A Heterogeneous Routing Game

Transcription

1 A Htrognous Routing Gam Farhad Farohi Walid Krichn Alxandr M. Bayn and Karl H. Johansson Abstract Most litratur on routing gams ma th assumtion that drivrs or vhicls ar of th sam ty and hnc xrinc th sam latncy or cost whn travling along th dgs of th ntwor. In contrast in this articl w roos a htrognous routing gam in which ach drivr or vhicl blongs to a crtain ty. Th ty dtrmins th cost of travling along an dg as a function of th flow of all tys of drivrs or vhicls ovr that dg. W xamin th xistnc of a Nash quilibrium in this htrognous routing gam. W study th conditions for which th roblm of finding a Nash quilibrium can b osd as a convx otimization roblm and is thrfor numrically tractabl. Numrical simulations ar rsntd to validat th rsults. I. INTRODUCTION Th roblm of dtrmining Nash quilibria in routing gams and bounding thir infficincy has bn xtnsivly studid 5. Howvr most of ths studis assum that drivrs or vhicls ar of th sam ty and hnc thy xrinc th sam latncy or cost whn using an dg in th ntwor. This is rimarily motivatd by transortation ntwors for which th drivrs only worry about th travl tim (and indd undr th assumtion that all th drivrs ar qually snsitiv to th latncy) or act routing in communication ntwors whr all th acts that ar using a articular lin xrinc th sam dlay. Howvr in mor gnral traffic ntwors this assumtion might not b ralistic. For instanc du to ful consumtion havyduty vhicls and cars might xrinc diffrnt costs for using th road vn if thir travl tims ar qual. In 6 this hnomnon has bn studid in atomic congstion gams in which th havy-duty vhicls xrinc an incrasd fficincy whn a highr numbr of havy-duty vhicls ar rsnt on th sam road (bcaus of a highr latooning ossibility and thrfor a highr ful fficincy) whil this may not b tru for cars. Th roblm of charactrizing Nash quilibria in htrognous routing gams has bn studid rviously 7 2. For instanc th roblm of routing a finit numbr of customrs with a givn suly rat was considrd in 7 F. Farohi and K. H. Johansson ar with ACCESS Linnaus Cntr School of Elctrical Enginring KTH Royal Institut of Tchnology SE- 44 Stocholm Swdn. s: {farohiallj}@.th.s W. Krichn is with th dartmnt of Elctrical Enginring and Comutr Scincs Univrsity of California at Brly CA USA. walid@cs.brly.du A. M. Bayn is with th dartmnt of Elctrical Enginring and Comutr Scincs and th dartmnt of Civil and Environmntal Enginring Univrsity of California at Brly CA USA. bayn@brly.du Th wor of F. Farohi and K. H. Johansson was suortd by grants from th Swdish Rsarch Council th Knut and Alic Wallnbrg Foundation and th iqflt rojct. Th wor of W. Krichn and A. M. Bayn was suortd by grants from th California Dartmnt of Transortation Googl and Noia. 8. Th authors in 9 studid th cas in which th classs of drivrs ract diffrntly to th imosd tolls for th road. In th snsitivity of th agnt to th latncy was adjustd by a multilicativ wight dnding on th class to which th drivr blongs. Howvr to our nowldg ths studis adjust ithr th snsitivity of th agnts to th obsrvd latncis or th tolls through a multilicativ wight and do not addrss mor gnral classs of cost functions. In this articl w roos a gnral htrognous routing gam in which th drivrs or th vhicls might blong to mor than on ty. Th ty of ach vhicl dtrmins th maing that calculats its cost for using an dg basd on th flow of all tys of drivrs or vhicls ovr that dg. W rov th xistnc of a Nash quilibrium undr mild conditions for gnral htrognous routing gams. To do so w rov that th introducd routing gam is quivalnt to an abstract gam with finit numbr of layrs in which ach layr corrsonds to on of th tys. For th cas in which only two tys of usrs ar articiating in th gam w charactriz ncssary and sufficint conditions for finding a otntial function for this abstract gam undr which th roblm of finding a Nash quilibrium for th gam is quivalnt to solving an otimization roblm. W also rsnt a st of tolls that can satisfy ths conditions. Th rst of th articl is organizd as follows. W formulat th htrognous routing gam in Sction II. In Sction III w rov that a Nash quilibrium may indd xist in this routing gam. W rsnt a st of ncssary and sufficint conditions to guarant th xistnc of a otntial function for this gam in Sction IV. In Sction V a st of tolls is rsntd to satisfy th aformntiond conditions. Finally w conclud th articl and rsnt dirctions for futur rsarch in Sction VI. A. Notation II. A HETEROGENEOUS ROUTING GAME Lt R and Z dnot th sts of ral and intgr numbrs rsctivly. Furthrmor dfin Z a {n Z n a} and R a {x R x a}. W us th notation N to dnot {... N}. All th othr sts ar dnotd by calligrahic lttrs such as R. Scifically C consists of all -tims continuously diffrntiabl functions. Lt X R n b a st such that X. A maing f : X R is calld ositiv dfinit if f(x) for all x X and f(x) imlis that x. W us th notation G (V E) to dnot a dirctd grah with vrtx st V and dg st E V V. Each ntry (i j) E dnots an dg from vrtx i V to vrtx j V. A dirctd ath of lngth z from vrtx i to vrtx j is a

2 st of dgs {(i i ) (i i 2 )... (i z i z )} E such that i i and i z j. B. Problm Formulation Hr w roos an xtnsion of th routing gam introducd in to admit mor than on ty of layrs. To b scific w assum tys θ blong to a finit st Θ. Lt us assum that a dirctd grah G (V E) is givn which modls th transortation ntwor. W also hav a st of sourc dstination airs {(s t )} K for som constant K Z. Each air (s t ) is calld a commodity. W us th notation P to dnot th st of all admissibl aths ovr th grah G that connct vrtx s V (i.. th sourc of this commodity) to vrtx t V (i.. th dstination of this commodity). Lt P K P. W assum that ach commodity K nds to transfr a flow qual to (F θ) θ Θ R Θ. W us th notation f θ R to dnot th flow of layrs of ty θ Θ that us a givn ath P. W us th notation f (f θ ) Pθ Θ R P Θ to dnot th aggrgat vctor of flows 2. A flow vctor f R P Θ is fasibl if P f θ F θ for all K and θ Θ. W us th notation F to dnot th st of all fasibl flows. To nsur that th st of fasibl flows is not an mty st w assum that P if F θ for any θ Θ. Notic that th constraints associatd with ach ty ar indndnt of th rst. Thrfor th flows of a scific ty can b changd without braing th fasibility of th flows associatd with th rst of th tys. A drivr or vhicl of ty θ Θ that travls along an dg E xrincs a cost qual to θ ((φ θ ) θ Θ) whr for any θ Θ φ θ dnots th total flow of drivrs of ty θ that ar using this scific dg i.. φ θ P: f θ. This cost can for xaml ncomass aggrgats of th latncy ful consumtion tc. For notational convninc w assum that w can chang th ordr with which th dg flows φ θ can aar as argumnts of th cost function θ ((φ θ ) θ Θ). A drivr of ty θ Θ from commodity K that uss ath P (for conncting s to t ) xrincs a total cost of l θ (f) θ ((φ θ ) θ Θ). Hr ach layr is an infinitsimal art of th flow that tris to minimiz its own cost (i.. ach layr is inclind to choos th ath that has th last cost). Now basd on this fact w can dfin th Nash quilibrium. DEFINITION 2.: (NASH EQUILIBRIUM IN HETEROGE- NEOUS ROUTING GAME) A flow vctor f (f θ ) Pθ Θ is a Nash quilibrium if for all K and θ Θ f θ > for a ath P imlis that l θ (f) l θ (f) for all P. This dfinition imlis that for a commodity K and ty θ Θ all aths with a nonzro flow for drivrs of Throughout this articl and in th contxt of th routing gam w us th trms layrs drivrs usrs and vhicls intrchangably to dnot an infinitsimal art of th flow that stratgically tris to minimiz its own cost for using th road. 2 Not that thr is a on-to-on corrsondnc btwn th lmnts of P Θ and th st of intgrs {... P Θ }. ty θ hav qual costs and th rst (i.. aths with a zro flow for drivrs of ty θ) hav largr than or qual costs. EXAMPLE : (ROUTING GAME WITH PLATOONING IN- CENTIVES) For this xaml w fix th st of tys as Θ {c t} whr t dnots trucs (or quivalntly havyduty vhicls) and c dnots cars (or quivalntly light vhicls). Lt th dg cost functions b charactrizd as c (φ c φ t ) ξ (φ c + φ t ) t (φ c φ t ) ξ (φ c + φ t ) + ζ (φ c + φ t )γ (φ t ) whr maings ξ : R R ζ : R R and γ : R R dnot th latncy for using dg as a function of th total flow of vhicls ovr that dg th ful consumtion of trucs as a function of th total flow and th invrs of th ful fficincy of th trucs as a function of th flow of trucs rsctivly. Ths costs actually imly that cars only obsrv th latncy ξ (φ c +φ t ) whn using th roads (which is only a function of th total flow ovr that dg and not th individual flows of ach ty). Howvr th cost associatd with trucs ncomasss an additional trm which modls thir ful consumtion. Following this intrrtation γ (φ t ) is a dcrasing function sinc by having a highr flow of trucs ovr a givn road (i.. largr φ t ) ach truc gts a highr robability for latooning (and as a rsult a highr chanc of dcrasing its ful consumtion). W ma th following standing assumtion rgarding th dg latncy functions for all th tys. ASSUMPTION 2.: For all θ Θ and E th dg cost function θ satisfis th following rortis: (i) θ C ; (ii) θ is ositiv dfinit; (iii) θ θ (u (φ θ ) θ Θ\{θ})du is a convx function in φ θ ; Assumtion 2. (iii) can b rlacd with th assumtion that θ ((φ θ ) θ Θ) is an incrasing function in φ θ (s 2 for th cas in which Θ ). W start by roving th xistnc of a Nash quilibrium and thn study th comutational comlxity of finding an quilibrium. III. EXISTENCE OF THE NASH EQUILIBRIUM To rov th xistnc of a Nash quilibrium w first show that th roblm of charactrizing a Nash quilibrium of th htrognous routing gam is quivalnt to charactrizing a ur stratgy Nash quilibrium of an abstract gam with finit numbr of layrs. For th sa of simlicity of rsntation and without loss of gnrality (sinc Θ is finit) w can assum that Θ {θ... θ N } whr N Θ. Now lt us dfin th abstract gam. DEFINITION 3.: Considr a gam with N layrs in which layr i N corrsonds to ty θ i Θ in th htrognous routing gam. Th action of layr i is dnotd by a i (f ) P which blongs to th action st A i { (f ) P R P f F P }.

3 Additionally th utility of layr i is dfind as U i (a i a i ) E θ i (u (φ θj ) θj Θ\{θ i})du () whr a i rrsnts th actions of th rst of layrs (a j ) j N \{i} and φ P: f for ach i N. Clarly an action rofil a N j A j is a ur stratgy Nash quilibrium of this abstract gam if for all i N U i (a i a i ) U i (ā i a i ) ā i A i. Th following rsult stablishs an intrsting rlationshi btwn th introducd high lvl gam and th undrlying htrognous routing gam. LEMMA 3.2: A flow vctor (f θ ) Pθ Θ is a Nash quilibrium of th htrognous routing gam if and only if ((f ) P... (f θ N ) P) is a ur stratgy Nash quilibrium of th abstract gam introducd in Dfinition 3.. Proof: Notic that ((f ) P... (f θ N ) P) bing a ur stratgy Nash quilibrium of th abstract gam is quivalnt to th fact that for all i N a i (f ) P is th bst rsons of layr i to th tul of actions a i ((f θj ) P) θj Θ\{θ i} or quivalntly whr φ θj a i arg min (f θ i ) P s.t. E P: θ i f f P f P. P: f θj (u (φ θj ) θj Θ\{θ i})du φ E F K for all j N \{i}. Notic that du to Assumtion 2. (iii) this roblm is indd a convx otimization roblm. Lt us dfin th Lagrangian as L i ((φ ) E (f ) P) E θ i + v i E K w i (u (φ θj P: P λ i f f P ) θj Θ\{θ i})du f φ F whr (v) i E R E (w i ) K R K and (λ i ) P R P ar Lagrang multilirs. Now using Karush Kuhn Tucr thorm otimality conditions ar φ L i ((φ ) E (f ) P) (φ (φ θj ) θj Θ\{θ i}) v i E (2) and f L i ((φ ) E (f ( ) v i ) P) w i λ i P K. Additionally th comlimntary slacnss conditions (for inquality constraints) rsults in λ i f i for all P. Hnc for all and P w hav ((f θ ) Pθ Θ) (φ (φ θj ) θj Θ\{θ i}) (3) v i by (2) w i + λ i. by (3) Thrfor for any 2 P if f f 2 > w hav λ λ 2 (bcaus of th comlimntary slacnss conditions) which rsults in ((f θ ) Pθ Θ) w i 2 ((f θ ) Pθ Θ). Furthrmor for any 3 P such that f 3 w gt (bcaus of dual fasibility i.. th Lagrang λ 3 multilirs associatd with inquality constraints must b ositiv) which rsults in 3 ((f θ ) Pθ Θ) w i + λ 3 w i ((f θ ) Pθ Θ). This comlts th roof. THEOREM 3.3: Th htrognous routing gam admits at last on Nash quilibrium. Proof: Following th rsult of Lmma 3.2 roving th statmnt of this thorm is quivalnt of showing th fact that th abstract gam introducd in Dfinition 3. admits at last on ur stratgy Nash quilibrium. First notic that A i i N is a non-mty convx and comact subst of th Euclidan sac R P. Scond U i (a i a i ) is continuous in all its argumnts (bcaus it is dfind as an intgral of a ral-valud masurabl function). Finally bcaus of Assumtion 2. (iii) U i (a i a i ) is a convx function in a i. Now w can us th clbratd rsult of 4 for abstract conomis (a gnralization of a gam) to show that th abstract gam introducd in Dfinition 3. admits at last on ur stratgy Nash quilibrium. IV. FINDING A NASH EQUILIBRIUM A family of gams that ar rlativly asy to analyz ar otntial gams. In this sction w giv conditions for whn th introducd abstract gam is a otntial gam. DEFINITION 4.: (POTENTIAL GAME 5) Th abstract gam introducd in Dfinition 3. is a otntial gam and

4 admits a otntial function V : N i A i R if for all i N V (a i a i ) V (ā i a i ) U i (a i a i ) U i (ā i a i ) a i ā i A i and a i j N \{i} A j. Th nxt lmma rovids a ncssary condition for th xistnc of a otntial function in C 2. LEMMA 4.2: If th abstract gam introducd in Dfinition 3. admits a otntial function V C 2 thn 2 φ θj ((φ θ ) θ Θ) φ θj ((φ θ ) θ Θ) for all i j N and 2 P. Proof: Sinc V ((f ) P... (f θ N ) P) is a otntial function for th abstract gam it satisfis V ((f ) P ((f θj ) P) θj Θ\{θ i}) V (( f ) P ((f θj ) P) θj Θ\{θ i}) U i ((f ) P ((f θj ) P) θj Θ\{θ i}) U i (( f ) P ((f θj ) P) θj Θ\{θ i}) which rsults in th idntity in (4) rsntd on to of th nxt ag in which δ ij dnots th Kroncr indx (or dlta) dfind as δ ij if i j and δ ij othrwis. Hnc w gt V ((f θ ) Pθ Θ) f θ i f E θ i ((φ θ ) θ Θ). (u(φ θj ) θj Θ\{θ i})du Now bcaus of Clairaut-Schwarz thorm w now that th following quality must hold sinc V C 2 2 V ((f θ ) Pθ Θ) Lt us calculat f f θj 2 2 V ((f θ ) Pθ Θ) f f θj and similarly 2 2 V ((f θ ) Pθ Θ) 2 f θj 2 f φ θj 2 θ V ((f ) Pθ Θ) f f V ((f θ θ j 2 f θj 2 f. (5) ) Pθ Θ) f θj 2 θj ((φ θ ) θ Θ) 2 ((φ θ ) θ Θ) φ ((φ θ 2 Substituting (5) and (6) into (7) rsults in ((φ θ ) θ Θ) for all 2 P and θ i θ j Θ. φ ) θ Θ) φ θj θj ((φ θ ) θ Θ) (6). (7) Intrstingly w can rov that this condition is also a sufficint condition for th xistnc of a otntial function (that blongs to C 2 ) for th introducd abstract gam whnvr only two tys of layrs ar articiating in th htrognous routing gam. LEMMA 4.3: Assum that Θ 2. If 2 φ for all 2 P thn (φ φ ) V ((f ) P (f ) P) E + φ θ θ 2 θ 2 θ (φ φ ) (u φ )du (φ u 2 )du 2 u (t u)dtdu is a otntial function for th abstract gam introducd in Dfinition 3.. Proof: Notic that for all P w gt (8) which is rsntd on to of th nxt ag. Now lt us dfin Ψ((φ ) E (φ ) E ) θ 2 φ W hav Ψ((φ ) E (φ ) E ) f ˆ ˆ φ for all ˆ P. Noticing that φ E w gt (φ u) (φ u) du. u (φ φ ) (φ φ ) du u Ψ((φ ) E (φ ) E ) Ψ((φ φ ˆ P: ˆ E. ˆ P: ˆ f ˆ for all ) E (φ ) E ) f ˆ Thus Ψ((φ ) E (φ ) E ) Ψ((φ ) E ). Stting Ψ((φ ) E (φ ) E ) (s dfinition abov) insid (8) rsults in V ((f θ ) Pθ Θ) f (φ φ ) U ((f ) P (f f ) P) whr th artial drivativs of U can b comutd from its dfinition in (). Lt (f ) P and ( f ) P b arbitrary oints in st of actions A i. Furthrmor lt r : A i b a continuously diffrntiabl maing (i.. r C ) such that r() ( f ) P and r() (f ) P which rmains insid A i R P for all t ( ). W dfin grah(r) as th collction of all ordrd airs (t r(t)) for all t (9)

5 V ((f θ ) Pθ Θ) f V ((f lim + ɛδ ) P ((f θj ) P) θj Θ\{θ i})v ((f ) P ((f θj ) P) θj Θ\{θ i}) ɛ ɛ U i ((f lim + ɛδ ) P ((f θj ) P) θj Θ\{θ i})u i ((f ) P ((f θj ) P) θj Θ\{θ i}) ɛ ɛ U i((f θ ) Pθ Θ). f (4) V ((f θ ) Pθ Θ) f f ( E θ (φ φ ) + (φ φ ) + θ 2 (u φ )du + (φ u 2 )du 2 θ 2 φ θ 2 (φ u 2 )du 2 φ θ 2 u θ 2 θ (φ u)du (φ u) (φ u) u du. u ) (t u)dtdu (8) which dnots a continuous ath that conncts (f ) P and ( f ) P. W now at last on such maing xists bcaus A i is a simly connctd st for all i N. Hnc w hav V (a a 2 ) grah(r) a dr ar V (a a 2 ) r(t) a dt ar(t) t d dt V (r(t) a 2) dt V (r() a 2 ) V (r() a 2 ) V ((f ) P (f ) P) V (( f ) P (f ) P) whr th scond to last quality is a dirct consqunc of th fundamntal thorm of calculus Not that this quality holds irrsctiv of th slctd ath. Thrfor V ((f ) P (f ) P) V (( f ) P (f ) P) V (a a 2 ) a dr ar grah(r) grah(r) U (a a 2 ) a U ((f ) P (f ) P) U (( dr by (9) ar f ) P (f ) P) Similarly w can also rov () rsntd on to of th nxt ag which rsults in V ((f ) P (f f ) P) (φ φ ) U 2((f ) P (f f ) P) and consquntly V ((f ) P (f ) P) V ((f ) P ( f U 2 ((f ) P (f ) P) U 2 ((f ) P ( ) P) f ) P). This concluds th roof. Now combing th rvious two lmmas rsults in th main rsult of this sction. THEOREM 4.4: Assum that Θ 2. Th abstract gam admits a otntial function V C 2 if and only if 2 φ (φ φ ) φ (φ φ ) for all 2 P. Proof: Th roof asily follows from Lmmas 4.2 and 4.3. Not that th otntial function rsntd in Lmma 4.3 blongs to C 2 du to Assumtion 2. (i). Following a basic rorty of otntial gams it is asy to rov th following corollary which shows that th rocss of finding a Nash quilibrium of th htrognous routing gam is quivalnt to solving an otimization roblm. COROLLARY 4.5: Assum that Θ 2. Furthrmor lt φ (φ φ ) φ (φ φ ) for all P. If f (f θ ) Pθ Θ is a solution of th otimization roblm min V ((f ) P (f ) P) s.t. f φ and P: f P P: F and f P f f R P f φ E F K whr V ((f ) P (f ) P) is dfind in Lmma 4.3 thn f (f θ ) Pθ Θ is a Nash quilibrium of th htrognous routing gam.

6 V ((f θ ) Pθ Θ) f f ( E θ θ θ 2 (u φ )du + (φ u 2 )du 2 φ (u φ )du + (φ φ ) θ 2 θ u φ (t φ )dt θ ) (t u)dtdu () Fig An xaml of grah G (V E). TABLE I PARAMETERS OF THE HETEROGENEOUS ROUTING GAME. α () aa α () ab α () bb β a () β () b Proof: Th roof is consqunc of th fact that a minimizr of th otntial function is a ur stratgy Nash quilibrium of a otntial gam 5. EXAMPLE 2: Lt us considr a numrical xaml with grah G (V E) illustratd in Figur. Hr w assum that Θ {a b}. W also hav thr aths i { i } for i 2 3. Th dg cost functions ar tan to b affin functions of th form l a i (φ a i φ b i ) α aa (i) φ a i + α (i) ab φb i + β a (i) l b i (φ a i φ b i ) α (i) ba φa i + α (i) bb φb i + β (i) b. If α (i) ba α(i) ab for all i 2 3 th condition of Corollary 4.5 is satisfid. In this cas w can calculat th otntial function as 3 V ((φ a i ) 3 i(φ b i ) 3 i) 2 α(i) aa (φ a i ) 2 +(α (i) ab φb i +β a (i) )φ a i i α (i) ab φa i φ b i + 2 α(i) bb (φb i ) 2 + (α (i) ba φa i + β (i) b )φb i. Noticing that solving a non-convx quadratic rogramming roblm might b numrically intractabl in gnral w focus on th cas in which V ((φ a i ) 3 i (φb i ) 3 i ) is a convx function. Following th argumnt of 3. 7 w now that V ((φ a i ) 3 i (φb i ) 3 i ) is a convx function if and only if α aa (i) 2 α(i) ba 2 α(i) ab α (i) bb i 2 3. Lt us ic th aramtrs for th routing gam according to Tabl I. Furthrmor w choos F a 5 and F b. Aftr solving th otimization roblm in Corollary 4.5 w gt (f a i ) 3 i ( ) (f b i ) 3 i (...) (l a i (f)) 3 i ( ) and (l b i (f)) 3 i ( ) which hnc is a Nash quilibrium. Notic that so far w hav rovd that a minimizr of th otntial function is a Nash quilibrium but not th othr way round. Now w ar rady to rov this whnvr th otntial function is convx. Howvr this rsult is rovd at th ric of a mor consrvativ condition. COROLLARY 4.6: Lt Θ 2 and (φ φ ) φ φ for all E. Furthrmor assum that th otntial function (φ φ ) V ((f ) P (f ) P) dfind in Lmma 4.3 is a convx function. Thn f (f θ ) Pθ Θ is a Nash quilibrium of th htrognous routing gam if and only if it is a solution of th convx otimization roblm min s.t. V ((f ) P (f ) P) f φ and P: f P P: f F and f P f and f P. Proof: Lt us dfin th Lagrangian as L V ((f ) P (f 2 + i E 2 K i v i w i ) P) P: f P f φ F φ E F K 2 λ i f i i P whr (v) E R E (v) 2 E R E (w ) K R K (w 2) K R K (λ ) P R P and (λ2 ) P R P ar Lagrang multilirs. Using Karush Kuhn Tucr conditions otimality conditions ar φ φ L (φ φ ) + θ 2 θ 2 u θ 2 φ (φ u)du v (φ φ ) v ( + φ L (φ u 2 )du 2 (φ u) (φ u) u (φ φ ) v E θ φ θ φ ) du (u φ )du + (φ φ ) (t φ )dt v 2 (φ φ ) v 2 E (a) (b)

7 and f f L v w λ P (2a) L v 2 w 2 λ 2 P. (2b) In addition th comlimntary slacnss conditions for inquality constraints rsult in λ f and λ 2 f 2 for all P. Hnc for all and P w hav (f) Thus if f f gt λ and λ (φ φ ) v i by () w i + λ i. by (2) > using comlimntary slacnss w which rsults in (f) (f) wi. Additionally for all P whr f w hav (bcaus of dual fasibility) which rsults in λ (f) wi + λ wi (f). This is th dfinition of a Nash quilibrium. EXAMPLE (CONT D): Lt us xamin th imlications of Corollary 4.6 in studying latooning incntivs. W can asily calculat that c (φ c φ t ) t (φ c φ t ) φ c φ t dξ (u) du dξ (u) du uφ c (3) +φ t uφ c +φ t + dζ (u) du uφ c +φ t γ (φ t ). (4) Assuming that γ (φ t ) (sinc othrwis both tys ar quivalnt) for th condition in Corollary 4.6 to hold w hav dζ (u)/du for all u or quivalntly w hav ζ (u) c for all u. Intuitivly this condition translats to th fact that th ful consumtion of th trucs is indndnt of total flow of vhicls φ c + φ t (which might only b valid in fr-flow traffic sinc th total flow is not dictating th vlocity of th trucs). Noting that if th roblm of finding a Nash quilibrium in th htrognous routing gam is numrically intractabl it might b highly unlily for th drivrs to figur out a Nash quilibrium in finit tim (lt alon an fficint on) and utiliz it which might rsult in wasting arts of th transortation ntwor rsourcs. Thrfor a natural qustion that coms to mind is whthr it is ossibl to guarant th xistnc of a otntial function for a htrognous routing gam by imosing aroriat tolls. V. IMPOSING TOLLS TO GUARANTEE THE EXISTENCE OF A POTENTIAL FUNCTION Lt us assum that a drivr or vhicl of ty θ Θ must ay a toll τ θ ((φ θ ) θ Θ) for using an dg E whr (as statd arlir) φ θ P: f θ. Thrfor a drivr that is using ath P ndurs a total cost of l θ (f) + τ θ (f) whr τ (f) is th total amount of mony that this drivr must ay for using ath and can b calculatd as τ θ (f) τ θ ((φ θ ) θ Θ). Th dfinition of a Nash quilibrium should thn b slightly modifid to account for th tolls. A flow vctor f (f θ ) Pθ Θ is a Nash quilibrium for th routing gam with tolls if for all K and θ Θ whnvr f θ > for som ath P thn l θ (f) + τ θ (f) l θ (f) + τ θ (f) for all P. Bfor stating th main rsult of this sction not that w can hav both distinguishabl and indistinguishabl tys. This charactrization is of scial intrst whn considring th imlmntation of tolls. For distinguishabl tys w can imos individual tolls for ach ty. Howvr for indistinguishabl tys th tolls ar indndnt of th ty. To giv an xaml if Θ {cars trucs} w can imos diffrnt tolls for ach grou of vhicls whil if Θ {atint drivrs imatint drivrs} w cannot. W trat ths two cass saratly. PROPOSITION 5.: (DISTINGUSHABLE TYPES) Assum that Θ 2. Th abstract gam admits th otntial function if V ((f ) P (f ) P) θ ( (u φ E + θ 2 ( (φ θ 2 θ τ (φ φ ) φ for all E. u ) + τ (u φ ))du u 2 ) + τ (φ u 2 ))du 2 ( (t u) + τ (t u))dtdu τ (φ φ ) φ (φ φ ) φ (φ φ φ Proof: Not that introducing th tolls τ θ (φ φ ) has th sam imact on th routing gam as rlacing th dg cost functions in th original htrognous routing gam from θ (φ φ ) to θ (φ φ ) + τ θ (φ φ ). Thans to Lmma 4.3 th abstract gam basd uon this nw htrognous routing gam admits th otntial function V if ( (φ φ ) + τ φ (φ φ )) ( (φ φ ) + τ φ (φ φ )) ). With rarranging th trms in this quality w can xtract th condition in th statmnt of th roosition. PROPOSITION 5.2: (INDISTINGUSHABLE TYPES) Assum that Θ 2. Th abstract gam admits th otntial function V C 2 in Proosition 5. with τ (φ φ )

8 τ (φ φ ) τ (φ φ τ (φ φ φ ) ) if τ (φ φ ) φ (φ φ ) φ (φ φ φ for all E. Proof: Th roof immdiatly follows from using Proosition 5. with th constraint that th tolls may not dnd on th ty i.. τ (φ φ ) τ (φ φ ) τ (φ φ ). EXAMPLE (CONT D): Lt us xamin th ossibility of finding a st of tolls that satisfis th conditions of Proositions 5. and 5.2. Th first cas is th distinguishabl tys. Substituting (3) and (4) into th condition of Proosition 5. rsults in τ c (φ c φ t ) φ t τ (φ t c φ t ) φ c dζ (u) du uφ c +φγ t (φ t ). (5) Following siml algbraic calculations w can chc that th tolls τ c (φ c φ t ) and τ (φ t c φ t ) γ (φ t )(κ ζ (φ c φ t )) for som aroriatly chosn constant κ R satisfy (5). Anothr xaml of aroriat tolls is τ (φ t c φ t ) and t τ c (φ c φ t dζ (u) ) du γ uφ c +q (q)dq. Ths two sts of tolls crtainly will hav diffrnt imlications on th bhavior of cars and trucs. Th scond cas that w study is th indistinguishabl tys. For that cas w study solv th artial diffrntial quation τ (φ c φ t ) φ t τ (φ c φ t ) φ c dζ (u) du uφ c +φγ t (φ t ). Noting th rsmblanc of this artial diffrntial quation with ons studid in 7 Ch. 4 w can dvis th tolls τ (φ c φ t ) dζ (u) φ t du uφ c γ +φ t (q)dq. In gnral w can rov th following corollary concrning th ty-indndnt tolls. COROLLARY 5.3: (INDISTINGUSHABLE TYPES) Assum that Θ 2. Th abstract gam admits a otntial function V C 2 if th imosd tolls ar of th following from τ (φ φ ) c + θ 2 ) f (q φ +φ q)dq+ψ (φ +φ ) whr c R ψ C and f (x y) (y x)/y (y x)/x for all E. Proof: Th roof is an alication of th rsult of 7 Ch. 4 to Proosition 5.2. Throughout this subsction w assumd that all th drivrs ortray similar snsitivity to th imosd tolls. This is indd a sourc of consrvatism scially whn daling with routing gams in which th htrognity is causd by th fact that th drivrs ract diffrntly to th imosd tolls. Crtainly an avnu for futur rsarch is to dvlo tolls for a mor gnral stu. VI. CONCLUSIONS In this articl w roosd a htrognous routing gam in which th layrs may blong to mor than on ty. Th ty of ach layr dtrmins th cost of using an dg as a function of th flow of all tys ovr that dg. W rovd that this htrognous routing gam admits at last on Nash quilibrium. Additionally w gav a ncssary and sufficint condition for th xistnc of a otntial function for th introducd routing gam which indd imlis that w can transform th roblm of finding a Nash quilibrium into an otimization roblm. Finally w dvlod tolls to guarant th xistnc of a otntial function. Possibl futur rsarch will focus on bounding th fficincy of a Nash quilibrium. REFERENCES J. G. Wardro Som thortical ascts of road traffic rsarch in Procdings of th Institut of Civil Enginrs: Enginring Divisions vol T. Roughgardn Routing gams in Algorithmic gam thory (N. Nisan T. Roughgardn E. Tardos and V. V. Vazirani ds.) vol. 8 Cambridg Univrsity Prss T. Roughgardn and É. Tardos How bad is slfish routing? Journal of th ACM (JACM) vol. 49 no D. Brass A. Nagurny and T. Waolbingr On a aradox of traffic lanning Transortation scinc vol. 39 no B. Awrbuch Y. Azar and A. Estin Th ric of routing unslittabl flow in Procdings of th thirty-svnth annual ACM symosium on Thory of comuting ACM F. Farohi and K. H. Johansson A gam-thortic framwor for studying truc latooning incntivs in Procdings of th 6th Intrnational IEEE Annual Confrnc on Intllignt Transortation Systms 23. To Aar. 7 S. Engvall M. Göth-Lundgrn and P. Värbrand Th htrognous vhicl-routing gam Transortation Scinc vol. 38 no R. Baldacci M. Battarra and D. Vigo Routing a htrognous flt of vhicls in Th Vhicl Routing Problm: Latst Advancs and Nw Challngs (B. Goldn S. Raghavan and E. Wasil ds.) vol. 43 of Orations Rsarch/Comutr Scinc Intrfacs Sringr G. Karaostas and S. G. Kolliooulos Edg ricing of multicommodity ntwors for htrognous slfish usrs in Procdings of th 45th Annual IEEE Symosium on Foundations of Comutr Scinc vol L. Flischr K. Jain and M. Mahdian Tolls for htrognous slfish usrs in multicommodity ntwors and gnralizd congstion gams in Procdings of th 45th Annual IEEE Symosium on Foundations of Comutr Scinc D. Fotais G. Karaostas and S. Kolliooulos On th xistnc of otimal taxs for ntwor congstion gams with htrognous usrs in Algorithmic Gam Thory (S. Kontogiannis E. Koutsouias and P. G. Sirais ds.) vol of Lctur Nots in Comutr Scinc Sringr Brlin Hidlbrg 2. 2 P. Marcott and D. L. Zhu Existnc and comutation of otimal tolls in multiclass ntwor quilibrium roblms Orations Rsarch Lttrs vol. 37 no S. P. Boyd and L. Vandnbrgh Convx Otimization. Cambridg Univrsity Prss K. J. Arrow and G. Dbru Existnc of an quilibrium for a comtitiv conomy Economtrica: Journal of th Economtric Socity D. Mondrr and L. S. Shaly Potntial gams Gams and conomic bhavior vol. 4 no S. T. Tan Multivariabl Calculus. Broos Col Publishing Comany A. D. Polianin V. F. Zaitsv and A. Moussiaux Handboo of First Ordr Partial Diffrntial Equations. Diffrntial and Intgral Equations and Thir Alications Taylor & Francis 22.