Efficient Carpooling and Toll Pricing for Autonomous Transportation
EEfficient Carpooling and Toll Pricing for AutonomousTransportation
Saurabh Amin, Patrick Jaillet, Manxi Wu ∗ Abstract
In this paper, we address the existence and computation of competitive equilibriumin the transportation market for autonomous carpooling first proposed by [18]. Atequilibrium, the market organizes carpooled trips over a transportation network in asocially optimal manner and sets the corresponding payments for individual riders andtoll prices on edges. The market outcome ensures individual rationality, stability ofcarpooled trips, budget balance, and market clearing properties under heterogeneousrider preferences. We show that the question of market’s existence can be resolved byproving the existence of an integer optimal solution of a linear programming problem.We characterize conditions on the network topology and riders’ disutility for carpoolingunder which a market equilibrium can be computed in polynomial time. This char-acterization relies on ideas from the theory of combinatorial auctions and minimumcost network flow problem. Finally, we characterize a market equilibrium that achievesstrategyproofness and maximizes welfare of individual riders.
Autonomous transportation has the potential to significantly transform urban mobility whenthe technology becomes mature enough for real-world deployment. A significant fleet ofdriverless cars could be utilized to organize carpooled trips at a much cheaper price and ina more flexible manner relative to the current mobility services that rely on human drivers.Naturally, this technology would reshape the riders’ incentives to make trips and sharecars. Whether autonomous driving technology will relieve or aggravate congestion cruciallydepends on how riders will be incentivized to participate in efficient carpooled trips that areconstrained by socially optimal tolls. Thus, to fully exploit the potential of self-driving cars,we need to address the complementarity between efficient carpooling and optimal tolling forriders with heterogeneous preferences. ∗ S. Amin is with the Laboratory for Information and Decision Systems, P. Jaillet is with the Departmentof Electrical Engineering and Computer Science, Laboratory for Information and Decision Systems, andOperations Research Center, M. Wu is with the Institute for Data, Systems, and Society, MassachusettsInstitute of Technology (MIT), Cambridge, MA, USA, { amins, jaillet, manxiwu } @mit.edu For example, when toll prices are zero on all roads, all riders will choose to take the shortest route inthe network, and the traffic load will exceed the capacity. As the toll prices of edges on this route increase,riders will be incentivized to take carpooled trips in order to split the toll prices (or switch to longer routes). a r X i v : . [ c s . G T ] F e b n [18], the authors introduced a competitive market model to study riders’ incentives toparticipate in autonomous carpooled trips and share the road capacity in a socially optimalmanner. In this model, the transportation authority sets toll prices on edges, and ridersorganize carpooled trips and make payments to split the toll prices and trip costs. An out-come is defined by organized trips, riders’ payments, and edge tolls. A market equilibrium isdefined as an outcome that satisfies the following conditions: individual rationality, stability,budget balance, and market clearing . The authors show that a market equilibrium may notalways exist, but when it does the equilibrium carpooled trips are socially optimal . Thisresult essentially follows from the first welfare theorem for competitive markets. Our goalin this paper is to address the question of existence of market equilibrium and provide toolsto compute a desirable equilibrium.Building on [18], we consider that each rider’s value of carpooled trips is equal to thevalue of completed trip, minus the travel time cost and a disutility from carpooling thatdepends on the size of rider group in the carpool. We make three contributions for thissetting: (1) We derive sufficient conditions under which market equilibrium exists; (2) Weprovide a computational approach to efficiently compute equilibrium outcomes; and (3) Wecharacterize an equilibrium in which riders truthfully report their preferences to a neutralplatform that facilitates market implementation.Market equilibrium is challenging to analyze because trip organization is essentially acoalition formation problem, in which the riders form carpooled groups and split paymentsin a manner which ensures that toll prices clear the market. Both trip organization and tollpricing are crucially influenced by the network topology since any trip on a certain routeconsumes a unit capacity of all edges in that route. Consequently, the toll price on an edgecan impact the usage of all edges on the route. The classical methods in mechanism designand coalition games cannot be readily applied to address these features. To address thischallenge, we develop a new approach that draw ideas from combinatorial auction theoryand network flow optimization.We now discuss our approach and main results. In Sec. 3, we analyze the linear program-ming relaxation of the optimal trip organization problem and its dual program. We find thatthe problem of existence of market equilibrium can be equivalently posed as the problemof existence of an integer optimal solution for the relaxed linear program. Consequently,our goal becomes that of finding conditions under which the primal linear program has anoptimal integer solution. Moreover, when these conditions hold, by strong duality of linearprogramming, optimal solutions of the primal and dual programs provide us an equilibriumoutcome of the market.We show that when the network is series-parallel and riders have homogeneous levelsof carpool disutility, the primal program is guaranteed to have an integer optimal solution,and thus market equilibrium exists (Sec. 4). The condition that the network is series-parallel allows us to compute a set of routes with integer capacities such that the optimaltrip organization in the sub-network formed by these routes is also optimal for the originalnetwork. These routes can be computed by a greedy algorithm that selects routes in theincreasing order of travel time and greedily allocates network capacity to them. Intuitively,the algorithm select routes in a manner that minimizes the total travel time and carpooldisutility costs for all trips on a series-parallel network. This intuition however does notapply to non-series-parallel networks. In fact, we provide an example to demonstrate that2arket equilibrium may not exist in a Wheatstone network.On the other hand, the condition that riders have homogeneous carpool disutility allowsus to augment the trip value function of each route into a function that is monotonic in ridergroups and satisfies the gross substitutes condition ([11, 10, 15]). The augmented trip valuefunction and the route set obtained from aforementioned greedy algorithm can be used todefine an equivalent “economy”, in which the riders are viewed as “indivisible goods” andeach unit capacity on the routes is viewed as an “agent”. Using this definition, we showthat the existence of market equilibrium on the sub-network is mathematically equivalent tothe existence of a Walrasian equilibrium in the economy, which is guaranteed by the grosssubstitutes condition [14].The issue of equilibrium computation in the autonomous carpooling market is addressedin Sec. 5. Our approach for computing optimal carpooled trips takes two steps: firstly, wecompute the set of optimal routes using the greedy algorithm; and secondly, we compute theoptimal trips on these routes by using well-known Kelso-Crawford algorithm that providesoptimal good allocation in Walrasian equilibrium of the equivalent economy. Moreover,riders’ equilibrium utilities and toll prices can be computed from the dual linear programusing a separation-based method that also relies on the gross substitutes condition.Finally, we identify a particular market equilibrium under which riders truthfully reporttheir preferences to a platform (in our context, this is a a neutral entity which facilitatesthe market implementation). We find that in this equilibrium, riders’ payments are equal totheir externalities on other riders, and hence are equivalent to the payments in the classicalVickery-Clark-Grove mechanism. This equilibrium also has the advantage of achieving thehighest rider utilities among all market equilibria, and only collecting the minimum totaltoll prices. Related literature
Autonomous vehicle market design and competition.
The paper [21] studied theimpact of competition between two ride-hailing platforms on their choices of autonomousvehicle fleet sizes, prices and wages of human drivers. The authors of [16] studied the pricesin ride-hailing markets, where an uncertain aggregate demand is served by a fixed fleet ofautonomous vehicles and elastic supply of human drivers. They argue that the only designthat unambiguously reduces the service prices corresponds to the setting when the provisionof autonomous carpooled trips occurs in a competitive environment. This finding aligns wellwith our focus on a competitive autonomous carpooling market. We show that by exploitingthe complementarity between carpooling and road pricing, we can achieve an equilibriumoutcome that is socially optimal (when sufficient conditions for equilibrium existence aresatisfied).
Human-driven ride-hailing platforms.
A rich body of literature exists on matchingand pricing schemes in ride-hailing platforms that rely on supply of human-driven cars.These work includes online matching ([2, 19]), dynamic and spatial pricing ([3, 8, 9, 7]),and stochastic control and queuing ([12, 1, 4]). A key challenge in these problems comesfrom the two-sided nature of matching between riders and human drivers. In contrast, theautonomous carpooling market that we consider focuses on forming carpooling groups amongriders with heterogeneous preferences with constraints imposed by car size, route capacity,3dge tolls, and network structure.
Consider a traffic network modeled as a directed graph with a single origin-destination pair.The set of edges in the network is E , and the capacity of each edge e ∈ E is a positive integer q e ∈ N + . The set of routes is R , where each route r ∈ R is a sequence of edges that form adirected path from the origin to the destination. We denote the travel time of each edge e as t e >
0, and the travel time of each route r as t r = (cid:80) e ∈ r t e . A finite set of riders m = 1 , . . . , M want to take autonomous carpool trips to travelfrom the origin to the destination. A trip is defined as a tuple ( b, r ), where b is the groupof riders taking route r during the trip. The maximum number of riders in any groupmust be below the capacity of individual car, denoted A . Thus, the set of rider groups is B ∆ = (cid:8) M | | b | < A (cid:9) , and the set of trips is ( b, r ) ∈ B × R . If the group b in a trip ( b, r ) isa singleton set { m } , then rider m takes a solo trip on route r . Otherwise riders in b share apooled trip. Each trip ( b, r ) occupies a unit capacity for all edges in route r .The value of each trip ( b, r ) for a rider m ∈ b , denoted as v mr ( b ), is given by: v mr ( b ) = α m − β m · t r − γ m ( | b | ) · t r , ∀ b ∈ { B | b (cid:51) m } , ∀ m ∈ M, ∀ r ∈ R. (1)Thus, riders have heterogeneous trip values: The parameter α m is rider m ’s value of arrivingat the destination, β m is rider m ’s value of time, and γ m ( | b | ) is rider m ’s disutility of sharingthe pooled trip with rider group of size | b | for a unit travel time. That is, rider m ’s value ofeach trip ( b, r ) equals to their value of arriving at the destination nets the cost of trip timeand the carpool disutility.The carpool disutility γ m ( | b | ) represents the rider m ’s inconvenience of sharing the vehiclewith other riders in the carpool group, potentially due to the need to share space withothers and time spent on taking detours and walking to pick-up location. This disutilityonly depends on the group sizes rather than the identity of riders in the group, riders’ valuesare identical for any two trips ( b, r ) and ( b (cid:48) , r ) with the same group sizes (i.e. | b | = | b (cid:48) | ) andthe same route r . We consider that the carpool disutility γ m ( | b | ) ≥ | b | = 1 , . . . , A ,and the disutility of solo trip is zero, i.e. γ m (1) = 0 for all m ∈ M . Thus, all riders prefer totake solo trips rather than pooling with other riders. Additionally, the marginal disutility γ m ( | b | ) − γ m ( | b | −
1) is non-decreasing in the group size | b | for all | b | = 2 , . . . , A , i.e. the Thus, in our setting, each edge has an L-shaped cost function: cost is a constant when the edge load isbelow the edge capacity, and becomes extremely high once the load exceeds capacity. In the context of trafficcongestion: when the traffic load is below the road capacity, all vehicles pass through the segment at thefree-flow speed. However, when the traffic load exceeds the capacity, the travel time significantly increasesdue to congestion. In our market mechanism, the toll prices are set to ensure that the load of each edge doesnot exceed its capacity. All individuals in the set b of an autonomous carpool trip are riders. On the other hand, in human-drivencarpool trips, we need to designate a driver in the set b , and match riders with drivers. For simplicity, we assume that cars are of homogeneous capacity. b, r ) is non-decreasing in the originaltrip size | b | .The cost of each trip includes the fuel charge and the cost of car’s wear and tear. Wesimply assume that the cost of each trip ( b, r ) ∈ B × R is c r ( b ) = δ | b | t r , and δ ≥ b, r ) is the summation of the trip values for riders in b netsthe cost of trip: V r ( b ) = (cid:88) m ∈ b v mr ( b ) − c r ( b ) = (cid:88) m ∈ b α m − (cid:88) m ∈ b β m t r − (cid:88) m ∈ b γ m ( | b | ) t r − δ | b | t r , ∀ b ∈ B, ∀ r ∈ R. (2) We now discuss how an efficient autonomous carpooling market can be organized. A trans-portation authority sets non-negative toll prices τ = ( τ e ) e ∈ E ∈ R | E |≥ on edges in the network,where τ e is the toll price of edge e . Riders form carpool trips. The trip vector is a binary vec-tor x = ( x r ( b )) r ∈ R,b ∈ B ∈ { , } | B |×| R | , where x r ( b ) = 1 if trip ( b, r ) is organized and x r ( b ) = 0if otherwise. A trip vector x must satisfy the following feasibility constraints: (cid:88) r ∈ R (cid:88) b (cid:51) m x r ( b ) ≤ , ∀ m ∈ M, (3a) (cid:88) r (cid:51) e (cid:88) b ∈ B x r ( b ) ≤ q e , ∀ e ∈ E, (3b) x r ( b ) ∈ { , } , ∀ b ∈ B, ∀ r ∈ R, (3c)where (3a) ensures that no rider takes more than 1 trip, and (3b) ensures that the totalnumber of trips that use any edge e ∈ E does not exceed the edge capacity.Additionally, each rider m ∈ M makes a payment p m for covering the cost of their tripand the toll prices of the taken edges. The payment vector is p = ( p m ) m ∈ M .An outcome of the carpooling market is represented by the tuple ( x, p, τ ). Given any( x, p, τ ), the utility of each rider m ∈ M equals to the value of the trip that m takes minusthe payment: u m = (cid:88) r ∈ R (cid:88) b (cid:51) m v mr ( b ) x r ( b ) − p m , ∀ m ∈ M. (4)We next define four properties of the market outcomes, namely individual rationality , stability , budget balance , and market clearing . Firstly, an outcome ( x, p, τ ) is individuallyrational if riders’ utilities are non-negative: u m ≥ , ∀ m ∈ M. (5)That is, no rider has an incentive to opt-out of the market.Secondly, an outcome ( x, p, τ ) is stable if there is no rider group in B that can gain higherutilities by organizing trips that are not included in x . Note that the total utility of all riders5n any group b for organizing a trip ( b, r ) cannot exceed the value of the trip minus the tollprice for route r , i.e. V r ( b ) − (cid:80) e ∈ r τ e . Thus, a stable market outcome ( x, p, τ ) requires thatthe total utilities of riders in b obtained using (4) is higher or equal to the total utility thatcan be obtained from any feasible trip ( b, r ): (cid:88) m ∈ b u m ≥ V r ( b ) − (cid:88) e ∈ r τ e , ∀ b ∈ B, ∀ r ∈ R. (6)Thirdly, an outcome ( x, p, τ ) is budget balanced if the total payments of each organizedtrip is equal to the sum of the toll prices and the cost of the trip; and moreover a rider’spayment is zero if they are not part of any organized trip, i.e. x r ( b ) = 1 , ⇒ (cid:88) m ∈ b p m = (cid:88) e ∈ r τ e + c r ( b ) , ∀ b ∈ B, ∀ r ∈ R, (7a) x r ( b ) = 0 , ∀ r ∈ R, ∀ b (cid:51) m, ⇒ p m = 0 , ∀ m ∈ M. (7b)Fourthly, an outcome ( x, p, τ ) is market-clearing if there are zero tolls on all edges whosecapacity limits are not met: (cid:88) r (cid:51) e (cid:88) b ∈ B x r ( b ) < q e , ⇒ τ e = 0 , ∀ e ∈ E. (8)We define market equilibrium as an outcome that satisfies all four properties: Definition 1
A market outcome ( x ∗ , p ∗ , τ ∗ ) is an equilibrium if it is individually rational,stable, budget balanced and market clearing. The autonomous carpooling market assumes a competitive environment in that riders arefree to join any trip and occupies a unit capacity on any route as long as their total paymentscover the trip cost and toll prices. From an implementation viewpoint, the process of triporganization and payment can be facilitated by introducing a market platform. In such animplementation, each rider m ∈ M reports their preference parameters (cid:16) α m , β m , ( γ m ( d )) Ad =1 (cid:17) to the platform, and the platform assigns riders to trips according to the trip vector x ∗ .Then, riders make payments according to p ∗ to the platform, and the platform pays for thetoll prices τ ∗ and trip costs on the riders’ behalf. When the vector ( x ∗ , p ∗ , τ ∗ ) is a marketequilibrium, riders follow the trip assigned by the platform, the payments cover the tollprices and trip costs, and toll prices are non-zero only on edges where the load meets thecapacity. In paper [18], the authors argued that such a transportation market can be mappedinto a standard competitive market, where the market equilibrium defined in Definition 1 is A stable market outcome ( x, p, τ ) is Pareto optimal in that no rider’s utility can be improved by orga-nizing different trips that are not in x without decreasing the utilities of other riders. For simplicity, we assume that this platform is a simple non-strategic market mediator and does notcharge a fee for organizing trips. However, a non-negative constant fee can be added to the model withoutchanging the results. The computed market equilibrium depends on the reported preference parameters ( α, β, γ ). For simplic-ity, we drop the dependence of ( x ∗ , p ∗ , τ ∗ ) with respect to these parameters in notation. x ∗ , p ∗ , τ ∗ ) exists, from the first welfaretheorem, we conclude that the trip vector x ∗ necessarily maximizes the total social welfare(Theorem 1 in [18]); i.e., x ∗ is an optimal solution of the following optimal trip organizationproblem: max x S ( x ) = (cid:88) b ∈ B (cid:88) r ∈ R V r ( b ) x r ( b ) s.t. x satisfies (3a) – (3c), (IP)where S ( x ) is the social welfare of all trips given by x . In this section, we show that there exists a market equilibrium if and only if the linearrelaxation of the optimal trip organization problem (IP) has integer optimal solutions. Wealso show that the equilibrium outcomes can be derived from the optimal solutions from thelinear relaxation and its dual program. We first introduce the linear relaxation of (IP) and its dual formulation. The primallinear program is as follows:max x S ( x ) = (cid:88) b ∈ B (cid:88) r ∈ R V r ( b ) x r ( b ) ,s.t. (cid:88) r ∈ R (cid:88) b (cid:51) m x r ( b ) ≤ , ∀ m ∈ M, (LP.a) (cid:88) r (cid:51) e (cid:88) b ∈ B x r ( b ) ≤ q e , ∀ e ∈ E, (LP.b) x r ( b ) ≥ , ∀ b ∈ B, ∀ r ∈ R. (LP.c)Note that the constraint x r ( b ) ≤ u = ( u m ) m ∈ M for constraints (LP.a) and τ = ( τ e ) e ∈ E forconstraints (LP.b), the dual program of (LP) can be written as follows:min u,τ U ( u, τ ) = (cid:88) m ∈ M u m + (cid:88) e ∈ E q e τ e s.t. (cid:88) m ∈ b u m + (cid:88) e ∈ r τ e ≥ V r ( b ) , ∀ b ∈ B, ∀ r ∈ R, (D.a) u m ≥ , τ e ≥ , ∀ m ∈ M, ∀ e ∈ E. (D.b) Theorem 1
A market equilibrium ( x ∗ , p ∗ , τ ∗ ) exists if and only if (LP) has an optimalinteger solution. Any optimal integer solution x ∗ of (LP) is an equilibrium trip vector, and All results in this section hold for arbitrary trip values V = ( V r ( b )) b ∈ B,r ∈ R . ny optimal solution ( u ∗ , τ ∗ ) of (D) is an equilibrium utility vector and an equilibrium tollvector. The equilibrium price vector p ∗ is given by: p m ∗ = (cid:88) r ∈ R (cid:88) b (cid:51) m x ∗ r ( b ) v mr ( b ) − u m , ∀ m ∈ M. (11)Thus, the question of existence of market equilibrium is equivalent to resolving whetherthere exists an integer optimal solution for the LP relaxation of the optimal trip problem.This result follows from the fact that the four properties of market equilibrium, namelyindividual rationality, stability, budget balance, and market clearing are equivalent to theconstraints of (LP) and (D), and the complementary slackness conditions. From strongduality, a market equilibrium exists if and only if the optimality gap between the linearrelaxation (LP) and the integer problem (IP) is zero. Hence, the linear relaxation (LP) musthave an integer optimal solution, which is the equilibrium trip vector x ∗ .Theorem 1 turns the problem of finding sufficient conditions on the existence of marketequilibrium to finding conditions under which (LP) has optimal integer solutions. Moreover,it enables us to compute market equilibrium as optimal solutions of (LP) and (D).As a consequence, we obtain that the total toll prices of shorter routes (routes with lowertravel time) must be no less than that of the longer ones (routes with higher travel time). Corollary 1
In any market equilibrium ( x ∗ , p ∗ , τ ∗ ) , for any r, r (cid:48) ∈ R such that t r ≥ t r (cid:48) , (cid:80) e ∈ r τ ∗ e ≤ (cid:80) e ∈ r (cid:48) τ ∗ e . This result is intuitive since for all rider groups, taking a shorter route results in a highertrip value than taking a longer route. Therefore, the toll price (which is charged per unitcapacity) of shorter routes must be no less than that of longer routes.
We characterize the sufficient conditions on network topology and trip values under whichthe there exists a market equilibrium. We first present an example when market equilibriumdoes not exist on a wheatstone network.
Example 1
Consider the wheatstone network as in Fig. 1. The capacity of each edge inthe set { e , e , e , e } is 1, and the capacity of edge e is 4. The travel time of each edge isgiven by t = 1, t = 3, t = 3, t = 1, and t = 0.The maximum capacity of vehicle is A = 2. Three riders m = 1 , , α m = 7, value of time β m = 1, zero carpool disutility, i.e. γ m ( d ) = 0 for any d = 1 , m ∈ M , and zerotrip cost parameter, i.e. δ = 0.We define the route e - e as r , e - e - e as r , and e - e as r . Then, trip values are: V ( m ) = V ( m ) = 3, and V ( m ) = 5 for all m ∈ M ; V ( m, m (cid:48) ) = V ( m, m (cid:48) ) = 6, and V ( m, m (cid:48) ) = 10 for all m, m (cid:48) ∈ M . The unique optimal solution of the linear program (LP)on this network is x ∗ (1 ,
2) = x ∗ (2 ,
3) = x ∗ (1 ,
3) = 0 .
5, and S ( x ∗ ) = 11. That is, (LP) doesnot have an integer optimal solution, and market equilibrium does not exist (Theorem 1).8igure 1: Wheatstone networkWe define a network to be series-parallel if a Wheatstone structure as in Example 1 isnot embedded. Definition 2 (Series-Parallel (SP) Network [17])
A network is series-parallel if theredo not exist two routes that pass through an edge in opposite directions. Equivalently, anetwork is series-parallel if and only if it is constructed by connecting two series-parallelnetworks either in series or in parallel for finitely many iterations.
Our next theorem shows that market equilibrium is guaranteed to exist if the network isseries-parallel (i.e. the Wheatstone structure is not embedded) and riders have homogeneouscarpool disutilities.
Theorem 2
Market equilibrium ( x ∗ , p ∗ , τ ∗ ) exists if the network is series-parallel and allriders have identical carpool disutility parameters, i.e. γ m ( d ) = γ ( d ) , ∀ d = 1 , . . . , A, ∀ m ∈ M. (12)Recall from Theorem 1 that showing the existence of market equilibrium is equivalent toproving that (LP) has an integer optimal solution. Our proof of Theorem 2 has three parts:Firstly, we compute an integer route capacity vector k ∗ = ( k ∗ r ) r ∈ R , where R ∗ ∆ = { R | k ∗ r > } is the set of routes that are assigned positive capacity and k ∗ r is the integer capacity of eachroute r . We show that when the network is series-parallel, any optimal trip vector for thesub-network with routes R ∗ and capacity vector k ∗ is also an optimal trip vector for theoriginal network (Lemma 1). Thus, to prove Theorem 2, we only need to show that thereexists an optimal integer solution of trip organization on the sub-network with capacityvector k ∗ . Secondly, we argue that mathematically the problem of trip organization on thesub-network with capacity vector k ∗ can be viewed as a problem of allocating goods in aneconomy with indivisible goods, and the existence of integer optimal solution is equivalentto the existence of Walrasian equilibrium in the economy (Lemmas 2 – 3). Finally, we showthat when riders have homogeneous carpool disutility parameters, the trip value functionssatisfy gross substitutes condition. This condition is sufficient to ensure the existence ofWalrasian equilibrium in the equivalent economy (Lemmas 2 – 6). These three parts ensurethat the trip organization problem on the sub-network with capacity vector k ∗ has an integeroptimal solution, and this solution is also an integer optimal solution of (LP). We can thusconclude that a market equilibrium exists.The rest of this section elaborates on these ideas and presents the lemmas correspondingto each of the three parts. The proofs of these lemmas are included in Appendix A.9art 1. We first compute the route capacity vector k ∗ by a greedy algorithm (Algorithm 1).The algorithm begins with finding a shortest route of the network r min , and sets its capacityas k ∗ r min = min e ∈ r min q e , which is the maximum possible capacity that can be allocated to r min . After allocating the capacity k ∗ r min to route r min , the residual capacity of each edgeon r min is reduced by k ∗ r min . We then repeat the process of finding the next shortest routeand allocating the maximum possible capacity to that route until there exists no route withpositive residual capacity in the network.Note that in each step of Algorithm 1, the capacity of at least one edge is fully allocatedto the route that is chosen in that step. Therefore, the algorithm must terminate in less than | E | number of steps. The algorithm returns the capacity vector k ∗ , where R ∗ = { R | k ∗ r > } isthe set of routes allocated with positive capacity, and the capacity of each r ∈ R ∗ is k ∗ r . Theremaining routes in R \ R ∗ are set with zero capacity. Since the network is series-parallel, thetotal capacity given by the output of the greedy algorithm equals to the network capacity C ([5]), i.e. (cid:80) r ∈ R ∗ k ∗ r = C .Moreover, the shortest path of the network in each step can be computed by Dijkstra’salgorithm with time complexity of O ( | N | ), where | N | is the number of nodes in the network.Therefore, Algorithm 1 has time complexity of O ( | N | | E | ). Algorithm 1:
Greedy algorithm for computing route capacity
Initialize:
Set ˜ q e ← q e , ∀ e ∈ E ; k r ← , ∀ r ∈ R ; ˜ E ← E ;( t min , r min ) ← ShortestRoute ( ˜ E ); while t min < ∞ do k ∗ r min ← min e ∈ r min ˜ q e ; for e ∈ r min do ˜ q e ← ˜ q e − k ∗ r min ; if ˜ q e = 0 then ˜ E ← ˜ E \ { e } ; endend ( t min , r min ) ← ShortestRoute ( ˜ E ); endReturn k ∗ Next, we consider the sub-network comprised of routes in R ∗ with corresponding routecapacities given by k ∗ . Analogous to (LP), the linear relaxation of optimal trip organizationproblem on this sub-network is given by:max x S ( x ) = (cid:88) b ∈ B (cid:88) r ∈ R V r ( b ) x r ( b ) ,s.t. (cid:88) r ∈ R (cid:88) b (cid:51) m x r ( b ) ≤ , ∀ m ∈ M, (LP k ∗ .a) (cid:88) b ∈ B x r ( b ) ≤ k ∗ r , ∀ r ∈ R, (LP k ∗ .b) x r ( b ) ≥ , ∀ b ∈ B, ∀ r ∈ R, (LP k ∗ .c)10here (LP k ∗ .a) ensures that each rider is in at most one trip, and (LP k ∗ .b) ensures that thetotal number of trips in each route r does not exceed the route capacity k ∗ r given by k ∗ . Lemma 1
If the network is series-parallel, then any optimal solution of (LP k ∗ ) is an opti-mal solution of (LP) . To prove Lemma 1, we first prove that any feasible solution of (LP k ∗ ) is also a feasiblesolution of (LP) by showing that the capacity vector k ∗ computed from Algorithm 1 satisfies (cid:80) r (cid:51) e k ∗ r ≤ q e for all e ∈ E . Thus, the optimal value of (LP k ∗ ) is no higher than that of(LP).Next, we argue that for a series-parallel network, the optimal value of (LP k ∗ ) is no lessthan that of (LP); hence, any optimal solution of (LP k ∗ ) must also be an optimal solutionof (LP). To prove this argument, we show that for any optimal solution ˆ x ∗ of (LP), we canconstruct another trip vector x ∗ such that x ∗ is feasible in (LP k ∗ ), and S ( x ∗ ) ≥ S (ˆ x ∗ ). Such avector x ∗ can be constructed from ˆ x ∗ by re-assigning rider groups ˆ b ∈ ˆ B ∆ = { B | (cid:80) r ∈ R ˆ x r (ˆ b ) > } – the set of rider groups with positive weights in ˆ x – to routes in k ∗ . Then, for eachˆ b ∈ ˆ B , the trip value (cid:16) ˆ b, r (cid:17) as in (2) can be written as V r (ˆ b ) = (cid:80) m ∈ ˆ b α m − g (ˆ b ) t r , where g (ˆ b ) = (cid:80) m ∈ ˆ b β m + (cid:80) m ∈ ˆ b γ m ( | ˆ b | ) + δ | ˆ b | is each group ˆ b ’s sensitivity to route travel time.Moreover, we define the weight of each group ˆ b under the vector ˆ x ∗ as f (ˆ b ) = (cid:80) r ∈ R ˆ x ∗ r (ˆ b ).To construct the new vector x ∗ , we start with re-assigning weights of the rider groups in ˆ B one-by-one in decreasing order of their sensitivities to the shortest route in R ∗ until thecapacity of the shortest route given k ∗ is fully utilized. Then, we proceed to assign theweights of the remaining rider groups in ˆ B to the second shortest route in R ∗ . This processis repeated until either all weights of rider groups in ˆ B are re-assigned to routes or all routes’capacities in k ∗ are used-up. Since the total weight of ˆ x is less than or equal to the networkcapacity C , and the total capacity given by k ∗ equals to C , all weights of ˆ B given by ˆ x mustget assigned to routes in R ∗ when the algorithm terminates. Additionally, the constructedtrip vector x ∗ is a feasible solution of (LP k ∗ ).This re-assignment process enables rider groups with higher sensitivity of travel time totake shorter routes. This ensures that the constructed x ∗ satisfies the inequality S ( x ∗ ) ≥ S (ˆ x ) when the network is series-parallel. We prove this by mathematical induction: First, S ( x ∗ ) ≥ S (ˆ x ) holds trivially on any single link network. Second, if this inequality holds onany two series-parallel networks, then it also holds on the network that is constructed byconnecting the two sub-networks in series or in parallel. Since any series-parallel networkis constructed by connecting single link networks in series or in parallel for a finite numbertimes, S ( x ∗ ) ≥ S (ˆ x ) must hold for any series-parallel network. Hence, we can conclude thatthe optimal value of (LP k ∗ ) is no less than that of (LP), and any optimal solution of (LP k ∗ )must also be an optimal solution of (LP).In part 1, Lemma 1 ensures that if (LP k ∗ ) has an integer optimal solution, then thatsolution must be an optimal integer solution of (LP). It remains to show that (LP k ∗ ) indeedhas an integer optimal solution.Part 2. In this part, we first construct an augmented trip value function that is monotonicin the rider group. Then, we construct an auxiliary network comprised of parallel routeswith unit capacities based on the set of routes given by k ∗ . We show that (LP k ∗ ) has an11nteger optimal solution if and only if the linear relaxation of the trip organization problemon the auxiliary network with the augmented value function has integer optimal solution.Moreover, the trip organization problem on the auxiliary network with the augmented valuefunction is equivalent to an allocation problem in an economy with indivisible goods. Theexistence of optimal integer solution is equivalent to the existence of Walrasian equilibriumin this economy.To begin with, we introduce the definition of monotonic trip value function as follows: Definition 3 (Monotonicity)
For each r ∈ R , the trip value function V r is monotonic iffor any b, b (cid:48) ∈ B , V r ( b ∪ b (cid:48) ) ≥ V r ( b ) . Monotonicity condition requires that adding any rider group b (cid:48) to a trip ( b, r ) does not reducethe trip’s value. The monotonicity condition may not be always satisfied in general becauseof two reasons: First, if the size of riders | b ∪ b (cid:48) | > A , then the trip ( b ∪ b (cid:48) , r ) is infeasible,and the trip value is not defined. Second, even when | b ∪ b (cid:48) | ≤ A , the value V r ( b ∪ b (cid:48) ) maybe less than V r ( b ) when the carpool disutility is sufficiently high.We augment V : B × R → N to a monotonic value function V : ¯ B × R → N , where¯ B ∆ = 2 M is the set of all rider subsets (including the rider subsets with sizes larger than A).The value of V r (¯ b ) can be written as follows: V r (¯ b ) ∆ = max b ⊆ ¯ b, b ∈ B V r ( b ) , ∀ r ∈ R, ∀ ¯ b ∈ ¯ B. (14)That is, the value of any rider group ¯ b ∈ ¯ B on route r equals to the maximum value of afeasible trip ( b, r ) where rider group b is a subset of ¯ b . The augmented value function V satisfies the monotonicity condition.We refer h r (¯ b ) ∆ = arg max b ⊆ ¯ b, b ∈ B V r ( b ) as the representative rider group of ¯ b for route r .From (2), we can re-write the augmented trip value function V as a linear function of traveltime: V r (¯ b ) = (cid:88) m ∈ h r (¯ b ) α m − (cid:88) m ∈ h r (¯ b ) β m t r − (cid:88) m ∈ h r (¯ b ) γ m ( | h r (¯ b ) | ) t r − δ | h r (¯ b ) | t r , ∀ ¯ b ∈ ¯ B, ∀ r ∈ R. (15)Next, we construct an auxiliary network given the set of routes R ∗ with capacity vector k ∗ output from Algorithm 1. Specifically, we convert each route r ∈ R ∗ with integer capacity k ∗ r to the same number of parallel routes each with a unit capacity in the auxiliary network.We denote the route set of the auxiliary network as L = ∪ r ∈ R ∗ L r , where each set L r is theset of routes converted from route r in the original network.We now consider the trip organization problem on the auxiliary network with the aug-mented trip value function. For each l ∈ L and each ¯ b ∈ ¯ B , we define (cid:0) ¯ b, l (cid:1) as an augmentedtrip. In this trip, the rider group h r (¯ b ) takes route l of the auxiliary network, while theremaining riders m ∈ ¯ b \ h r (¯ b ) are not included in the trip. We denote the augmentedtrip vector as y = (cid:0) y l (¯ b ) (cid:1) ¯ b ∈ ¯ B,l ∈ K ∈ { , } | ¯ B |× L , where y l (¯ b ) = 1 if the augmented trip (cid:0) ¯ b, l (cid:1) is organized, and y l (¯ b ) = 0 if otherwise. The value of the augmented trip is defined as W l (¯ b ) = V r (¯ b ) for any ¯ b ∈ ¯ B , any l ∈ L r and any r ∈ R ∗ .For any y ∈ { , } | ¯ B |× L , we can compute a trip vector for the original optimal triporganization problem x = χ ( y ) ∈ { , } | B |× R such that the actually organized trips given12y x = χ ( y ) are the same as that given by y . In particular, for each route r ∈ R ∗ , andeach augmented trip (cid:0) ¯ b, l (cid:1) ∈ ¯ B × L r such that y l (¯ b ) = 1, we choose a representative ridergroup ˆ b ∈ h r (¯ b ) and set x r (ˆ b ) = 1 for the original trip (cid:16) ˆ b, r (cid:17) that represents the organizedaugmented trip (cid:0) ¯ b, l (cid:1) . We set x r ( b ) = 0 for all other trips. The trip vector x = χ ( y ) can bewritten as follows: ∀ r ∈ R ∗ , ∀ (cid:0) ¯ b, l (cid:1) s.t.y l (¯ b ) = 1 , ∃ ˆ b ∈ h r (¯ b ) , s.t. x r (ˆ b ) = 1 , and x r ( b ) = 0 , ∀ b ∈ B \ { ˆ b } (16)Hence, we write the linear relaxation of optimal trip organization problem on the auxiliarynetwork with the augmented trip value function as follows:max y S ( y ) = (cid:88) ¯ b ∈ ¯ B (cid:88) l ∈ L W l (¯ b ) y l (¯ b ) ,s.t. (cid:88) l ∈ L (cid:88) ¯ b (cid:51) m y l (¯ b ) ≤ , ∀ m ∈ M, (LP-y.a) (cid:88) ¯ b ∈ ¯ B y l (¯ b ) ≤ , ∀ l ∈ L, (LP-y.b) y l (¯ b ) ≥ , ∀ ¯ b ∈ ¯ B, ∀ l ∈ L, (LP-y.c) Lemma 2
The linear program (LP k ∗ ) has an integer optimal solution if and only if (LP-y) has an integer optimal solution. Moreover, if y ∗ is an integer optimal solution of (LP-y) ,then x ∗ = χ ( y ∗ ) as in (16) is an optimal integer solution of (LP k ∗ ) . This lemma shows that finding an optimal integer solution of (LP k ∗ ) is equivalent tofinding an optimal integer solution of (LP-y).We finally show that the augmented trip organization problem is mathematically equiva-lent to an economy G with indivisible goods, and the existence of market equilibrium in ourcarpooling market is equivalent to the existence of Walrasian equilibrium of the economy.In G , the set of indivisible “goods” is the rider set M and the set of agents is the route set L in the auxiliary network. Each agent l ’s value of any good bundle ¯ b ∈ ¯ B is equivalentto the augmented trip value function W l (¯ b ). Moreover, each good m ’s price is equivalent torider m ’s utility u m . The vector of good allocation is y , where y l (¯ b ) = 1 if good bundle ¯ b is allocated to agent l . Given any y , for each l ∈ L , we denote the bundle of goods that isallocated to l as ¯ b l , i.e. y l (¯ b l ) = 1. If no good is allocated to l (i.e. (cid:80) ¯ b ∈ ¯ B y l (¯ b ) = 0), then¯ b l = ∅ . The Walrasian equilibrium of economy G is defined as follows: Definition 4 (Walrasian equilibrium [14])
A tuple ( y ∗ , u ∗ ) is a Walrasian equilibriumif (i) For any l ∈ L , ¯ b l ∈ arg max ¯ b ∈ ¯ B W l (¯ b ) − (cid:80) m ∈ ¯ b l u m , where ¯ b l is the good bundle that isallocated to l given y ∗ (ii) For any m ∈ M that is not allocated to any agent, (i.e. (cid:80) l ∈ L (cid:80) ¯ b (cid:51) m y ∗ l (¯ b ) = 0 ), u m ∗ = 0 .
13n fact, we can show that (LP-y) has integer optimal solution if and only if Walrasianequilibrium exists in this equivalent economy:
Lemma 3
The linear program (LP-y) has integer optimal solution if and only if a Walrasianequilibrium ( y ∗ , u ∗ ) exists in the equivalent economy. Furthermore, y ∗ is an integer optimalsolution of (LP-y) , and x ∗ = χ ( y ∗ ) as in (16) is an optimal integer solution of (LP k ∗ ) . In part 2, from Lemmas 2 – 3, we turn the problem of proving the existence of inte-ger optimal solution in (LP k ∗ ) to proving that the equivalent economy G has Walrasianequilibrium.Part 3. In this final part, we show that if the carpool disutility parameter γ m is homogeneousacross all m ∈ M , then Walrasian equilibrium exists in the economy G constructed in Part2. To begin with, we introduce the following definition of gross substitutes condition onthe augmented value function V . In this definition, we utilize the notion of marginal valuefunction V r (¯ b (cid:48) | ¯ b ) = V r (¯ b ∪ ¯ b (cid:48) ) − V r (¯ b ) for all r ∈ R and all ¯ b, ¯ b (cid:48) ⊆ M . Definition 5 (Gross Substitutes [20])
For each r ∈ R , the augmented trip value func-tion V r is said to satisfy gross substitutes condition if(a) For any ¯ b, ¯ b (cid:48) ⊆ ¯ B such that ¯ b ⊆ ¯ b (cid:48) and any i ∈ M \ ¯ b (cid:48) , V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ) .(b) For all groups ¯ b ∈ ¯ B and any i, j, k ∈ M \ ¯ b , V r ( i, j | ¯ b ) + V r ( k | ¯ b ) ≤ max (cid:8) V r ( i | ¯ b ) + V r ( j, k | ¯ b ) , V r ( j | ¯ b ) + V r ( i, k | ¯ b ) (cid:9) . (18)In Definition 5, (a) requires that the augmented value function V is submodular, i.e. themarginal valuation of (cid:0) ¯ b, r (cid:1) decreases in the size of group ¯ b . Additionally, the gross substi-tutes condition also requires that the augmented value function satisfy (b) . This conditionensures that the sum of marginal values of { i, j } and k is not strictly higher than that ofboth i, { j, k } and j, { i, k } .The following lemma shows that when all riders have a homogeneous carpool disutility,the augmented trip value function V satisfies gross substitutes condition. Lemma 4
The augmented value function V r satisfies gross substitutes for all r ∈ R if ridershave homogeneous carpool disutility: γ m ( d ) ≡ γ ( d ) for all d = 1 , . . . , A and all m ∈ M . In the economy G , since each agent l ’s value function W l (¯ b ) = V r (¯ b ) for all ¯ b ∈ ¯ B and all l ∈ L r , the agents’ value functions W satisfy gross substitutes under the condition in Lemma4. Moreover, from (15), the value functions W are also monotonic. From the following result,we know that a Walrasian equilibrium exists in economy with value functions that satisfymonotonicity and gross substitutes conditions. Lemma 5 ([6]) If W l satisfies the monotonicity and gross substitutes conditions for all l ∈ L , then Walrasian equilibrium ( y ∗ , u ∗ ) exists. Based on Lemmas 3, 4 and 5, we conclude the following:14 emma 6
The linear program (LP k ∗ ) has an optimal integer solution if all riders havehomogeneous carpool disutilities, i.e. γ m ( d ) ≡ γ ( d ) for all m ∈ M and all d = 1 , . . . , A . Lemma 6 shows that (LP k ∗ ) has an optimal integer solution. From Lemma 1, we knowthat this solution is also an optimal integer solution of (LP). Therefore, we can concludeTheorem 2 that market equilibrium exists when the network is series parallel and riders havehomogeneous carpool disutilities. In Sec. 5 and 6, we assume that the sufficient conditionsin Theorem 2 hold, and market equilibrium exists. In this section, we present an algorithm for computing the market equilibrium ( x ∗ , p ∗ , τ ∗ ).The ideas behind the algorithm are based on Theorems 1 – 2 and their proofs. Computing optimal trip vector x ∗ . To begin with, one can obtain the optimal tripvector x ∗ following the proof of Theorem 2. In particular, we compute the route capacityvector k ∗ from Algorithm 1. From Lemma 1, we know that the optimal trip assignmentvector x ∗ is an optimal integer solution of (LP k ∗ ). Moreover, from Lemmas 2 – 6, we knowthat: (i) x ∗ can be derived from optimal solution y ∗ on the auxiliary network with theaugmented trip value function W ; and (ii) y ∗ is the same as the optimal good allocation inWalrasian equilibrium of the equivalent economy G . We introduce the following well-knownKelso-Crawford algorithm (Algorithm 2) for computing Walrasian equilibrium y ∗ . Algorithm 2:
Kelso-Crawford Auction [14]
Initialize:
Set u m ← ∀ m ∈ M ; ¯ b l ← ∅ , ∀ l ∈ L ; while TRUE dofor l ∈ L do J l ← arg max J ⊆ M \ ¯ b l φ l ( J | ¯ b l ) ∆ = (cid:8) W l ( J ∪ ¯ b l ) − (cid:80) m ∈ ¯ b l u m − (cid:80) m ∈ J ( u m + (cid:15) ) (cid:9) if J l = ∅ , ∀ l ∈ L then break else Arbitrarily pick ˆ l with J ˆ l (cid:54) = ∅ ;¯ b ˆ l ← ¯ b ˆ l ∪ J ˆ l ;¯ b ˆ l ← ¯ b ˆ l \ J ˆ l , ∀ l (cid:54) = ˆ l ; u m ← u m + (cid:15), ∀ m ∈ J ˆ l . Return (cid:0) ¯ b l (cid:1) l ∈ L Algorithm 2 begins with all riders having zero utilities u m = 0 and all routes in theauxiliary network being empty ¯ b l = ∅ . In each iteration, we compute the set of riders J l who are currently unassigned to route l and maximize the function φ l ( J l | ¯ b l ). The function φ l ( J l | ¯ b l ) equals to the trip value minus the riders’ utilities when the set J l is added to ¯ b l . Ifthere exists a route ˆ l ∈ L with J ˆ l (cid:54) = ∅ , then we assign riders in J ˆ l to one of such route ˆ l , andincrease the utilities of these riders by a small number (cid:15) .15lgorithm 2 terminates when J l = ∅ for all l ∈ L . Given any (cid:15) < | M | , when thealgorithm terminates, all routes are assigned with the rider set that maximizes its trip valueminus riders’ utilities. The trip vector based on (cid:0) ¯ b l (cid:1) l ∈ L is given by: y ∗ l (¯ b l ) = 1 , and y ∗ l (¯ b ) = 0 , ∀ ¯ b ∈ ¯ B \ { ¯ b l } , ∀ l ∈ L. (19)The following lemma shows that y ∗ is optimal under the conditions of monotonicity andgross substitutes. Lemma 7 ([14])
For any (cid:15) < | M | , if the augmented value function W satisfies mono-tonicity and gross substitutes condition, then y ∗ as in (19) is an optimal integer solution of (LP-y) . Recall from Lemma 4, we know that when all riders have identical carpool disutility, i.e. γ m ( d ) = γ ( d ) for all d = 1 , . . . , A , then the augmented trip value function V satisfies grosssubstitutes condition. Since W l (¯ b ) = V r (¯ b ) for all l ∈ L r and all r ∈ R , W also satisfiesgross substitutes condition. Therefore, y ∗ is a Walrasian equilibrium good allocation vectorin the equivalent economy G , and from Lemmas 1 – 3, the vector x ∗ = χ ( y ∗ ) as in (16) is anoptimal trip vector in market equilibrium.In each iteration of Algorithm 2, we need to compute the set J l ∈ arg max J ⊆ M \ ¯ b l φ l ( J l | ¯ b l )for each l ∈ L . Since the value function W l (¯ b ) is monotonic and satisfies gross substitutescondition, J l can be computed by a greedy algorithm, in which riders are added to the set J l one by one in decreasing order of the difference between the rider’s marginal trip value W l ( m | ¯ b l ∪ J l ) = W l ( { m } ∪ ¯ b l ∪ J l ) − W l (¯ b l ∪ J l ) and their utility u m ([14]). Since W l (¯ b ) = V r (¯ b )as in (14), and all riders have identical carpool disutility parameter, we can write W l (¯ b ) asfollows: W l (¯ b ) = V r (¯ b ) = (cid:88) m ∈ h r (¯ b ) η mr − θ ( | h r (¯ b ) | ) t r , ∀ l ∈ L r , ∀ ¯ b ∈ ¯ B, where η mr ∆ = α m − β m t r and θ ( | h r (¯ b ) | ) = | h r (¯ b ) | γ ( | h r (¯ b ) | ) + δ | h r (¯ b ) | . The representative ridergroup h r (¯ b ) for any trip (cid:0) ¯ b, r (cid:1) ∈ ¯ B × R can be constructed by selecting riders from ¯ b indecreasing order of η mr . The last selected rider ˆ m (i.e. the rider in h r (¯ b ) with the minimumvalue of η mr ) satisfies: η ˆ mr ≥ (cid:0) θ ( | h r (¯ b ) | ) − θ ( | h r (¯ b ) | − (cid:1) t r . That is, adding rider ˆ m to the set h r (¯ b ) \ { ˆ m } increases the trip value. Additionally, η mr < (cid:0) θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | ) (cid:1) t r , ∀ m ∈ ¯ b \ h r (¯ b ) . We can compute the set J l ← arg max J ⊆ M \ ¯ b l φ l ( J | ¯ b l ) in each iteration of Algorithm 2 usingAlgorithm 3. In this algorithm, we first compute the size of the representative rider group˜ h = | h r (¯ b l ) | , then we add riders not in ¯ b l into J l greedily according to their marginal tripvalue minus utility. Note that for computing marginal trip value, we do not need to compute16he augmented trip value function W l (¯ b ), but simply need to keep track of the representativerider group size ˜ h . Algorithm 3:
Computing J l Initialize:
Set J l ← ∅ , ˜ h ←
0, ˜ b l ← ¯ b l ; while TRUE do ˆ m ← arg max m ∈ ˜ b l η ml ; if η ˆ ml < (cid:16) θ (˜ h + 1) − θ (˜ h ) (cid:17) t l then break else ˜ h ← ˜ h + 1, ˜ b l ← ˜ b l \ { ˆ m } while TRUE do ˆ j ← arg max j ∈ S \ ( ¯ b l ∪ J l ) η jl − u j ; if η jl − u j < (cid:16) θ (˜ h + 1) − θ (˜ h ) (cid:17) t l then break else ˜ h ← ˜ h + 1, J l ← J l ∪ { ˆ j } Return J l We next discuss the time complexity of Algorithm 2. The time complexity of computing J l as in Algorithm 3 is O ( | M | ) for each l ∈ L (each rider is counted at most once in Algorithm3). Additionally, we know from Sec. 4 that the sum of route capacities given k ∗ equals tothe maximum capacity of the network C . Thus | L | = C , and the time complexity of eachiteration of Algorithm 2 is O ( | M | C ). Moreover, riders’ utilities are non-decreasing and atleast one rider increases their utility by (cid:15) in each iteration. Besides, riders’ utilities can notexceed the maximum trip value V max , because otherwise J l = ∅ for all l ∈ L regardless ofthe assigned set ¯ b l ; thus Algorithm 2 must terminate before the utility exceeds V max . Wecan conclude that Algorithm 2 terminates in less than M V max /(cid:15) iterations, and its timecomplexity is O (cid:0) V max (cid:15) | M | C (cid:1) .We summarize that x ∗ is computed in the following two steps: Step 1:
Compute the optimal route capacity vector k ∗ from Algorithm 1. Step 2:
Compute y ∗ from Algorithm 2. Derive the optimal trip organization vector x ∗ = χ ( y ∗ ). Computing equilibrium payments p ∗ and toll prices τ ∗ . Given the optimal tripvector x ∗ , we compute the set of rider payments p ∗ and toll prices τ ∗ such that ( x ∗ , p ∗ , τ ∗ ) isa market equilibrium. Recall from Theorem 1, the riders’ utilities and toll prices ( u ∗ , τ ∗ ) inany market equilibrium are optimal solutions of the dual program (D). Sec. 4 constructedthe augmented trip value function V , which satisfies monotonicity and gross substitutesconditions. Following the same proof ideas as in Theorem 1, we can show that the utilityvector u ∗ and toll prices τ ∗ also can be solved from the following dual program with the This step can be omitted if the network is parallel with vector k ∗ = ( q r ) r ∈ R . u,τ U ( u, τ ) = (cid:88) m ∈ M u m + (cid:88) e ∈ E q e τ e s.t. (cid:88) m ∈ ¯ b u m + (cid:88) e ∈ r τ e ≥ V r (¯ b ) , ∀ ¯ b ∈ ¯ B, ∀ r ∈ R, (D.a) u m ≥ , τ e ≥ , ∀ m ∈ M, ∀ e ∈ E. (D.b)The linear program (D) has | M | + | E | number of variables and | R |×| ¯ B | number of constraints.This linear program can be solved by the ellipsoid method. In each iteration of this method,we need to solve a separation problem to decide whether or not a solution ( u, τ ) is feasible,and if not find the constraint that it violates. Since the trip value function V is monotonic andsatisfies the gross substitutes condition, we can solve the separation problem using Algorithm3. For each route r ∈ R , we compute ¯ b r ∈ arg max ¯ b ∈ ¯ B { V r (¯ b ) − (cid:80) m ∈ ¯ b u m } using Algorithm3. Then, by checking whether or not (cid:80) m ∈ ¯ b r u m + (cid:80) e ∈ r τ e ≥ V r (¯ b r ), we can determine if theconstraint (D.a) is satisfied for all route r ∈ R . In this way, we solve the separation problemin time polynomial in | M | and | R | . Thus, the optimal solution of (D) can also be solved byellipsoid method in time polynomial in | M | and | R | .Finally, given any optimal solution ( u ∗ , τ ∗ ), the riders’ payment vector p ∗ can be obtainedfrom (4). Thus, we obtain ( x ∗ , p ∗ , τ ∗ ) as a market equilibrium.Notice that the set of equilibrium utility and toll prices ( u ∗ , τ ∗ ) may not be single-ton. From strong duality theory, we know that the sum of riders’ equilibrium utilities andtoll prices, must equal to the optimal social welfare given the organized trips in x ∗ , i.e. (cid:80) m ∈ M u m ∗ + (cid:80) e ∈ E q e τ ∗ e = S ( x ∗ ). Therefore, different market equilibria can result in differ-ent splits of social welfare between the riders’ utilities and the collected toll prices. Next, wehighlight a specific market equilibrium that provides the maximum share of social welfare toriders and collects the minimum tolls. In this section, we consider the situation where the market is facilitated by a platform thatimplements a market equilibrium based on the reported preferences of each rider. Twoquestions arise in this situation: The first is whether or not riders truthfully report theirpreference parameters to the platform. The second is which market equilibrium is imple-mented and how it determines the splits between riders’ utilities and collected tolls. Weshow that there exists a strategyproof market equilibrium under which riders truthfully re-port their preferences. Moreover, this market equilibrium also achieves the maximum utilityfor all riders and the total toll is the minimum.We first introduce the definition of strategyproofness. To distinguish between the truepreference parameters and the reported preference parameters, we denote the reported pa-rameters as α (cid:48) and β (cid:48) . The corresponding market equilibrium is denoted (cid:0) x ∗ (cid:48) , p ∗ (cid:48) , τ ∗ (cid:48) (cid:1) . Theutility vector under market equilibrium with the true preference parameters (resp. reportedpreference parameters) u ∗ (resp. u ∗ (cid:48) ) can be computed as in (4). We assume that riders have homogeneous carpool disutility that is known by the platform. efinition 6 (Strategyproofness) A market equilibrium ( x ∗ , p ∗ , τ ∗ ) is strategyproof if forany preference parameters α (cid:48) (cid:54) = α and β (cid:48) (cid:54) = β , u m ∗ ≥ u m ∗ (cid:48) for all m ∈ M . We next define the Vickery-Clark-Grove (VCG) Payment vector. For each m ∈ M ,we denoted x − m ∗ as the optimal trip vector when rider m is not present. The social wel-fare for riders in M \ { m } given the optimal trip vector x − m ∗ is denoted S − m ( x − m ∗ ) = (cid:80) b ∈ B (cid:80) r ∈ R V r ( b ) x − m ∗ r ( b ) , and the social welfare for riders in M \ { m } with x ∗ is S − m ( x ∗ ) = S ( x ∗ ) − (cid:80) b (cid:51) m (cid:80) r ∈ R v mr ( b ) x ∗ r ( b ). Definition 7
A VCG payment vector p † = (cid:0) p m † (cid:1) m ∈ M is given by: p m † = S − m ( x − m ∗ ) − S − m ( x ∗ ) , ∀ m ∈ M. (21)In VCG payment vector (21), each rider m ’s payment is the difference of the total tripvalues for all other riders with and without rider m , i.e. p m † is the externality of each rider m on all other riders. Under the optimal trip vector x ∗ and the VCG payment vector p † , theutility vector u † = (cid:0) u m † (cid:1) m ∈ M is given by: u m † (4) = (cid:88) b (cid:51) m (cid:88) r ∈ R V r ( b ) x ∗ r ( b ) − p m † (21) = S ( x ∗ ) − S − m ( x ∗− m ) , ∀ m ∈ M. (22)That is, the utility of each rider m ∈ M is the difference of the optimal social welfare withand without rider m . Lemma 8 ([22])
A market equilibrium is strategyproof if the payment vector is p † . The next theorem shows that there exists a toll price vector such that the market equi-librium payment vector is p † and the riders’ utility vector is u † . This market equilibriumis strategyproof. Moreover, all riders’ utilities are higher than that under any other marketequilibrium, and the total collected tolls is the minimum. Theorem 3
There exists a toll price vector τ † such that (cid:0) x ∗ , p † , τ † (cid:1) is a market equilibrium,and is strategyproof. Moreover, for any other market equilibrium ( x ∗ , p ∗ , τ ∗ ) , u m † ≥ u m ∗ , ∀ m ∈ M, (cid:88) e ∈ E q e τ † e ≤ (cid:88) e ∈ E q e τ ∗ e . We denote the set of u ∗ in the optimal solutions of the dual problem (D) as U ∗ . FromTheorem 1, we know that any utility vector u ∗ is an equilibrium utility vector if and only ifthere exists a toll price vector τ ∗ such that ( u ∗ , τ ∗ ) is an optimal solution of (D), i.e. u ∗ ∈ U ∗ .To show that u † is the maximum equilibrium utility vector, we need to prove that u † is themaximum component in the set U ∗ .We proceed in three steps: Firstly, Lemma 9 shows that the set U ∗ is equivalent to theset of utility vectors in the optimal solution set of the dual program of (LP k ∗ ). Secondly,the set of optimal utility vectors in the dual program of (LP k ∗ ) is the same as the set ofprices in Walrasian equilibrium of the equivalent economy constructed in Sec. 4 (Lemma10). Finally, the set of good prices in Walrasian equilibrium is a complete lattice, and themaximum component is u † as in (22) (Lemma 11).We now present the formal statements of these lemmas and their proof ideas. The proofsare included in Appendix B. 19 emma 9 A utility vector u ∗ ∈ U ∗ if and only if there exists vector λ ∗ = ( λ ∗ r ) r ∈ R such that ( u ∗ , λ ∗ ) is an optimal solution of the following linear program: min u,λ (cid:88) m ∈ M u m + (cid:88) r ∈ R ∗ k ∗ r λ r ,s.t. (cid:88) m ∈ b u m + λ r ≥ V r ( b ) ∀ r ∈ R ∗ , ∀ b ∈ B, (D k ∗ .a) u m ≥ , λ r ≥ , ∀ m ∈ M, ∀ r ∈ R ∗ , (D k ∗ .b) where λ r is the dual variable of constraint (LP k ∗ .b) for each r ∈ R . In (D k ∗ ), the dual variable λ r can be viewed as the toll price set on each route r ∈ R ∗ .We note that (D k ∗ ) is less restrictive than (D), which is the dual program on the originalnetwork, in two respects: Firstly, constraints (D k ∗ ) are only set for the set of routes R ∗ ofthe sub-network rather than on all routes in the whole network. Secondly, the toll prices λ in (D k ∗ ) are set on routes instead of on edges as in τ of (D). Any edge toll price vectorcan be equivalently represented as toll prices on routes by summing the tolls of all edges onany route. Therefore, given any feasible solution ( u, τ ) of (D), ( u, λ ) where λ r = (cid:80) e ∈ r τ e foreach r ∈ R ∗ is also feasible in (D k ∗ ).We can check that for any optimal solution ( u ∗ , τ ∗ ) of (D), the vector ( u ∗ , λ ∗ ) – where λ ∗ r = (cid:80) e ∈ r τ ∗ e for each r ∈ R ∗ – must also be optimal in (D k ∗ ). That is, the set U ∗ is a subsetof the optimal utility vectors in (D k ∗ ). This result follows from strong duality theory andLemma 1: From the strong duality theory, the optimal values of the objective function in(D k ∗ ) (resp. (LP)) equals to the optimal value of the primal problems (LP) (resp. (LP k ∗ )).From Lemma 1, we know that the optimal trip organization vector is the same in both (LP)and (LP k ∗ ). Thus, the optimal value of (D) is the same as that of (LP k ∗ ). Since the value ofthe objective function with ( u ∗ , τ ∗ ) equals to that with ( u ∗ , λ ∗ ), we know that ( u ∗ , λ ∗ ) mustbe an optimal solution of (D k ∗ ).Furthermore, we can show that for any optimal solution u ∗ of (D k ∗ ), there must exist anedge toll vector τ ∗ such that ( u ∗ , τ ∗ ) is an optimal solution of (D). That is, any equilibriumutility vector with route toll prices on the sub-network can also be induced by edge toll priceson the original network. This result relies on the fact that the network is series parallel, andit is proved by mathematical induction.Lemma 9 enables us to characterize the riders’ utility set U ∗ using the less restrictive dualprogram (D k ∗ ). Recall that in Sec. 4, we have shown that the trip organization problem onthe constructed augmented network with the augmented value function is equivalent to aneconomy with indivisible goods (Lemma 3). The next lemma shows that the set U ∗ is thesame as the set of Walrasian equilibrium prices in the equivalent economy. Lemma 10
A utility vector u ∗ ∈ U ∗ if and only if there exists y ∗ such that ( y ∗ , u ∗ ) is aWalrasian equilibrium of the economy. Moreover, since the augmented trip value function W is monotonic and satisfies grosssubstitutes condition, the set of Walrasian equilibrium price vectors is a lattice, and has amaximum component. 20 emma 11 ([11]) If the value function W satisfies the monotonicity and gross substitutesconditions, then the set of Walrasian equilibrium prices is a lattice and has a maximumcomponent u † = (cid:0) u m † (cid:1) m ∈ M as in (22) . From Lemmas 9 – 11, we know that u † is the maximum component in the set U ∗ . Thatis, there exists a toll price vector τ † = (cid:0) τ † e (cid:1) e ∈ E such that (cid:0) u † , τ † (cid:1) is an optimal solution of(D), and hence (cid:0) x ∗ , p † , τ † (cid:1) is a market equilibrium. Additionally, from Lemma 8, we knowthat this market equilibrium is strategyproof. Moreover, all riders achieve the maximumequilibrium utilities in the equilibrium. Since (cid:80) m ∈ M u m ∗ + (cid:80) e ∈ E q e τ ∗ e = S ( x ∗ ) for anymarket equilibrium ( x ∗ , u ∗ , τ ∗ ), this also implies that the total amount of tolls (cid:80) e ∈ E q e τ † e thatis collected in market equilibrium (cid:0) x ∗ , p † , τ † (cid:1) is the minimum. We thus conclude Theorem3. Finally, we discuss the computation of the market equilibrium (cid:0) x ∗ , p † , τ † (cid:1) . In particular,the optimal trip assignment x ∗ can be computed in two steps described in Sec. 5 usingAlgorithms 1 – 2. Then, we re-run Algorithm 2 given k ∗ and rider set M \ { m } to compute x − m ∗ for each m ∈ M . We compute the utility vector u † (resp. payment vector p † ) as in(22) (resp. (21)).For any e ∈ E , we set τ † e = 0 if (cid:80) b ∈ B (cid:80) r (cid:51) e x ∗ r ( b ) < q e . From (D), we know that τ † is anyvector that satisfies the following constraints: (cid:88) e ∈ r τ † e = max ¯ b ∈ ¯ B V r (¯ b ) − (cid:88) m ∈ ¯ b u m † , ∀ r ∈ R ∗ , (cid:88) e ∈ r τ † e ≥ max ¯ b ∈ ¯ B V r (¯ b ) − (cid:88) m ∈ ¯ b u m † , ∀ r ∈ R \ R ∗ . (24)Finding a vector τ † that satisfies constraints in (24) is equivalent to solving a linear programwith a constant objective function and feasibility constraints (24). This linear program canbe computed by the ellipsoid method, in which the separation problem in each iteration is tocheck whether or not the toll price vector τ † satisfies the feasible constraints in (24). Sincethe augmented trip value function V satisfies monotonicity and gross substitutes condition,we can compute the right-hand-side value of the constraint in (24) using Algorithm 3 in time O ( | M | ) for each r ∈ R . That is, the separation problem in each iteration can be computedin polynomial time of | M | and | R | . Therefore, a toll vector τ † that satisfies (24) can becomputed in polynomial time of | M | and | R | . In this article, we studied the existence and computation of market equilibrium for organizingsocially efficient carpooled trips over a transportation network using autonomous cars. Wealso identified a market equilibrium that is strategyproof and maximizes riders’ utilities.Our approach can be used to analyze incentive mechanisms for sharing limited resources innetworked environment.One interesting direction for future work is to characterize equilibrium in a transporta-tion market when riders belong to different classes that are differentiated by their carpool21isutility levels. In this situation, riders with different carpool disutilies may be groupedinto trips that are organized using different vehicle sizes to reflect the riders’ car sharingpreferences.A more general problem is to design market with both autonomous and human-drivencarpooled trips, wherein riders may have different preferences of over these service types. Apre-requisite to the design of such a market is quantitative evaluation of how autonomous andhuman-driven vehicles differ in terms of their utilization of road capacity and the incurredroute travel times [13]. Analysis of differentiated pricing and tolling schemes correspondingto trip assignments between the two service types is an interesting and relevant problem forfuture work.
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Proof of Section 3
Proof of Theorem 1.
First, we proof that the four conditions of market equilibrium ( x ∗ , p ∗ , τ ∗ )ensures that x ∗ satisfies the feasibility constraints of the primal (LP), ( u ∗ , τ ∗ ) satisfies thedual (D), and ( x ∗ , u ∗ , τ ∗ ) satisfies the complementary slackness conditions. The vector u ∗ isthe utility vector computed from (4).(i) Feasibility constraints of (LP). Since x ∗ is a feasible trip vector, x ∗ must satisfy theFeasibility constraints of (LP).(ii) Feasibility constraints of (D). From the stability condition (6), individual rationality(5), and the fact that toll prices are non-negative, we know that ( u ∗ , τ ∗ ) satisfies thefeasibility constraints of (D).(iii) Complementary slackness condition with respect to (LP.a). If rider m is not assigned,then (LP.a) is slack with the integer trip assignment x ∗ for some rider m . The budgetbalanced condition (7b) shows that p ∗ m = 0. Since rider m is not in any trip and thepayment is zero, the dual variable (i.e. rider m ’s utility) u m ∗ = 0. On the other hand,if u m ∗ >
0, then rider m must be in a trip, and constraint (LP.a) must be tight. Thus,we can conclude that the complementary slackness condition with respect to the primalconstraint (LP.a) is satisfied.(iv) Complementary slackness condition with respect to (LP.b). Since the mechanism ismarket clearing, toll price τ e is nonzero if and only if the load on edge e is belowthe capacity, i.e. the primal constraint (LP.b) is slack for edge e ∈ E . Therefore,the complementary slackness condition with respect to the primal constraint (LP.b) issatisfied.(v) Complementary slackness condition with respect to (D.a). From (7a), we know thatfor any organized trip, the corresponding dual constraint (D.a) is tight. If constraint(D.a) is slack for a trip ( b, r ), then the budget balance constraint ensures that trip isnot organized. Therefore, the complementary slackness condition with respect to theprimal constraint (D.a) is satisfied.We can analogously show that the inverse of (i) – (v) are also true: the feasibility con-straints of (LP) and (D), and the complementary slackness conditions ensure that ( x ∗ , p ∗ , τ ∗ )is a market equilibrium. Thus, we can conclude that ( x ∗ , p ∗ , τ ∗ ) is a market equilibrium if andonly if ( x ∗ , u ∗ , τ ∗ ) satisfies the feasibility constraints of (LP) and (D), and the complementaryslackness conditions.From strong duality theory, we know that the equilibrium trip vector x ∗ must be anoptimal integer solution of (LP). Therefore, the existence of market equilibrium is equivalentto the existence of an integer optimal solution of (LP). The optimal trip assignment is anoptimal integer solution of (LP), and ( u ∗ , τ ∗ ) is an optimal solution of the dual problem (D).The payment p ∗ can be computed from (4). (cid:3) Proof of Corollary 1.
Consider any two routes r, r (cid:48) ∈ R such that t r ≥ t r (cid:48) . Given x ∗ , wedenote the rider group that takes route r as b r . If no rider group is assigned to route r , then24e denote b r = ∅ . From (7a), we have (cid:88) m ∈ b r u ∗ m + (cid:88) e ∈ r τ ∗ e = V r ( b r ) . Additionally, since ( u ∗ , τ ∗ ) satisfies constraint (D.a), we know that (cid:88) m ∈ b r u ∗ m + (cid:88) e ∈ r (cid:48) τ ∗ e ≥ V (cid:48) r ( b r ) . Therefore, we must have: (cid:88) e ∈ r (cid:48) τ ∗ e − (cid:88) e ∈ r τ ∗ e ≥ V (cid:48) r ( b r ) − V r ( b r ) = (cid:32) (cid:88) m ∈ b r α m − (cid:88) m ∈ b r β m t r (cid:48) − (cid:88) m ∈ b r γ m ( | b r | ) t r (cid:48) − δ | b r | t r (cid:48) (cid:33) − (cid:32) (cid:88) m ∈ b r α m − (cid:88) m ∈ b r β m t r − (cid:88) m ∈ b r γ m ( | b r | ) t r − δ | b r | t r (cid:33) = (cid:32) (cid:88) m ∈ b r β m + (cid:88) m ∈ b r γ m ( | b r | ) + δ | b r | (cid:33) ( t r − t r (cid:48) ) ≥ . (cid:3) B Proof of Section 4.
Proof of Lemma 1.
Consider any (fractional) optimal solution of (LP), denoted as ˆ x . Wedenote ˆ f ( b ) = (cid:80) r ∈ R ˆ x r ( b ) as the flow of group b , and (cid:98) F = (cid:80) b ∈ B ˆ f ( b ) is the total flows. Sinceˆ x is feasible, we know that (cid:98) F ≤ C , where C is the maximum capacity of the network. Foreach b ∈ B , we re-write the trip valuation as follows: V r ( b ) = z ( b ) − g ( b ) t r , ∀ ( b, r ) ∈ B × R, where g ( b ) = (cid:80) m ∈ b β m + (cid:80) m ∈ b γ ( | b | ) + δ | b | , and z ( b ) = (cid:80) m ∈ b α m .The set of all groups with positive flow in ˆ x is (cid:98) B ∆ = { ˆ b ∈ B | ˆ f (ˆ b ) > } . We denote thenumber of rider groups in ˆ B as n , and re-number these rider groups in decreasing order of g (ˆ b ), i.e. g (ˆ b ) ≥ g (ˆ b ) ≥ · · · ≥ g (ˆ b n ) . We now construct another trip vector x ∗ by the following procedure: Initialization:
Set route set ˜ R = R ∗ , route capacity ˜ q r = k ∗ r for all ∀ r ∈ ˜ R , and initial zeroassignment vector x ∗ r ( b ) ← r ∈ R and all b ∈ B For j = 1 , . . . , n : (a) Assign rider group ˆ b j to a route ˆ r in ˜ R , which has the minimum travel time among allroutes with flow less than the capacity, i.e. ˆ r ∈ arg min r ∈{ ˜ R | (cid:80) b ∈ B x ∗ r ( b ) < ˜ q r } { t r } .25b) If (cid:80) b ∈ B x ∗ r ( b ) + ˆ f (ˆ b j ) ≤ ˜ q r , then x ∗ r (ˆ b j ) = ˆ f (ˆ b j ).(c) Otherwise, assign x ∗ r (ˆ b j ) = ˜ q r − (cid:80) b ∈ B x ∗ r ( b ), and continue to assign the remainingweight of rider group ˆ b j to the next unsaturated route with the minimum cost. Repeatthis process until the condition in (b) is satisfied, i.e. the total weight ˆ f ( b j ) is assigned.We can check that (cid:80) b (cid:51) m (cid:80) r ∈ R x ∗ r ( b ) = (cid:80) b (cid:51) m ˆ f ( b ) ≤ k ∗ .a) is satisfied.Additionally, since in the assignment procedure, the total weight assigned to route r is lessthan or equal to k ∗ r , we must have (cid:80) b ∈ B x ∗ r ( b ) ≤ k ∗ r for all r ∈ R , i.e. (LP k ∗ .b) is satisfied.Thus, x ∗ is a feasible solution of (LP k ∗ ).It remains to prove that x ∗ is optimal of (LP k ∗ ). We prove this by showing that V ( x ∗ ) ≥ V (ˆ x ). The objective function S ( x ∗ ) can be written as follows: (cid:88) r ∈ R (cid:88) b ∈ B V r ( b ) x ∗ r ( b ) = (cid:88) r ∈ R (cid:88) b ∈ B z ( b ) x ∗ r ( b ) − (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x ∗ r ( b ) . (25)We note that since (cid:80) r ∈ R k ∗ r = C and (cid:80) r ∈ R (cid:80) b ∈ B ˆ x r ( b ) ≤ C , the algorithm must terminatewith all groups in ˆ x being assigned. Therefore, (cid:80) r ∈ R x ∗ r ( b ) = ˆ f ( b ) = (cid:80) r ∈ R ˆ x r ( b ) for all b ∈ B . Therefore, (cid:88) r ∈ R (cid:88) b ∈ B z ( b ) x ∗ r ( b ) = (cid:88) b ∈ B z ( b ) ˆ f ( b ) = (cid:88) r ∈ R (cid:88) b ∈ B z ( b )ˆ x r ( b ) (26)Then, V ( x ∗ ) ≥ V (ˆ x ) is equivalent to (cid:80) r ∈ R (cid:80) b ∈ B g ( b ) t r x ∗ r ( b ) ≤ (cid:80) r ∈ R (cid:80) b ∈ B g ( b ) t r ˆ x ∗ r . Toprove this, we show that x ∗ minimizes the term (cid:80) r ∈ R (cid:80) b ∈ B g ( b ) t r x ∗ r ( b ) among all feasible x that induces the same flow of groups as ˆ x , i.e. x ∗ ∈ arg min x ∈ X ( ˆ f ) (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x r ( b ) , (27)where X ( ˆ f ) ∆ = ( x r ( b )) r ∈ R,b ∈ B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:80) r ∈ R x r ( b ) = ˆ f ( b ) , ∀ b ∈ B, (cid:80) b ∈ B (cid:80) r (cid:51) e x r ( b ) ≤ q e , ∀ e ∈ E,x r ( b ) ≥ , ∀ r ∈ R, ∀ b ∈ B. (28)We prove (27) by mathematical induction. To begin with, (27) holds trivially on anysingle-link network. We ext prove that if (27) holds on two series-parallel sub-networks G and G , then (27) holds on the network G that connects G and G in series or in parallel.In particular, we analyze the cases of series connection and parallel connection separately: (Case 1) Series-parallel Network G is formed by connecting two series-parallel sub-networks G and G in series.We denote the set of routes in subnetwork G and G as R and R , respectively. Since G and G are connected in series, the set of routes in network G is R ∆ = R × R . For any flowvector ˆ f , we define the set of trip vectors on G that satisfy the constraint in (27) as X ( ˆ f ).We also define the trip vector that is obtained from the above-mentioned procedure basedon ˆ f as x ∗ . 26ince the two sub-networks are connected in sequence, the group flow vectors in G and G are also ˆ f . Analogously, we define the set of trip vectors on sub-network G (resp. G )that satisfies the constraint in (27) as X ( ˆ f ) (resp. X ( ˆ f )). We can check that X ( ˆ f )(resp. X ( ˆ f )) is the set of trip vectors in X ( ˆ f ) that is restricted on network G (resp. G ). That is, for any x ∈ X ( ˆ f ), we can find x ∈ X ( ˆ f ) (resp. x ∈ X ( ˆ f )) such that (cid:80) r ∈ R x r r ( b ) = x r ( b ) (resp. (cid:80) r ∈ R x r r ( b ) = x r ( b )) for all b ∈ B and all r ∈ R (resp. r ∈ R ). Since the two subnetworks are connected sequentially, we have the follows: (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x r ( b ) = (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r (cid:32) (cid:88) r ∈ R x r r ( b ) (cid:33) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r (cid:32) (cid:88) r ∈ R x r r ( b ) (cid:33) = (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x r ( b ) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x r ( b ) . (29)We also denote the trip vector that is obtained from the above-mentioned procedurebased on ˆ f in G (resp. G ) as x ∗ (resp. x ∗ ). We now argue that (cid:80) r ∈ R x ∗ r r ( b ) = x ∗ r ( b )for all b ∈ B and all r ∈ R . For the sake of contradiction, assume that there exists b ∈ B such that (cid:80) r ∈ R x ∗ r r ( b ) (cid:54) = x ∗ r ( b ) for at least one r ∈ R . We denote ˆ b as onesuch group with the maximum g (ˆ b ). Since the total flow of ˆ b is ˆ f (ˆ b ) in both x ∗ and x ∗ ,if (cid:80) r ∈ R x ∗ r r (ˆ b ) (cid:54) = x ∗ r (ˆ b ) on one r ∈ R , the same inequality must hold for another r (cid:48) ∈ R . Without loss of generality, we assume that t r < t r (cid:48) . Since any group b that areassigned before ˆ b ( g ( b ) < g (ˆ b )) satisfy (cid:80) r ∈ R x ∗ r r ( b ) = x ∗ r ( b ) for all r ∈ R , we know thatthe available route capacities ˜ f in the round of assigning ˆ b in procedure (i) – (iii) satisfy (cid:80) r ∈ R ˜ f r r = ˜ f r for all r ∈ R . Therefore, if (cid:80) r ∈ R x ∗ r r (ˆ b ) < x ∗ r (ˆ b ), then x ∗ is notobtained by procedure (i) – (iii) on G because r is not saturated with x ∗ in the roundof assigning ˆ b , and more flow of ˆ b should be moved from r (cid:48) to r to saturate route r . Wecan analogously argue that if (cid:80) r ∈ R x ∗ r r (ˆ b ) > x ∗ r (ˆ b ), then x ∗ is not obtained from thealgorithm for G . In either case, we have arrived at a contradiction. We can analogouslyargue that (cid:80) r ∈ R x ∗ r r ( b ) = x ∗ r ( b ) for all b ∈ B and all r ∈ R . Therefore, (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x ∗ r ( b ) = (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r (cid:32) (cid:88) r ∈ R x ∗ r r ( b ) (cid:33) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r (cid:32) (cid:88) r ∈ R x ∗ r r ( b ) (cid:33) = (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x ∗ r ( b ) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r x ∗ r ( b ) (30)If (27) holds on both sub-networks (i.e. x ∗ ∈ arg min x ∈ X ( ˆ f ) (cid:80) r ∈ R (cid:80) b ∈ B g ( b ) t r x r ( b )and x ∗ ∈ arg min x ∈ X ( ˆ f ) (cid:80) r ∈ R (cid:80) b ∈ B g ( b ) t r x r ( b )), then from (29) – (30), we know that(27) also holds in network G . (Case 2) Series-parallel Network G is formed by connecting two series-parallel networks G and G in parallel.Same as case 1, we denote R (resp. R ) as the set of routes in G (resp. G ). Then, theset of all routes in G is R = R ∪ R .Given any ˆ f , we compute x ∗ from the procedure (i) – (iii) in network G . We denote f ∗ = (cid:80) r ∈ R (cid:80) b ∈ B x ∗ r ( b ) (resp. f ∗ = (cid:80) r ∈ R (cid:80) b ∈ B x ∗ r ( b )) as the total flow assigned to subnetwork27 (resp. G ) given x ∗ . We now denote x ∗ (resp. x ∗ ) as the trip vector x ∗ restricted onsub-network G (resp. G ), i.e. x ∗ = (cid:0) x ∗ r ( b ) (cid:1) r ∈ R ,b ∈ B (resp. x ∗ = (cid:0) x ∗ r ( b ) (cid:1) r ∈ R ,b ∈ B ). Wecan check that x ∗ (resp. x ∗ ) is the trip vector obtained by the procedure (i) – (iii) giventhe total flow f ∗ (resp. f ∗ ) on network G (resp. G ).Consider any arbitrary split of the total flow ˆ f to the two sub-networks, denoted as (cid:16) ˆ f , ˆ f (cid:17) , such that ˆ f ( b ) + ˆ f ( b ) = ˆ f ( b ) for all b ∈ B . Given ˆ f (resp. ˆ f ), we denote thetrip vector obtained by procedure (i) – (iii) on sub-network G (resp. G ) as ˆ x ∗ (resp. ˆ x ∗ ).We also define the set of feasible trip vectors on sub-network G (resp. G ) that induce thetotal flow ˆ f (resp. ˆ f ) given by (28) as X ( ˆ f ) (resp. X ( ˆ f )). Then, the set of all tripvectors that induce ˆ f on network G is X ( ˆ f ) = ∪ ( ˆ f , ˆ f )( X ( ˆ f ) , X ( ˆ f )).Under our assumption that (27) holds on sub-network G and G with any total flow, weknow that given any flow split (cid:16) ˆ f , ˆ f (cid:17) , (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r ˆ x ∗ r ( b ) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r ˆ x ∗ r ( b ) ≤ (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r ˆ x r ( b ) + (cid:88) r ∈ R (cid:88) b ∈ B g ( b ) t r ˆ x r ( b ) , ∀ ˆ x ∈ X ( ˆ f ) , ˆ x ∈ X ( ˆ f ) . Therefore, the optimal solution of (27) must be a trip vector (ˆ x ∗ , ˆ x ∗ ) associated with aflow split (cid:16) ˆ f , ˆ f (cid:17) . It thus remains prove that any (ˆ x ∗ , ˆ x ∗ ) associated with flow split (cid:16) ˆ f , ˆ f (cid:17) (cid:54) = ( f ∗ , f ∗ ) cannot be an optimal solution (i.e. can be improved by re-arrangingflows).For any (cid:16) ˆ f , ˆ f (cid:17) (cid:54) = ( f ∗ , f ∗ ), we can find a group b j such that ˆ f ( b j ) (cid:54) = f ∗ ( b j ) (hence-forth ˆ f ( b j ) (cid:54) = f ∗ ( b j )). We denote b ˆ j as one such group with the maximum g ( b ), i.e.ˆ f ( b j ) = f ∗ ( b j ) for any j − , . . . , ˆ j −
1. Since groups b , . . . , b ˆ j − are assigned before group b ˆ j according to procedure (i) – (iii), we know that x ∗ r ( b j ) = x ∗ r ( b j ) and x ∗ r ( b j ) = x ∗ r ( b j )for all r ∈ R , all r ∈ R and all j = 1 , . . . , ˆ j −
1. Since ˆ f ( b ˆ j ) (cid:54) = f ∗ ( b ˆ j ), the trip vectorin x ∗ and x ∗ must be different from that in x ∗ . Without loss of generality, we assumethat ˆ f ( b ˆ j ) > f ∗ ( b ˆ j ) and ˆ f ( b ˆ j ) < f ∗ ( b ˆ j ). Then, there must exist routes ˆ r ∈ R andˆ r ∈ R such that x ∗ ˆ r ( b ˆ j ) > x ∗ ˆ r ( b ˆ j ) and x ∗ ˆ r ( b ˆ j ) < x ∗ ˆ r ( b ˆ j ). Moreover, since x ∗ assigns group b ˆ j to routes with the minimum travel time cost that are unsaturated after assigning groups b , . . . , b ˆ j − , we have t ˆ r < t ˆ r . If route ˆ r is unsaturated given ˆ x ∗ , then we decrease ˆ x ∗ ˆ r ( b ˆ j )and increase ˆ x ∗ ˆ r ( b ˆ j ) by a small positive number (cid:15) >
0. We can check that the objective func-tion of (27) is reduced by (cid:15) ( t ˆ r − t ˆ r ) (cid:15)g ( b ˆ j ) >
0. On the other hand, if route ˆ r is saturated,then group b ˆ j +1 must be assigned to ˆ r because it is assigned right after group b ˆ j . Then, wedecrease x ∗ ˆ r ( b ˆ j ) and x ∗ ˆ r ( b ˆ j +1 ) by (cid:15) >
0, increases x ∗ ˆ r ( b ˆ j +1 ) and x ∗ ˆ r ( b ˆ j ) by (cid:15) (i.e. exchangea small fraction of group b ˆ j with group b ˆ j +1 ). Note that g ( b ˆ j ) > g ( b ˆ j +1 ) and t ˆ r > t ˆ r . Wecan thus check that the objective function of (27) is reduced by (cid:15) ( t ˆ r g ( b ˆ j ) − t ˆ r g ( b ˆ j +1 )) (cid:15) > x ∗ , ˆ x ∗ ) that reduces the objec-tive function of (27). Hence, for any flow split (cid:16) ˆ f , ˆ f (cid:17) (cid:54) = ( f ∗ , f ∗ ), the associated tripvector (ˆ x ∗ , ˆ x ∗ ) is not the optimal solution of (27). The optimal solution of (27) must beconstructed by procedure (i) – (iii) with flow split ( f ∗ , f ∗ ), i.e. must be x ∗ .28e have shown from cases 1 and 2 that if x ∗ is an optimal solution of (27) on two series-parallel sub-networks, then x ∗ is an optimal solution on the connected series-parallel network.Moreover, since (27) holds trivially when the network is a single edge, and any series-parallelnetwork is formed by connecting series-parallel sub-networks in series or parallel, we canconclude that x ∗ obtained from procedure (i) – (iii) minimizes the objective function in (27)for any flow vector ˆ f on any series-parallel network.From (25), (26) and (27), we can conclude that V ( x ∗ ) ≥ V (ˆ x ∗ ). Hence, x ∗ must be anoptimal solution in (LP k ∗ ). (cid:3) Proof of Lemma 2.
First, for any feasible x in (LP k ∗ ), consider a vector y such that forany ( r, b ) ∈ { B × R | x r ( b ) = 1 } , y l ( b ) = 1 for one l ∈ L r and y l (¯ b ) = 0 for any other (cid:0) ¯ b, l (cid:1) .We can check that y is feasible in (LP-y) and S ( x ) = S ( y ). On the other hand, for anyfeasible y in (LP-y), there exists x = χ ( y ) as in (16) such that x is feasible in (LP k ∗ ) and S ( x ) = S ( y ). Thus, (LP k ∗ ) and (LP-y) are equivalent in that for any feasible solution ofone linear program, there exists a feasible solution that achieves the same social welfare inthe other linear program.Therefore, (LP k ∗ ) has an integer optimal solution if and only if (LP-y) has an integeroptimal solution, and for any integer optimal solution y ∗ of (LP-y), x = χ ( y ∗ ) as in (16) isan optimal solution of (LP k ∗ ). (cid:3) Proof of Lemma 3.
We write the dual program of (LP-y) as follows:min u,µ (cid:88) m ∈ M u m + (cid:88) l ∈ L µ l ,s.t. (cid:88) m ∈ ¯ b u m + µ l ≥ W l (¯ b ) ∀ ¯ b ∈ ¯ B, ∀ l ∈ L, (D-y.a) u m ≥ , µ l ≥ , ∀ m ∈ M, ∀ l ∈ L. (D-y.b)For any Walrasian equilibrium ( y ∗ , u ∗ ), we consider the vector µ ∗ = ( µ ∗ l ) l ∈ L as follows: µ ∗ l = max ¯ b ∈ ¯ B W l (¯ b ) − (cid:88) m ∈ ¯ b u m ∗ , ∀ l ∈ L. (32)From the definition of Walrasian equilibrium, we know that y ∗ is a feasible solution of(LP-y), and ( u ∗ , µ ∗ ) is a feasible solution of (D-y). We now show that ( y ∗ , u ∗ , µ ∗ ) satisfiescomplementary slackness condition of (LP-y) and (D-y).- Complementary slackness condition for (LP-y.a): Condition (ii) in Definition 5 ensuresthat rider m ’s utility is positive if and only if (LP-y.a) is tight (i.e. rider m joins atrip).- Complementary slackness condition for (LP-y.b): If no rider group takes route l ∈ L ,i.e. (LP-y.b) is slack and ¯ b l = 0, then µ ∗ l as in (32) is zero. On the other hand, µ ∗ l > b l (cid:54) = 0. Hence, (LP-y.b) must be tight.- Complementary slackness condition for (D-y.a): From condition (i) in Definition 5, weknow that y ∗ l (¯ b l ) = 1 if and only if ¯ b l ∈ arg max ¯ b ∈ ¯ B W l (¯ b ) − (cid:80) m ∈ ¯ b u m ∗ , i.e. constraint(D-y.a) is tight. 29rom strong duality, we know that y ∗ must be an integer optimal solution of (LP-y) and( u ∗ , µ ∗ ) must be an optimal solution of (D-y). Therefore, we can conclude that a Walrasianequilibrium ( y ∗ t, u ∗ ) exists in the equivalent economy G if and only if (LP-y) has an optimalinteger solution. (cid:3) Proof of Lemma 4.
Since all riders have homogeneous carpool disutility, we can simplify thetrip value function from (15) as follows: V r (¯ b ) = (cid:88) m ∈ h r (¯ b ) η mr − θ ( | h r (¯ b ) | ) t r , where η mr ∆ = α m − β m t r and θ ( | h r (¯ b ) | ) = | h r (¯ b ) | γ ( | h r (¯ b ) | ) + δ ( | h r (¯ b ) | ).Before proving that the augmented trip value function V r (¯ b ) satisfies (a) and (b) inDefinition 5, we first prove the following statements that will be used later: (i) The function θ ( | h r (¯ b ) | ) is non-decreasing in | h r (¯ b ) | because the marginal carpooldisutility is non-decreasing in the group size. (ii) The representative rider group for any trip (cid:0) ¯ b, r (cid:1) ∈ ¯ B × R can be constructed byselecting riders from ¯ b in decreasing order of η mr . The last selected rider (cid:96) (i.e. the rider in h r (¯ b ) with the minimum value of η mr ) satisfies: η (cid:96)r ≥ (cid:0) θ ( | h r (¯ b ) | ) − θ ( | h r (¯ b ) | − (cid:1) t r . (33)That is, adding rider (cid:96) to the set h r (¯ b ) \ { (cid:96) } increases the trip valuation. Additionally, η mr < (cid:0) θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | ) (cid:1) t r , ∀ m ∈ ¯ b \ h r (¯ b ) . (34)Then, adding any rider in ¯ b \ h r (¯ b ) to h r (¯ b ) no longer increases the trip valuation. (iii) | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | for any two rider groups ¯ b (cid:48) , ¯ b ∈ B such that ¯ b (cid:48) ⊇ ¯ b . Proof of (iii).
Assume for the sake of contradiction that | h r (¯ b (cid:48) ) | < | h r (¯ b ) | . Consider the rider (cid:96) ∈ arg min m ∈ h r (¯ b ) η mr . The value η (cid:96)r satisfies (33). Since | h r (¯ b (cid:48) ) | < | h r (¯ b ) | , ¯ b (cid:48) ⊇ ¯ b , and weknow that riders in the representative rider group h r (¯ b (cid:48) ) are the ones with | h r (¯ b (cid:48) ) | highest η mr in ¯ b (cid:48) , we must have (cid:96) / ∈ h r (¯ b (cid:48) ). From (34), we know that η (cid:96)r < (cid:0) θ ( | h r (¯ b (cid:48) ) | + 1) − θ ( | h r (¯ b (cid:48) ) | ) (cid:1) t r .Since the marginal carpool disutility is non-decreasing in the rider group size, we can checkthat θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | ) is non-decreasing in | h r (¯ b ) | . Since | h r (¯ b (cid:48) ) | < | h r (¯ b ) | , we have | h r (¯ b (cid:48) ) | ≤ | h r (¯ b ) | −
1. Therefore, η (cid:96)r < (cid:0) θ ( | h r (¯ b (cid:48) ) | + 1) − θ ( | h r (¯ b (cid:48) ) | ) (cid:1) t r ≤ (cid:0) θ ( | h r (¯ b ) | ) − θ ( | h r (¯ b ) | − (cid:1) t r , which contradicts (33) and the fact that (cid:96) ∈ h r (¯ b ). Hence, | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | .We now prove that V satisfies (i) in Definition 5. For any ¯ b, ¯ b (cid:48) ⊆ M and ¯ b ⊆ ¯ b (cid:48) , considertwo cases: Case 1: i / ∈ h r ( { i } ∪ ¯ b (cid:48) ). In this case, h r (¯ b (cid:48) ∪ i ) = h r (¯ b (cid:48) ), and V ( i | ¯ b (cid:48) ) = V (¯ b (cid:48) ∪ i ) − V (¯ b (cid:48) ) = 0.Since V satisfies monotonicity condition, we have V ( i | ¯ b ) ≥
0. Therefore, V ( i | ¯ b ) ≥ V ( i | ¯ b (cid:48) ). Case 2: i ∈ h r ( { i }∪ ¯ b (cid:48) ). We argue that i ∈ h r ( { i }∪ ¯ b ). From (33), η ir ≥ (cid:0) θ ( | h r (¯ b (cid:48) ) | ) − θ ( | h r (¯ b (cid:48) ) | − (cid:1) t r .Since ¯ b (cid:48) ⊇ ¯ b , we know from (iii) that | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | . Hence, η ir ≥ (cid:0) θ ( | h r (¯ b ) | ) − θ ( | h r (¯ b ) | − (cid:1) t r ,and thus i ∈ h r ( { i } ∪ ¯ b ). 30e define (cid:96) (cid:48) ∆ = arg min m ∈ h r (¯ b (cid:48) ) η mr and (cid:96) ∆ = arg min m ∈ h r (¯ b ) η mr . We also consider two thresh-olds µ (cid:48) = (cid:0) θ ( | h r (¯ b (cid:48) ) | + 1) − θ ( | h r (¯ b (cid:48) ) | ) (cid:1) t r , and µ = (cid:0) θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | ) (cid:1) t r . Since¯ b (cid:48) ⊇ ¯ b , from (iii), we have | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | and thus µ (cid:48) ≥ µ . We further consider foursub-cases: (2-1) η (cid:96) (cid:48) r ≥ µ (cid:48) and η (cid:96)r ≥ µ . From (33) and (34), h r ( { i } ∪ ¯ b (cid:48) ) = h r (¯ b (cid:48) ) ∪ { i } and h r ( { i } ∪ ¯ b ) = h r (¯ b ) ∪ { i } . The marginal value of i is V r ( i | ¯ b (cid:48) ) = η ir − µ (cid:48) , and V r ( i | ¯ b ) = η ir − µ . Since µ (cid:48) ≥ µ , V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ). (2-2) η (cid:96) (cid:48) r < µ (cid:48) and η (cid:96)r ≥ µ . Since i ∈ h r ( { i }∪ ¯ b (cid:48) ) in Case 2 , we know from (33) and (34) that h r ( { i }∪ ¯ b (cid:48) ) = h r (¯ b (cid:48) ) \{ (cid:96) (cid:48) }∪{ i } and h r ( { i }∪ ¯ b ) = h r (¯ b ) ∪{ i } . Therefore, V r ( i | ¯ b (cid:48) ) = η ir − η (cid:96) (cid:48) r and V r ( i | ¯ b ) = η ir − µ . We argue in this case, we must have | h r (¯ b (cid:48) ) | > | h r (¯ b ) | . Assume for the sakeof contradiction that | h r (¯ b (cid:48) ) | = | h r (¯ b ) | , then µ (cid:48) = µ and η (cid:96) (cid:48) r ≥ η (cid:96)r because ¯ b (cid:48) ⊇ ¯ b . However,this contradicts the assumption of this subcase that η (cid:96) (cid:48) r < µ (cid:48) = µ ≤ η (cid:96)r . Hence, we must have | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | + 1. Then, from (33), we have η (cid:96) (cid:48) r ≥ (cid:0) θ ( | h r (¯ b (cid:48) ) | ) − θ ( | h r (¯ b (cid:48) ) | − (cid:1) t r ≥ µ .Hence, V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ). (2-3) η (cid:96) (cid:48) r ≥ µ (cid:48) and η (cid:96)r < µ . From (33) and (34), h r ( i ∪ ¯ b (cid:48) ) = h r (¯ b (cid:48) ) ∪ { i } and h r ( { i } ∪ ¯ b ) = h r (¯ b ) \ { (cid:96) (cid:48) } ∪ { i } . Therefore, V r ( i | ¯ b (cid:48) ) = η ir − µ (cid:48) and V r ( i | ¯ b ) = η ir − η (cid:96)r . Since µ (cid:48) ≥ µ ≥ η (cid:96)r , weknow that V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ). (2-4) η (cid:96) (cid:48) r < µ (cid:48) and η (cid:96)r < µ . From (33) and (34), h r ( { i } ∪ ¯ b (cid:48) ) = h r (¯ b (cid:48) ) \ { (cid:96) (cid:48) } ∪ { i } , and h r ( { i } ∪ ¯ b ) = h r (¯ b ) \ { (cid:96) } ∪ { i } . Therefore, V r ( i | ¯ b (cid:48) ) = η ir − η (cid:96) (cid:48) r and V r ( i | ¯ b ) = η ir − η (cid:96)r . If | h r (¯ b (cid:48) ) | = | h r (¯ b ) | , then we must have η (cid:96) (cid:48) r ≥ η (cid:96)r , and hence V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ). On the otherhand, if | h r (¯ b (cid:48) ) | ≥ | h r (¯ b ) | + 1, then from (33) we have η (cid:96)r ≥ (cid:0) θ ( | h r (¯ b (cid:48) ) | ) − θ ( | h r (¯ b (cid:48) ) | − (cid:1) t r ≥ µ > η (cid:96)r . Therefore, we can also conclude that V r ( i | ¯ b (cid:48) ) ≤ V r ( i | ¯ b ).From all four subcases, we can conclude that in case 2, V r ( i | ¯ b ) ≥ V r ( i | ¯ b (cid:48) ).We now prove that V satisfies condition ( ii ) of Definition 5 by contradiction. Assume forthe sake of contradiction that (18) is not satisfied. Then, there must exist a group ¯ b ∈ ¯ B ,and i, j, k ∈ M \ ¯ b such that: V r ( i, j | ¯ b ) + V r ( k | ¯ b ) > V r ( i | ¯ b ) + V r ( j, k | ¯ b ) , ⇒ V r ( j | i, ¯ b ) > V r ( j | k, ¯ b ) , (35a) V r ( i, j | ¯ b ) + V r ( k | ¯ b ) > V r ( j | ¯ b ) + V r ( i, k | ¯ b ) , ⇒ V r ( i | j, ¯ b ) > V r ( i | k, ¯ b ) . (35b)We consider the following four cases: Case A: h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) ∪{ j } and h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) ∪{ j } . In thiscase, if | h r (cid:0) ¯ b ∪ { i } (cid:1) | ≥ | h r (cid:0) ¯ b ∪ { k } (cid:1) | , then V r ( j | i, ¯ b ) ≤ V r ( j | k, ¯ b ), which contradicts (35a).On the other hand, if | h r (cid:0) ¯ b ∪ { i } (cid:1) | < | h r (cid:0) ¯ b ∪ { k } (cid:1) | , then we must have h r (cid:0) ¯ b ∪ { i } (cid:1) = h r (¯ b )and h r (cid:0) ¯ b ∪ { k } (cid:1) = h r (¯ b ) ∪ { k } . Therefore, V r ( i | j, ¯ b ) = 0, and (35b) cannot hold. We thusobtain the contradiction. Case B: | h r (cid:0) ¯ b ∪ { i, j } (cid:1) | = | h r (cid:0) ¯ b ∪ { i } (cid:1) | and | h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) | . We furtherconsider the following four sub-cases: (B-1). h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) and h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) . In this case, V r ( j | i, ¯ b ) = V r ( j | k, ¯ b ) = 0. Hence, we arrive at a contradiction against (35a). (B-2). h r (cid:0) ¯ b ∪ { i, j } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { i } (cid:1) and h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) . In this case, when j is added to the set ¯ b ∪ { i } , j replaces a rider, denoted as (cid:96) ∈ ¯ b ∪ { i } . Since (cid:96) is replaced, wemust have η (cid:96)r ≤ η mr for any m ∈ h r (¯ b ∪ { j } ). If (cid:96) = i , then h r (¯ b ∪ { i, j } ) = h r (¯ b ∪ { j } ). Hence, V r ( i | j, ¯ b ) = 0, and we arrive at a contradiction with (35b). On the other hand, if (cid:96) (cid:54) = i , then31 is a rider in group ¯ b . This implies that (cid:96) ∈ ¯ b should be replaced by j when j is added to theset { k } ∪ ¯ b , which contradicts the assumption of this case that h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) . (B-3). h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) and h r (cid:0) ¯ b ∪ { j, k } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { k } (cid:1) . Analogousto case B-2 , we know that h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { j } (cid:1) and η jr ≥ η kr . Moreover, since h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) , we must have η jr ≤ η ir . Therefore, V r (¯ b ∪ { i, j } ) = V r (¯ b ∪ { i } ),and V r ( i | j, ¯ b ) = V r (¯ b ∪ { i } ) − V r (¯ b ∪ { j } ). Since η jr ≤ η ir and η jr ≥ η kr , we know that V r ( i | k, ¯ b ) = V r (¯ b ∪ { i } ) − V r (¯ b ∪ { k } ) ≥ V r (¯ b ∪ { i } ) − V r (¯ b ∪ { j } ) = V r ( i | j, ¯ b ), whichcontradicts (35b). (B-4). h r (cid:0) ¯ b ∪ { i, j } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { i } (cid:1) and h r (cid:0) ¯ b ∪ { j, k } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { k } (cid:1) . In this case, if h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { j } (cid:1) , then V r ( i | j, ¯ b ) = V r ( i, j, ¯ b ) − V r ( j, ¯ b ) = V r ( j, ¯ b ) − V r ( j, ¯ b ) = 0,which contradicts (35b). On the other hand, if h r (cid:0) ¯ b ∪ { i, j } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { j } (cid:1) , then one rider (cid:96) ∈ ¯ b must be replaced by j when j is added into the set ¯ b ∪ { i } , i.e. h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b \ { (cid:96) } ∪ { i, j } (cid:1) . Hence, η (cid:96)r ≤ η ir and η (cid:96)r ≤ η jr . If η (cid:96)r ≤ η kr , then under the assump-tion that | h r (cid:0) ¯ b ∪ { j, k } (cid:1) | = | h r (cid:0) ¯ b ∪ { k } (cid:1) | and h r (cid:0) ¯ b ∪ { j, k } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { k } (cid:1) , we must have h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b \ { (cid:96) } ∪ { j, k } (cid:1) . Then, we can check that V r ( j | i, b ) = V r ( j | k, b ), whichcontradicts (35a).On the other hand, if η (cid:96)r > η kr , then h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { j } (cid:1) . In this case, V r ( i | j, ¯ b )is the change of trip value by replacing (cid:96) with i , and V r ( i | k, ¯ b ) is the change of trip valueby replacing k with i . Since η kr < η (cid:96)r , we must have V r ( i | j, ¯ b ) < V r ( i | k, ¯ b ), which contradicts(35b). Case C: h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) ∪ { j } and | h r (cid:0) ¯ b ∪ { j, k } (cid:1) | = | h r (cid:0) ¯ b ∪ { k } (cid:1) | . Wefurther consider the following sub-cases: (C-1). h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) . In this case, η jr ≤ η mr for all m ∈ h r (¯ b ∪ { k } ), and η jr < θ ( | h r (¯ b ∪ { k } ) + 1 | ) − θ ( | h r (¯ b ∪ { k } ) | ). Since h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) ∪ { j } , we knowthat η jr ≥ θ ( | h r (¯ b ∪{ i } )+1 | ) − θ ( | h r (¯ b ∪{ i } ) | ). Since carpool disutility is non-decreasing in ridergroup size, for η jr to satisfy both inequalities, we must have | h r (¯ b ∪{ i } ) | < | h r (¯ b ∪{ k } ) | . Then,we must have h r (¯ b ∪{ i } ) = h r (¯ b ) and h r (¯ b ∪{ k } ) = h r (¯ b ) ∪{ k } . Therefore, V r ( i, j, ¯ b ) = V r ( j, ¯ b )and V r ( i, k, ¯ b ) = V r ( k, ¯ b ). Hence, V r ( i | j, ¯ b ) = V r ( i | k, ¯ b ) = 0, which contradicts (35b). (C-2). h r (cid:0) ¯ b ∪ { j, k } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { k } (cid:1) . Since | h r (cid:0) ¯ b ∪ { j, k } (cid:1) | = | h r (cid:0) ¯ b ∪ { k } (cid:1) | , j replacesa rider (cid:96) in ¯ b ∪{ k } , and η (cid:96)r ≤ (cid:96) mr for all m ∈ ¯ b ∪ k . If (cid:96) = k , then h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { j } (cid:1) .Therefore, V r ( j | i, ¯ b ) = η jr − (cid:0) θ ( | h r (¯ b ∪ { i } ) | + 1) − θ ( | h r (¯ b ∪ { i } ) | ) (cid:1) and V r ( j | k, ¯ b ) = η jr − η kr .If η kr ≤ θ ( | h r (¯ b ∪ { i } ) | + 1) − θ ( | h r (¯ b ∪ { i } ) | ), then (35a) is contradicted. Thus, η kr >θ ( | h r (¯ b ∪ { i } ) | + 1) − θ ( | h r (¯ b ∪ { i } ) | ). Since k is replaced by j when j is added to ¯ b ∪ { k } , wemust have η kr < θ ( | h r (¯ b ∪ { j } ) | + 1) − θ ( | h r (¯ b ∪ { j } ) | ). For η kr to satisfy both inequalities, wemust have | h r (¯ b ∪{ j } ) | > | h r (¯ b ∪{ i } ) | . Hence, h r (¯ b ∪{ j } ) = h r (¯ b ) ∪{ j } and h r (¯ b ∪{ i } ) = h r (¯ b ).Then, V r ( i | j, ¯ b ) = V r (¯ b ∪ { i, j } ) − V r (¯ b ∪ { j } ) = 0, which contradicts (35b).On the other hand, if (cid:96) ∈ ¯ b , then we know from (34) that η (cid:96)r < θ ( | h r (cid:0) ¯ b ∪ { k } (cid:1) | +1) − θ ( | h r (cid:0) ¯ b ∪ { k } (cid:1) | ). Additionally, since h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) ∪ { j } , we know from(33) that η (cid:96)r ≥ θ ( | h r (cid:0) ¯ b ∪ { i } (cid:1) | + 1) − θ ( | h r (cid:0) ¯ b ∪ { i } (cid:1) | ). If η (cid:96)r satisfies both inequalities,then we must have | h r (cid:0) ¯ b ∪ { i } (cid:1) | < | h r (cid:0) ¯ b ∪ { k } (cid:1) | . Therefore, h r (cid:0) ¯ b ∪ { i } (cid:1) = h r (¯ b ). Then, V r ( i | j, ¯ b ) = 0, which contradicts (35b). Case D: | h r (cid:0) ¯ b ∪ { i, j } (cid:1) | = | h r (cid:0) ¯ b ∪ { i } (cid:1) | and h r (cid:0) ¯ b ∪ { j, k } (cid:1) = h r (cid:0) ¯ b ∪ { k } (cid:1) ∪ { j } . Wefurther consider the following sub-cases: (D-1). h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) . In this case, analogous to (C-1) , we know that32 h r (¯ b ∪ { k } ) | < | h r (¯ b ∪ { i } ) | . Therefore, h r (¯ b ∪ { k } ) = h r (¯ b ) and h r (¯ b ∪ { i } ) = h r (¯ b ) ∪{ i } . Therefore, η kr < η ir . Additionally, since h r (cid:0) ¯ b ∪ { i, j } (cid:1) = h r (cid:0) ¯ b ∪ { i } (cid:1) , η jr < η ir . Then, V r ( i | j, ¯ b ) = V r ( i, ¯ b ) − V r ( j, ¯ b ) and V r ( i | k, ¯ b ) = V r ( i, ¯ b ) − V r (¯ b ). Since V is monotonic, V r ( j, ¯ b ) ≥ V r (¯ b ) so that V r ( i | j, ¯ b ) ≤ V r ( i | k, ¯ b ), which contradicts (35b). (D-2). h r (cid:0) ¯ b ∪ { i, j } (cid:1) (cid:54) = h r (cid:0) ¯ b ∪ { i } (cid:1) . Since | h r (cid:0) ¯ b ∪ { i, j } (cid:1) | = | h r (cid:0) ¯ b ∪ { i } (cid:1) | , j replacesthe rider (cid:96) ∈ ¯ b ∪ { i } such that η (cid:96)r ≤ η mr for all m ∈ h r (¯ b ∪ { i } ). If (cid:96) = i , then analogousto case C-2 , we know that if (35b) is satisfied, then | h r (¯ b ∪ { j } ) | < | h r (¯ b ∪ { k } ) | . Hence, h r (¯ b ∪ { j } ) = h r (¯ b ) and V ( j | i, ¯ b ) = 0, which contradicts (35a).On the other hand, if (cid:96) ∈ ¯ b , then again analogous to case C-2 , we know that | h r (cid:0) ¯ b ∪ { k } (cid:1) | < | h r (cid:0) ¯ b ∪ { i } (cid:1) | . Therefore, h r (cid:0) ¯ b ∪ { k } (cid:1) = h r (¯ b ), and h r (cid:0) ¯ b ∪ { i } (cid:1) = h r (¯ b ) ∪ { i } . Then, V r ( j | i, ¯ b ) = V r (¯ b \ { (cid:96) } ∪ { i, j } ) − V r ( i, ¯ b ), and V r ( j | k, ¯ b ) = V r (¯ b ∪ { j } ) − V r (¯ b ). Since (cid:96) (cid:54) = i , V r ( i | j, ¯ b ) = V r (¯ b \ { (cid:96) } ∪ { i, j } ) − V r ( j, ¯ b ) = η ir − η (cid:96)r . Additionally, since h r ( i, ¯ b ) = h r (¯ b ) ∪ { i } , V r ( i | k, ¯ b ) = V r ( i, ¯ b ) − V r (¯ b ) = η ir − ( θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | )). Since h r (¯ b ∪ { i } ) = h r (¯ b ) ∪ { i } and (cid:96) ∈ ¯ b , we know from (33) that η (cid:96)r ≥ θ ( | h r (¯ b ) | + 1) − θ ( | h r (¯ b ) | ). Therefore, V r ( i | j, ¯ b ) ≤ V r ( i | k, ¯ b ), which contradicts (35b).From all above four cases, we can conclude that condition (ii) of Definition 5 is satisfied.We can thus conclude that V satisfies gross substitutes condition. (cid:3) C Proof of Section 6
Proof of Lemma 9.
We first show that for any optimal utility vector u ∗ ∈ U ∗ , there exists avector λ ∗ such that ( u ∗ , λ ∗ ) is an optimal solution of (D k ∗ ). Since u ∗ ∈ U ∗ , there must exista toll price vector τ ∗ such that ( u ∗ , τ ∗ ) is an optimal solution of (D). Consider λ ∗ = ( λ ∗ r ) r ∈ R ∗ as follows: λ ∗ r = (cid:88) e ∈ r τ ∗ e , ∀ r ∈ R ∗ . (36)Since ( u ∗ , τ ∗ ) is feasible in (D), we can check that ( u ∗ , λ ∗ ) is also a feasible solution of (D k ∗ ).Moreover, since ( x ∗ , u ∗ , τ ∗ ) satisfies complementary slackness conditions with respect to (LP)and (D), ( x ∗ , u ∗ , λ ∗ ) also satisfies complementary slackness conditions with respect to (LP k ∗ )and (D k ∗ ). Therefore, ( u m ∗ , λ ∗ ) is an optimal solution of (D k ∗ ).We next show that for any optimal solution ( u ∗ , λ ∗ ) of (D k ∗ ), we can find a toll pricevector τ ∗ such that ( u ∗ , τ ∗ ) is an optimal solution of (D) (i.e. u ∗ ∈ U ∗ ) on the originalnetwork. We prove this part by mathematical induction: First, if the network has a singleedge e , then τ ∗ e = λ ∗ e is the toll price vector. Second, if the network is parallel, then τ ∗ e = λ ∗ e for all parallel edges e ∈ E is the toll price vector. Third, if the argument holds on two seriesparallel networks G and G , then there exist vectors τ ∗ and τ ∗ such that ( u ∗ , τ ∗ ) and( u ∗ , τ ∗ ) are optimal solutions of (D) restricted on the sub-network G and G , respectively.Then, we can check that τ ∗ = ( τ ∗ , τ ∗ ) is feasible in (D) and achieves the same objectivefunction as the sum of that restricted in each one of the two sub-networks when the twonetworks are connected in series or in parallel. From Lemma 1, we know that the optimalvalues of the both dual problems equal to the optimal social welfare given x ∗ . Thus, ( u ∗ , τ ∗ )is also an optimal solution of (D). (cid:3) roof of Lemma 10. For any u ∗ ∈ U ∗ , we define λ ∗ = ( λ ∗ ) r ∈ R as follows: λ ∗ r = max ¯ b ∈ ¯ B V r (¯ b ) − (cid:88) m ∈ ¯ b u m ∗ , ∀ r ∈ R ∗ . Analogous to the proof of Lemma 3, we can show that ( y ∗ , u ∗ ) is a Walrasian equilibriumif and only if ( y ∗ , u ∗ , λ ∗ ) satisfies the feasibility constraints of (LP k ∗ ) and (D k ∗ ) and thecomplementary slackness conditions. Therefore, ( u ∗ , λ ∗ ) must be an optimal solution of(D k ∗ ) and u ∗ ∈ U ∗ . (cid:3)(cid:3)