User Equilibrium in a Disrupted Network with Real-Time Information and Heterogeneous Risk Attitude

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1 University of Massachusetts Amherst Amherst Masters Theses February User Equilibrium in a Disrupted Network with Real-Time Information and Heterogeneous Risk Attitude Ryan J. Pothering University of Massachusetts Amherst Follow this and additional works at: Part of the Civil Engineering Commons Pothering, Ryan J., "User Equilibrium in a Disrupted Network with Real-Time Information and Heterogeneous Risk Attitude" (2012). Masters Theses February Retrieved from This thesis is brought to you for free and open access by Amherst. It has been accepted for inclusion in Masters Theses February 2014 by an authorized administrator of Amherst. For more information, please contact

2 US ER EQU ILIBR IUM IN A D IS R UP TED NET W OR K W ITH R EA L-T IM E IN FOR M AT ION AND H ETER OGENEOUS R IS K ATT IT UDE A Th es i s P res en t ed B y RYAN J. P OTHER ING S ub m i t t ed t o t h e Grad u at e S ch o o l o f t h e Un i v ers i t y o f M as s ach u s et t s Am h ers t i n p art i al fu l fi l l m en t o f t h e req u i r em en t s fo r t h e d e gre e o f M AS TER OF C IV IL A N D ENV IR ONM EN TA L ENG INEER IN G M a y Dep art m en t o f C i v i l an d En v i ro n m en t al En gi n eeri n g

3 US ER EQU ILIBR IUM IN A D IS R UP TED NET W OR K W ITH R EA L-T IM E IN FOR M AT ION AND H ETER OGENEOUS R IS K ATT IT UDE A Th es i s P res en t ed B y RYAN J. P OTHER ING Ap p r o ve d a s t o st yl e a nd c on t e nt b y: _ So n g Ga o, Ch a i r _ J oh n Co ll ur a, M e mb e r Ri c h ar d Pal me r, De p a rt me n t He a d Ci vi l a n d En vi r o n me n t al En gi n e e ri n g

4 DED IC AT IO N I wo u l d l i k e t o d ed i cat e t h i s t h es i s t o m y l o v i n g an d s up p o rt i v e fam i l y fo r ch eeri n g m e o n t h ro u gh t h i s i n t en s e yet v e r y r e wardi n g ex p eri en ce. S p e ci al d ed i c at i o n go es t o m y p ar en t s, S h ar o n an d G eo rge, wh o s e a d v i ce h as al wa ys s u cce e d ed i n p u s h i n g m e i n t h e ri gh t d i rect i o n, an d t o m y s i s t er, J es s i ca, wh o h el p ed gi v e m e a p erfe ct d o s e o f l i gh t -h ea rt ed en co u ra g e m en t t o m ak e m y t h es i s a fu n ex p eri en ce wi t h l o t s o f l au gh s.

5 AC KNOW LED GEM EN TS I wo u l d l i k e t o t h an k m y ad v i s er, D r. S o n g Gao, f or t h e p l en t i fu l an d o u t s t an d i n g ad v i ce s h e h as p ro v i d ed m e s i n ce t h e b e gi n n i n g. Her p at i en c e an d fai t h i n h er s t u d en t s i s i m m eas u rab l e an d en co u ra ged m e t o p u t m y b es t fo o t fo rward an d p ro v i d e h er wi t h m y b es t wo rk. I wo u l d l i k e t o a ck n o wl e d ge Dr. J o h n C o l l u ra fo r serv i n g o n m y t h es i s co m m i t t ee an d s h o wi n g i n t e res t i n t h e wo rk o f each s t u d en t i n t h i s p ro gr am. I wo u l d al s o l i k e t o ack n o wl ed g e t h e UM as s Tr an s p ort at i o n C en t er fo r al l o wi n g m e t o co n d u ct m y res e ar ch an d p ro v i d i n g an y as s i s t an ce o r ad v i ce wh en n e ed ed. Xu an Lu al s o d es erv es s p eci al a ck n o wl ed g em en t as he r d o ct o r at e wo rk s erv es as t h e b as i s fo r t h i s t h es i s. Hel p i n g h er s et up ex p eri m en t s es s i o n s an d an al ys i s d at a fro m t h e ex p eri m en t u l t i m at el y l e d m e t o ch o o s i n g m y t h e s i s t op i c. Fi n al l y, I wo u l d l i k e t o t h an k al l m y fri en d s fo r t h e i r s up p o rt t h ro u gh o u t m y t h es i s an d u n d ers t an d i n g wh en I wo u l d b e u n av ai l ab l e w h en m y t h es i s n e ed ed at t en t i o n. i v

6 ABS TR AC T US ER EQU ILIBR IUM IN A D IS R UP TED NET W OR K W IT H R EA L-T IM E IN FOR M AT ION A ND HETER O GENEOUS R IS K ATT ITU DE M AY R YAN J. P OTHER ING, B.S., UN IVER S ITY OF M AS S AC HUS ETT S AM HER S T M.S., UN IVER S IT Y OF M AS S AC HUS ETTS AM HER S T Di rect ed b y: D r. S o n g G ao Th e t raffi c n et wo rk i s s ub j ect t o ran d o m d i s rup t i o ns, s u ch as i n ci d en t s, b ad weat h er, o r o t h er d ri v ers ran d o m b eh av i o r. A t r a v el er s ro u t e ch o i ce b eh a v i o r i n s u ch a n et wo rk i s t h u s affect ed b y t h e p rob ab i l i t i es o f s uch d i s rup t i o n s, h i s / h er at t i t u d e t o ward s ri s k, an d real -t i m e i n fo r m at i o n o n rev eal ed t raffi c co n d i t i o n s t h at co u l d po t en t i al l y red u c e t h e l ev el o f u n cert ai n t y d u e t o t h e d i s rup t i o n s. As t h e ro ad n et wo rk s p er f o rm an ce i s d e - t erm i n ed co l l ect i v el y b y al l t rav el ers ch o i c es, i t i s al s o affect ed b y t h es e fact o rs. Th i s t h es i s feat u res t h e d ev el o p m en t o f a m u l t i -cl as s user eq u i l i b ri u m m o d el b as ed o n h et ero - gen eo u s ri s k at t i t u d e d i s tri b u t i o n s an d a u s er eq u il i b ri u m m o d el b as ed o n v ari o u s d i s rup - t i o n p rob ab i l i t i es an d i n fo rm at i o n p en et rat i o n r at e s t h at can b e u s ed t o p erf o rm s en s i t i v i t y an al ys es fo r a t raf fi c n et wo rk. Th e m et h o d o f s u c ces s i v e av er a ge (M S A ) i s u s ed t o s o l v e fo r t h e eq u i l i b ri u m co n d i t i o n s. Lab o rat o r y ex p eri m en t al d at a are u s ed t o c al i b rat e t h e ri s k at t i t u d e m o d el. A s am p l e s en s i t i v i t y an al ys i s i s p e rfo rm ed t o s h o w t h e d i s r up t i o n an d i n - fo rm at i o n p en et rat i o n ef f ect s o n n et wo rk p er fo rm an ce. In i t i al cal i b r at i o n s s h o w p ro m i s - i n g res u l t s fo r ro u t e fl o w p red i ct i o n s i n a co n g es t ed n et wo rk wi t h res p e ct t o h et ero gen e - o u s at t i t u d e. W i t h res p e ct t o d i s rup t i o n p rob ab i l i t y an d i n fo rm at i o n ac ce s s, h av i n g t o o v

7 h i gh i n fo rm at i o n p en et ra t i o n wi l l n o t im p ro v e t h e net wo rk s p erfo rm an ce, wh i l e h av i n g a s m al l d i s rup t i o n p rob ab i l i t y can i m p ro v e t raf fi c co n d i t i o n s i n t h e n et wo rk. v i

8 T ABLE OF CONT ENT S PAGE ACK NOW LEDGEM ENTS i v ABST RACT v L IST OF T ABLES vi i i L IST OF FIGU RES i x CHAPT ER 1. INT RODUCT ION Re s ea r ch Obj ec ti ve s Co n tr i bu ti o n s Lit e ra t ur e Re vi e w Us er Eq ui li b ri u m M o d e l s Ri s k Att it u d e M o d el s Re al -T i me In f o r ma t i on M o d el s EX PERIM ENT DESIG N M ODELIN G USER EQ U IL IB R IUM W IT H HET EROGENEOUS R IS K ATT IT UDE Us e r Eq u il ib ri u m Co n d i ti o n s wi t h He t er o ge n e o us Ri s k At t it u de So l ut io n Al go r it h m M o de l Cal ib r at io n Di sa ggr e ga t e An a l ys is Aggr e ga t e An a l ys i s M ODELIN G USER EQ U IL IB R IUM W IT H ADAPT IV E ROUT E CHOICE U NDER REAL - T IM E INFORM AT IO N Us e r Eq u il ib ri u m Co n d i ti o n s wi t h Re a l -T i me In f or ma t i o n So l ut io n Al go r it h m Ap p li c at io n s CONCLUS IONS AND FUT URE W ORK REFERENCES v i i

9 LIS T OF TAB LES Tab l e P age 1. Di s ag g re gat e R es u l t s fo r S es s i o n A Di s ag g re gat e R es u l t s fo r S es s i o n A Di s ag g re gat e R es u l t s fo r S es s i o n A Di s ag g re gat e R es u l t s fo r S es s i o n A A g gre g at e R i s k At t i t u d e Di s t ri b u t i o n P red i ct i o n R es u l t s A g gre g at e P at h Fl o w P red i ct i o n s an d R M S N R es u l t s v i i i

10 LIS T OF F IG UR ES Fi gu re P age 1. Ex p eri m en t R o ad Net wo rk R i s k At t i t u d es wi t h R es p ect t o Ex p ect ed Di s u t i l it y P red i ct ed R i s k P aram e t er Di s t ri b u t i o n fo r S es s i on A P red i ct ed R i s k P aram e t er Di s t ri b u t i o n fo r S es s i on A P red i ct ed R i s k P aram e t er Di s t ri b u t i o n fo r S es s i on A P red i ct ed R i s k P aram e t er Di s t ri b u t i o n fo r S es s i on A S am p l e Net wo rk fo r S en s i t i v i t y An al ys i s To t al Tr av el Ti m e An a l ys i s u n d er Vari o u s Di s ru p t i o n P rob ab i l i t i es To t al Tr av el Ti m e An a l ys i s u n d er Vari o u s M P R i x

11 C HAP TER 1 IN TR ODUC T IO N P eop l e m ak e ro u t e ch o i c es ev er y d a y i n an y gi v e n road n et wo rk f o r v ari o u s reas o n s (m i n i m i z e t rav el t i m e, av o i d co n ges t i o n, et c.). In fo rm at i o n ab o u t t h e t ra ffi c co n d i - t i o n s i n t h e n et wo rk, e.g., t h at p ro v i d ed b y an ad van ced t r av el e r i n fo r m at i o n s ys t em (AT IS ), co u l d p o t en t i al l y en ab l e t rav el ers t o m ak e b et t er ro u t e ch o i ces t h at can h el p t h em s at i s f y t h ei r go al s m o re e ffect i v el y. Ho wev e r, wh at m o s t d o n o t real i z e i s t h at t h ei r ro u t e ch o i ce an d t h e ro u t e ch o i ces o f o t h ers i n t h e n et work h av e an i m p act o n t h e o v eral l p e r- fo rm an c e o f t h e n et wo rk. Th e wel l -k n o wn Bra es s P arad o x s t at es t h at ad d i n g c ap a ci t y t o a n et wo rk wi t h t h e i n t en t i o n t o al l ev i at e p o o r t raf f ic co n d i t i o n s can act u al l y d ecr eas e t h e n et wo rk s e ffi ci en c y as t rav el ers wi l l co n s i d er m i ni m iz i n g t h ei r p ers o n al t rav el t i m e wi t h o u t co n s i d eri n g t h e e ffect s i t h as o n t h e n et wo rk. Li k e wi s e, ad d i n g m o r e i n fo rm at i o n ab o u t t raffi c co n d i t i o n s i n t h e n et wo rk co u l d c re ate s i m i l ar eff ect s. Ad d i t i o n al l y, an o t h e r i m p o rt an t fact o r t h at co u l d affect t h e p erfo rm a n ce o f a n et wo rk i s t rav el ers ri s k at t i t u d e wh en m ak i n g a ro u t e ch o i ce. Trav el ers s en s i t i v i t y t o t raffi c co n d i t i o n s o n a n et wo rk d et erm i n es wh et h er o r n o t t h e y wi l l t ak e a ch an ce o n m i n i m iz i n g t h ei r t rav el t i m e at t h e p o s s i b l e co s t of gre at er p ers o n al d e l a y, av o i d t h e s i t u at i o n al t o get h er, o r rem ai n i n d i ffe ren t t o t h e m at t er. Th es e o u t l o o k s affe ct ev er y t rav el er s ro u t e ch o i c e, a n d wh en ap p l i ed t o t h e e nt i re p op u l at i o n, h av e an eff ect o n t h e effi ci en c y o f t h e n et wo rk. 1

12 1.1 Research Objectives Th e t h es i s p res en t s t h e d ev el op m en t o f t wo m o d el s th at p red i ct u s er eq u i l i b ri u m co n d i t i o n s fo r a gi v en n et wo rk b as ed o n h et er o gen eou s ri s k at t i t u d e an d i n fo rm at i o n acc es s, res p ect i v el y. He t e r o ge n e o u s r i sk at ti t u de in c o rp o r at e s mu l ti p le r i sk a t ti t u de s ( r i sk - se e k i n g, r i sk - a ve r se, o r r i sk - n e u t ra l ) a c r o ss a ll use r s i n a r o u t e c h o ic e m o d e l, u nl ike h o m o ge n e o u s r i sk a t ti t ud e, wh i c h o n l y i n c o rp o r a t e s o n e. Th e ri s k at t i tu d e m o d el i s cal i b rat ed u s i n g ex p eri m en t al d at a fro m t h e ro u t e ch o i c e s t u d y p er fo rm e d i n Lu et al. ( ). Th e t r av el er i n f o rm at i o n m o d el h as t h e o ret ical ap p l i c at i o n s o n t h e eff ect s o f i n fo rm at i o n acc es s an d d i s rup t i o n p rob ab i l i t y o n t he p erfo rm an c e o f t h e ro ad n et wo rk. 1.2 Contributions Th ere i s a gap i n t h e d e v el op m en t o f t raffi c eq u i l ib ri u m m o d el s i n a st och as t i c n et wo rk. Trav el ers i n h er en t l y d i ffer i n t e rm s o f th ei r ri s k at t i t u d e an d a cc e s s t o real -t i m e i n fo rm at i o n. Th ere ar e s o m e eq u i l i b ri u m mo d el s t h at i n co rp o rat e ri s k at t i t u de (h o m o ge n eo u s o r h et e r o gen eo u s ) an d t h o s e t h at i n co rp o rat e i n fo rm at i o n acces s an d d i s rup t i o n p rob ab i l i t y ( M i rch an d an i an d S o ro u s h, ; Lo an d Tu n g, ; Gao, ; Uk k u s u ri et al., ). Ho wev e r, t h ere h as b e e n n o res e arch o n t h e d e v el op m en t an d cal i b rat i o n o f an eq u i l i b ri u m m o d el t h at i n cl u d es bo t h h et ero gen eo u s ri s k at t i t u d es an d i n fo rm at i o n acc es s. Th e res ea rch p rop o s ed co n t ri b u tes t o t h e s t art o f t h e art b y t ack l i n g t wo i m p o rt an t s ub -p rob l em s wh o s e s o l u t i o n s can b e l at er co m b i n ed t o b u i l d s u ch a m o d el : 1. Dev el op m en t an d c al i b ra t i o n o f a u s er eq u i l i b ri u m mo d el wi t h h et ero g en eo u s ri s k at t i t u d e b as ed o n t h e ex p ect ed u t i l i t y t h eo r y. 2. Dev el op m en t o f a m o d el s h o wi n g t h e co m b i n ed effect o f i n fo rm at i o n p en e t rat i o n an d d i s rup t i o n p rob ab i l i t y o n i n d i v i d u al ro u t e ch o ice an d n et wo rk p er fo rm an ce. 2

13 1.3 Literature Review Fo r t h e p u rp o s e o f t h i s res ea rch, i t i s i m p o rt a n t t o res ear ch i m p o rt an t s t u d i es co n cern i n g m o d el i n g u s er eq u i l i b ri u m, h et ero ge n eo u s ri s k at t i t u d e, an d ro u t e ch o i ce m o d el s b as ed o n i n fo rm a t i o n acces s User Equilibrium Models In an y g en er al n et wo rk, t rav el ers w an t t o ch o o s e t h e ro u t e t h at b es t s u i t s t h ei r n eed s ( e. g. m i n i m i z e t ra v el t i m e, av o i d co n ges t i o n, et c.); h o wev e r, t h e y d o n o t al wa ys ch o o s e t h e b es t ro u t e. Al s o, wh i l e a ro u t e m a y n o t al wa ys b e t h e b es t ch o i c e fo r a t rav el er, i t d o es h av e a ch an ce t o b e. P rob ab i l i s t i c ch o i ce th eo r y as s u m es t h at b o t h t rav el ers w an t t o ch o o s e t h e b es t ro u t e an d each ro u t e h as a c h an ce o f b ei n g t h e b es t. Ap p l i cat i o n o f p rob ab i l i s t i c ch o i ce t h eo r y can t h en b e u s ed i n t h e d ev el op m en t o f t ra ff i c as s i gn m en t m o d el s t h at can creat e s t o ch as t i c u s er eq u i l i b ri u m (S UE) co n d i t i o n s (Dag a n z o an d S h effi, ). Un d e r S UE, fl o w s are as s i gn ed t o ro u t es b as ed o n t h e p rob ab i l i t y t h at ea ch ro u t e wi l l b e ch o s en. Lo gi t m o d el s (Di al, ; Fi s k, ; Bel l, ; M ah er, ) as s u m e t h e erro r t erm s o f e ach ro u t e ch o i ce a re i n d ep en d e nt l y an d i d en t i cal l y d i s t ri b u t ed Gu m b el v ari ab l es, al l o wi n g fo r a cl o s ed fo rm p rob ab i l i t y (Ben - Ak i v a an d Le rm a n, ). Th e d i s t ri b u t i o n as s u m p t i o n can b e t ro ub l es o m e fo r ro u te ch o i c e m o d el i n g, h o wev er, as ran d o m erro rs o f d i ff ere n t o v erl ap p i n g ro u t es ca n hav e h i gh co rr el at i o n s an d d i ffer en t v ari an c es (S h effi, ). P rob i t m o d el s as su m e t ravel t i m e d i s rup t i o n s fo l l o w m u l t i v ari at e n o rm al d is t ri b u t i o n s, al l o wi n g fo r fl ex i b l e v ari an ce a n d co v ari an c e rel at i o n s h i p s. Th i s m ak e s t h e p rob i t m o d el eas i l y ap p l i cab l e fo r ro u t e ch o i ce p r ed i ct i o n s. Th e m o d el l ack s a cl o s e d fo rm p rob ab i l i t y l i k e l o git m o d el s an d c an h av e l i m i t ed l arg e- 3

14 s cal e ap p l i c at i o n s d u e t o ru n n i n g c o s t s (Ni e, ). Th es e m o d el s i n t h ei r o ri gi n al fo rm s, h o w e v er, d o no t t ak e i n t o ac co u n t d i s rup t i o n s t o t h e n et wo rk ex p l i ci t l y an d t h e u n d er l yi n g t rav el t i m es a re d et erm i n i s t i c Risk Attitude Models Th e n u m b er o f s t u d i es d o n e o n m o d el i n g ri s k at t i t ud e i s ex t en s i v e, yet t h e ev al u at i o n m et h o d s fo r r i s k at t i t u d e s t u d i es can res u l t i n d i fferen t cl as s i fi cat i o n s o f ri s k at t i t u d e (S l o v i c ; MacC ri m m o n an d Weh ro n g ). A r eas o n fo r t h i s i s b ecau s e ri s k at t i t u d e m a y n o t b e d et ect ab l e b y j u s t l o o k i n g at peop l e s ch o i ces. Ot h er s i t u at i o n al fact o rs c an cau s e d i ffe ren t cl as s i fi cat i o n s (S ch o em ak er ). Web e r et al. ( ) s u rm i s e fro m t h ei r s t u d y t h at t h ree m et h o d s t h at can h el p m eas u re ri s k at t i t u d e i n v ari o u s wa ys : ex p ect ed u t i l i t y, r e l at i v e ri s k at t i t u d e, an d p ercei v ed ri s k at t i t u d e. Ut i l i t y i s t h e rel at i v e b e n efi t o r u s efu l n es s an ob j ect h as fo r an i n d i v i d u al. W h en ap p l i ed t o a s et o f ri s k y al t ern at i v es, s u ch as a ga m b l e, t h i s b en efi t i s k n own as ex p ect ed u t i l i t y. W h en an al yz i n g a s et o f ch o i ces, i t i s h ypo t h es i z ed t h at an i n d i vi d ual wi l l p i ck an al t ern at i v e t h at m ax i m i zes t h e ex p ect ed u t i l i t y. The at t i t u d e o f an i n d i v i d u al can b e d et erm i n ed b y t h e ch o i c es m ad e d u ri n g ea ch ga m b l e prop o s ed an d fo rm u l at i n g a u t i l i t y fo rm u l a fo r t h at p ers o n. P rat t ( ) an d Arro w ( ) h el p ed d efi n e t h e c h aract eri z at i o n o f ri s k at t i t u d es t h ro u gh u s e o f u t i l i t y fo rm u l as. Th ere ar e t wo as s u m p t i o n s as s o ci at ed wi t h ex p ect ed u t i l i t y t h e o r y: 1 ) ri s k p r efe ren c es can b e d es cri b ed b y a u t i l i t y fu n ct i o n k n o wn o n l y b y t h e m o d el er an d 2 ) at t i t u d es t o ward s ri s k can b e r at i o n al i z ed b y t h e ex p ect ed u t i l i t y fu n ct i o n. Ex p ect ed u t i l i t y c an b e v er y u s e fu l i n i d en t i f yi n g an d d es c ri b i n g p eop l e s ch o i c e p at t ern s o v er a s p eci fi ed p e ri o d, bu t q u an t i f yi n g t h e p e rs o n s ri s k at t i t u d e 4

15 i s rel at i v el y i n s i gn i fi can t (Web er et al ). On e t h eo r y t h at c an i l l u s t rat e t h e d i s ad v an t a g es in ex p ect ed u t i l i t y t h eo r y i s cal l e d cu m u l at i v e p ro s p ect t h eo r y (C P T), wh i ch s t at es t h at i n d i v i d u al s h av e a b i as ed p ercep t i o n o f t h e p rob ab i l i t i es i n a gi v en ch o i ce. Ac co rd i n g to C PT, p eop l e t en d t o o v er-es t i m at e s m al l p rob ab i l i t i es an d u n d er-es t i m at e l arg e p rob a bi l i t i es (Kah n n em an an d Tv ers k y ). R el at i v e ri s k s h o ws t h at t h e d i ff eren ces i n ri s k a t t i t u d e wi t h res p ect t o e x p ect ed u t i l i t y m a y b e a r es u l t o f t h e d i ffer en ces i n m argin al v al u es. R i s k p referen c es m a y r em ai n u n ch an ged wi t h t h i s ap p ro ach, r es u l t i n g i n a m o re st ab l e p er cep t i o n o f ri s k. Us i n g rel at i v e ri s k can m eas u re p eop l e s at t i t u d es t o wards u n c ert ai n o u t co m es rat h er t h an cert ai n o u t co m es (D ye r a n d S ari n ). P ercei v ed ri s k at t i t u d e i s t h e as s u m p t i o n t h at deci s i o n m ak ers a re at t r a ct ed o r rep el l ed b y al t ern at i v e c h o i ces t h at t h e y fe el are ri s k i er t h an ch o i ces t h ey fe el are l es s ri s k y (Web e r an d Bo t t o m ). P ercei v ed ri s k at t i tu d e h as s i gn i fi can t l y s t ro n ge r cro s s - s i t u at i o n al s t ab i l i t y t h an b o t h ex p ect ed u t i l i t y and rel at i v e ri s k. Th i s m et h o d i s p refer ab l e fo r m eas u ri n g p eop l es t en d en c y t o ch o o s e b et w een ri s k y an d s afe ch o i c es (Web er an d M i l l i m an ). W h i l e t h ere are n u m ero u s s t u d i es o n v ari o u s mo d el s t o d et erm i n e p eop l es ri s k at t i t u d e, t h ere i s n o t m u ch res e arch d et ai l i n g t h e cal i b rat i o n o f v a ri o u s ri s k at t i t u d e d i s t ri b u t i o n s t o a fi n d a gen e ral t ren d i n t rav el e r b eh av i o r. Th er e are, h o wev er, m an y ex p eri m en t s wi t h res u l t s t h at s h o w a p o s s i b l e p at t ern i n t rav el er ri s k at t i t u d e. Web er an d Bo t t o m ( ) d et erm i n ed p eop l es ri s k at t i t u d es b y ex am i n i n g t h ei r ch o i ces o n t h eo ret i cal l o t t eri es wi t h v ari o u s p rob ab i l i t i es. U s i n g C P T, t h e y fo u n d t h at 7 6 % o f al l p art i ci p an t s we re ei t h e r ri s k -av ers e o r ri s k -n eu t r al. An o t h er s t u d y, d e P a l m a an d P i card 5

16 ( ), co n cl u d ed t h at 6 6 % o f p art i ci p an t s w ere r i sk -av ers e o r ri s k -n eu t ral an d 3 3 % wer e ri s k -s eek i n g. Web er an d M i l l i m an ( ) u s ed ex p ected u t i l i t y t h eo r y t o ch ar act eri z e t rav el er ri s k at t i t u d e i n o n e o f t h ei r ex p eri m en t s. Th e ex p eri m en t req u i red p art i ci p an t s t o ch o o s e v a ri o u s co m m u t i n g t i m es fo r a t r ai n t o d et erm i n e t h ei r u t i l i t y fu n ct i o n s fo r ri s k as s es s m en t. B y ob s e rv i n g t h e s h ap e o f e ach p art i ci pan t s u t i l i t y fu n ct i o n, i t was fo u n d t h at a m aj o ri t y o f p art i ci p an t s wer e ri s k -av e rs e o r ri s k -n eu t r al wh en t h e co m m u t i n g t i m es were s l o wer t h an o r eq u a l t o t h e av er a ge. W h en c o m m u t i n g t i m es we re fa s t er o r eq u al t o t h e av e ra g e, h o w ev er, a m aj o ri t y o f p art i ci p an t s were fo u n d t o b e ei t h er ri s k -s eek i n g ri s k - n eu t ral. Th e r es u l t s o f t h es e s t u d i es an d o t h ers (Bru i n s m a et al ; Lam an d S m al l ) s h o w a s t ro n g i n d i cat i o n t h at p eop l e t en d t o b e ri s k -av e rs e Real-Time Traveler Information Models P re-t ri p an d en -ro u t e i n fo rm at i o n can al l o w t rav ele rs t o p l an an d ad ap t t h ei r t ri p t o eff ect i v el y m eet t h ei r n e ed s. Tr av el e rs wh o h av e acc es s t o t ra ffi c i n fo rm at i o n are m o r e l i k el y t o fo l l o w t h e p ro v i d ed p re-t ri p an d en -r o u t e t raffi c i n fo rm at i o n (Ab d el -At y an d Ab d al l a, ). En - ro u t e r eal -t i m e t r av el i n fo r m at io n al l o ws t rav el ers t o m ak e ro u t e ch o i ces at d eci s i o n n o d e s b as ed o n cu rren t co n d i t i on s t o av o i d d el a y (M cqu een et al., ). Wi t h rega rd s t o e n -ro u t e s h o rt -t e rm ch o i c es, p ro v i d i n g q u al i t at i v e i n fo rm at i o n i s m o re b en efi ci al fo r t r a v el ers t h an q u an t i t at i v e i n fo rm at i o n (Ab d el -At y an d Ab d al l a, ). Uk k u s ru i et al. ( ) co n cl u d ed t h at an y chan ge i n u s er b eh av i o r d u e t o re al -t i m e i n fo rm at i o n m u s t b e acco u n t fo r i n t raffi c as s i gn m en t m o d el s. 6

17 M o d el s t h at i n co rp o rat e i n fo rm at i o n acces s can g i v e fu rt h er i n s i gh t i n t o trav el e r ro u t e ch o i ce b eh av i o r. Gao et al. ( ) d e s cri b e two ro u t e ch o i c e m o d el s t h at i n co rp o rat e re al -t i m e i n fo rm at i o n : ad ap t i v e p at h m od el s an d s t rat e gi c ro u t e ch o i ce m o d el s. Ad ap t i v e p at h mo d el s as s u m e ro u t e ch o i c es are a s eri es o f p at h ch o i ces at ev er y d eci s i o n n o d e. Th i s can acco u n t fo r d i v ers i o n f ro m t h e i n i t i al p at h b ut d o es n o t p l an ah ead fo r up co m i n g i n fo rm at i o n. S t rat egi c ro u t e ch oi ce m o d el s a re b as ed o n a ru l e t h at m ap s s t o ch as t i c n et wo rk co n d i t i o n s t o ro u t i n g d eci si o n s. Th i s m o d el as s u m es t rav el e rs h av e ex p ect at i o n s fo r e n -ro u t e t r av el i n fo rm at i o n. It al s o as s u m es t h a t t rav el e rs a re p ro act i v e wh en p l an n i n g ro u t es. 7

18 C HAP TER 2 EXP ER IM EN T DES IGN Ex peri m ent R oad Net work Th e d ev el op m en t an d ca l i b rat i o n o f t h e u s er eq u i l ib ri u m m o d el s were b a s ed o n d at a ob t ai n ed fro m t h e ro u t e ch o i ce ex p eri m en t featu red i n Lu et al. ( ). Th e ex p eri m en t was co m p o s ed o f ei gh t s es s i o n s, e ach i n vo l v i n g t h e p art i ci p at i o n o f 1 6 i n d i v i d u al s. Du ri n g ea ch s es s i o n, p art i ci p an t s w er e i n s t ru ct ed t o m ak e ro u t e ch o i c es f ro m wo rk t o h o m e i n t h e n et wo rk, s h o wn i n Fi gu re 1, o n a d a y-t o -d a y b a s i s. Th e y w ere s h o wn t h e free- fl o w t rav el t i m es o f each ro u t e b ef ore t h e s t art o f t h e ex p eri m en t an d were t o l d t h at t h e h i gh wa y h as a d i s rup t i o n p rob ab i l i t y o f W h i l e n o t i n fo rm ed o f t h e d u rat i o n o f t h e ex p eri m en t t o p rev en t b i as, p art i cip an t s m ad e ro u t e ch o i ces fo r d a ys. Aft er ev e r y p art i ci p an t m ad e a ro u t e ch o i ce fo r a gi v en d a y, t h e y w ere s h o wn t h e t rav el t i m e fo r t h ei r ch o s en ro u t e. 8

19 Th e ex p eri m en t was s p l i t i n t o t wo s cen ari o s, each co m p o s ed o f fo u r s es s i o n s. Th e fi rs t s cen a ri o, k n o wn as t h e i n ci d en t cas e, h ad p art i ci p an t s m ak e ro u t e c h o i ces wi t h o u t b ei n g i n fo rm ed i f t h e h i gh w a y w as ex p eri en ci n g an in ci d en t t h at d a y t h a t wo u l d great l y i n creas e t h e t rav el t i m e o n t h at p at h. Th e s eco n d scen ari o, k n o wn as t h e i n f o rm at i o n cas e, p ro v i d ed an i n fo rm at i o n n o d e fo r t rav el e rs w h o ch o se t h e co n n e ct o r. On ce at t h e i n fo rm at i o n n o d e, p a rt i ci p an t s wer e t o l d wh et h e r o r n o t t h e h i gh w a y wa s ex p eri en ci n g s o m e fo rm o f d i s rup t i o n t h at d a y. Th e p art i ci p an t s were t h en ab l e t o m ak e an i n fo rm ed d eci s i o n b as ed o n t h e t r affi c co n d i t i o n s o n t h e h i gh wa y. 9

20 C HAP TER 3 M ODE LING US ER EQU ILIBR IUM W ITH HE TER OGENEOUS R IS K ATT ITUDE Th e fo l l o wi n g s e ct i o n d et ai l s t h e d ev el op m en t o f t he u s er eq u i l i b ri u m m o d el t h at i n co rp o rat es h et ero g en e o u s ri s k at t i t u d e i n i t s p red i ct i o n s. Th i s ch ap t er wi l l d et ai l t h e co n d i t i o n s fo r u s er eq u i l i b ri u m wi t h res p ect t o h etero gen eo u s ri s k at t i t u d e, t h e s o l u t i o n al go ri t h m u s ed t o d eri v e t h e u s er eq u i l i b ri u m, an d t h e as s u m p t i o n s u s ed t o cal i b rat e t h e m o d el. 3.1 User Equilibrium Conditions with Heterogeneous Risk Attitude Th e co n v en t i o n al u s e r eq u i l i b ri u m co n d i t i o n i n a st at i c an d d et erm i n i s t i c n et wo rk i s gen e ral i z ed t o t h e s t o ch as t i c n et wo rk wi t h h et ero gen eo u s u s er ri s k at t i t u d e (wi t h n o real -t i m e i n fo rm at i o n ) i n t h i s t h es i s as fo l l o ws : a t u s er eq u i l i b ri u m, al l u s ed p at h s h av e t h e s am e an d m i n i m u m ex p ect ed d i s u t i li t y fo r each ori gi n -d es t i n at i o n p ai r an d ri s k at t i t u d e cl as s. Fo r t h e n e t wo rk i n Fi gu re 1, u s er e q u i l i b ri u m fo r t h e i n ci d en t cas e i s m et wh en t h e ex p ect ed d i s u t i l i t i es fo r al l t h ree p at h s (t h e art e ri al, h i gh w a y, a n d d et o u r) ar e eq u al an d m i n i m i z ed. It was p rev i o u s l y s t at ed t h at wh en an i n d i v i d u al faces a s et o f ch o i ces, h e/ s h e wi l l p i ck an al t ern at i v e t h at m ax i m iz es t h e ex p ect ed u t il i t y. Ho w ev e r, i n t h e co n t ex t o f ro u t e ch o i ces i n a ro ad n et wo r k, t rav el ers ar e as s u m ed t o m i n i mi z e t h ei r ex p ect ed d i s u t i l i t y, o r t h e rel at i v e co s t a ch o i c e h as fo r an i n d i v i d u al. Th e re as o n ex p ect ed d i s u t i l i t y i s u s ed i n s t ead o f ex p ect ed u t i l i ty i s t h at re ga rd l es s o f c h o i ce, a ro u t e i s as s o ci at ed wi t h a co s t (e. g. t rav el t i m e). 1 0

21 Fi gu re 1: R i s k At t i t udes wi t h R espect t o Ex pect ed Di s ut i l i t y Th e cu rv at u re o f t h e d i s u t i l i t y fu n ct i o n o f ro u t e t rav el t i m es wi l l ch ara ct e ri z e t h e t rav el er s ri s k at t i t u d e as s h o wn i n Fi gu r e 2. Th e ex p ect ed d i s u t i l i t y fo r a t rav el er o n a ch o s en ro u t e i s c al cu l at e d b y t h e f o l l o wi n g eq u at i on : minutes Th e c-v al u e i n t h e eq u at i o n h el p s d en o t e a t r av el e r's ri s k at t i t u d e. R i s k n eu t ral i t y can b e rep r es en t ed wi t h d i s u t i l i t y as t h e s t rai gh t l i n e s h o wn i n Fi gu r e 2, a n d t h e m argi n a l d i s u t i l i t y i s co n s t an t. Un d er ri s k n eu t ral i t y, a t rav el er 's ex p ect ed d i s u t i l i t y fo r a ro u t e i s eq u al t o t h at ro u t e 's t rav el t i m e. Th er efo r e, p eop le wh o a re ri s k n eu t ral wi l l h av e a c- v al u e o f 1.0. Trav el ers e x h i b it i n g a ri s k -av ers e at t i t u d e can b e rep r es en t ed b y t h e co n v ex fu n ct i o n, wh e re t h e m argi n al d i s u t i l i t y i s i n c reas in g. As t h e m argi n al d i s u t i l i t y i s 11

22 i n creas i n g, ri s k - av e rs e t r av el ers wi l l h av e c -v al u e s gr eat e r t h an 1.0. Th e co n cav e fu n ct i o n b es t rep res en t s ri s k -s e e k i n g b eh av i o r, as t h e m argi n al d i s u t i l i t y i s d ecre as i n g. R i s k s eek i n g i n d i v i d u al s wi l l h av e c-v al u es t h at are gr eat er t h an z ero b u t l es s t h an 1.0. Th e c- v al u e i s a p ar am et er t h at m u s t b e cal i b rat ed f ro m dat a. No t e t h at u s er eq u i l i b ri u m i s u s u al l y u s ed t o d es cri b e a s t e ad y s t at e o f a t raffi c n et wo rk. In a n et wo rk s u b j ect t o ran d o m cap a ci t y red u ct i o n, h o w t o d efi n e a s t ead y s t at e i s b y i t s el f a res e ar ch q u es t i o n. In t h i s t h es i s, t h e m ean t rav el t i m e (o r d i s u t i l i t y) t ak en o v er a l a rge n u m b er o f d a ys i s u s ed t o d es cri b e t h e s t at e o f t h e n et wo rk, as b y d efi n i t i o n t h e t rav el t i m e i s a r an d o m v ari ab l e d i s t ri b u t ed o v er d a ys, an d o n e c an n o t ex p ect a fi x ed t rav el t i m e fro m d a y t o d a y. 3.2 Solution Algorithm Th e s o l u t i o n al go ri t h m u s ed t o d e ri v e eq u i l i b ri u m ro u t e ch o i ces (t r affi c fl o ws ) i n t h e n et wo rk i s b as ed o n t h e m et h o d o f s u c ces s i v e ave ra ges (M S A). M S A i s an i t erat i v e p ro ces s t h at wi l l h eu ri s t i cal l y s o l v e fo r eq u i l i b riu m co n d i t i o n s (S h effi an d P o wel l ). Th e al go ri t h m wi l l ru n m u l t i p l e i t erat i o n s an d d i s tri b u t e fl o ws t o an op t i m al p at h wi t h t h e s m al l es t ex p ect ed d i s u t i l it y fo r ea ch ri s k gro up. Th e fi rs t i t erat i o n fo r a g i v en ri s k gro up b e gi n s b y l o o k i n g at t h e f ree fl o w t rav el t i m e o f ev er y p at h i n t h e n et wo rk. Th e p at h wi t h t he l o wes t fre e fl o w t rav el t i m e wi l l t h en h av e t h e en t i re d em an d o f t h e n et wo rk. Fo r t h e n ex t i t erat i o n, t h e ex p ect ed d i s u t i l i t y fo r each p at h wi l l b e cal cu l at ed. Th e p at h wi t h t h e l o wes t ex p ect ed d i s u t i l i t y wi l l t h en b eco m e t h e n ew op t i m al ro u t e ch o i ce. Fl o ws f ro m al l p at h s ar e t h en re- a l l o cat ed b y t h e fo l l o wi n g al go ri t h m. 1 2

23 1. For All Paths: 1 h h h 2. Aggregate Re-allocated Flow: h h 3. Add Aggregate Flow to Optimal Routing Policy: + h 1 3

24 On ce al l t h e fl o ws h av e b een re -al l o cat ed t o t h e opt i m al ro u t e ch o i ce fo r a gi v en ri s k gro up, t h e ex p ect e d d i s u t i l it y fo r ea ch ro u t e ch o i ce wi l l b e re- c al cu l at ed. Th e d i ffer en ce b et ween t h e c u rren t an d p rev i o u s i t erat io n s ex p ect ed d i s u t il i t i es fo r each ro u t e ch o i ce wi l l b e cal cu l at ed. If t h e ab s o l u t e v al u e o f t h i s d i fferen ce i s l es s t h an o r eq u al t o t h e d es i red l i m i t acro s s a l l p at h s fo r each ri s k gro up, t h en t h e M S A an al ys i s wi l l s t op an d eq u i l i b ri u m co n d i t i o n s a re m et. Ot h erwi s e, t h e i t erat i o n p ro ces s wi l l b e c arri ed t h ro u gh u n t i l t h e l i m it co n d i t i o n is m et. 3.3 Model Calibration Two d i ffe ren t t yp es o f cal i b rat i o n s wer e p e rf o rm ed t o d ev el op ri s k at t i t u d e p aram et e rs an d d i s t ri b u t i o n s. Th e fi rs t is a d is ag gre gat e an al ys i s t o d eri v e a ri s k p aram et e r fo r ea ch p art i ci p an t b as ed o n t h e i nd i v i du al l ev el ro u t e ch o i ces o v er t h e d u rat i o n o f t h e ex p eri m en t. Th e s eco n d cal i b rat i o n i s an ag gre g at e an al ys i s t h at cal i b rat es t h e ri s k p aram et er d i s t ri b u t i o n am o n g al l t h e p art ici p an t s o f a gi v en s es s i o n b as ed o n ag g re gat e p at h fl o ws av e ra ged o v er t h e s t e ad y p e r i od o f t h e ex p eri m en t. Th e d i s a g gr e gat e an al ys i s i s u s ed t o p ro v i d e s up p o rt fo r t h e ri s k p aram et e r d i s t ri b u t i o n as s u m p t i o n s i n t h e ag g re gat e an al ys i s. Du e t o a l i m i t ed n u m b er o f co m bi n at i o n s o f p aram e t er v al u es, t h e d i s ag gr e gat e an al ys i s i s n o t yet co m p l et e b u t s t i l l ab l e t o p ro v i d e res u l t s Disaggregate Analysis Th e d i s a g gr e gat e cal i b r at i o n an al ys i s at t em p t s t o ch ar act e ri z e an ex p eri m en t s es s i o n s ri s k at t i t u d e d i s t ri b u t i o n b y ch a ract e ri z i n g ea ch p art i ci p an t s ro u t e ch o i c e b eh av i o r an d as s i gn i n g a c-v al u e b as ed o n t h ei r d ail y ro u t e ch o i ces. T h i s i s d o n e b y 1 4

25 as s es s i n g wh at c-v al u e a n d ad d i t i o n al p aram et e rs p ro d u ce t h e m o s t ac cu ra t e ro u t e ch o i c e p red i ct i o n s fo r each i n d i v i d u al. Th e i n d i v i d u al mo del i n g i s d o n e b y an al yz i n g each d a y s ro u t e ch o i ce, cal cu l at i n g t h e ex p ect ed d i s u t i l i t y f o r each p o s s i b l e ro u t e c h o i ce fo r e ach d a y, cal cu l at i n g t h e p rob ab i l i t y o f an i n d i v i d u al ch o o s i n g a s p eci fi c ro u t e u s i n g a l o gi t m o d el, an d cal cu l at i n g t h e l i k el i h o o d t h e p red i ct e d ro u t e ch o i c es ar e co rre ct b as ed o n t h e gi v en p aram et ers. Th e i np u t fi l e co n t ai n s t h e ro u t e ch o i ce i n fo rm at i on o n a p art i ci p an t o f an ex p eri m en t s es s i o n, i n clu d i n g t h e p art i ci p an t s r o ut e ch o i ce, t h e ro u t e s t rav el t i m e, an d wh et h er o r n o t t h ere w as an i n ci d en t o n t h e h i gh wa y fo r e ach d a y o f t h e ex p eri m en t. Al l 6 4 p art i ci p an t s i n t h e i n ci d en t cas e w er e an al yz ed t h ro u gh t h i s m et h o d. W h en a p art i ci p an t s fi l e i s re ad i n t o t h e p ro gra m, t h e y ar e as s i gn ed wi t h t h re e p aram et e rs t h at wi l l v ar y t h ro u gh o u t t h e cal i b r at i on : a c-v al u e, an art eri al b i as (α ), an d a p rob ab i l i s t i c s cal e ( λ). T h e c -v al u e wi l l h el p d et er m i n e h o w ri s k -s eek i n g o r ri s k -av ers e an i n d i v i d u al i s. Th e art eri al b i as wi l l cap t u re an y bi a s fo r t h e s afe art e ri al n o t acco u n t ed fo r b y t h e ex p ect ed u t i l i t i es, fo r ex am p l e, t o o ffs et an y co m p l i cat i o n s cau s ed b y p art i ci p an t s s el ect i n g t h e art e ri al j u s t b ecau s e l es s cl i ck s w ere req u i re t o fi n i s h t h at d a y s ro u t e ch o i ce s el ect i o n. Fi n al l y, t h e l o g i t m o d el s cal e i s u s e d to d es cri b e t h e s en s i t i v i t y o f t h e ch o i ce t o t h e d i ffer en c e i n ex p ect e d d i s u t i l i ti es. On ce al l t h e p aram et er s are as s i gn ed t o an i n d i v i du al, t h e fi rs t s t ep i n t h e cal i b rat i o n al go ri t h m i s t o an al yz e t h ei r ro u t e ch o i ces fo r ea ch d a y o f t h e e x p eri m en t. Fo r each d a y o f t h e ex p eri m en t, t h e m o d el d et erm i n es whet h er o r n o t t h e art e ri al, d et o u r, o r t h e h i gh wa y w as ch o s en t h at d a y. If t h e h i gh w a y was ch o s en o n t h at d a y, t h e m o d el al s o n o t es wh et h er o r n o t t h e re was an i n ci d en t o n t h e hi gh w a y t h at d a y. W h i l e n o t i n g each 1 5

26 d a y s ro u t e ch o i ce, t h e m o d el wi l l al s o reco rd t h e t o t al n u m b er o f t i m es a ro u t e h as b een ch o s en b y an i n d i v i d u al as t h e m o d el an al yz es t h e rem ai n i n g d a ys o f t h e e x p eri m en t. Th e m o d el wi l l al s o reco rd t h e t o t al n u m b er o f t i m es the h i gh w a y w as ch o s en wh en t h ere w as an i n ci d en t an d wh en t h e re was n o i n ci d en t. Aft er t h e d a y s ro u t e ch o i ce was reco rd ed, e ach ro u t e s av era g e ex p ect ed d i s u t i l i t y was cal cu l at ed fo r t h at d a y. C al cu l at i n g t h e ex p ected d i s u t i l i t y fo r t h e a rt eri al an d t h e d et o u r fo r each d a y u s e t h e s am e p ro c ed u re wh i l e cal cu l at i n g t h e ex p ect e d d i s u t i l i t y fo r t h e h i gh w a y i s m o re co m p l ex. If t h e a rt eri al o r t h e d et o u r w as ch o s en, t h en t h e ex p ect ed d i s u t i l i t y i s cal cu l at ed n o rm al l y u s i n g t h e ch o s en r o u t e s t rav el t i m e. If t h e ro u t e h as b ee n ch o s en o n p r ev i o u s d a ys, t h en i t s ex p ect ed d i s u t i l it y i s av e ra g e wi t h i t s p rev i o u s d i s u t i l i ti es t o cre at e a n e w av e ra ge ex p ect ed d i s u ti l i t y fo r t h at d a y. If a ro u t e h as n o t b een ch o s en at al l, t h en i t s ex p ect ed d i s u t i l i t y i s cal cu l a t ed u s i n g i t s fr ee- fl o w t r a v el t i m e. Th i s i s u n d er t h e as s u m p t i o n t h at p art i ci p an t s rem em b er b ei n g s h o wn e ach ro u t e s fr ee -fl o w t rav el t i m e at t h e b e gi n n i n g o f t h e s es s i o n. Th e r ou t e s ex p ect ed d i s u t i l i ty b as ed o n i t s free -fl o w t rav el t i m e wi l l s erv e as i t s av er a ge ex pect ed d i s u t i l i t y u n t i l i t h as b een ch o s en b y t h e i n d i v i d u al. If an i n d i v i d u al ch o o s es a p r ev io u s l y u n ch o s en ro u t e fo r t h e fi rs t t i m e, t h en i t s ex p ect ed d i s u t il i t y wi l l b eco m e t h e n ew aver a ge ex p ect ed d i s u t i l i t y, as t h e i n d i v i d u al h as m o re ac cu rat e i n fo rm at i o n o n t h e r o ut e t h an t h ei r i n i t i al as s u m p t i o n. W h i l e t h e p ro ces s fo r an al yz i n g t h e h i gh w a y s av er a ge ex p ect ed d i s u t i l i t y i s s i m i l ar t o t h e p r ev i o u s m et h o d u s ed fo r t h e art e ri al an d t h e d et o u r, t h e i n ci d en t p rob ab i l i t y fo r t h e n et wo rk cre at es d i ffer en t co n d it i o n s fo r cal cu l at i n g t h e ex p ect ed d i s u t i l i t y. As t h e h i gh w a y i s s t o ch as t i c, t h ere a re fo u r co n d i t i o n s fo r c al cu l at i n g t h e ex p ect ed d i s u t i l i t y fo r t h e h i gh wa y, d ep en d i n g o n th e n u m b er o f t i m es t h e h i gh w a y w as 1 6

27 ch o s en an d wh et h e r o r n o t an i n ci d en t o c cu rr ed o n th at d a y. If t h e h i gh wa y h as n o t b een ch o s en, t h en t h e ex p ect e d d i s u t i l i t y i s cal cu l at ed u s i n g t h e fo l l o wi n g fo rm u l a: 1 + h h h h Th e fre e-fl o w t r av el t i m e (2 0 m i n u t es ) wh en t h e re was n o i n ci d en t w as p ro v i d ed fo r p art i ci p an t s at t h e b egi n n i n g o f t h e ex p eri m en t, b u t t h ere was n o d es cri p t i o n t o d es cri b e h o w l a rge t h e fr ee-fl o w t r av el t i m e wo u l d i n creas e i f t h ere w as a n i n ci d en t. It i s as s u m ed t h at p art i ci p an t s co u l d t h i n k o f s om e u n reas o n ab l y l o n g t rav el t i m e t o v i s u al iz e t h e h i gh w a y d u ri n g an i n ci d en t, s o u s i n g an i n ci d e nt free -fl o w t r av el t i m e o f m i n u t es was ch o s en fo r an al ys i s. Th e ex p ect ed d i s u t i l i t y cal cu l at ed wi t h t h es e fr ee-fl o w t r av el t i m es i s u s ed as t h e av er a ge ex p ect ed d i s u t i l i t y u nt i l t h e h i gh wa y i s ch o s en. On ce t h e h i gh wa y i s c h o s en, t h e ex p ect ed d i s u t i l i t y i s cal cu l at ed u n d er t h ree p o s s i b l e co n d i t i o n s. On e co n d i t i o n i s i f t h e h i gh w a y h as o n l y b een ch o s en wh en t h er e h as n o t b een an i n ci d en t, wh ere t h e ex p ect ed d i s u t i l i ty eq u at i o n i s : 1 + h h h h 1 7

28 Th e n ex t co n d i t i o n i s i f t h e h i gh wa y h as o n l y b e e n ch o s en wh en t h e re h as b een a n i n ci d en t. It s eq u at i o n i s g i v en as : 1 + h h h h Fi n al l y, t h e l as t ex p ect e d d i s u t i l it y eq u at i o n fo r t h e h i gh wa y ap p l i es w h en t h e h i gh wa y h as b e en ch o s en d u ri n g an i n ci d en t an d n o i n ci d en t at l eas t o n ce, cal cu l at ed wi t h t h e fo l l o wi n g eq u at i o n : 1 + Th e an al ys i s o f a p art i ci p an t s ro u t e ch o i ce an d t he cal cu l at i o n o f i t s resp ect i v e ex p ect ed d i s u t i l i t y co n t i n u e t h ro u gh o u t t h e d u rat i on o f t h e d a y ex p e ri m en t, b u t t h e ro u t e ch o i ce p red i ct i o n p ro ces s o n t h e m o d el s t art s o n t h e 3 1 st s et o f ca l cu l at i o n s. Th i s rep res en t s t h e en d o f t h e ex p l o rat i o n p eri o d p ar t i ci p an t s ex p eri en ce d u ri n g t h e fi rs t 3 0 d a ys o f t h e ex p eri m en t. Th e n ex t 9 0 ro u t e ch o i c e an al ys es fo r t h e i n d i v i d u al re fl ect t h e i d ea t h at i n d i v i d u al s are m ak i n g ro u t e ch o i ces t h a t refl ect t h ei r ri s k at t i t u d e. Th e fi rs t s t ep i n t h e p r ed i ct i o n p ro ces s i s t o d et erm i n e t h e p rob ab i l i t y t h at an i n d i v i d u al wi l l ch o o s e a s p eci fi c ro u t e. A l o gi t m od el i s u s ed t o h el p d et erm i n e t h e p rob ab i l i t y a ro u t e wi l l b e ch o s en ea ch d a y. Th e p rob ab i l i t i es ar e b as ed o n t h e p rev i o u s d a y s av er a ge ex p ect ed d i s u t i l i t y fo r ea ch ro u t e. T h e art e ri al b i as i s ad d ed t o t h e av er a ge ex p ect ed d i s u t i l i t y t o d et erm i n e i f t h e i n d i v i d u al wi l l b e m o re p referen t i al t o t h e art eri al 1 8

29 t h an t h e o t h e r ro u t es i f e v er yt h i n g el s e i s eq u al. Th e av era g e ex p ect ed d i s u t i l i t y fo r ea ch ro u t e i s m u l t i p l i ed b y t h e s cal e an d n e gat ed as t h e ex p ect ed d i s u t i l i t y i s an a s s o ci at ed co s t an d s up p o s ed t o b e m i n im i z ed. Th e l o gi t m o d el eq u ati o n s fo r ro u t e ch o i c e p rob ab i l i t i es are s h o wn b el o w: Th e n ex t an d fi n al s t ep i n t h e p red i ct i o n p ro ces s is t o cal cu l at e t h e i n d i v i d u al s l o g l i k el i h o o d t h at t h e y wi l l ch o o s e t h e ch o s en ro u t es o v er t h e 9 0 -d a y p er i o d, fo u n d b y s u m m i n g t h e n at u ral l o g s o f t h e l i k el i h o o d o f ch o o si n g t h e ch o s en ro u t es fo r ea ch o f t h e 9 0 d a ys i n t h e p red i ct i o n p eri o d. Th e g en er al fo rm ul a fo r t h i s cal cu l at i o n i s s h o wn b el o w: ln h ; 31 h Th e t o t al l o g l i k el i h o o d i s u s ed t o co m p are h o w well t h e d i s a g gre g at e an al ys i s p red i ct s an i n d i v i d u al s r o u t e ch o i ce. Th e cl o s er t h e l o g l i k el i h o o d i s t o z e ro (l i k el i h o o d cl o s er t o 1 ), t h en t h e b et t er t h e m o d el h as p red i c ted an i n d i v i d u al s ro u t e ch o i ce wi t h a gi v en s et o f ri s k p ar am et er, art eri al b i as an d s cale p aram et ers. Th e l i k el i h o o d fo r an 1 9

30 i n d i v i d u al i s cal cu l at ed f o r al l p o s s i b l e p ar am et er gro up i n gs b et we en t h e c -v al u e, a rt eri a l b i as, an d l o gi t s c al e. Th e c-v al u e i n t h e g ro up i n g wi t h t h e t o t al l i k el i h o o d th at i s cl o s es t t o z ero wi l l b e u s ed t o rep res en t t h at p ers o n i n co n s t ru ct i o n o f t h e ri s k at t i t ud e d i s t ri b u t i o n fo r t h e en t i re ex p eri m en t s es s i o n. Calibration Results Th i s s ect i o n p res en t s t h e d i s ag g re g at e an al ys i s res u l t s fo r ea ch s es s i o n i n t h e i n ci d en t cas e. Tab l es 1, 2, 3, an d 4 b el o w s h o w s t he i n d i v i d u al ri s k an a l ys i s o f ev er y p art i ci p an t i n s es s i o n s A1, A2, A3, an d A4, res p ect i v el y. Fo r ea ch i n d i v i d u al, t h ei r p red i ct ed c-v al u e, art eri a l b i as, l o gi t s cal e, an d o p t i m al l i k el i h o o d are p re s en t ed. Fi gu res 3, 4, 5, an d 6 res p ect i v el y p res en t t h e ri s k d i s t rib u t i o n s o f A1, A2, A 3, an d A4, as p red i ct ed b y t h e d i s a g g re gat e an al ys i s. Tab l e 1 : Di s a g gre g at e R e s u l t s fo r S es s i o n A1 User c α λ Likelihood A1u A1u2* A1u A1u A1u A1u A1u A1u8* A1u A1u A1u A1u A1u A1u A1u A1u

31 5 4 Frequency 3 2 A C-Value Fi gu re 2 : P red i ct ed R i s k P aram et er Di s t ri b u t i o n fo r S es s i o n A1 Tab l e 2 : Di s a g gre g at e R e s u l t s fo r S es s i o n A2 User c α λ Likelihood A2u A2u A2u A2u4 1.5 Multiple 0 A2u A2u A2u A2u A2u A2u A2u A2u A2u A2u14* A2u A2u

32 Frequency C-Value A2 Fi gu re 3 : P red i ct ed R i s k P aram et er Di s t ri b u t i o n fo r S es s i o n A2 Tab l e 3 : Di s a g gre g at e R e s u l t s fo r S es s i o n A3 User c α λ Likelihood A3u1* A3u A3u A3u A3u A3u A3u7* A3u A3u A3u10* A3u11* A3u A3u A3u A3u15* A3u

33 12 10 Frequency A C-Value Fi gu re 4 : P red i ct ed R i s k P aram et er Di s t ri b u t i o n fo r S es s i o n A3 Tab l e 4 : Di s a g gre g at e R e s u l t s fo r S es s i o n A4 User c α λ Likelihood A4u A4u A4u A4u4* A4u A4u A4u A4u A4u A4u A4u A4u A4u A4u A4u A4u

34 8 7 6 Frequency A C-Value Fi gu re 5 : P red i ct ed R i s k P aram et er Di s t ri b u t i o n fo r S es s i o n A4 Th e res u l t s s u g ges t t h at m o s t o f t h e i n d i v i d u al s i n t h e s es s i o n a re ri s k -s ee k i n g, b u t t h ere ar e cert ai n i n d i v i d u al s wh o s e op t i m al l i k el i ho o d s s eem ed q u es t i o n ab l e, as i n d i cat ed b y t h e s t a rred u s er IDs. Th e re as o n t h es e l i k el i h o od s are q u es t i o n ab l e i s t h at t h ei r v al u es are s o cl o s e t o t h e l o g l i k el i h o o d o f a p u rel y ran do m gu es s (as s i gn i n g 1 / 3 ch o i c e p rob ab i l i t y t o ea ch al t ern at i v e), , s u g ges t i n g t h at t h e m o d el d o es n o t ex p l ai n t h e p art i ci p an t s b eh av i o r b et t er t h an a p u r el y ran d o m gu es s. Th ere a re a fe w reas o n s as t o wh y t h e d i s a g g re ga t e cal i b rat i o n m a y n o t co rrect l y ch ara ct eri z e t h e i n d i v i d u al ri s k at t i t u d es fo r al l p art i ci p an t s, p a rt i cu l arl y f o r s es s i o n A3. Th e s i m p l es t ex p l an at i o n co u l d b e t h at t h e ran g e o f v al u es fo r t h e a rt eri al b i as an d t h e l o gi t s cal e i s l i m i t ed. Th e art eri al b i as r an ges b et ween an d i n i n crem en t s o f fact o rs o f 1 0, i n cl u d i n g z ero. Th e l o gi t s c al e h as t h e s am e r an ge an d i n crem en t s, ex cl u d i n g z ero. Th e op t i m al b i as an d s c al e v al u e s co u l d b e wi t h i n t h e l i m i t s o f t h e ran g e o r ev en o u t s i d e t h e as s u m ed ran ge. Ex p eri m en t at i o n wi t h t h e ran g e o f v al u es co u l d 2 4

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