- Email: [email protected]
Utility, entropy and a “paradox” of traffic flow
UTILITY, ENTROPY AND A “PARAJIOX” OF TRAFFIC FLOW-I
Centrede rcchercheSLIT les transports,Universitbde Mont&, C.P.6128,St&on A, Mti,
CanadaID7 3J7
(Received22 Nova&r 1979)
.
1.WlWWtXON
cost of each path. This is also advocated by She@and
attention has
been paid in the trans- Dagnxo, who use tbis quantity to resolve .the apparent port&on reaearcbliterature to the intelpretatiott of paradox of tbe stocbestic assignmentme&ml. ConsldetaMe
counter-intuitive flow and travel times that result from the application of particular tr&ic assignment models after the addition of a link to tbe network or the mod&cationof its iength. Of note is the so called Braess paradox (Braess, 1968;Murcldand, 1970)which occurs for networks with flow dependent link costs (travel times); the addition of a link may result in a “Useroptimized” flow distribution over the routes of the network which increases the total travel costs. This phenomenon is weil understood, since &ding the useroptimixed flows is equivalent to solving a minirnilatinn problem with an objective function which does not equal the total travel costs. Also, it has been observed that the application of multipath stochastic assignment methods to network with 6xed fink costs (Van Falkenhausen, 1966;Burrell, 1968, Dial, 1971; Gunmusoa, 1972) leads to counterintuitive Sows and average travel costs. The use of the logistic function in such methods results in the assignment of unreasonably high flows to overlapping routes (Schneider, 1973;Burrell, 1976;Florian and Fox, 1976). This deficiency may be remedied by the use of probit path choice, as pointed out by Daganxoand Sbefli(1978). al$u~&zes$le problems may not yet be handled by
Thepurposeoftbisnoteistoo&radiffercntutility basedderivationoftllestoci?Acass@lmentnrmllodof Dialandto8howthattheexpectedJmr&edminimum traveltime,thatisderivedfromrandomat@tyuwory arguments,maybesbowntobepartofadtndvariablc ofaconstrainedmaxi&a&probkmwitbmobjective function which is the average utility of travellers using the available routes of the network. Sii this objective function is &imately related to entropy maxMx&n, somegenerafcon&sionsaredrawnontberelation betweentheprope&sofmodelsbasedonutilitymaximixationprin@esandonentropymaxGx&nSome ofthesecouclusionsareimplicitintheworkofWiNiams (1977) Theparadoxofstocba&croutechoice,a.%presented by Sheffi and Daganxo, is briefly revisited in tbe next
More recently,‘She@and Daganxo(1978)pointed out that tbe reduction of the cost of a linkor the addition of a link may result in flows (obtained from the application of Dial’smethod) that increase the overall travel time, a phenomenon similar to Braess’ paradox. Williams(1977) uoted that tbis phenomenon may occur with stochastic choice models that are not based on the logit fun&on. (See also Harris and Tanner, 1974;Lerman, 1975:Ben Akiva and Lerman, 1977.)Wiis proposed the use of tbe expected pcrccivcd minimumtravel cost, a quantity which may be shown to be monotonicwith respect to the
F;exp(-W
tl’his research was suppmtd inari&~~u$ftom~ Nat~~~~ISciencesandEnginekq by a pnnt from the CensctboNttckml de Dcscnvolvimento profeaor at P.U.C. do Rio de hneiro on leave from the U& vernitbde Montrhl.
S&OIL Z.TEBWlWlllWP-oS-MMV
Consider a uetwork uu&tinq of two distinct and disjoint routes of kngtb cl and q. The logit fbw allocationsinDial%a&ithmassignstoroute1thefraction ,“xp(- k,) I
oftotaltripsaudtoroute2ti4erema&rofthetrips. The average travel time (or expected cost per trip) E is given by the expression
which is not a monotone function of c2 or cl. C achieves its maximumat (q-c,) - 1.28/Oand decrases if c2 is inaefi@ for 5xed cl if
c2-c,>y. Thus, for a given cl, if c2 is decreased (link improve-
321
M. FJLimw
321)
meat) and both the initial and improved values satisfy
(2L then E &eases, a most counterAntuitiveproperty. ThispecuMtyisduestrictlytothelogitfIowalloc&n ruk. The second paradoxicalproperty pointed out by !&is andDaganxoisrelatedtolinkorrouteadditions.supposethatathirdrouteisaddedaruIthecostofthisroute isc~,c~>mox(c,,c3.~rwtcmoybedistind~ the0therroutesornlaysharesomelkks.)Then,the averagetimeC kueasesaswell,siucetlkpro&bUyof usingthislolqerrouteisnonxero!Thispcculisriryisnot necessarilyduetotheIogitiIow&lk&ionruk,but wouldocourwith@lytraacaw@uantpro&uFebMed on stochastic lIow alkcatka derived from random utility theory principles:the probabilhy that the longer route (31 isperceivintastheshor&etrouteisnonzero,duetothe
rationalbehaviorwhich implks the nlaxhakn of his net UtiIity IV,, . . ,, T2, . . . , T.1- z ck Ti, where @ is tbcMlmberoftripstbrt&willtalreonroutek.Clearly, z Tk- T. If the utility function is linear or convex, he will ulwoys select one route, since the net utility will bs by an extremal choice. On the other hand, if this utility function is concave, then he may select, over . . maxmad
theSdd$dOdOftiUW,fROWtbUOIlCroute,hlpWtiif his utility function is the integmted log function, U= U,,+~[B.T,-/,%trdr], introduced by &CkfiUlM
c&r,
Web (1911) for a utility based derivation of the gravity trip distriMion problem, the vector of trips {E} taken by an individualis the solution to the probkm md
StOClMStiCdihbUtb~Ofths~enon. ThC@S&+StillCOJlV~~bytbGSepropaties -wbere&etotaitraveltimehlllCdtoevaluatethe viebi&yofnemrkexpmS~tbS~logicat corrclmioaWOUldbeto&~W~E~. TIIUS, k%nlSiStencywith ralhm
TTk =T
utiiity the4xy impoSes t&c miniam
travel time,
i$j:g-9
TkaO, k=l,....~ i= np (t,}
(31
It is easy to show, by using the w Ku&Tuckertheorythat
andt,istheperceivedcostofroutejofmeasuredcost CbThe main result is that, ignoringsome constants,
Williams provides the most general derivation of this result.Thesa~pqertyof(4)isthatitismoiu+ to&ally increasing with respect to the measured travel cost q, thus its use would do away with the first paradoximlbehaviorofthek#itfkwanoattknNk;the less satkfying properQ of (4) is that it is motmtoncially decreasing with the sixe of the choice set. Thus. it may onlybeusedtocomparetwodistinctlinkadditionsand cannotbeusedtocomperealinkaddiiwiththe “Q-nothing” alternative. Be that as it may, the use of (4) as an evaluation measure is consistent with the random utility derivation of tk stochastic tra&.ass@nent methods. In the following section we show that ti lo& based stocha& assignmentmethod and (4) may be derived by usingade termi&& utility n&mix&m approach.
-ck Tk = exp r+
0 approach or
(8)
where u is the dual variabk assockted with eqn (5). Thus, the frequency, or proportkn, of trips taken on routekis
(9)
fk$$G
isa constant equal to (I/&). If we suppose that the route choice of an individualmay be observed, then 49 isinprincipkaparameterwhkhmaybecalii from data. Now, the rest of the argumentationis tmnquuent and [email protected] travelkrsbetweentheoriginandthedestinationbe govemedbythesameutilityfunctkaThus,theresulting daily avenge llows hr the availabk routes L are given by where 8
hk
‘8’
exP(-ek)
7em- w
,dlMtCSk
(10)
Consider the following interpretation of the route and the link flows u. will be given by choiceprobkm as perceived by an individualwho takes daily trips from an origin (zone) to a destination @one), (11) such as the work trips. Let us suppose that over a period of time he will take T trips, for which he may select among n routes and that this selection is governed by a where 8.k is an indicatorfunction which equals 1 if src II
utility, cllwopy and a ‘)smdOx” of trafficnow
329
belongs to routek and nil otherwise.Now,it is possible
&rived, merely underline the fact that the functional choiis made in various theoretical approaches often dwcvf the respectability of the theorctic.alundcrpii of the method. Thus, one may paradoxically conclude that the assumptionthat the error terms associated with the perception of the route costs, by cl/ tmvelers, have Max Ho+ 7 I(B+ WI - IMJ, In 4 + Ul- ch, (12) an extreme type 1 distribution with zero mean and variance one, in the random utility approach, is somehow equivalent to supposing that all ~ruueflersare governed subjectto by the same integrated log utility function in the dcterministie utility approach; furthermore both approaches (13) are somehow equivalent to using the entropy maximixa4 =g 7 tion approach, which makes no assumption whatsoever h,rO (14) about the sources of variability and genera&r in principle, a descriptive model. However, in each ease, the theoreticalinterpretationof each modelis diffcrent in spite and (10). A plau&k intcrpWation of the objective function of the identical mathematicalstructure. (12)is that, due to the derivationof (lo), it is tquivaknt to For instance, when Shc5 and Daganzointerpret(4) by the max&atkn of the net average aggr@e (daily) supposing that “if one interviewed the users of the utility of travelkrs g and thus, the dual variabk asso- system one would find that the total prrcaiued cost per ciated with coWraint (13) is a proper measure of the user would have decreased!” and that “users of the change in bencflt to these travellers when, the set of system will base their choice on minim&ion of peravailabk routes, the route costs or the total demand ceived cost, the basic principle of random utility models”, they trust that all travclkrs behave in fact as change. random utilii theory supposes them to do. Smcc the It is relatively easy to show that benefit measure may be derived from other utility maximization arguments,it is clearthat other interpretations arc possible. On the other hand,whateverthe derivation,the counand thus, by substituting(10)for hk we obtain ter-intuitivepropertiesof these modelsprevail,and their use is often just&d by computationalfacility,an often misusedattributeof a transportation planningmodel. u=-~lnZexp(-ec,)+Osg-1)/g. (16) I
to find (see also Fisk, 1978)an equivalent maximization probkm which is cquivaknt to solving (10) and (11). In partkukr
Sincethe last termof (16)is constantif the total demand Rcn Akiva M. and Lerman S. (1977)-gate travel and g remainscoastant,it may hc disrcgardcdwhen mcasurmobility choke models and measures of accsssiity. Paper ing changes in benefitdue changesof routes and their pmstnted at the 3rd In!. Conf Tmucl ModcZZkg, Adclaidc, Australia. costs. Thus,we have found the samemeasureof benefit as (4) by using a deterministicutility maximi&on ap- BurrcllJ. (1968)lWipath Route Assignmentand its Application to Capacity Restraint. Pnx. 4rk Znf. Symp. 37uory of Tm$ic proach. Flow, Karlsmhc, Germany. Note that the same resultmay be obtainedby nplac- Rurrcll J. (1976)Multipath route assignment: A comparisonof two methods. Dujic EquZZibrfnmMeWfs @lited by M. ing (11)with
WhiChiStllC maximhtion of the entropy of the route flows. This is rather well known by now, however it is rckvant to emphasixcthat the derivation of (17)dots not make any assumptionsregardingthe source of variability of route choke and doss not offer any causal intcrprctation. The implications of the multiple ways in which the logit based route choice model may be derived arc explored brk5y in the next section.
The multiple ways in which the logistic route choice model and its proper evaluation measures may bc
Plorian), Lecture Notes in Rconomicsand MathematicalSystems, Vol. 118.Springer-Verlag.Berlin. Bracss D. (1968)Ubcr ein Paradox der Verkehrsplanung.Untemehemenforschung 12.258-268. Daganzo C. F. and ShelIi Y. (1978)On stochastic models of trafhcassigmncnt. Trarspn. Sci. 11. Dial R. (1971)A prob&ihitic multipathtratk assignmentmodel whichobviates path enunreratior. Tmnspn Rus. IO,339-341. Pii C. (1979) Some ZkueZopnmrtsin Eqvllibnim Tm#c As&uncnr. l’&licatio+lX Ccntrc de rccherchs sur ks tta&ports, UnivcrsitCde Mont&al.To appear in Transpn Res. Plorian, Id. and Fox B. (1976)On the probabiliiticoriginof Dial’s multipathtraRicassignment model. Tmnspn Res. 10.339-341. Gtm&3on S. (1972)An algorithmfor muliipath tratRc assignment. Fnx. P7RC. U&an Tmfic Model Research Seminar, London. IIarris A. and Tanner J. (1974)Transport demand models based on personal characteristics. Pmt. 6rZ1Znt Symp. Tmnsport&on und Tm#ic Theory, Sydney. Lerman S. (1975)A Lbaggmgate Behaniond Model of U&an MoMity Azcisions. M.I.T..C.T.S.Reports75-5. McFadden D. (1973)Conditionai logit analysis of quantitative
ht. bRIAN
330
Rwiwmdca. @dltod by P. dwicobe4der.lnpmtiarof ZkOdN.AadbieRws.NOWYalt Td&; D&1970) horn paradox oftn@c !bw. 7hnapn
sdddh z-- .
x.&j
Rokbility bahi2uh i0 t40h~orirr. -BiwmBddpaPp.748-
hi.
So@ Y. and Dapmo C. F. (1978)Anorbor “Paradox”ofTratEc Flow. lhnapa Raa.l2,43-46. Von Falkenba~~~a 8. NMii~.T&c Assip+ by a t3teeb~tlc Model. m. 4th Iat. colt/. QwIutid ScmcG pp. 413&l. WilliamsIi. C. W. L (1977)OII the Forma& of Tnvd Dwud Modelsand Ecownic Evrbsltion Mof User BendIts En~~d#A9,2s5-344.
Centrede rcchercheSLIT les transports,Universitbde Mont&, C.P.6128,St&on A, Mti,
CanadaID7 3J7
(Received22 Nova&r 1979)
.
1.WlWWtXON
cost of each path. This is also advocated by She@and
attention has
been paid in the trans- Dagnxo, who use tbis quantity to resolve .the apparent port&on reaearcbliterature to the intelpretatiott of paradox of tbe stocbestic assignmentme&ml. ConsldetaMe
counter-intuitive flow and travel times that result from the application of particular tr&ic assignment models after the addition of a link to tbe network or the mod&cationof its iength. Of note is the so called Braess paradox (Braess, 1968;Murcldand, 1970)which occurs for networks with flow dependent link costs (travel times); the addition of a link may result in a “Useroptimized” flow distribution over the routes of the network which increases the total travel costs. This phenomenon is weil understood, since &ding the useroptimixed flows is equivalent to solving a minirnilatinn problem with an objective function which does not equal the total travel costs. Also, it has been observed that the application of multipath stochastic assignment methods to network with 6xed fink costs (Van Falkenhausen, 1966;Burrell, 1968, Dial, 1971; Gunmusoa, 1972) leads to counterintuitive Sows and average travel costs. The use of the logistic function in such methods results in the assignment of unreasonably high flows to overlapping routes (Schneider, 1973;Burrell, 1976;Florian and Fox, 1976). This deficiency may be remedied by the use of probit path choice, as pointed out by Daganxoand Sbefli(1978). al$u~&zes$le problems may not yet be handled by
Thepurposeoftbisnoteistoo&radiffercntutility basedderivationoftllestoci?Acass@lmentnrmllodof Dialandto8howthattheexpectedJmr&edminimum traveltime,thatisderivedfromrandomat@tyuwory arguments,maybesbowntobepartofadtndvariablc ofaconstrainedmaxi&a&probkmwitbmobjective function which is the average utility of travellers using the available routes of the network. Sii this objective function is &imately related to entropy maxMx&n, somegenerafcon&sionsaredrawnontberelation betweentheprope&sofmodelsbasedonutilitymaximixationprin@esandonentropymaxGx&nSome ofthesecouclusionsareimplicitintheworkofWiNiams (1977) Theparadoxofstocba&croutechoice,a.%presented by Sheffi and Daganxo, is briefly revisited in tbe next
More recently,‘She@and Daganxo(1978)pointed out that tbe reduction of the cost of a linkor the addition of a link may result in flows (obtained from the application of Dial’smethod) that increase the overall travel time, a phenomenon similar to Braess’ paradox. Williams(1977) uoted that tbis phenomenon may occur with stochastic choice models that are not based on the logit fun&on. (See also Harris and Tanner, 1974;Lerman, 1975:Ben Akiva and Lerman, 1977.)Wiis proposed the use of tbe expected pcrccivcd minimumtravel cost, a quantity which may be shown to be monotonicwith respect to the
F;exp(-W
tl’his research was suppmtd inari&~~u$ftom~ Nat~~~~ISciencesandEnginekq by a pnnt from the CensctboNttckml de Dcscnvolvimento profeaor at P.U.C. do Rio de hneiro on leave from the U& vernitbde Montrhl.
S&OIL Z.TEBWlWlllWP-oS-MMV
Consider a uetwork uu&tinq of two distinct and disjoint routes of kngtb cl and q. The logit fbw allocationsinDial%a&ithmassignstoroute1thefraction ,“xp(- k,) I
oftotaltripsaudtoroute2ti4erema&rofthetrips. The average travel time (or expected cost per trip) E is given by the expression
which is not a monotone function of c2 or cl. C achieves its maximumat (q-c,) - 1.28/Oand decrases if c2 is inaefi@ for 5xed cl if
c2-c,>y. Thus, for a given cl, if c2 is decreased (link improve-
321
M. FJLimw
321)
meat) and both the initial and improved values satisfy
(2L then E &eases, a most counterAntuitiveproperty. ThispecuMtyisduestrictlytothelogitfIowalloc&n ruk. The second paradoxicalproperty pointed out by !&is andDaganxoisrelatedtolinkorrouteadditions.supposethatathirdrouteisaddedaruIthecostofthisroute isc~,c~>mox(c,,c3.~rwtcmoybedistind~ the0therroutesornlaysharesomelkks.)Then,the averagetimeC kueasesaswell,siucetlkpro&bUyof usingthislolqerrouteisnonxero!Thispcculisriryisnot necessarilyduetotheIogitiIow&lk&ionruk,but wouldocourwith@lytraacaw@uantpro&uFebMed on stochastic lIow alkcatka derived from random utility theory principles:the probabilhy that the longer route (31 isperceivintastheshor&etrouteisnonzero,duetothe
rationalbehaviorwhich implks the nlaxhakn of his net UtiIity IV,, . . ,, T2, . . . , T.1- z ck Ti, where @ is tbcMlmberoftripstbrt&willtalreonroutek.Clearly, z Tk- T. If the utility function is linear or convex, he will ulwoys select one route, since the net utility will bs by an extremal choice. On the other hand, if this utility function is concave, then he may select, over . . maxmad
theSdd$dOdOftiUW,fROWtbUOIlCroute,hlpWtiif his utility function is the integmted log function, U= U,,+~[B.T,-/,%trdr], introduced by &CkfiUlM
c&r,
Web (1911) for a utility based derivation of the gravity trip distriMion problem, the vector of trips {E} taken by an individualis the solution to the probkm md
StOClMStiCdihbUtb~Ofths~enon. ThC@S&+StillCOJlV~~bytbGSepropaties -wbere&etotaitraveltimehlllCdtoevaluatethe viebi&yofnemrkexpmS~tbS~logicat corrclmioaWOUldbeto&~W~E~. TIIUS, k%nlSiStencywith ralhm
TTk =T
utiiity the4xy impoSes t&c miniam
travel time,
i$j:g-9
TkaO, k=l,....~ i= np (t,}
(31
It is easy to show, by using the w Ku&Tuckertheorythat
andt,istheperceivedcostofroutejofmeasuredcost CbThe main result is that, ignoringsome constants,
Williams provides the most general derivation of this result.Thesa~pqertyof(4)isthatitismoiu+ to&ally increasing with respect to the measured travel cost q, thus its use would do away with the first paradoximlbehaviorofthek#itfkwanoattknNk;the less satkfying properQ of (4) is that it is motmtoncially decreasing with the sixe of the choice set. Thus. it may onlybeusedtocomparetwodistinctlinkadditionsand cannotbeusedtocomperealinkaddiiwiththe “Q-nothing” alternative. Be that as it may, the use of (4) as an evaluation measure is consistent with the random utility derivation of tk stochastic tra&.ass@nent methods. In the following section we show that ti lo& based stocha& assignmentmethod and (4) may be derived by usingade termi&& utility n&mix&m approach.
-ck Tk = exp r+
0 approach or
(8)
where u is the dual variabk assockted with eqn (5). Thus, the frequency, or proportkn, of trips taken on routekis
(9)
fk$$G
isa constant equal to (I/&). If we suppose that the route choice of an individualmay be observed, then 49 isinprincipkaparameterwhkhmaybecalii from data. Now, the rest of the argumentationis tmnquuent and [email protected] travelkrsbetweentheoriginandthedestinationbe govemedbythesameutilityfunctkaThus,theresulting daily avenge llows hr the availabk routes L are given by where 8
hk
‘8’
exP(-ek)
7em- w
,dlMtCSk
(10)
Consider the following interpretation of the route and the link flows u. will be given by choiceprobkm as perceived by an individualwho takes daily trips from an origin (zone) to a destination @one), (11) such as the work trips. Let us suppose that over a period of time he will take T trips, for which he may select among n routes and that this selection is governed by a where 8.k is an indicatorfunction which equals 1 if src II
utility, cllwopy and a ‘)smdOx” of trafficnow
329
belongs to routek and nil otherwise.Now,it is possible
&rived, merely underline the fact that the functional choiis made in various theoretical approaches often dwcvf the respectability of the theorctic.alundcrpii of the method. Thus, one may paradoxically conclude that the assumptionthat the error terms associated with the perception of the route costs, by cl/ tmvelers, have Max Ho+ 7 I(B+ WI - IMJ, In 4 + Ul- ch, (12) an extreme type 1 distribution with zero mean and variance one, in the random utility approach, is somehow equivalent to supposing that all ~ruueflersare governed subjectto by the same integrated log utility function in the dcterministie utility approach; furthermore both approaches (13) are somehow equivalent to using the entropy maximixa4 =g 7 tion approach, which makes no assumption whatsoever h,rO (14) about the sources of variability and genera&r in principle, a descriptive model. However, in each ease, the theoreticalinterpretationof each modelis diffcrent in spite and (10). A plau&k intcrpWation of the objective function of the identical mathematicalstructure. (12)is that, due to the derivationof (lo), it is tquivaknt to For instance, when Shc5 and Daganzointerpret(4) by the max&atkn of the net average aggr@e (daily) supposing that “if one interviewed the users of the utility of travelkrs g and thus, the dual variabk asso- system one would find that the total prrcaiued cost per ciated with coWraint (13) is a proper measure of the user would have decreased!” and that “users of the change in bencflt to these travellers when, the set of system will base their choice on minim&ion of peravailabk routes, the route costs or the total demand ceived cost, the basic principle of random utility models”, they trust that all travclkrs behave in fact as change. random utilii theory supposes them to do. Smcc the It is relatively easy to show that benefit measure may be derived from other utility maximization arguments,it is clearthat other interpretations arc possible. On the other hand,whateverthe derivation,the counand thus, by substituting(10)for hk we obtain ter-intuitivepropertiesof these modelsprevail,and their use is often just&d by computationalfacility,an often misusedattributeof a transportation planningmodel. u=-~lnZexp(-ec,)+Osg-1)/g. (16) I
to find (see also Fisk, 1978)an equivalent maximization probkm which is cquivaknt to solving (10) and (11). In partkukr
Sincethe last termof (16)is constantif the total demand Rcn Akiva M. and Lerman S. (1977)-gate travel and g remainscoastant,it may hc disrcgardcdwhen mcasurmobility choke models and measures of accsssiity. Paper ing changes in benefitdue changesof routes and their pmstnted at the 3rd In!. Conf Tmucl ModcZZkg, Adclaidc, Australia. costs. Thus,we have found the samemeasureof benefit as (4) by using a deterministicutility maximi&on ap- BurrcllJ. (1968)lWipath Route Assignmentand its Application to Capacity Restraint. Pnx. 4rk Znf. Symp. 37uory of Tm$ic proach. Flow, Karlsmhc, Germany. Note that the same resultmay be obtainedby nplac- Rurrcll J. (1976)Multipath route assignment: A comparisonof two methods. Dujic EquZZibrfnmMeWfs @lited by M. ing (11)with
WhiChiStllC maximhtion of the entropy of the route flows. This is rather well known by now, however it is rckvant to emphasixcthat the derivation of (17)dots not make any assumptionsregardingthe source of variability of route choke and doss not offer any causal intcrprctation. The implications of the multiple ways in which the logit based route choice model may be derived arc explored brk5y in the next section.
The multiple ways in which the logistic route choice model and its proper evaluation measures may bc
Plorian), Lecture Notes in Rconomicsand MathematicalSystems, Vol. 118.Springer-Verlag.Berlin. Bracss D. (1968)Ubcr ein Paradox der Verkehrsplanung.Untemehemenforschung 12.258-268. Daganzo C. F. and ShelIi Y. (1978)On stochastic models of trafhcassigmncnt. Trarspn. Sci. 11. Dial R. (1971)A prob&ihitic multipathtratk assignmentmodel whichobviates path enunreratior. Tmnspn Rus. IO,339-341. Pii C. (1979) Some ZkueZopnmrtsin Eqvllibnim Tm#c As&uncnr. l’&licatio+lX Ccntrc de rccherchs sur ks tta&ports, UnivcrsitCde Mont&al.To appear in Transpn Res. Plorian, Id. and Fox B. (1976)On the probabiliiticoriginof Dial’s multipathtraRicassignment model. Tmnspn Res. 10.339-341. Gtm&3on S. (1972)An algorithmfor muliipath tratRc assignment. Fnx. P7RC. U&an Tmfic Model Research Seminar, London. IIarris A. and Tanner J. (1974)Transport demand models based on personal characteristics. Pmt. 6rZ1Znt Symp. Tmnsport&on und Tm#ic Theory, Sydney. Lerman S. (1975)A Lbaggmgate Behaniond Model of U&an MoMity Azcisions. M.I.T..C.T.S.Reports75-5. McFadden D. (1973)Conditionai logit analysis of quantitative
ht. bRIAN
330
Rwiwmdca. @dltod by P. dwicobe4der.lnpmtiarof ZkOdN.AadbieRws.NOWYalt Td&; D&1970) horn paradox oftn@c !bw. 7hnapn
sdddh z-- .
x.&j
Rokbility bahi2uh i0 t40h~orirr. -BiwmBddpaPp.748-
hi.
So@ Y. and Dapmo C. F. (1978)Anorbor “Paradox”ofTratEc Flow. lhnapa Raa.l2,43-46. Von Falkenba~~~a 8. NMii~.T&c Assip+ by a t3teeb~tlc Model. m. 4th Iat. colt/. QwIutid ScmcG pp. 413&l. WilliamsIi. C. W. L (1977)OII the Forma& of Tnvd Dwud Modelsand Ecownic Evrbsltion Mof User BendIts En~~d#A9,2s5-344.