Social Resource Allocation in a Mobility System with Connected and Automated Vehicles: A Mechanism Design Problem

Social Resource Allocation in a Mobility System with Connected and A utomated V ehicles: A Mechanism Design Problem Ioannis V asileios Chremos, Student Member , IEEE , and Andreas A. Malikopoulos , Sen ior Member , IEEE Abstract — In this paper , we in vestigate the socia l resour ce allocation in an emerging mobility system consisting of con- nected and automated v ehicles (CA Vs) using mechanism design. CA Vs prov ide the most intriguing opp ortunity f or enabling tra velers to m onitor mobility sy stem conditions efficiently and make better decisions. Ho wever , th i s new re ality wil l in fluence tra velers’ tendency-of-trav el a nd might give rise to rebound effects, e.g., increased-vehicle-miles trav eled. T o tackle this phenomenon, we propose a mechanism design formulation that prov ides an efficient social res ource allocation of tra vel time fo r all tra velers. Our focus is on the socio-technical aspect of th e problem, i . e., by designing app ropriate socio-economic incentives, we seek to prev ent p otential rebound effects. In particular , we pr opose an economically inspired mechanism to influence the impact of t h e trav elers’ decision-making on the well-being of an emerging mobility system. I . I N T R O D U C T I O N Now aday s, it is nearly impossible to commu te in a major urban area withou t the frustration of a traf fic jam or conges- tion. Congestion leads to more acciden ts and alter cations, and, m ost imp ortantly , con g estion is one of the key con- tributors that dam ages the environment (e.g . , air pollutio n caused by the h uge n umbers o f id ling engines). It is highly expected th at emerging mo b ility systems, e.g ., conne c ted and automa ted vehicles (CA Vs), will b e ab le to eliminate congestion and incr e ase mo bility efficiency in terms o f energy a n d tra vel time [1]. Ho wever , urb an social life has been greatly associated with the technological impact of the car , which compels us to reassess th e relation ship betwe e n automob ility an d social life [2], [3]. Thus, it is v ital to study the impact of CA Vs in a socio-techn ical context focu sing on the social dynam ics. The m ost n ovel an d definin g o f all the formida b le characteristics of the em erging mobility system is its socio- economic co mplexity . Fu ture mobility system s will enable h uman-vehicle inte r action and allow en hanced and universal accessibility . Ev id ent from similar techn ological rev olution s (e.g ., the impact of elev ators on building desig n and social class hierar c hies [4]) , h uman social per spectiv e and view can have a tremendo us effect on h ow technolo gical innovations are utilized and implemente d . Similarly , CA Vs are expected to beco me a so cially disrup tiv e innovation with vast tec h nolog ica l, comm ercial, and r egulatory implicatio ns. For example, in the form of rebound effects, the be n efits of conv enien ce and safety could potentially lead p eople to travel This research was s upported in part by A RP AE’ s NE XTCAR program under the awar d number DE-AR0000796 and by the Delaw are E nerg y Institut e (DEI). The authors are with the Department of Mechanical Engineer - ing, Uni versity of Delawa re, Ne wark, DE 19716 USA (ema ils: ichremos@udel .edu; and reas@udel.edu. ) more frequen tly using th eir car , a nd thu s, in crease the traffic volume in the transportation network. Even though CA Vs are not co m mercially av ailable ye t, the motivati on behin d our research is the f o llowing: “systems with intertwined social and techn o logical dime n sions are n ot gu aranteed to exhibit an op tim al perfo rmance. ” In p revious work, we modeled the hu man social interaction with CA Vs as a social dilemma in a gam e -theoretic appro a ch [5]. W e in vestigated th e socia l-mobility dilemma , i.e ., the binary decision-makin g of trav elers between commuting with a CA V or u sin g pu b lic tr a nsportation . In this paper, using mechanism d esign, we model th e routing of travelers in a transportatio n network with CA Vs as a social resou r ce allocation pro blem. Mechanism design th eory has em erged to mathemati- cally model, analy ze, and solve info rmationally decen tr al- ized p roblems in volving systems of mu ltiple rational an d intelligent agents [ 6]. M echanism design is co ncerned with methodo logies that implement system-wid e o ptimal solu tions to a my riad of p roblems - pro b lems in which the strategically interacting agents can hide th eir true pref erences f or better individual bene fits, thu s hu rting th e overall efficiency of the system. It has been widely u sed in areas like communica tio n networks [7], power markets [8], and so cial ne tworks [9]. A m echanism m ay be defined as a mathem atical structur e that mo dels institutions th rough which econom ic activity is guided and coor dinated [1 0]. W e are using the notio n of a mechanism in this sense, tho ugh the econo mic activity we aim to control is the allo cation of travel time among travelers in a mobility system. Ou r pro posed mechanism pr esupposes a central au th ority (e.g ., centr a l co mputer) th at gathers all routing requ ests a n d tra vel time de mands from CA Vs aro und a city with a ro ad infrastru cture th at supp o rts connected and autom ated traffic. In this c o ntext, we build an d design approp riate protocols and interfaces (e.g ., tolls, subsidies) for a central traffic m anagemen t com puter, which will gu arantee the realization of the desired outcome, i.e., m aximizing soc ial welfare and eliminating cong estion. The auth ors in [ 10] form ulated a resou r ce allocation problem with in the framework o f mechanism de sign. Th is work led to a spark of research as mech anism de sig n has been used extensively in c ommun ic a tion networks in the for m of decen tralized resou rce allocation prob lem s [7], [11], an d also in tra nsportation [12], [13]. I n such pr o blems, th e ma in methodo logy is to ap ply the V ickrey-Clarke-Groves (VCG) mechanisms, which ar e direct mech anisms that achieve a socially-optim al solution as a dominan t strategy . Because the VCG m echanisms have certain limitation s ( e .g., th ey are not budget ba la n ced), there h av e been attem pts to u se different a p proach es to solve the mech anism design pro b lem. For example, b y a d opting the Nash equilibrium (NE) as the solution concept of the mech anism, a surr ogate optimiza tio n method can be u sed wher e th e network man ager asks the agents to repor t a bundle of messages that appr oximate their priv ate inform ation [11], [ 14]. In this paper, we fo cus on th e social p e r spectiv e of the emerging mobility systems with CA Vs. It is wid ely accep ted that CA Vs will rev olution ize urba n m obility and th e way people co mmute. An example would b e for CA Vs to make empty trips, i.e., no travelers, to av oid p arking, and thu s add extra con gestion in th e network [1 5]. In a d dition, CA Vs could potentially affect driv ers’ behavior and have an imp act on tra ffic perfor m ance in gene ral [16]. The question of the actual imp a ct of CA Vs on travel, energy , an d car b on demand has attracted co nsiderable attention [1 7]. Depen ding on d ifferent environmental indicato r s, the au thors in [18] provided a p ractical mic r oecono mic environmental r e bound effect mo del. So far , ther e has bee n research o n the effects of a considerate pen etration of shared CA Vs in a major metropo litan area [ 1 9]. Ho wever , m o st studies on CA Vs have focu sed on how to coo rdinate CA Vs in d ifferent traffic scenarios [2 0], [21]. In this paper, we in vestigate the travel tim e p rovisioning in transporta tio n networks with CA Vs and strategic travelers. The main con tribution of this paper is th e d ev elop m ent of an inf o rmationally decentralized travel time social allocation mechanism with strategic travelers p o ssessing the fo llowing proper ties: (a) existence of at least one Nash equilibriu m (NE), (b) budget ba lanced at equilib rium, ( c) individually rational, (d) strong ly implemen table at NE, and (e) fea sib le at o r off of eq uilibrium . Ano th er contribution of the pap er is that the design of ou r mecha n ism’ s tolls f or the travelers’ utilization of th e network’ s resourc e s is intuitive enou gh to provide a good understandin g of the practical implementatio n of the mecha n ism. The remaind er of the pap er is organ ized as f o llows. In Section I I, we present the m athematical formu lation of ou r propo sed mech anism. I n Section III, we provid e its forma l specification, and then , in Section IV, we for m ally show that our pro posed m e chanism has pr operties ( a ) - (e) . Fin a lly , in Section V, we d r aw some conclu ding rema r ks and discuss potential avenues for fu ture research. I I . M A T H E M A T I C A L F O R M U L A T I O N W e co nsider a tran sp ortation network represen ted b y a graph G = ( V , E ) , where V = { 1 , . . . , V } cor respond s to the index set of vertices and E = { 1 , . . . , E } the in dex set of dir ected ed g es. Each ed ge e ∈ E has a fixed capacity , i.e., c e ∈ R > 0 , e.g., a high cap acity c e correspo n ds to a h ighway while a low capacity correspon ds to an urban ro ad. Th ere are n ∈ N ≥ 2 trav elers represen ted by the set I = { 1 , 2 , . . . , n } . Each tr av eler i is associated with an origin -destination pair ( o i , d i ) ∈ V × V . The utilization of the ro ads in G is do ne by the u se of CA Vs, wh e re each CA V correspo n ds to on e trav eler . W e consider 1 00% pen etration rate of CA Vs. Definition 1 . A traveler i ∈ I seeks to commu te f r om o i to d i via a given and fixed route p i ( o i , d i ) at prefer red tr avel time, den oted by θ i ∈ Θ i = [0 , + ∞ ) . In game theoretic terms, θ i is the type of traveler i . W e den ote the type profile of all travelers by θ = ( θ 1 , θ 2 , . . . , θ n ) . In addition, each edge e ∈ E in the network is character- ized by θ e which rep resents the minimum po ssible travel time that any traveler ca n experience if edge e ∈ E is an empty (uncon gested) road. This allows us to take into account rural or ur ban road s of different traffic capacities in the tr ansportation network G . Next, each trav eler i ∈ I has a c o st fu nction v i which expresses th e “ c o mmute-satisfaction” that traveler i expe r i- ences from com muting in ( o i , d i ) with travel time θ i . W e expect v i and θ i to be tr aveler i ’ s private in formation (i.e . , unknown to the n etwork manager ). Assumption 1. Assume that v i : R ≥ 0 → R is continuo usly differ entiable, strictly concave, and strictly decr easing in θ i with v i (0) = 0 . Next, we denote by t i the mon etary p ayment m ade by trav eler i to th e network manag er . W e h ave t i ∈ R , i.e., a positive t i means th a t traveler i pays a toll an d a negative t i means that i re c ei ves a monetary subsidy . Thus, in our mechanism, traveler i ’ s total u tility is given by u i ( θ i , t i ) = v i ( θ i ) − t i . (1) W e co n sider th at all travelers ar e ration a l and in tellig ent decision-ma kers in th e system. Each traveler i ∈ I has two objectives: (i) to r each their destination, and (ii) to maxim ize their own utility . A social co nsequence of the tr avelers’ behavior is that there is an individual disregard of the overall good o f the system an d it is natu ral to expect that a t least one ed ge e ∈ E will exceed its maxim um cap acity . If the network manag er does not intervene, th en con gestion is to be expected. So, u sing ap propr iate mo netary pa y ments, th e network manager can incentivize tr avelers to report truthfully their typ e θ i and allocate travel time o n each edge e ∈ E in such a way that all travelers are satisfied and congestion is prevented. T o achieve this, th e network manager’ s o bjective is to ma ximize the overall “social welfare” of the network and ensure that the n etwork r emains conge stion -free. Th e social welfare fu nction is defined as the P i ∈I v i ( θ i ) and denoted b y W . W e cho ose to define th e socia l welfare as the sum of th e utilities of all travelers because we follow the utilitarian princip les, i.e., we measure the co llecti ve be nefits gained by the trav elers in the transportation network. Next, n ote that the travelers’ strategic beh avior indicates a n a tural com petition over the utilization o f the e dges. Definition 2. Giv en e ∈ E , we d efine the following sets: (i) the set S e of all travelers tha t ed ge e is part of the ir route that con nects o i and d i , and (ii) th e set R i of traveler i ’ s edges that consist o f th e ir route p i ( o i , d i ) . Before we co n tinue, we in troduce th e notion of re verse value o f time , say parame ter α i ∈ R ≥ 1 , that can vary amon g each traveler i ∈ I . The social pa r ameter α i ∈ ( α, α ) , wh ere α ≥ 1 , can b e interp reted as fo llows. If α i → α, traveler i is willing to to lerate a slightly high er travel time, while if α i → α , traveler i is n ot willing to tolera te a high er travel time. W e assume that each traveler i ∈ I can be classified based on socio- economic demo graphic data (e.g ., mobility choices and trav el tend encies, civil status an d income) [2 2]. Problem 1. The cen tralized social-welfare max imization problem is presented b elow: max θ e i X i ∈I X e ∈R i v i ( θ e i ) , subject to: θ e i ≥ θ e , ∀ e ∈ E , ∀ i ∈ I , (2) X i ∈S e α i · θ e i ≤ c e , ∀ e ∈ E , (3) where θ e i is the travel time of traveler i on edg e e with θ i = P e ∈R i θ e i , and v i ( θ i ) = P e ∈R i v i ( θ e i ) ; inequalities (2) ensure that each traveler i ’ s trav el tim e θ e i on all ed ges e ∈ E is non-negative but not zero at any case; an d in equality (3) expr esses th e network’ s capacity on each edge e ∈ E . By Assumption 1, it is im perative to imp ose a n etwork threshold on the feasible values of each traveler i ’ s travel time. W e can achieve this in (2) by on ly ac cepting travelers’ trav el times that are above θ e . Also, we interpr et θ i = 0 to be the case of trav eler i no t seeking to commu te in stead of wishing to comm ute in zero time. Problem 1 would b e a stand ard conve x optimization pro b- lem if the strategic travelers were expec te d to rep o rt their priv ate in formation truthfully . As this is unreason able to expect f rom strategic decision -makers, th e network man ager in ord er to solve Pro blem 1 is tasked to elicit the necessary informa tio n using monetar y in centives. A. The Mechanism Desig n Pr oblem In ou r form ulation, we use th e NE as our solution concep t. Howe ver , a NE requires complete informatio n. But, we can interpret a NE as the fixed p oint of an iter ati ve pro cess in an incomp lete infor mation setting [23], [24]. This is in accordan ce with J. Nash’ s interpretatio n of a NE, i.e. , the complete inform ation NE c a n be a po ssible equilib r ium of an iterative lear ning pro c e ss. In this section, we present the f u ndamen tals of an indirect and d ecentralized resour ce allocatio n mechan ism f ollowing the fr a mew ork presen ted in [10]. First, w e need to specify a set of messages that all trav elers h av e access and a r e able to use in order to co mmunicate informa tio n. Based on this infor mation, travelers make dec isions wh ich affect the reaction of the network man a ger . Onc e th e com m unication between th e n etwork mana ger and the travelers is complete, we say that the mech anism induc e s a gam e; strategic travelers then comp ete for the network’ s reso u rces. In this line of reasoning , we define fo rmally below what we mean by indirect m echanism and induced ga me. An indirect mechanism can be described as a tuple of two compon ents, n amely h M , g i . W e write M = ( M 1 , M 2 , . . . , M n ) , where M i defines the set of p ossible messages of traveler i . T h us, the travelers’ co mplete m es- sage space is M = M 1 × · · · × M n . T he com ponen t g is the o utcome f unction defin ed by g : M → O which m aps each message profile to the outpu t space O = { ( θ 1 , . . . , θ n ) , ( t 1 , . . . , t n ) | θ i ∈ R ≥ 0 , t i ∈ R } , i. e., the set of all possible travel time allocations to the travelers and the monetary pay ments (e.g., to ll, subsidies) made or received by the travelers. Th e outcome fun ction g deter mines the outcome, n amely g ( µ ) f or any given m e ssag e pro file µ = ( m 1 , . . . , m n ) ∈ M . The pay ment func tio n t i : M → R determines the monetar y p ayment made or receiv ed by a trav eler i ∈ I . Definition 3. A mechan ism h M , g i together with the utility function s ( u i ) i ∈I induce a game h M , g , ( u i ) i ∈I i , where each utility u i is ev aluated at g ( µ ) for each traveler i ∈ I . Definition 4. Co n sider a game h M , g , ( u i ) i ∈I i . The so- lution concept of NE is a message profile µ ∗ such th a t u i ( g ( m ∗ i , m ∗ − i )) ≥ u i ( g ( m i , m ∗ − i )) , for all m i ∈ M i and for each i ∈ I , where m − i = ( m 1 , . . . , m i − 1 , m i +1 , . . . , m n ) . Definition 5. Let the utility of no participa tion of a traveler i ∈ I to b e g iv en by u i (0 , 0) = v i (0) = 0 . Th en, we say that a mec h anism is individually rational if u i ( g ( µ ∗ )) ≥ 0 , for all i ∈ I , an d all NE µ ∗ ∈ M . I I I . P RO P O S E D M E C H A N I S M In this section, we show how th e n etwork man ager can design m o netary incentives wh ich achieve the d esirable go al, i.e., align everyone’ s decisions by incentivizing them to sen d social-welfare supportin g messages. But first, we need to establish the in formatio n al stru cture o f our me c h anism. Th e network man ager h as com plete knowledge of the network’ s topolog y and resources and travelers k now only their own utility which they rep o rt priv ately to th e network m a nager . Before we continu e, let u s de fine explicitly a traveler’ s message. For each i ∈ I , message m i ∈ M i is given by m i = ( ˜ θ i , τ i ) , wh ere ˜ θ i = ( ˜ θ e i : e ∈ R i ) is the repo r ted preferr ed travel time of traveler i , and τ i = ( τ e i : e ∈ R i ) is the price traveler i is willin g to p ay fo r ˜ θ i along their ro ute. Definition 6. Th e average price of all travelers that com pete to u tilize ed ge e ∈ E oth er than traveler i is given b y τ e − i = P j ∈S e : j 6 = i τ e j |S e |− 1 . Next, fo r each traveler i an d fo r each edge e ∈ E of their ro u te, we endow a fair share for each edge e ∈ E , i.e., c e / |S e | . This can help us design th e m onetary paym e n ts that each traveler is asked to pay . Using Definition 6 , we propo se the following payments, for a particu la r edge e ∈ E , t e i ( µ ) = τ e − i ·  α i · ˜ θ e i − c e |S e |  + ( τ e i − ν e ) 2 + τ e − i · ( τ e i − τ e − i ) · c e − X i ∈S e α i · ˜ θ e i ! 2 . (4) The first term in (4) is the m onetary p ayments (e.g. , to ll, subsidies) made or received b y traveler i corresp onding to their travel time allocation ˜ θ e i on edge e ∈ E . In tuitiv ely , this m eans that traveler i will pay a toll that is deter m ined by the other travelers’ r ecommen dations and only for the excess of th e fair share of travelers over a particular ed ge. Using this f ormulatio n, ther e is no inc entiv e for traveler i to lie in an attempt to red u ce th eir pay m ent to the network. The seco n d term in ( 4) co rrespond s to a penalty that traveler i will pay if she r eports a different p rice τ e i from ν e , where ν e represents the Lag range mu ltiplier cor respond ing to the capacity constraint defined forma lly next. The third term in (4), collectively incentivizes all travelers to b id the same price p er un it of travel time and to utilize the f ull capa c ity of each edge e ∈ E . Thus, given any message profile µ , th e total mo netary paymen t t i ( µ ) for traveler i is t i ( µ ) = X e ∈R i t e i ( µ ) + φ i ( ˜ θ i ) , (5) where φ i is a mo netary incentive th at encour a g es traveler i to repo rt a reason a b le tr av el time d e mand respectin g roa d rules and the network’ s efficiency g oals. In detail, we have φ i ( ˜ θ i ) =          γ , ∃ e ∈ R i , s.t. ˜ θ e i > θ e and |S e | = 1 , 0 , ∃ e ∈ R i , s.t. ˜ θ e i > θ e and |S e | ≥ 2 , δ, ∃ e ∈ R i , s.t. ˜ θ e i = θ e and |S e | ≥ 2 , 0 , ∃ e ∈ R i , s.t. ˜ θ e i = θ e and |S e | = 1 , (6) where γ , δ ∈ R > 0 represent the imp o sition o f very hig h penalties. It is n ecessary to impo se such penalties since for the first case in (6), traveler i vio lates the goal of efficiency in th e network a nd fo r th e thir d case in (6), trav eler i vio lates the go al of road safety . In the severe case of ˜ θ e i < θ e , we have φ i ( ˜ θ i ) = + ∞ . I V . P R O P E RT I E S O F T H E M E C H A N I S M In this sectio n, we p resent the properties of ou r pro posed mechanism. Lemma 1. Pr oblem 1 h as a uniqu e o p timal solution . Pr oof. The objective function of Problem 1 is a sum of sev- eral strictly concave fun ctions. Hence, it is strictly concave. Thus, th e necessary KKT con ditions ar e also sufficient for optimality . Since the fe asible r egion is non -empty , convex, and comp act, we conclude th at Proble m 1 has always a unique op timal solution . Lemma 2 . A so lu tion to Pr oblem 1 is uniqu e a n d optimal if, and only if, it satisfies the feasibility con ditions of Pr oblem 1 and there exist Lagrange multipliers λ = ( λ e i : e ∈ E ) i ∈I and ν = ( ν e ) e ∈E that satisfy the following con ditions: ∂ v i ( θ e i ∗ ) ∂ θ e i + λ e i ∗ − X e ∈R i α i · ν ∗ e = 0 , (7) λ e i ∗ · ( θ e i ∗ − θ e ) = 0 , ∀ e ∈ E , ∀ i ∈ I , (8) ν ∗ e · X i ∈S e α i · θ e i ∗ − c e ! = 0 , ∀ e ∈ E , ( 9 ) λ e i ∗ , ν ∗ e ≥ 0 , ∀ e ∈ E , ∀ i ∈ I . (10) Pr oof. First, le t us derive the Lag r angian of Problem 1: L ( θ, λ, ν ) = X i ∈I X e ∈R i v i ( θ e i ) + X i ∈I X e ∈E λ e i · ( θ e i − θ e ) − X e ∈E ν e · X i ∈S e α i · θ e i − c e ! . (11) From (1 1), it is easy to d erive th e KKT condition s, i.e., ∂ v i ( θ e i ∗ ) ∂ θ e i + λ e i ∗ − X e ∈R i α i · ν ∗ e = 0 , (12) λ e i ∗ · ( θ e i ∗ − θ e ) = 0 , ∀ e ∈ E , ∀ i ∈ I , (13) ν ∗ e · X i ∈S e α i · θ e i ∗ − c e ! = 0 , ∀ e ∈ E , (14) λ e i ∗ , ν ∗ e ≥ 0 , ∀ e ∈ E , ∀ i ∈ I . (15) Since th e KKT condition s are necessary and su fficient to guaran tee th e optima lity of any allocation of travel time that satisfies the m , it is enoug h to find λ e i ∗ and ν ∗ e such that the above con ditions are satisfied. Theorem 1 (Feasibility) . F or any message p r ofile µ , the corr espond ing travel time allocation θ is a fea sib le point of P r oblem 1. Pr oof. Consider any traveler i and denote by C the constraint set of Prob lem 1. Then, for a r eported pr eferred travel tim e ˜ θ i , the travel time θ i of Problem 1 generated by th e outco me function is equal to (i) ˜ θ i if ˜ θ i ∈ C ; or (ii) θ 0 i if ˜ θ i / ∈ C , where ˜ θ i = ( ˜ θ e i : e ∈ R i ) , an d θ 0 i is the po int on the b ound a ry of C (i.e., we ign ore the “u nreasona b le” demand of traveler i and allocate only the por tio n of th e resour ce that is available). By constructio n , it follows immediately th at if ˜ θ i ∈ C , then the a llocation θ i is feasible fo r a ny trav eler i ∈ I . In the case of ˜ θ i / ∈ C , the allocation is on the boun dary of C , he n ce it is still feasible as the constraint set of Prob le m 1 is closed. Thus, th e result follows. Lemma 3. Let µ ∗ be a NE of the in d uced g ame. Then , we have τ e i ∗ = ν ∗ e , fo r a ll i ∈ I a n d each e ∈ R i . In ad dition, it follows that τ e − i = P j ∈S e : j 6 = i τ e j |S e |− 1 = τ e i ∗ . Pr oof. Suppo se th e re is one trav eler, say i , that deviates from the NE message profile µ ∗ and instead repo rts the m essage m i = ( ˜ θ ∗ i , τ i ) . This d eviation to be justifiable has to pr ovid e a h ig her utility to traveler i ∈ I . But, we h av e v i ( ˜ θ ∗ i ) − t i ( m ∗ i , m ∗ − i ) ≥ v i ( ˜ θ ∗ i ) − t i ( m i , m ∗ − i ) . (16 ) Next, we substitute (4) into (16). For ease of notational exposition, let ξ =  c e − P i ∈S e α i · ˜ θ e i ∗  2 . Thus, X e ∈R i ( τ e i ∗ − ν ∗ e ) 2 + τ e − i ∗ · ( τ e i ∗ − τ e − i ∗ ) · ξ ≤ X e ∈R i ( τ e i − ν ∗ e ) 2 + τ e − i ∗ · ( τ e i − τ e − i ∗ ) · ξ . (1 7) Since traveler i b e haves as a utility-m aximizer, we need to minimize the right hand side of ( 1 7). Thu s, th e b est price is τ e i = τ e − i ∗ , and also the so lution of the min imization problem min ( τ e i ) P e ∈R i ( τ e i − ν ∗ e ) 2 . The r efore, at µ ∗ , we have τ e i ∗ = ν ∗ e , for all e ∈ E and for all i ∈ I and τ e − i = P j ∈S e : j 6 = i τ e j |S e |− 1 = τ e i ∗ follows imm ediately . Lemma 4 . Let µ ∗ be a NE of th e ind uced g ame. Then, for every traveler i ∈ I , we ha ve φ i ( ˜ θ ∗ i ) = 0 . Pr oof. W e p rove th is by co ntradiction . Supp o se the re exists a NE me ssag e µ ∗ = ( m ∗ i = ( ˜ θ ∗ i , τ ∗ i )) i ∈I such that φ i ( ˜ θ ∗ i ) 6 = 0 for traveler i ∈ I . By (6), we o nly have two cases to consider: let ˜ θ e i ∗ > θ e with |S e | = 1 (the proo f f or the o th er case is similar). Suppose trav eler i deviates from the NE with message m i = (( ˜ θ e i = θ e : e ∈ R i ) , τ ∗ i ) . By Defin ition 4, we h av e u i ( g ( m i , m ∗ − i )) ≤ u i ( g ( m ∗ )) . (18) Substitute (1), (4), and (6) into (18) an d then L emma 3 giv es [ v i ( ˜ θ i ) − v i ( ˜ θ i ∗ )] − X e ∈S e α i · ν ∗ e · ( ˜ θ e i − ˜ θ e i ∗ ) + φ i ( ˜ θ ∗ i ) ≤ 0 , (19) where by A ssum ption 1, th e first difference term of ( 19) is negativ e; likewise the difference of ( ˜ θ e i − ˜ θ e i ∗ ) is positive. Thus, it fo llows th a t, since φ i ( ˜ θ ∗ i ) ≫ 0 , traveler i rig h tfully deviates from the NE µ ∗ = ( m ∗ i = ( ˜ θ ∗ i , τ ∗ i )) i ∈I such that φ i ( ˜ θ ∗ i ) 6 = 0 as (19) cannot be t ru e (by construction of (6)). Since the case o f ˜ θ e i < θ e , where φ i ( ˜ θ i ) = + ∞ is straightfor ward to show , and th e p r oof is c omplete. Theorem 2 (Budg et Balanc e) . Let the message pr ofile µ ∗ be a NE of the indu ced game. The pr opo sed mechan ism at µ ∗ does not r equire any exter na l or internal moneta ry payments, i.e., P i ∈I t i ( µ ∗ ) = 0 fo r all µ ∗ . Pr oof. Summin g (4) over all tr av elers y ields P i ∈I t i ( µ ∗ ) = P i ∈I  P e ∈R i t e i ( µ ∗ )  = P e ∈H P i ∈S e t e i ( µ ∗ ) , wh ere H is the set of compe titive edges in the network (i.e. , any edg e utilized by more than two trav elers). Henc e, X e ∈H X i ∈S e τ e − i ∗ ·  α i · ˜ θ e i ∗ − c e |S e |  + ( τ e i ∗ − ν ∗ e ) 2 + τ e − i ∗ · ( τ e i ∗ − τ e − i ∗ ) · c e − X i ∈S e α i · ˜ θ e i ∗ ! . (20) By L emma 3, we ha ve for all e ∈ H , P i ∈I t i ( µ ∗ ) = P e ∈H ν ∗ e ·  P i ∈S e α i · ˜ θ e i ∗ − c e  , wh ich is e qual to zero by the KKT co nditions in Lemma 2. Theorem 3 (In dividually Rational) . The pr oposed mecha- nism is individually rationa l. In p articular , ea ch traveler pr efers the outcom e of any NE of the induce d ga me to the outcome of no p articipation . Pr oof. Let the me ssage profile µ ∗ be an arb itrary NE o f th e induced g ame. W e nee d to show that u i ( µ ∗ ) ≥ u i (0) = 0 for each trav eler i (see D e finition 5). Consider the message m i = ( ˜ θ i , τ i ) with ˜ θ i = 0 and τ i = ( τ e i = ν e : e ∈ R i ) . That is, tr aveler i deviates with m i while the other trav elers adhere to the NE µ ∗ . By Definition 4, we have the fo llowing: u i ( g ( µ ∗ )) ≥ u i ( g ( m i , m ∗ − i )) = v i (0) − X e ∈R i τ e − i ∗ ·  0 − c e |S e |  = X e ∈R i ν ∗ e ·  c e |S e |  ≥ 0 . (21) Thus, fr om (21), th e re su lt follows. In ou r next result, we show that o ur mechanism is strong ly implementab le at NE. Strong implemen tation en su res th a t the efficient allocation of tr av el time to the travelers is implemented b y all equilibria of the induced gam e [2 5]. Theorem 4 (Str ong Imp le m entation) . At a n a rbitrary NE µ ∗ of the induc ed game, th e a llocation travel time ( ˜ θ ∗ i ) i ∈I is equal to the optimal solutio n ( θ ∗ i ) i ∈I of Pr oblem 1 for each i ∈ I . Pr oof. Suppo se µ ∗ is a NE of the indu ced game. Then, b y Lemma 3, it follows that τ e i ∗ = τ e − i ∗ = ν ∗ e for each e ∈ R i . Next, con sider som e traveler i th at particip a tes in the mechanism and has prefer red tr avel time θ i . The utility of trav eler i for such an allocation is given b y u i ( g ( m i , m ∗ − i )) = v i ( θ i ) − t i ( m i , m ∗ − i ) , (22) where t i ( m i , m ∗ − i ) = P e ∈R i ν ∗ e  α i · θ e i − c e |S e |  . By Def- inition 4, it f ollows that at NE no tr av eler should have an incentive to deviate. Henc e, th e ma ximization of traveler i ’ s utility (2 2) must b e attained at the NE travel tim e allocation , i.e., θ ∗ i = ˜ θ ∗ i . Th e Nash-ma x imization problem is ˜ θ e i ∗ = arg ma x θ e i " X e ∈R i v i ( θ e i ) − X e ∈R i ν ∗ e  α i · θ e i − c e |S e |  # , (23) subject to the exact same c onstraints as in Pro blem 1. Now , it is easy to d e riv e th e KKT cond itions that will give the optimal “Nash solu tion. ” By Lemma 2, the KKT conditions are necessary and su fficient to gu arantee the optimality of any travel time allo cation ( θ i ) i ∈I that satisfies them . Thus, it is sufficient to show that there exist ap propr iate Lagra n ge multipliers λ e i ∗ and ν ∗ e such that ( 7) - ( 9) are satisfied. By setting λ e i = 0 and ν e = τ e i for all e ∈ E , by differentiation of (4) with respect to θ e i and τ e i , we get ∂ v i ( ˜ θ e i ∗ ) ∂ ˜ θ e i = X e ∈R i α i · ν ∗ e , ∀ i ∈ I , (24) ν ∗ e · X i ∈S e α i · ˜ θ e i ∗ − c e ! = 0 , ∀ e ∈ E . (25) It is straightfo r ward to see th at ( 24) and ( 25) are iden tical to (7) an d (9), respectively . Condition (8) in both pr o blems holds trivially . Consequen tly , the solu tion ˜ θ ∗ = ( ˜ θ ∗ 1 , . . . , ˜ θ ∗ n ) of (2 4) an d (25) along with the specification of the p ayment function s (4) are equiv alent to the o ptimal unique so lution of Problem 1. Thus, at a ny NE µ ∗ , we g et an identical allocation g ( µ ∗ ) = ( ˜ θ ∗ 1 , . . . , ˜ θ ∗ n , t ∗ 1 , . . . , t ∗ n ) that is equal to the optima l solution o f Prob lem 1 , an d the proof is complete. Theorem 5 (Existen ce) . Let θ ∗ be the optimal solution of Pr oblem 1 and ν ∗ e be the corresponding Lagrange multipliers of th e KKT conditions. If for each i ∈ I , m ∗ i = ( ˜ θ ∗ i = θ ∗ i , τ ∗ i ) , where τ ∗ i = ( τ e i ∗ = ν ∗ e : ∀ e ∈ R i ) a nd φ i ( ˜ θ ∗ i ) = 0 for a ll i ∈ I . Then the message µ ∗ = ( m ∗ i ) i ∈I is a NE of the in duced game. Pr oof. W e show that the m essage profile µ ∗ = ( m ∗ i ) i ∈I where m ∗ i = ( ˜ θ i = θ ∗ i , τ ∗ i ) is a NE. By Lemm a 2 , it follows that θ ∗ along with th e ap propr iate Lagr ange multiplier s satisfies th e KK T c o nditions of Prob lem 1 and is the only feasible allocatio n. For any traveler i , the u tility at message µ ∗ is u i ( g ( µ ∗ )) = v i ( ˜ θ ∗ i ) − P e ∈R i ν ∗ e ·  α i · ˜ θ e i ∗ − c e |S e |  . Now , supp ose tr av eler i deviates f rom µ ∗ by ch a n ging their message while all the othe r travelers adhere to the message µ ∗ (thoug h we would still have τ e − i ∗ = ν ∗ e ). W e have u i ( g ( m i , m ∗ − i )) ≤ v i ( ˜ θ ′ i ) − X e ∈R i ν ∗ e ·  α i · ˜ θ e i ′ − c e |S e |  ≤ max ˜ θ ′ i " v i ( θ ′ i ) − X e ∈R i ν ∗ e ·  α i · ˜ θ e i ′ − c e |S e |  # . (26) The ma x imization pr o blem (26) is eq u iv alent to (2 3). As the message µ ∗ clearly satisfies the KKT co n ditions of (23), we have θ i = ˜ θ i = ˜ θ ′ i , wh ich in turn implies: u i ( g ( m i , m ∗ − i )) ≤ v i ( ˜ θ ∗ i ) − X e ∈R i ν ∗ e ·  α i · ˜ θ e i − c e |S e |  , (27) where th e righ t hand side o f ( 2 7) is equ al to u i ( g ( µ ∗ )) , for all m ∗ i and all i ∈ I . Th erefore, m essage µ ∗ is a NE. V . C O N C L U D I N G R E M A R K S In this paper, we f ormulated th e ro uting o f strategic trav elers that use CA Vs in a transporta tio n ne twork as a social resource alloca tion mechanism d esign proble m . Consider ing a Na sh-implemen tation appr oach, we showed tha t our pro- posed inform ationally decentra lize d mec h anism efficiently allocates trav el time to all travelers th at seek to co mmute in the network. Our mech anism induces a g a me which at least one equilib rium p revents congestion (a significant rebo und effect), while also attain ing the pr operties of individually rationality , budget balanced , stro n gly implem entability . On- going work inclu des cond ucting a simu lation-based a n alysis under different traffic scenar ios to showcase the practica l implications of our me chanism. E x tending and enhanc in g the tr av eler-behavioral model, motiv ated b y a social-mobility survey can b e a worthwhile und ertaking as a futur e research direction allowing the study of the relatio n ship of em e rging mobility and the intricacies of human decision -making . R E F E R E N C E S [1] A. A. Malik opoulos, C. G. Cassandras, and Y . Zhang, “ A decentral ized ener gy-optimal contro l framew ork for connected automat ed vehic les at signal-f ree intersection s, ” Automatica , vol. 93, pp. 244–256, 2018. [2] M. Sheller and J. U rry , “The city and the car , ” Internationa l journal of urban and re gional resear ch , vol. 24.4, pp. 737–757, 2000. [3] D. Bissell, T . Birtchnel l, A. Elliott, and E. L. Hsu, “ Auton omous automobil ities: The social impacts of driv erless vehic les, ” Curre nt Sociolo gy , 2018. [4] A. Bernard, Lifted: A Cultural History of the E lev ator . Ne w Y ork Uni versity Press, 2014. [5] I. V . Chremos, L . E. Beav er , and A. A. Malikopoulo s, “ A game- theoret ic sociotechnic al analysis of the rebound effec ts of connect ed and automat ed vehicl es, ” in Proc. 23rd Int. IEEE Conf . Intel l. T ransp . Syst. , 2020. [6] R. Myerson, “Per specti ves on mechanism design in economic theory , ” American Economic Revi ew , vol. 98.3, pp. 586–603, 2008. [7] A. Kakhbod and D. T enek etzis, “ An efficien t game form for unicast service provisio ning, ” IEEE Tr ansactions on Automatic Contr ol , vol. 57.2, pp. 392–404, 2011. [8] C. Silv a, B. F . W ollenb erg, and C. Z. Zheng, “ Application of m echa- nism design to electric po wer market s (republished ), ” IEEE Tr ansac- tions on P owe r Systems , vol. 16.4, pp. 862–869, 2001. [9] A. Dave, I. V . Chremos, and A. A. Malikopo ulos, “Social media and m isleadi ng information in a democrac y: A mechanism design approac h, ” arXiv pre print arXiv:2003.07192 , 2020. [10] L. Hurwicz and S. Reiter , Designing economic mechani sms . Ca m- bridge Uni versity Press, 2006. [11] R. Johari and J . N. T sitsikli s, “Efficie ncy of scalar-pa rameterize d mechanisms, ” Operat ions Resear ch , vol. 57.4, pp. 823–839, 2009. [12] M. V asirani and S. Ossowski, “ A marke t-inspired approach for in- tersect ion management in urban road traffic networks, ” Jou rnal of Artificial Intelli gence Researc h , vol. 43, pp. 621–659, 2012. [13] W . Z ou, M. Y u, and S. Mizo kami, “Mec hanism design for an incenti ve subsidy scheme for bus transport, ” Sustainability , vol. 11.6, p. 1740, 2019. [14] T . Alpcan and T . Bas ¸ ar , “ A game-theoreti c frame work for congesti on control in general topology networ ks, ” Proce edings of the 41st IE E E Confer ence on Decision and Contr ol , vol. 2, 2002. [15] G. H. de Almeida Correia and B. van Arem, “Solving the user opti- mum priv ately owned automated vehicle s assignment problem (UO- POA V AP): A model to explore the impacts of self-dri ving vehicle s on urban mobility , ” T ransporta tion R esearc h P art B: Methodolo gical , vol. 87, pp. 64–88, 2016. [16] E. Aria, J . Olstam, and C. Schwieteri ng, “In vesti gation of automated vehi cle eff ects on dri ver’ s behavior and traf fic performance , ” T rans- portation rese arc h pr ocedia , vol. 15, pp. 761–770, 2016. [17] Z. W adud, D. MacKenz ie, and P . Leiby , “Help or hindrance? the trav el, ener gy and carbon impacts of highly automated vehicle s, ” T ransport ation Resear ch P art A: P olicy and Practi ce , vol. 86, pp. 1– 18, 2016. [18] F . D. V i vanc o, J. Freire-Gonz ´ alez, R. Kemp, and E. van der V oet, “The remarkable en vironmenta l rebound effec t of electric cars: a microeco nomic approac h, ” Journal of Clean er Pr oduction , vol. 48.20, pp. 12 063 –12 072, 2014. [19] L. M. Martinez and J. M. V iega s, “ Assessing the impacts of depl oying a s hared self-dri ving urban mobility system: An agent-based model applie d to the city of Lisbon, Portugal, ” Internation al Jo urnal of T ransport ation Scienc e and T ec hnology , vol. 6.1, pp. 13–27, 2017. [20] J. Rios-T orres and A. A. Mali kopoulos, “ A survey on the coordination of connect ed and automated vehic les at intersect ions and merging at highwa y on-ramps, ” IEE E T ransac tions on Intellige nt T ransportat ion Systems , vol. 18.5, pp. 1066–1077, 2017. [21] L. Z hao and A. A. Malikop oulos, “Enhanc ed mobility with connecti v- ity and automati on: A re view of shared autonomous ve hicle systems, ” IEEE Intell igent T ransportat ion Systems Magazine , 2020. [22] D. Brownstone , A. Ghosh, T . F . Golob, C. Kazimi, and D. V an Amels- fort, “Driv ers’ willingne ss-to-pay to reduce trav el time: evide nce from the san die go I-15 congest ion pricing project, ” T ransportat ion Resear ch P art A: P olicy and Pract ice , vol. 37.4, pp. 373–387, 2003. [23] S. Reichel stein and S. Reiter , “Game forms with minimal message spaces, ” Econometric a , vol. 56.3, pp. 661—692, 1988. [24] T . Grove s and J . Ledyard, “Optimal allocation of public goods: A solution to the free rider problem, ” Econometrica , vol. 45.4, pp. 783– 809, 1977. [25] L. Mathev et, “Supermodular mechanism design, ” Theoret ical Eco- nomics , vol. 5. 3, pp. 403–443, 2010.