. The mco level eltes to the pth flows obseved by the modelle. RDSUE is fundmentlly diffeent fom the conventionl SUE. In RDSUE, ech use updtes the efeence point, i.e. chnges clss, nd theefoe, my chnge pth s well. In the specil cse whee the stochstic tems e unchnged with the updting no chnge of pth would occu (this ws poved in Delle Site nd Filippi, 20), chnges of pth my occu if the stochstic tems e ssumed to chnge with the updting. RDSUE is condition of potentil mico-shifts while t the mco level the flows do not chnge. In contst, in conventionl SUE ech use is in condition of est becuse she is ssigned to pth nd she sticks to this.

21 EJTIR 3(2), 203, pp Refeence-Dependent Stochstic Use Equilibium with Endogenous Refeence Points The RDSUE is fomulted s fixed point in the pth flows. It hs been poved tht the solution exists unde conditions usully stisfied in pctice. The RDSUE model is geneliztion to the cse of symmetic pefeences of the conventionl SUE model, since RDSUE educes to SUE when the bsolute vlues of the loss nd gin coefficients of the tvel time nd of the money expenditue ttibutes e equl. In such specil cse, the lgoithm poposed to solve RDSUE educes to the clssic pth flow-bsed MSA fo SUE. The enhnced behviol elism which is gined with RDSUE is pid t the pice of the loss of the uniqueness conditions, which e left fo futue esech. The fomultion of RDSUE in the ppe cn be extended to efeence-dependent oute choice models with diminishing sensitivity. Refeences Bekho S., Toledo T. (2005). Investigting pth-bsed solution lgoithms to the stochstic use equilibium poblem. Tnspottion Resech Pt B, 39(3), Ben-Akiv M., Bielie M. (999). Discete choice methods nd thei pplictions to shot-tem tvel decisions. In: R. Hll (ed.) Hndbook of Tnspottion Science, Kluwe Acdemic Publishes, Boston, pp Cntell G.E., Velonà P. (200). Assegnzioni Reti di Tspoto. Modelli di Punto Fisso. In Itlin: Tnspot Netwok Assignment: Fixed-Point Models. Fnco Angeli, Milno. Cscett E. (2009). Tnspottion Systems Anlysis. Models nd Applictions. Spinge, New Yok. Cscett E., Nuzzolo A., Russo F., Vitett A. (996). A modified logit oute choice model ovecoming pth ovelpping poblems: specifiction nd some clibtion esults fo inteubn netwoks. In: J.B. Lesot (ed.) Poceedings of the Intentionl Symposium on Tnspottion nd Tffic Theoy, Lyon, pp Chen A., Pvinvongvuth S., Xu X., Ryu S., Chootinn P. (202). Exmining the scling effect nd ovelpping poblem in logit-bsed stochstic equilibium models. Tnspottion Resech Pt A, 46(8), Chin A., Knetsch J.L. (2006). Vlues depend on the mesue: e mny tnspot poect vlutions seiously bised? Ppe pesented t the TRAIL Msteclss, 0 Apil 2006, Delft, The Nethelnds. De Boge B., Fosgeu M. (2008). The tde-off between money nd tvel time: test of the theoy of efeence-dependent pefeences. Jounl of Ubn Economics, 64(), 0-5. Delle Site P., Filippi F. (20). Stochstic use equilibium nd vlue-of-time nlysis with efeencedependent oute choice. Euopen Jounl of Tnspot nd Infstuctue Resech, (2), Hess S., Rose J.M., Henshe D.A. (2008). Asymmetic pefeence fomtion in willingness to py estimtes in discete choice models. Tnspottion Resech Pt E, 44(5), Khnemn D., Tvesky A. (979). Pospect theoy: n nlysis of decision unde isk. Econometic, 47(2), Knetsch J.L. (2007). Bised vlutions, dmge ssessments, nd policy choices: the choice of mesue mttes. Ecologicl Economics, 63(4), Muno A. (2009). Bounded Rtionlity nd Public Policy: Pespective fom Behvioul Economics, Spinge. Muno A., Sugden R. (2003). On the theoy of efeence-dependent pefeences. Jounl of Economic Behvio nd Ogniztion, 50(4), Nguyen S., Dupuis C. (984). An efficient method fo computing tffic equilibi in netwoks with symmetic tnspottion costs, Tnspottion Science, 8(2),

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