Whn Do Potntial Functions Exist in Htrognous Routing Gams? 1 Farhad Farokhi 2, Walid Krichn 3,4, Alxandr M. Bayn 4,5, and Karl H. Johansson 2 Abstract W study a htrognous routing gam in which vhicls might blong to mor than on ty. Th ty dtrmins th cost of travling along an dg as a function of th flow of various tys of vhicls ovr that dg. W rlax th assumtions ndd for th xistnc of a Nash quilibrium in this htrognous routing gam. W xtnd th availabl rsults to rsnt ncssary and sufficint conditions for th xistnc of a otntial function. W charactriz a st of tolls that guarant th xistnc of a otntial function whn only two tys of usrs ar articiating in th gam. W rsnt an ur bound for th ric of anarchy (i.., th worst-cas ratio of th social cost calculatd for a Nash quilibrium ovr th social cost for a socially otimal flow) for th cas in which only two tys of layrs ar articiating in a gam with affin dg cost functions. A htrognous routing gam with vhicl latooning incntivs is usd as an xaml throughout th articl to clarify th concts and to validat th rsults. 1 Introduction 1.1 Motivation Routing gams ar of scial intrst in transortation ntworks [2 4 and communication ntworks [5 7 bcaus thy allow us to study cass in which a dsirabl global bhavior (.g., achiving a socially otimal solution) can mrg from siml local stratgis (.g., imosing tolls on ach road basd on only local information). For th uros of this articl, routing gams can b dcomosd into two catgoris. In th first catgory, namly, homognous routing gams, drivrs or vhicls ar of th sam ty and, thrfor, xrinc th sam cost whn using an dg in th ntwork. Such an assumtion is rimarily motivatd by transortation ntworks for which th drivrs only worry about th travl tim (and indd undr th assumtion that all th drivrs ar qually snsitiv to latncy through considring thir avrag bhavior) or ackt routing in communication ntworks whr all th ackts that ar using a articular link xrinc th sam dlay or rliability. In th scond catgory, namly, htrognous routing gams (a.k.a., multi-class routing gams [8, 9), drivrs or vhicls blong to mor than on ty du to th following rasons: - Ful Consumtion: In a transortation ntwork, if w includ th ful consumtion of th vhicls in th cost functions, two vhicls (of diffrnt tys) may xrinc diffrnt costs for using a road vn if thir travl tims ar qual. For instanc, [1 studid this hnomnon in atomic congstion gams in which 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 ossibility of latooning and, thrfor, a highr ful fficincy, whil such an incrasd fficincy may not b tru for cars. For an xrimntal study of imrovmnts in th ful fficincy causd by latooning in havy-duty vhicls, s [11. 1 An arly vrsion of this articl was rsntd at th Annual Allrton Confrnc on Communication, Control, and Comuting [1. 2 F. Farokhi and K. H. Johansson ar with ACCESS Linnaus Cntr, School of Elctrical Enginring, KTH Royal Institut of Tchnology, SE-1 44 Stockholm, Swdn. Th work of F. Farokhi and K. H. Johansson was suortd by grants from th Swdish Rsarch Council, th Knut and Alic Wallnbrg Foundation, and th iqflt rojct. E-mails: {farokhi,kallj}@.kth.s 3 W. Krichn is with th dartmnt of Elctrical Enginring and Comutr Scincs, Univrsity of California at Brkly, CA, USA. E-mail: walid@cs.brkly.du 4 Th work of W. Krichn and A. M. Bayn was suortd by grants from th California Dartmnt of Transortation, Googl, and Nokia. 5 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 Brkly, CA, USA. E-mail: bayn@brkly.du 1
- Snsitivity to Latncy: It is known that drivrs on a road hav diffrnt snsitivitis to th latncy undr diffrnt circumstancs as wll as dnding on thir rsonality and background [12, 13. In addition, du to conomic advantags, havy-duty vhicls might b mor snsitiv to latncy in comarison to cars (bcaus thy nd to dlivr thir goods at scific tims). - Snsitivity to Tolls: Drivrs gnrally ract diffrntly to road tolls,.g., basd on th rason of th tri or thir socioconomic background. For instanc, in 21, by th rqust of th Swdish Institut for Transort and Communications Analysis, th consulting firm Inrgia comild a survy to stimat th valu of tim for th drivrs in Stockholm [14. According to that study, drivrs valud thir tim as.98, 3.3, and.19 SEK/min for work and school commuting tris, businss tris, and othr tris, rsctivly. Ths xamls motivat our intrst for studying htrognous routing gams in which th drivrs or th vhicls might blong to mor than on ty and thir ty dtrmins th cost of travling along an dg as a function of th flow of all tys of vhicls ovr that dg. 1.2 Rlatd Work In th contxt of transortation ntworks, routing gams wr originally studid in [3. This study also formulatd th dfinition of Nash quilibrium in routing gams 1. Latr, [17 showd that undr som mild conditions, th routing gam admits a otntial function and th minimizrs of this otntial function ar Nash quilibria of th routing gam. This rsult guarants th xistnc of a Nash quilibrium for th routing gam. Th roblm of bounding th infficincy of Nash quilibria has bn xtnsivly studid; s [15, 18 22 for a survy of ths rsults. Htrognous routing gams hav bn studid xtnsivly ovr th ast starting with ionring works in [8, 9. In ths studis, a routing gam with multi-class usrs wr introducd and th dfinition of quilibrium was givn. Furthrmor, in [9, th author introducd a sufficint condition for transforming th roblm of finding an quilibrium to that of an otimization (i.., quivalnt to th xistnc of a otntial function [23, 24). Th sufficint condition is that ovr ach dg, th incrasd cost of a usr of th first ty du to addition of on mor usr of th scond ty is qual to th incrasd cost of a usr of th scond ty du to addition of on mor usr of th first ty, i.., th usrs of th first and th scond ty influnc ach othr qually [9. This condition was considrd latr in [25 in which it was also notd that satisfaction of this symmtry condition may dnd on th units (.g., tim or mony) adotd for rrsnting th cost functions for th cas in which th usrs tys ar dtrmind by thir valu of tim (i.., a scalar factor that balancs th rlationshi btwn th latncy and th imosd tolls). This rsults is of scial intrst sinc th quilibrium dos not chang by using diffrnt units for th cost functions (if th latncy only dnds on th sum of th flows of various tys ovr th dg, not th individual flows, and th valu of tim aars linarly in th cost functions) [26. Ncssary and sufficint conditions for th xistnc of otntial functions in gams with finit numbr of layrs was rcntly invstigatd in [27; howvr, ths rsults wr not gnralizd to gams with a continuum of layrs as in htrognous routing gams. Th authors in [28 studid th xistnc of an quilibrium in htrognous routing gams vn if such a symmtry condition dos not hold. In contrast to ths articls that assumd a finit st of tys to which th usrs may blong, a walth of studis also considrd th cas in which th usrs may blong to a continuum of tys [29, 3. Th roblm of finding tolls for gnral htrognous routing gams as wll as th cas in which tys of th usrs is dtrmind by thir valu of tim hav bn considrd xtnsivly [31 36. For instanc, in [31, th roblm of dtrmining tolls on ach dg or ath for htrognous routing gams was studid. Guarants wr rovidd for that th socially otimal solution (also rfrrd to as systm-otimizing flow [9) is indd an quilibrium of th gam. Howvr, in that articl, th usrs wr assumd to b qually snsitiv to th imosd tolls. Th roblm of finding otimal tolls for routing gam in which th usrs valu of tim blongs to a continuum was studid in [32. 1 Throughout this articl, following th convntion of [15, 16, w us th trm Nash quilibrium to rfr to this quilibrium. S Rmark 2.1 for mor information rgarding this mattr. 2
1.3 Contributions of th Articl In this articl, w formulat a gnral htrognous routing gam in which th vhicls 2 might blong to mor than on ty. Th vhicl s ty dtrmins th maing that scifis th cost for using an dg basd on th flow of all tys of vhicls ovr that dg. W rov that th roblm of charactrizing th st of Nash quilibria for a htrognous routing gam is quivalnt to th roblm of dtrmining th st of ur stratgy Nash quilibria in a gam with finitly many layrs (in which ach layr rrsnts on of th tys in th original htrognous routing gam). Doing so, w can mloy classic rsults in gam thory and conomics litratur, scially rgarding th xistnc of an quilibrium in gams and abstract conomis [37,38 (which is an xtnsion of gams to a situation whr th actions of othr layrs can modify th st of fasibl actions for a layr), to show that undr mild conditions, a htrognous routing gam admits at last on Nash quilibrium. Thn, w rsnt a ncssary condition for th xistnc of a otntial function for th htrognous routing gam. W show that this condition is also sufficint for th cas in which only two tys of layrs ar articiating in th routing gam. In this cas, w show, following th otntial gam litratur [23, that th roblm of finding a Nash quilibrium in th htrognous routing gam can b osd as an otimization roblm (which is numrically tractabl if th otntial function is convx). Motivatd by th sufficint condition, in th rst of this articl, w focus on htrognous routing gams in which only two tys of usrs ar articiating. Not that in contrast to th rsults of [9, 25, hr, w rsnt a ncssary and sufficint condition for th xistnc of otntial function through which minimization w can rcovr an quilibrium. Howvr, th ric of roviding this tightr condition is that w can only trat routing gams with two distinct tys contrary to th sufficint condition in [9, 25. If th roblm of finding a Nash quilibrium in th htrognous routing gam is numrically intractabl 3, it might b unlikly for th drivrs to figur out a Nash quilibrium in finit tim (lt alon an fficint on) and utiliz it. This might rsult in infficint utilization of th transortation ntwork rsourcs. Thrfor, w rsnt a st of tolling olicis for distinguishabl tys (i.., a routing gam in which w may imos diffrnt tolls for diffrnt usr tys) and indistinguishabl tys (i.., whn w cannot imos ty-dndnt tolls) to guarant th xistnc of a otntial function for htrognous routing gams. Th ida of roosing tolls for indistinguishabl tys has bn rviously studid in [42. Howvr, in that study, th tolls wr introducd to minimiz th total travl tim and th total travl cost (as a bi-objctiv otimization roblm). In addition, in [42, th usrs tys corrsondd to socio-conomic charactristics and, thrfor, th cost functions of various tys of usrs wr th latncy (which th function of th total flow and not individual flows of ach ty) lus th tolls multilid by th valu of tim. Finally, bcaus a Nash quilibrium is tyically infficint (i.., it dos not minimiz th social cost function 4 ), w study th ric of anarchy 5 (a masur of th infficincy of a Nash quilibrium which can b dfind as th worst-cas ratio of th social cost at a Nash quilibrium ovr th social cost at a socially otimal flow). W rov that for th cas in which a convx otntial function xists, th ric of anarchy is boundd from abov by two for affin dg cost functions, that is, th social cost of a Nash quilibrium can b at most twic as much as th cost of a socially otimal solution. 1.4 Articl Organization Th rst of th articl is organizd as follows. W formulat th htrognous routing gam in Sction 2. In Sction 3, 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 in Sction 4. In Sction 5, a st of tolling olicis is rsntd to satisfy th aformntiond conditions. 2 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. 3 In gnral, th roblm of finding a ur stratgy Nash quilibrium is not numrically tractabl;.g., [39 41. 4 W us a utilitarian social cost function (i.., summation of th individual cost functions of all th layrs) as oosd to a Rawlsian social cost function (i.., th worst-cas cost function of th layrs); s [43,. 413 for mor information rgarding th diffrnc btwn ths two catgoris of social cost functions. W rsnt th dfinition of th social cost function in Sction 6. 5 Th notion of ric of anarchy was first introducd in [44, 45. Latr, it was utilizd in various gams including routing gams [15, 18, 46 48. 3
W bound th ric of anarchy for affin cost functions in Sction 6. A numrical xaml motivatd by a htrognous routing gam with latooning incntivs is studid in Sction 7. Finally, w conclud th articl and rsnt dirctions for futur rsarch in Sction 8. 2 A Htrognous Routing Gam 2.1 Notation 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}. For simlicity of rsntation, lt N Z 1. W us th notation N to dnot {1,..., N}. All othr sts ar dnotd by calligrahic lttrs. Scifically, C k consists of all k-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. A st-valud maing f : X Y is said to b continuous at x X if for vry y f(x ) and vry squnc {x k } k N such that lim k x k x, thr xists a squnc {y k } k N such that y k f(x k ) for all k N and lim k y k y. 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 st of dgs {(i, i 1 ), (i 1, i 2 ),..., (i z 1, i z )} E such that i i and i z j. 2.2 Problm Formulation W roos an xtnsion of th routing gam introducd in [3 to admit mor than on ty of layrs. To b scific, w assum that th ty of a layr θ blongs to a finit st Θ. Lt us assum that a dirctd grah G (V, E) modls th transortation ntwork and that a st of sourc dstination airs {(s k, t k )} K k for som constant K N ar givn. Each air (s k, t k ) is calld a commodity. W us th notation P k to dnot th st of all admissibl aths ovr th grah G that connct vrtx s k V (i.., th sourc of this commodity) to vrtx t k V (i.., th dstination of this commodity). Lt P K k1 P k. W assum that ach commodity k K nds to transfr a flow qual to (Fk 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 ) P,θ Θ R P Θ to dnot th aggrgat vctor of flows 6. A flow vctor f R P Θ is fasibl if P k f θ Fk θ for all k 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 k if Fk 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 braking th fasibility of th flows associatd with th rst of th tys. A vhicl of ty θ Θ that travls along an dg E xrincs a cost qual to l 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 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 aar as argumnts of th cost function l θ ((φ θ ) θ Θ). A drivr of ty θ Θ from commodity k K that uss ath P k (conncting s k to t k ) xrincs a total cost of l θ (f) l θ ((φ θ ) θ Θ). Each 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 modl, w can dfin th Nash quilibrium. Dfinition 2.1 (Nash Equilibrium in Htrognous Routing Gams) A flow vctor f (f θ ) P,θ Θ is a Nash quilibrium if for all k K and θ Θ, f θ > for a ath P k imlis that l θ (f) l θ (f) for all P k. Θ }. 6 Not that thr is a on-to-on corrsondnc btwn th lmnts of P Θ and th st of intgrs {1,..., P 4
This dfinition imlis that for a commodity k K and ty θ Θ, all aths with a nonzro flow for vhicls of ty θ hav qual costs and th rst (i.., aths with a zro flow for vhicls of ty θ) hav largr or qual costs. Rmark 2.1 Not that various articls us diffrnt nams for th quilibrium such as, usr-otimizing flow [9, 28, Wardro quilibrium [4, 28, 49, Wardro first rincil [4, and Nash quilibrium [15, 16. Th trm Wardro quilibrium is common, scially in transortation litratur, du to th ionring work of [3 as wll as th fact that th trm ur stratgy Nash quilibrium is rimarily utilizd in th contxt of gams with finitly many layrs [49. It is vital to not that th dfinition of Nash quilibrium in this ar is indd diffrnt from that of [49, which shows that by incrasing th numbr of usrs (in a gam with finitly many layrs), th Nash quilibrium convrgs to th Wardro quilibrium undr aroriat assumtions. Throughout this articl, following th convntion of [15, 16, w us th trm Nash quilibrium to rfr to this quilibrium. W mak th following standing assumtion rgarding th dg latncy functions for all th tys. Assumtion 2.1 For all θ Θ and E, th dg cost function l θ satisfis th following rortis: (i) lθ C 1 ; (ii) lθ is ositiv dfinit; (iii) θ l θ (u, (φ θ ) θ Θ\{θ})du is a convx function in φ θ for any givn (φ θ ) θ Θ\{θ}. Assumtion 2.1 (iii) is quivalnt to 7 : (iii) l θ (φ θ, (φ θ ) θ Θ\{θ}) is an incrasing function of φ θ for any givn (φ θ ) θ Θ\{θ}. W start by roving th xistnc of a Nash quilibrium and, thn, study th comutational comlxity of finding such an quilibrium. Howvr, bfor that, w rsnt an xaml of a htrognous routing gam in th nxt subsction. 2.3 Examl: Routing Gam with Platooning Incntivs Lt Θ {c, t}, whr t dnots trucks (or, quivalntly, havy-duty vhicls) and c dnots cars (or, quivalntly, light vhicls). Lt th dg cost functions b charactrizd as l c (φ c, φ t ) ξ (φ c + φ t ), l t (φ c, φ t ) ξ (φ c + φ t ) + ζ (φ c, φ t ), whr maings ξ : R R and ζ : R R R dnot rsctivly th latncy for using dg as a function of th total flow of vhicls ovr that dg and th ful consumtion of trucks as a function of th flow of ach ty. Ths costs 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 trucks ncomasss an additional trm which modls thir ful consumtion. Following this intrrtation, ζ (φ c, φ t ) is a dcrasing function in φ t sinc by having a highr flow of trucks ovr a givn road (i.., largr φ t ) ach truck gts a highr robability for collaboration such as latooning (and as a rsult, a highr chanc of dcrasing its ful consumtion). Lt us giv xamls of ths functions. Basd on th traffic data masurmnts availabl from [3,. 366 (s [1 for a cas study on th rlationshi btwn th avrag vlocity and th numbr of th vhicls on th road in Stockholm), w know that whnvr th traffic on a road is in fr-flow mod, w can modl th avrag vlocity of travling along that road as an affin function of th flow of vhicls ovr that dg according to v (φ c, φ t ) a (φ c + φ t ) + b. 7 Consult [18 for th roof of th quivalnc whn Θ 1. Th roof in th htrognous cas follows th sam lin of rasoning. 5
In this modl, b R and a R for E. Thrfor, if th lngth of dg E is qual to L R, w can calculat th latncy of using that dg as ξ (φ c + φ t ) L v (φ c, φ t ) L a (φ c + φ t. ) + b Now, in cass whr a (φ c + φ t ) b, w can us a linarizd 8 modl for th latncy ξ (φ c + φ t ) L L a (φ c + φ t b ). In addition, using [53, w know that th total ful consumtion of a truck which is travling with vlocity v for distanc L ovr a flat road can b modld by ζ (φ c, φ t ) c ( ) L 1 η ng ρ d 2 ρ aa a c D v (φ 2 c, φ t ) + mgc r, (1) whr η ng is th ngin fficincy, ρ d is th nrgy dnsity of disl ful, c D is th air drag cofficint, A a is th frontal ara of th truck, ρ a is th air dnsity, m is th mass of th truck, g is th gravitational acclration, and c r is th th roll rsistanc cofficint. In addition, w hav multilid th ful consumtion by c to balanc th trad-off btwn th latncy and ful consumtion in th aggrgat cost function of th trucks. Following [11, w know that th air drag cofficint c D dcrass if th trucks ar latooning (.g., two idntical trucks can achiv 4.7% 7.7% rduction in th ful consumtion causd by th air drag rduction whn latooning at 7 km/h dnding on th distanc btwn thm). Lt us modl ths changs as c D c D γ(φt ) whr γ : R [, 1 is th robability of forming latoons (which is a function of th flow of trucks φ t ) multilid by th imrovmnts in th air drag cofficint uon latooning. Lt us dfin aramtrs b 2 α L ρ a A a c D 2 η ng ρ d, β L mgc r η ng ρ d. Now, again if w linariz (1) around φ t, w gt ( ζ (φ c, φ t ) c α d ) du γ(u) ub 2 + 2c αγ()b a φ t + (2c αγ()b a ) φ c + ( c β + c αγ()b 2 ). Combing all ths trms rsults in l c (φ c, φ t ) L ( + L ) ( a b b 2 φ c + L ) a b 2 φ t, l t (φ c, φ t ) L ( + c β + c αγ()b 2 + L ) a b b 2 + 2c αγ()b a φ c ( ) + b2 +2c αγ()b a φ t. L a b 2 + c α dγ() du Notic that Assumtion 2.1 (i) and (ii) ar asily satisfid. Howvr, Assumtion 2.1 (iii) is only satisfid if L a b 2 This is indd tru bcaus of th obsrvation that Assumtion 2.1 (iii) and (iii) ar quivalnt. + c α dγ() du b2 + 2c αγ()b a. 3 Existnc of Nash Equilibrium In this sction, w show that th htrognous routing gam admits a Nash quilibrium. Bfor stating th rsult, w nd to introduc som concts from [37 which uss rsults of [38 to rov that an abstract conomy (an xtnsion of a gam) admits an quilibrium undr aroriat conditions 9. 8 Notic that such a linarization is crtainly not valid for a wid rang of traffic flows, howvr, it modls th latncy functions wll-nough for small flows. Th authors in [5, 51 roosd a icwis linar maing (basd on numrical data from th Toronto mtroolitan ara) for modling th latncy as a function of th flow of vhicls. This modl justifis using a linar modl for small flows (i.., at th bginning what thy call th fasibl rgion), howvr, it also oints out that a linar aroximation is not valid for larg flows. For a comrhnsiv comarison of diffrnt latncy maings (linar as wll as nonlinar ons), s [52. 9 Not that w could altrnativly follow th dfinition and rsults of [38 in a dirct mannr, howvr, in that cas, w nd mor background matrial rsntd which might b distracting to th audinc. 6
3.1 Existnc of Nash Equilibrium in Gams Lt us dfin an abstract conomy 1 as follows. Lt X i R n (for som n N) dnot th action st of layr i N in an abstract conomy with N layrs. W us th notation x i X i to dnot th action of layr i. In contrast to a gam, th fasibl st of actions that layr i can choos from is a function of actions of othr layrs x i (x j ) j i. Lt Z i : j i X j X i b a st-valud maing that dtrmins th st of fasibl actions for layr i. Th utility of layr i is govrnd by a ral-valud function U i : N j1 X j R. In this stu (oosd to th on rsntd in [37), w assum th layrs ar sking to minimiz thir utility. Dfinition 3.1 (Equilibrium of an Abstract Economy [37) x is an quilibrium oint of an abstract conomy if, for all i N, x i arg min x i Z i(x i ) U i (x i, x i ). For any i N, w say that Z i has a closd grah at x i j i X j if th st {(x j ) j N x i Z i (x i )} is a closd st. Now, w can stat th rsult of [37 rgarding th xistnc of such an quilibrium. Thorm 3.2 ([37) If, for ach i N, X i is a comact convx st, U i (x i, x i ) is continuous on N j1 X j and quasi-convx in x i for ach x i j i X j, Z i is a continuous st-valud maing that has a closd grah, and Z i (x i ) is a nonmty convx st for ach x i j i X j, thn th abstract conomy admits an quilibrium. Not that whn Z i (x i ) X i for all x i j i X j, and all i, w hav a gam with finitly many layrs. Thrfor, an abstract conomy can b considrd as a gnralization of a gam. Dfinition 3.3 (Pur Stratgy Nash Equilibrium in Gams with Finitly Many Playrs [54) x is a ur stratgy Nash quilibrium if, for all i N, x i arg min x i X i U i (x i, x i ). Thorm 3.2 rsults now in th following usful corollary. Corollary 3.4 If, for ach i N, X i is a comact convx st and U i (x i, x i ) is continuous on N j1 X j and quasi-convx in x i for ach x i j i X j, thn th gam admits a ur stratgy Nash quilibrium. Proof: Th roof follows from utilizing Thorm 3.2 whn, for all i N, Z i (x i ) X i for all x i j i X j. With this rsult in hand, w can go ahad and rov th xistnc of a Nash quilibrium in th htrognous routing gam. In th nxt subsction, w first rov that th roblm of finding a Nash quilibrium for th htrognous routing gam is quivalnt to th roblm of finding a ur stratgy Nash quilibrium in an abstract gam 11 with finitly many layrs. Thn, w us Corollary 3.4 to show that this gam admits a Nash quilibrium undr Assumtion 2.1. 3.2 Existnc of Nash Equilibrium in Htrognous Routing Gams For th sak of simlicity of rsntation and without loss of gnrality (sinc Θ is finit), w can assum that Θ {θ 1,..., θ N } whr N Θ. Now, lt us dfin th abstract gam. Dfinition 3.5 An abstract gam is 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 a i (f θi ) P which blongs to th action st { A i (f θi ) P R P } f θi F θi k. P k 1 An abstract conomy was originally dfind in [37. It is an xtnsion of a gam. 11 W us th trm abstract to mhasiz th fact that th introducd gam dos not hav any hysical intuition and it is simly a mathmatical conct dfind for roving th rsults of this ar. This xrssion should not b confusd with that of an abstract conomy. 7
Additionally, th utility of layr i is dfind as U i (a i, a i ) E θ i l θi (u, (φ θj ) θj Θ\{θ i})du, (2) whr a i rrsnts th actions of th rst of th layrs (a j ) j N \{i} and φ θi th dg flow of ty θ i for ach i N. P: f θi is Th following rsult stablishs an intrsting rlationshi btwn th introducd abstract gam and th undrlying htrognous routing gam. Lmma 3.6 A flow vctor (f θ ) P,θ 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. Proof: Notic that ((f ) P,..., (f θ N ) P) bing a ur stratgy Nash quilibrium (s Dfinition 3.3) of th abstract gam is quivalnt to that for all i N, a i (f θi ) P is th bst rsons of layr i to th tul of actions a i ((f θj ) P) θj Θ\{θ i} or, quivalntly, a i arg min (f θ i ) P s.t. E P: θ i f θi f θi P k f θi, P. l θi (u, (φ θj ) θj Θ\{θ i})du, φ θi, E, F θi k, k K, whr φ θj P: f θj for all j N \{i}. Notic that du to Assumtion 2.1 (iii), this roblm is indd a convx otimization roblm. Lt us dfin th Lagrangian as L i ((φ θi ) E, (f θi ) P) E θ i K k1 w i k l θi (u, (φ θj f θi P k ) θj Θ\{θ i})du + v i E F θi k P λ i f θi, P: f θi φ θi whr (v) i E R E, (wk i ) k K R K, and (λ i ) P R P ar Lagrang multilirs. Now, using Karush Kuhn Tuckr thorm [55,. 244, otimality conditions ar and f θi φ θi L i ((φ θi ) E, (f θi ) θi P) l (φ θi, (φ θj ) θj Θ\{θ i}) v i, E, (3) L i ((φ θi ) E, (f θi ) P) ( v i ) w i k λ i, P k, k K. (4) Additionally, th comlimntary slacknss conditions (for inquality constraints) rsult in λ i f i for all P. Hnc, for all k and P k, w hav l θi ((f θ ) P,θ Θ) l θi (φ θi, (φ θj ) θj Θ\{θ i}) v i by (3) w i k + λ i. by (4) 8
Thrfor, for any 1, 2 P k, if f θi 1, f θi 2 >, w hav λ θi 1 λ θi 2 (bcaus of th comlimntary slacknss conditions), which rsults in l θi 1 ((f θ ) P,θ Θ) w i k l θi 2 ((f θ ) P,θ Θ). Furthrmor, for any 3 P k such that f θi 3, w gt λ θi 3 (bcaus of dual fasibility, i.., th Lagrang multilirs associatd with inquality constraints must b non-ngativ), which rsults in l θi 3 ((f θ ) P,θ Θ) w i k + λ θi 3 w i k l θi 1 ((f θ ) P,θ Θ). This comlts th roof. Thorm 3.7 Undr Assumtion 2.1, th htrognous routing gam admits at last on Nash quilibrium. Proof: Following th rsult of Lmma 3.6, roving th statmnt of this thorm is quivalnt to showing th fact that th abstract gam introducd in Dfinition 3.5 admits at last on ur stratgy Nash quilibrium. First, notic that for all i N, A i 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 2.1 (iii), U i (a i, a i ) is a convx function in a i. Now, it follows from Corollary 3.4 that th abstract gam admits at last on ur stratgy Nash quilibrium. Rmark 3.1 Thorm 3.7 can b sn as an xtnsion of [28. In that study, th authors assum that th cost functions ar monoton, that is, l θ ((φ θ ) θ Θ) l θ θ (( φ ) θ Θ) for all θ Θ if φ θ θ φ if θ Θ; s [28,. 58. This condition, in turn, imlis that l θ ((φ θ ) θ Θ) is a non-dcrasing function of all its argumnts which is strongr than Assumtion 2.1 (iii). 4 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. Dfinition 4.1 (Potntial Gam [23) Th abstract gam is a otntial gam with otntial function V : N i1 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. Lmma 4.2 If th abstract gam admits a otntial function V C 2, thn [ l θj φ θi ((φ θ ) θ Θ) l θi φ θj ((φ θ ) θ Θ), 1 2 for all i, j N and 1, 2 P. Proof: Sinc V ((f ) P,..., (f θ N ) P) is a otntial function for th abstract gam, it satisfis, for all i N, V ((f θi ) P, ((f θj ) θi P) θj Θ\{θ i}) V (( f ) P, ((f θj ) P) θj Θ\{θ i}) U i ((f θi ) P, ((f θj ) P) θj Θ\{θ i}) U i (( f θi ) P, ((f θj ) P) θj Θ\{θ i}), 9
which rsults in th idntity V ((f θ ) P,θ Θ) f θi 1 V ((f θi lim ɛ U i ((f θi lim ɛ U i((f θ + ɛδ 1 ) P, ((f θj ) P) θj Θ\{θ i}) V ((f θi ) P, ((f θj ) P) θj Θ\{θ i}) + ɛδ ɛ ) 1 P, ((f θj ) P) θj Θ\{θ i}) U i ((f θi ) P, ((f θj ) P) θj Θ\{θ i}) ) P,θ Θ) f θi 1. in which δ ij dnots th Kronckr indx (or dlta) dfind as δ ij 1 if i j and δ ij othrwis. Hnc, w gt V ((f θ ) P,θ Θ) f θi 1 ɛ θ i f θi 1 E 1 lθ i ((φ θ ) θ Θ). l θi (u,(φ θj ) θj Θ\{θ i})du Now, bcaus of Clairaut-Schwarz thorm [56,. 167, w know that th following quality must hold sinc V C 2, 2 V ((f θ ) P,θ Θ) Lt us calculat and, similarly, f θi 1 f θj 2 2 V ((f θ ) P,θ Θ) f θi 1 f θj 2 2 V ((f θ ) P,θ Θ) f θj 2 f θi Substituting (6) and (7) into (8) rsults in [ 1 2 φ θj (5) 2 θ V ((f ) P,θ Θ). (6) f θj 2 f θi 1 f θi 1 f θi 1 [ V ((f θ [ θ l j 1 2 ) P,θ Θ) f θj 2 lθj ((φ θ ) θ Θ) 2 ((φ θ ) θ Θ) φ θi lθi ((φ θ 1 1 2 l θi ((φ θ ) θ Θ) φ θi, ) θ Θ) φ θj l θj ((φ θ ) θ Θ) (7). (8) for all 1, 2 P and θ i, θ j Θ. Intrstingly, w can rov that this condition is also a sufficint condition for th xistnc of a otntial function (that blongs to C 2 ) whnvr two tys of layrs ar articiating in th htrognous routing gam. Lmma 4.3 Assum that Θ 2. If [ for all 1, 2 P, thn 1 2 φ V ((f ) P, (f θ2 ) P) E l θ2 (φ, φ θ2 ) [ θ 1 φ θ2 θ 2 θ 1, l (φ, φ θ2 ), l (u 1, φ θ2 )du 1 + 1 θ 2 l θ2 (φ, u 2 )du 2 u 2 lθ 1 (u 1, u 2 )du 1 du 2
is a otntial function for th abstract gam. Proof: Notic that for all P, w gt V ((f θ ) P,θ Θ) f θ 1 f θ 1 ( E [ θ 1 [ l θ 1 (φ θ 1, φ θ 2 ) + l θ 1 (φ θ 1, φ θ 2 ) + θ φ 2 l θ 1 (u 1, φ θ 2 )du 1+ l θ 2 (φ θ 1, u 2)du 2 θ 2 φ θ 1 θ 2 θ φ 2 θ φ 1 θ φ 2 l θ 2 (φ θ 1, u 2)du 2 1 lθ (φ θ 1, u 2)du 2 u 2 [ φ θ 1 l θ 2 (φ θ 1, u 2) 1 lθ (φ θ 1, u 2) du 2. u 2 u 2 lθ 1 (u 1, u 2)du 1du 2 ) (9) Now, lt us dfin Ψ((φ ) E, (φ θ2 ) E ) θ 2 [ φ l θ2 (φ, u) l (φ, u) du. u W hav Ψ((φ ) E, (φ θ2 ) E ) [ l θ2 f θ2 ˆ φ (φ, φ θ2 ) l (φ, φ θ2 ), ˆ u for all ˆ P. Noticing that φ θ2 ˆ P: ˆ f θ2 ˆ Ψ((φ ) E, (φ θ2 ) E ) Ψ((φ φ θ2 ˆ P: ˆ for all E, w gt ) E, (φ θ2 ) E ), E. Thus, Ψ((φ ) E, (φ θ2 ) E ) Ψ((φ ) E, ). Stting Ψ((φ ) E, (φ θ2 ) E ) (s dfinition abov) insid (9) rsults in V ((f θ ) P,θ Θ) f f θ2 ˆ l (φ, φ θ2 ) U 1((f ) P, (f θ2 f ) P), (1) whr th artial drivativs of U 1 can b comutd from its dfinition in (2). Lt (f ) P and ( f ) P b arbitrary oints in st of actions A 1. Furthrmor, lt r : [, 1 A 1 b a continuously diffrntiabl maing (i.., r C 1 ) such that r() ( f ) P and r(1) (f ) P which rmains insid A 1 R P for all t (, 1). W dfin grah(r) as th collction of all ordrd airs (t, r(t)) for all t [, 1, which dnots a continuous ath that conncts (f ) P and ( f ) P. W know that at last on such maing xists bcaus A 1 is a simly connctd st for all i N. Hnc, w hav [ [ V (a 1, a 2 ) 1 V (a 1, a 2 ) grah(r) a 1 dr r(t) a1r a 1 dt a1r(t) t 1 [ d dt V (r(t), a 2) dt V (r(1), a 2 ) V (r(), a 2 ) V ((f ) P, (f θ2 ) P) V (( f ) P, (f θ2 ) P), 11
whr th scond to last quality is a dirct consqunc of th fundamntal thorm of calculus [56,. 1257. Not that this quality holds irrsctiv of th slctd ath. Thrfor, V ((f ) P, (f θ2 ) P) V (( Similarly, w can also rov f ) P, (f θ2 [ V (a 1, a 2 ) a 1 dr a1r [ U 1 (a 1, a 2 ) a 1 dr by (1) a1r grah(r) grah(r) ) P) U 1 ((f ) P, (f θ2 ) P) U 1 (( f ) P, (f θ2 ) P), V ((f θ ) P,θ Θ) f θ 2 f θ 2 E [ θ φ 1 ( [ θ φ 1 φ θ 2 l θ 1 (u 1, φ θ 2 )du 1+ θ φ 2 l θ 2 (φ θ 1, u 2)du 2 θ l θ 1 (u 1, φ θ 2 )du 1 + l φ 1 θ 2 (φ θ 1, φ θ 2 ) φ θ 2 θ φ 2 θ φ 1 u 2 lθ 1 (u 1, u 2)du 1du 2 ) l θ 1 (u 1, φ θ 2 )du 1 (11) l θ 2 (φ θ 1, φ θ 2 ), which rsults in and, consquntly, V ((f ) P, (f θ2 f θ2 ) P) V ((f ) P, (f θ2 ) P) V ((f ) θ2 P, ( f ) P) l θ2 (φ, φ θ2 ) U 2((f ) P, (f θ2 f θ2 U 2 ((f ) P, (f θ2 ) P) U 2 ((f ) P, ( ) P), f θ2 ) P). This concluds th roof. Now, combing th rvious two lmmas rsults in th main rsult of this sction. Thorm 4.4 Assum that Θ 2. Th abstract gam admits a otntial function V C 2 if and only if [ l θ2 (φ, φ θ2 ) l (φ, φ θ2 ), for all 1, 2 P. 1 2 φ 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 2.1 (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. φ φ θ2 Corollary 4.5 Assum that Θ 2. Furthrmor, lt [ l θ2 (φ, φ θ2 ) φ θ2 l (φ, φ θ2 ), 12
for all P. If f (f θ ) P,θ Θ is a solution of th otimization roblm min V ((f ) P, (f θ2 ) P), s.t. f φ and P: f P k P: f θ2 F k and f θ2 P k f, f θ2, P, φ θ2, E, F θ2 k, k K, whr V ((f ) P, (f θ2 ) P) is dfind in Lmma 4.3, thn f (f θ ) P,θ Θ is a Nash quilibrium of th htrognous routing gam. 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; s [23. 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. Corollary 4.6 Lt Θ 2 and φ l θ2 (φ, φ θ2 ) φ θ2 l (φ, φ θ2 ), for all E. Furthrmor, assum that th otntial function V ((f ) P, (f θ2 ) P), dfind in Lmma 4.3, is a convx function. Thn f (f θ ) P,θ 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 θ2 ) P), φ and P: f f P k P: f θ2 F k and f θ2 P k f, f θ2, P. φ θ2, E, F θ2 k, k K, Proof: S Andix A. Rmark 4.1 Not that Corollary 4.6 is rovd at th ric of a mor consrvativ condition bcaus th conditions in Corollary 4.5 rquirs th summation of th diffrncs btwn th drivativs of th cost functions to b qual to zro whil Corollary 4.6 nds th individual diffrncs to b qual to zro. Notic that Corollary 4.6 rovids th sam sufficint condition for charactrizing th st of all quilibria as in [9, 25, but ths rfrncs handl th gnral cas of Θ 2 (scifically, s Proosition 1 and Thorm 1 in [25). Thrfor, w can s that th rsntd condition in Corollary 4.5 is tightr than th rsults of [9,25 (sinc it is also a ncssary condition); howvr, it is only valid for Θ 2 in contrast. 4.1 Examl: Routing Gam with Platooning Incntivs Lt us xamin th imlications of Corollary 4.6 in th routing gam with latooning incntivs in Subsction 2.3. For th linarizd modl, w can asily calculat that l c (φ c, φ t ) φ t L a /b 2, (12) l t (φ c, φ t ) φ c L a /b 2 + 2c αγ()b a L a /b 2 + 2c αb a. (13) 13
whr th scond quality follows from γ () 1, which holds bcaus from th dfinition of th maing γ : R [, 1, w know that in this cas (i.., whn no trucks ar using that dg) th air drag cofficint is qual to its nominal valu. Thrfor, th condition of Corollary 4.6 dos not hold (unlss c ). Noting that if th roblm of finding a Nash quilibrium in th htrognous routing gam is numrically intractabl, it might b highly unlikly for th drivrs to figur out a Nash quilibrium in rasonabl tim (lt alon an fficint on) and utiliz it, which might rsult in wasting arts of th transortation ntwork 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. 5 Imosing Tolls to Guarant th Existnc of a Potntial Function 5.1 Dfinition and Rsults Lt us assum that a vhicl of ty θ Θ must ay a toll τ θ ((φ θ ) θ Θ) for using an dg E, whr φ θ P: f θ. Thrfor, a vhicl using ath P k ndurs a total cost of l θ (f)+τ θ (f), whr τ (f) is th total amount of mony that th vhicl 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 is slightly modifid to account for th tolls. Dfinition 5.1 (Nash Equilibrium in Htrognous Routing Gam with Tolls) A flow vctor f (f θ ) P,θ Θ is a Nash quilibrium for th routing gam with tolls if, for all k K and θ Θ, whnvr f θ > for som ath P k, thn l θ (f) + τ θ (f) l θ (f) + τ θ (f) for all P k. 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, trucks}, w can imos diffrnt tolls for ach grou of vhicls whil if Θ {atint drivrs, imatint drivrs}, w cannot. Notic that in th cas of indistinguishabl tys, on might argu that w cannot masur φ θi for ach θ i Θ individually (bcaus as w motivatd th ty of usr may not b idntifid from hysical traits). Howvr, w can us survys and historical data to xtract th statistics of ach ty (.g, to raliz what ratio of th actual flow blongs to ach ty) but whn calculating th tolls for ach usr w cannot forc that usr to articiat in a survy. W trat ths two cass saratly. Proosition 5.2 (Distinguishabl Tys) Assum that Θ 2. Th abstract gam admits th otntial function if V ((f ) P, (f θ2 ) P) E + [ θ 1 θ 2 θ 2 θ 1 ( l (u 1, φ θ2 ( l θ2 (φ ) + τ, u 2 ) + τ θ2 (u 1, φ θ2 ))du 1 (φ, u 2 ))du 2 u 2 ( l (u 1, u 2 ) + τ (u 1, u 2 ))du 1 du 2 for all E. τ (φ, φ θ2 ) φ θ2 θ2 τ (φ, φ θ2 θ2 ) l φ (φ, φ θ2 φ ) l (φ, φ θ2 φ θ2 ), Proof: S Andix B. 14
Proosition 5.3 (Indistinguishabl Tys) Assum that Θ 2. Th abstract gam admits th otntial function V C 2 in Proosition 5.2 with τ (φ, φ θ2 ) τ θ2 (φ, φ θ2 ) τ (φ, φ θ2 ) if for all E. τ (φ, φ θ2 ) φ θ2 τ (φ, φ θ2 θ2 ) l φ (φ, φ θ2 φ ) l (φ, φ θ2 φ θ2 ), Proof: Th roof immdiatly follows from using Proosition 5.2 with th constraint that th tolls may not dnd on th ty, i.., τ (φ, φ θ2 ) τ θ2 (φ, φ θ2 ) τ (φ, φ θ2 ). In gnral, w can rov th following corollary concrning th ty-indndnt tolls. Corollary 5.4 ( Indistingushabl Tys) Assum that Θ 2. Th abstract gam admits a otntial function V C 2 if th imosd tolls ar of th following form τ (φ, φ θ2 ) c + θ 2 f (q, φ + φ θ2 q)dq + ψ (φ + φ θ2 ), whr c R, ψ C 1, and f (x, y) l θ2 (y, x)/y l (y, x)/x for all E. Proof: S Andix C. 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. 5.2 Examl: Routing Gam with Platooning Incntivs Lt us xamin th ossibility of finding a st of tolls that satisfis th conditions of Proositions 5.2 and 5.3 for th htrognous routing gam introducd in Subsction 2.3. Distinguishabl Tys-Cas 1: Substituting (12) and (13) into th condition of Proosition 5.2 rsults in τ c (φ c, φ t ) φ t τ t (φ c, φ t ) φ c 2c αb a. (14) Following siml algbraic calculations, w can chck that th tolls τ c (φ c, φ t ) and τ t (φ c, φ t ) (2c αb a )φ c satisfy (14). Noticing that τ t (φ c, φ t ) bcaus by dfinition a R, ths trms can b intrrtd as subsidis aid to th trucks to comnsat for th ful that is wastd du to rsnc of th cars on that scific dg. Distinguishabl Tys-Cas 2: Anothr xaml of aroriat tolls is τ t (φ c, φ t ) and τ c (φ c, φ t ) ( 2c αb a )φ t. Now, w hav τ c (φ c, φ t ). In this cas, th cars ay dirctly for th incrasd ful consumtion of th trucks and, thrfor, thy ar inclind to travl along th dgs that trucks do not us. Indistinguishabl Tys: For this cas, using Corollary 5.4, it is asy to s that tolls τ (φ c, φ t ) (2c αb a )φ t work fin. W us ths tolls in th numrical xaml dvlod in Sction 7. 6 Pric of Anarchy for Affin Cost Functions In th routing gam litratur, it is a widly known fact that gnrally, a Nash quilibrium is infficint vn whn daling with homognous routing gams; s [15, 18, 22. To quantify this infficincy, many studis hav usd Pric of Anarchy (PoA) as a mtric. 15
6.1 Social Cost Function First, lt us dfin th social cost of a flow vctor f (f θ ) P,θ Θ as C(f) f θ l θ (f) P θ Θ φ θ l θ ((φ θ ) θ Θ), E θ Θ whr th scond quality can b asily obtaind by rarranging th trms. Using this social cost, w can dfin th otimal flow that w will us latr for comarison with th Nash quilibrium. Dfinition 6.1 (Socially Otimal Flow) f F is a socially otimal flow if C(f) C( f) for all f F. Dfinition 6.2 (PoA) Th ric of anarchy is dfind as PoA C(f Nash ) su f Nash N min f F C(f), whr N dnots th st of Nash quilibria of th htrognous routing gam. In this dfinition, w follow th convntion that 1. 6.2 Bounding th Pric of Anarchy for Two Tys with Affin Cost Functions Hr, w rsnt an ur bound for th infficincy of th Nash quilibrium in htrognous routing gams whn Θ 2. Th dg cost functions ar takn to b affin functions of th form l (φ, φ θ2 ) αθ 1θ 1 φ + αθ 1θ 2 φ θ2 + βθ 1, l θ2 (φ, φ θ2 ) αθ 2θ 1 φ + αθ 2θ 2 φ θ2 + βθ 2, whr αθ 1θ 1, αθ 1θ 2, αθ 2θ 1, αθ 2θ 2, βθ 1, βθ 2 R ar aramtrs of th routing gam for ach dg E. Notic that th condition αθ 1θ 1, αθ 1θ 2, αθ 2θ 1, αθ 2θ 2 R imlis that th cost of using an dg is incrasing in ach flow saratly (i.., whn a drivr of any ty switchs to an dg, sh cannot dcras th cost of th usrs on this nw dg) whil βθ 1, βθ 2 R imlis that th starting cost of using a road is non-ngativ. This assumtion is crtainly strongr than Assumtion 2.1. Subsction 2.3 rsnts a motivating xaml for affin cost functions. Thorm 6.3 Lt for all E. Thn, PoA 2. αθ 2θ 1 αθ 1θ 2 (15a) [ α θ 1 αθ 1θ 2 αθ 2θ 1 αθ, (15b) 2θ 2 Proof: First, not that if αθ 2θ 1 αθ 1θ 2 for all E, th condition of Corollary 4.5 is satisfid. Thrfor, w can asily calculat th otntial function as V (f) [ 1 2 α θ 1 θ 1 (φ θ 1 ) 2 +(αθ 1 θ 2 φ θ 2 +βθ 1 )φ θ 1 + 1 2 α θ 2 θ 2 (φ θ 2 ) 2 + (αθ 2 θ 1 φ θ 1 + βθ 2 )φ θ 2 αθ 1 θ 2 φ θ 1 φ θ 2 E [ 1 2 φθ 1 l θ 1 (φ θ 1, φ θ 2 ) + 1 2 β θ 1 φ θ 1 + 1 2 φθ 2 l θ 2 (φ θ 1, φ θ 2 ) + 1 2 β θ 2 φ θ 2 (16) E 1 2 C(f) + [ 1 βθ 2 1 φ θ 1 + βθ 2 φ θ 2, E Furthrmor, following th argumnt of [55,. 71, w know that th social cost function is a convx function if and only if (15b) is satisfid. Notic that (16) shows that th otntial function V is a convx function if th social cost function C is a convx function (bcaus th summation of a 16
Tabl 1: Paramtrs of th htrognous routing gam in th numrical xaml. 1 2 3 4 5 6 7 8 9 1 11 α aa 1. 2. 3. 1. 4..5 1. 1. 2. 1. 4. 1. α at.6.4.1.1.5.1.7.1.1.2.1.3 α tt 2. 3. 1..8 1. 1. 1.5 3. 1.7 3. 1. 1.3 β a 2. 2. 4.5 2. 2. 4.5 2. 2. 4.5 2. 2. 4.5 β t 4. 4. 1.5 4. 4. 1.5 4. 4. 1.5 4. 4. 1.5 convx function and a linar function is a convx function). Lt us us f and f to dnot th Nash quilibrium and th socially otimal flow, rsctivly. Now, w can rov inquality C( f) 2V ( f) by (16) and βθ 1, βθ 2 R 2V (f) by Corollary 4.6 2 ( θ φ 1 θ φ 2 l θ 1 (u 1, φ θ 2 )du 1 + l θ 2 (φ θ 1, u 2)du 2 E θ φ 2 θ φ 1 ) u l θ 1 (t, u)dtdu by Dfinition of V ( θ φ 1 θ φ 2 l θ 1 (u 1, φ θ 2 )du 1 + by αθ 1 θ 2, αθ 2 θ 1 R 2 E ( 2 2 E θ φ 1 + ( E E 2C(f). θ 2 [ l θ 1 (u 1, φ θ 2 [ l θ 2 (φ θ 1 l θ 2 (φ θ 1, u 2)du 2 ) ) + u 1 1 lθ (u 1, φ θ 2 u 1, u 2) + u 2 2 lθ (φ θ 1 u 2 φ θ 1 l θ 1 (φ θ 1, φ θ 2 ) + φ θ 2 l θ 2 (φ θ 1, φ θ 2 ) E ) du 1 ), u 2) du 2 ) by α θ 1 θ 1, α θ 2 θ 2 R (17) This comlts th roof. Notic that in many ractical situations (such as th on rsntd in Subsction 4.1 for routing gams with latooning incntivs), αθ 2θ 1 αθ 1θ 2. Thrfor, w may not b abl to us Thorm 6.3 to find an ur bound for th PoA. Howvr, as also discussd in Sction 5, in som cass, w might b abl to maniulat ths gains through aroriat tolls to mak sur (15a) holds. In addition, condition (15b) is quivalnt to th condition that αθ 1θ 1 αθ 2θ 2 αθ 2θ 1 αθ 1θ 2 for all E. This condition intuitivly mans that cost function of ach ty of vhicls is mor influncd by th flow of its own ty than th flow of th othr ty. This condition may not hold in gnral in transortation ntworks. In such cas, instad of using Corollary 4.6, w may us Corollary 4.5 in th roof of Thorm 6.3 (that is th only lac that w us th convxity of th otntial function which w rovd using th convxity of th social dcision function). Howvr, doing so, w cannot bound th ratio C(f Nash )/ min f C(f) for all f Nash N. Thrfor, instad of showing that PoA is boundd from abov by two, w can thn only show that th Pric of Stability 12 is ur boundd by two (bcaus w can show that th ratio is boundd by two for only on Nash quilibrium and not for all Nash quilibria). 7 Numrical Examl In this sction, w rsnt a numrical xaml motivatd by th routing gam with latooning incntivs in Subsction 2.3. W us th grah G (V, E) in Figur 1. W hav thr commoditis (s 1, t 1 ) (, 1), (s 2, t 2 ) (2, 3), and (s 3, t 3 ) (7, 8). Th corrsonding aths for th commoditis ar P 1 {{ 1 },{ 2, 4, 3 },{ 2, 7, 5 }}, P 2 {{ 1 },{ 9, 7, 8 },{ 9, 4, 6 }}, P 3 {{ 11, 1, },{ 11, 9, 7, 8, },{ 11, 9, 4, 6, }}. 12 Pric of Stability (PoS), or commonly known as th otimistic Pric of Anarchy, is dfind as inf f Nash N C(f Nash )/ min f F C(f); not that w us inf orator instad of su orator in this dfinition in contrast to that of Dfinition 6.2. S [57 for mor xlanation rgarding th diffrnc btwn PoS and PoA. 17
1 1 3 2 4 4 5 5 6 7 6 9 8 7 2 3 8 11 1 Figur 1: Transortation ntwork in th numrical xaml. Th dg cost functions ar takn to b affin functions of th form l c i (φ c i, φ t i ) α cc (i) φ c i + ᾱ (i) ct φ t i + β c (i), l t i (φ c i, φ t i ) α (i) tc φ c i + ᾱ (i) tt φ t i + β (i) t, whr th dfinitions and th hysical intuition of th aramtrs α (i) cc, α (i) tc, ᾱ (i) tt, ᾱ (i) ct, β c (i), β (i) t can b found in Subsction 2.3. Rcalling that ᾱ (i) tc ᾱ (i) ct (s Subsction 4.1), th condition of Corollary 4.5 is not satisfid. Thrfor, w us th tax τ i (φ c i, φ t i ) (2c αb a )φ t i which is dvlod in Subsction 5.2. This rsults in l c i (φ c i, φ t i ) + τ i (φ c i, φ t i ) α (i) cc φ c i + α (i) ct φ t i + β (i) c, whr α (i) ct function as 11 [ 1 V l t i (φ c i, φ t i ) + τ i (φ c i, φ t i ) α (i) tc φ c i + α (i) tt φ t i + β (i) t, ᾱ (i) ct + 2c αb a and α (i) tt i 2 α(i) cc (φ c i ) 2 +(α (i) ct φ t i +β (i) ᾱ (i) tt + 2c αb a. In this cas, w can calculat th otntial c )φ c i α (i) ct φ c i φ t i + 1 2 α(i) tt (φ t i ) 2 + (α (i) tc φ c i + β (i) t )φ t i. Noticing that solving a non-convx quadratic rogramming roblm might b numrically intractabl in gnral, w focus on th cas in which th otntial function is a convx function. Following th argumnt of [55,. 71, w know that th otntial function is a convx function if and only if [ α cc (i) 1 2 α(i) tc 1 2 α(i) ct α (i) tt, i {,..., 11}. Lt us ick th aramtrs for th routing gam according to Tabl 1. Furthrmor, w choos (F1 a, F1 b ) (5, 1), (F2 a, F2 b ) (3, 3), and (F3 a, F3 b ) (2, 4). Aftr solving th otimization roblm in Corollary 4.5, w can xtract th ath flows and ath cost functions shown in Tabl 2 which dmonstrat a Nash quilibrium (s Dfinition 5.1) 13. In addition, w can calculat C(f) C(f 1.137 2 Ur Bound of th PoA, ) whr f dnots th socially otimal flow. This shows that th social cost of th rcovrd Nash quilibrium is only 1.137 tims th cost of th socially otimal solutions. 8 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, 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 gnralizing ths rsults to highr numbr of tys or a continuum of layr tys. 13 S htt://dl.drobox.com/u/36867745/htrognousroutinggam.zi for th Python cod to simulat this numrical xaml. 18
Tabl 2: Th ath flow and ath cost function at a Nash quilibrium xtractd by minimizing th otntial function. f c f t P 1.. 4.97.79.3.21 P 2.4. 1.4.6 1.92 2.94 P 3 1.1..1..88 4. l c (f) lt (f) P 1 13.18 1.29 12.26 8.35 12.26 8.35 P 2 13.92 13.98 13.92 11.22 13.92 11.22 P 3 28.2 34.48 28.2 31.73 28.2 31.72 Rfrncs [1 F. Farokhi, W. Krichn, A. M. Bayn, and K. H. Johansson, A htrognous routing gam, in Procdings of th Annual Allrton Confrnc on Communication, Control, and Comuting, 213. [2 C. S. Fisk, Gam thory and transortation systms modlling, Transortation Rsarch Part B: Mthodological, vol. 18, no. 4 5,. 31 313, 1984. [3 J. G. Wardro, Som thortical ascts of road traffic rsarch, in Procdings of th Institut of Civil Enginrs: Enginring Divisions, no. 3,. 325 362, 1952. [4 M. J. Smith, Th xistnc, uniqunss and stability of traffic quilibria, Transortation Rsarch Part B: Mthodological, vol. 13, no. 4,. 295 34, 1979. [5 R. Bannr and A. Orda, Bottlnck routing gams in communication ntworks, IEEE Journal on Slctd Aras in Communications, vol. 25, no. 6,. 1173 1179, 27. [6 E. Altman, T. Boulogn, R. El-Azouzi, T. Jiménz, and L. Wyntr, A survy on ntworking gams in tlcommunications, Comutrs & Orations Rsarch, vol. 33, no. 2,. 286 311, 26. [7 A. Czumaj, Slfish routing on th intrnt, in Handbook of Schduling: Algorithms, Modls, and Prformanc Analysis (J. Y.-T. Lung, d.), Chaman & Hall/CRC Comutr & Information Scinc Sris, CRC Prss, 24. [8 M. Nttr, Equilibrium and marginal cost ricing on a road ntwork with svral traffic flow tys, in Procdings of th 5th Intrnational Symosium on th Thory of Traffic Flow and Transortation,. 155 163, 1971. [9 S. C. Dafrmos, Th traffic assignmnt roblm for multiclass-usr transortation ntworks, Transortation Scinc, vol. 6, no. 1,. 73 87, 1972. [1 F. Farokhi and K. H. Johansson, A gam-thortic framwork for studying truck latooning incntivs, in Procdings of th 16th Intrnational IEEE Annual Confrnc on Intllignt Transortation Systms,. 1253 126, 213. [11 A. A. Alam, A. Gattami, and K. H. Johansson, An xrimntal study on th ful rduction otntial of havy duty vhicl latooning, in Procdings of th 13th Intrnational IEEE Confrnc on Intllignt Transortation Systms,. 36 311, 21. [12 E. Strn and H. W. Richardson, Bhavioural modlling of road usrs: currnt rsarch and futur nds, Transort Rviws, vol. 25, no. 2,. 159 18, 25. [13 E. Strn, Ractions to congstion undr tim rssur, Transortation Rsarch Part C: Emrging Tchnologis, vol. 7, no. 2 3,. 75 9, 1999. 19