Route-cost-assignment with joint user and operator behavior as a many-to-one stable matching assignment game

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1 Route-cost-assignment with joint user and operator behavior as a many-to-one stable matching assignment game Saeid Rasulkhani, Joseph Y. J. Chow* C SMART University Transportation Center, Department of Civil & Urban Engineering, New York University, New York, NY, USA *Corresponding author joseph.chow@nyu.edu Abstract We propose a generalized market equilibrium model using assignment game criteria for evaluating transportation systems that consist of both operators and users: fixed route transit, multimodal systems, and Mobility-as-a-Service in general. The model finds stable pricing and matches between users in a network with a set of routes with line capacities. The proposed model gives a set of stable outcomes instead of single point pricing that allows operators to design their ticket pricing based on a desired mechanism or policy (as opposed to other matching studies that focus on operating a specific mechanism design). We look at three different examples to test our model. As a case study we evaluate taxi ridesharing in NYC. Based on the taxi data and study area in Lower Manhattan, the model justifies a market share of 71.8% of users that would take shared rides. Sharing taxi results in a 31.4% decrease in travel mileage. The model suggests that adopting a policy that can require passengers to walk up to 1 block away to meet with a shared taxi would require the operating cost savings to exceed the $0.44/traveler by another $0.13 if they do not wish to turn away any of these passengers. Keywords: transportation network assignment, Mobility-as-a-Service, stable matching, service network design, assignment game

2 1. Introduction Planning for mobility in a smart cities era requires an understanding beyond the route choices of travelers. With the increasing ubiquity of multiple forms of Mobility as a Service (MaaS) options to travelers (e.g. conventional fixed route transit, flexible transit, rideshare, carshare, microtransit, ridesourcing) provided by both public agencies and private operators known as transportation network companies (TNCs), travel forecast models need to focus on both the decisions of travelers and operators (Djavadian and Chow, 017a). For example, a person s decision to take one mobility service over another may depend on the travel performance of that service, but the performance in turn depends on the operator s cost allocation decisions to best serve their users. Table 1 illustrates the broad range of cost allocation decisions exemplified by different types of mobility systems. Table 1. Illustration of cost allocations Cost allocation Cost transfer Example systems Fare User Operator Public transit, taxi, on-demand ridesharing, vehicle sharing Wait time Operator User Public transit, taxi, on-demand ridesharing Access time Operator User Public transit, vehicle sharing Detour time User User Public transit, on-demand ridesharing Reservation time Operator User Vehicle sharing, on-demand ridesharing Capacity reliability Operator User Public transit, vehicle sharing Credit/discount for switching pickup/drop-off location Operator User Public transit, on-demand ridesharing, vehicle sharing Fare splitting User User Public transit, on-demand ridesharing In this table, cost transfers refer to the direction of cost allocation: for example, a fare is a cost to a user that is transferred as a benefit to the operator. When planning for these systems, modelers need to forecast how operators decide their cost allocations in order to forecast the route flows. The success and failure of various systems (e.g. Kutsuplus in Helsinki (Kelly, 016), CarGo in San Diego (Krok, 016)) depend on forecasting the ridership, which is linked to the cost allocation decisions of those systems and the structure of other mobility options in their respective regions. State-of-the-art techniques tend to keep these two decisions separate. For example, traffic or transit equilibrium models are designed without any operator response to obtain route flows that satisfy Wardrop user equilibrium principles. Using conventional transportation assignment methods, highly complex bilevel models with upper level Nash equilibrium are needed (e.g. Zhou et al., 005). There are models to forecast certain equilibrium patterns like taxi-passenger matching (Yang et al., 010) and ride-sourcing supply-demand equilibrium (Zha et al., 016). The problem is that such methods do not allow city agencies to evaluate across multiple service types to compare alternatives and substitution effects between service designs. System optimization models like vehicle sharing (Chow and Sayarshad, 014) or ridesharing optimization (Masoud and Jayakrishnan, 017; Wang et al., 017) assume inelastic user demand. Those are generally normative models meant to be decision support tools for the operators, not as policy analysis tools for public agencies. An overview of this gap in the literature is given by Djavadian and Chow (017a, b), who also propose simulation-based methods to evaluate such systems. However, a major drawback is that sensitivity analyses cannot be conducted and the fundamental structure of the interactions is not clearly understood. For instance, the stability of a

3 certain cost allocation strategy may be simulated, but the tool does not provide analytical thresholds to consider perturbations in a strategy or for transferability to other instances. We propose a new transportation network assignment model framework based on stable matching for this purpose; it simultaneously considers user route behavior, service operator route selection decisions, and the resulting cost allocations and pricing mechanisms needed to reach a stable state. Stable matching theory (Gale and Shapley, 196) shows equilibrium between two disjoint sets (buyers and sellers). If transport services are the sellers and travelers are the buyers, finding their supply-demand equilibrium is a stable matching problem in a generalized sense. Two primary contributions are made in this study: a methodological one and an empirical one. A model formulation and solution method are proposed and the model s stable matching properties are analyzed using a simple illustrative example. We then demonstrate the applicability of the model to evaluate a public policy: allowing taxis to share rides in New York City (which was recently adopted through a partnership with ride matching provider Via (Hu, 017)), and show how our model can provide new insights that prior studies (e.g. Santi et al., 014; Ma et al., 017; Alonso-mora et al., 017) may have missed.. Review of assignment games The stable matching problem has a long literature. Gale and Shapley (196) first studied the problem through two applications: the marriage problem for one-to-one matching, and the college admissions problem for many-to-one matching. Shapley and Shubik (1971) formulated a linear program (LP) called the assignment game for matching problems that have transferable utilities. They defined the core as a region derived from the dual variables corresponding to the LP of the assignment game where players do not have incentive to change their matched partner(s). Consider two disjoint sets denoted by P for buyers and Q for sellers. A buyer i P that matches with a seller j Q earns a utility of U ij. The item has a cost of production of c j for seller j. The difference between utility and cost of production can be interpreted as payoff a ij = max(0, U ij c j ). The assignment game is essentially a game of outcome splitting. It is a cooperative game (P, Q, a) wherein the players get payoffs by forming coalitions with each other. Each pair of i and j who make a coalition win a payoff with the value of a ij. The goal in the assignment problem is to match sellers and buyers in a way that the generated payoff from their coalitions is maximized. Note that this is not an equivalent to Wardrop s user equilibrium and system optimal concepts which refer to competition between users when payoffs depend on their collective choices. Here we are looking at cooperation between buyers and sellers such that the outcome is consistent with behavior. A successful match means the seller transfers the utility to the buyer with a price p. Based on this price each side s profit from the match can be understood. Each buyer who satisfies her utility earns a profit from the difference between utility and the price of the item, u i = U ij p. Each seller profits from the price sold minus the cost of production, v j = p c j. A review of the basic principles of the assignment game is provided in Roth and Sotomayor (1990). The model formulation is shown here in Eq. (1) (4). 3

4 max a ij x ij i P j Q s.t. (1) x ij i P q j j Q () x ij w i j Q i P (3) x ij {0,1} j Q, i P (4) In the model, x ij is a binary variable which shows whether buyer i and seller j are matched or not. The parameters q j and w i are the quotas of each side. If q j and w i are equal to one, the assignment game becomes a one-to-one or simple assignment game. Because this corresponds to a transportation problem which conforms to the Unimodularity Theorem, the optimal solution of the LP relaxation has also a binary solution. In the simple assignment game, an outcome ((u, v); x) is feasible if u i and v j are non-negative and i P u i + j Q v j = i P a ij x ij. A feasible payoff is stable if u i + v j = a ij when x ij = 1, j Q and u i + v j a ij when x ij = 0. The core is the set of solutions of the dual corresponding to the assignment game. In multiple partner assignment game (Sotomayor, 199), the profit received by buyer i (seller j) from matching to seller j (buyer i) is defined as u ij (v ij ). The unmatched buyers and sellers are assumed to be matched to a dummy seller or buyer, respectively (x ij0 and x i0 j), with a payoff of zero. If C(i, x) and C(j, x) are defined as the set of matched players to i and j, respectively, under optimal assignment of x, then u i is the minimum of u ij for each j C(i, x) and v j is the minimum of v ij for each i C(j, x). Stability of payoffs in the many-to-many assignment games implies that the feasible outcome ((u, v); x) is stable if u ij + v ij = a ij when x ij = 1 and u i + v j a ij when x ij = 0. Also u i and v j are non-negative for all i and j. Sotomayor (199) showed circumstances where the multiple partner game can be converted to an equivalent one-to-one assignment game to find the core. Doing so, she showed that the set of stable matches in multiple partner games lie within (not necessarily equivalent to) the core. Sotomayor (1999) found that the set of stable outcomes in the multiple partner game is a convex and compact lattice, and there exists a single optimal point for each side (buyer, seller) leading to buyer-optimal or seller-optimal outcomes. If we can model the matching of travelers as buyers to transport operated routes as sellers of transferable utility, then we can use the properties of the many-to-one assignment game to identify the stable outcomes as well as the corresponding user-optimal and operator-optimal outcomes within that space. The main challenge is defining a model structure where sellers are defined as routes with line capacities. We use the term line in reference to service lines (see Schöbel, 01) in a network in which multiple lines may share a link but the capacity effect of interest is within the vehicle, not at the link level. This is the case with MaaS systems: for example, bike sharing systems have capacity effects at stations and station-pairs, not at which arterials are being used to 4

5 ride a bike down. Further, two different users can match a route using two different node pairs, but share the same line level quotas or capacities, all while maintaining a unique stable outcome space. These points will be illustrated in more detail in Section 3. While there are transportation applications using stable matching or cooperative game theory as shown in Table, none have considered both of the following: (1) assignment of travelers onto a route that can be composed of a sequence of nodes with line capacities, and () route-level cost allocation decisions of operators. Table. Summary of sample literature on transportation applications of stable matching and cooperative games. Reference Type of network Allocation decision Mechanism Matsubayashi et al. (005) Node pair two types of cost, Set of stable cost allocation in constant + coalition core Players own capacities Agarwal and Ergun (008) network on links and also There is no cost allocation demands then cooperate Anshelevich et al. (013) Bipartite Utility as a random value, cost of stability N.A. Two types of cost Hernandez and Peeta Route (LTL trucks) policy: fixed cost + (014) variable cost There is no cost allocation Wang et al. (017) Dai and Chen (015) Furuhata et al. (015) Stiglic et al. (015) Hezarkhani et al. (016) Aghajani and Kalantar (017) Masoud et al. (017) Qian et al. (017) Rosenthal (017) Hara and Hata (017) Route Nodes N.A. Route Route N.A. Node and routes (matching pp ridesharing system) Network of NYC and taxi data Network Node Nourinejad and Roorda (016) Network Alonso-Mora et al. (017) Network Dynamic ridesharing, matching individual cars to each other Finding profit allocation inside core, efficiency measure Online cost sharing mechanism Finding meeting point with minimizing cost Cooperative truck load delivery Maximizing parking owner profit Rider is charged for costs Incentives to taxi riders to share their cab Cooperative game to cost allocating in rapid transit Auction for car/bike sharing Decentralized and centralized models for dynamic ridesharing Algorithm for dynamic ridesharing matching and relocation of idle cars Cost is divided equally between two matched pair Profit allocation is done by egalitarian view Different types: equally splitting, incremental etc. Matching demand to routes through matching problem Gain sharing is done in a procedure to be fare Seller side optimal Two types of cost: fixed cost (deviation from main route), variable cost (distance based) Best incentive that maximize efficiency of GRG Equally sharing Giving incentives to avoid imbalance problem Single-shot first-price Vickrey auction price No cost allocation 5

6 3. Proposed model 3.1. Definitions and model formulation Unlike conventional transportation assignment models, the proposed model outputs not only traveler route flows and link performances, but also the space of stable cost allocations at the route level (fare prices, other generalized traveler cost transfers like additional walking or waiting time, etc.). It explicitly considers the incentive behavior of both travelers and operators. The model is used to evaluate service operating policies illustrated in Table 1 by determining stability of a policy and its cost allocation bounds for given demand patterns. The most basic formulation setting is a static many-to-one assignment game. Consider a graph G(N, A) in which there is a set R of mobility operators, where each route r R is assumed to represent a separate seller, and a set S of users looking for service. A dummy user k is created to match with routes that are not matched with any user. A route r R is assumed to have only one sequence of links a A r between any pair of nodes served, where A r A are disjoint sets. For example, for a 4-node network, one route may be , and another may be A user that travels from node 1 to node 4 would have to visit, 3, and 4 if matched with the first route, and only node 4 if matched with the latter. Variations can be modeled. For example, direct services like taxis may be represented by single link routes. A route can also be replaced with a cycle without altering the model. For operators that serve a network of multiple overlapping routes under this model setting, we assume that there is no cost transfer between routes, and hence each route is treated as a decentralized operator. If the operator is a single centralized decision-maker, then the matching would be made between the set of users with a single seller. The line level capacities would remain. Operators may be tolled road operators, public transport providers, taxis, or TNCs, among others mentioned earlier. One route (one) may be matched to multiple travelers (many). A traveler is assumed to use only one route to get from one node to another. The following input parameters are needed. A match between route r and a user or set of homogeneous users s S imposes a generalized travel cost to the user, t sr, that depends on the origin-destination (OD) of user s. This cost may be different for one user than another. For example, a user traveling from node 1 to node 4 on route has a travel cost of visiting, 3, and then 4, whereas a traveler from node to 3 on the same route would only incur a travel cost of visiting 3. The cost is fixed (there is no crowding effect on the line, just a hard capacity). Each user gains a utility U sr when matched to a route r such that the net payoff is a sr = max {0, U sr t sr }. When we set the utility for the next best travel option outside of the available options among R to be zero, the U sr is defined so the payoff a sr represents the savings from that outside option. For a traveler with no other travel options (or if R comprehensively includes all possible travel options), the next best travel option is simply to not make the trip. If we define b sr to be the minimum benefit acceptable for operator r to match with user s, and g sr as the minimum acceptable benefit for user s to match with route r, then the payoff value is a sr = max {0, U sr t sr g sr b sr }. These minimum acceptable values just make the payoff values smaller and all the rest of characteristics of the proposed model remain same. As a result, for the rest of this paper we assume the minimum acceptable profit for both users and operators is equal to zero. Each route has an operating cost C r which requires the operator payoff allocation to exceed. For private operators, the fare payment portion of the allocation needs to exceed C r as some of the 6

7 other user payoffs like travel time savings may not be transferable to offset operating cost. The cost of route will be divided between the users of that route (c sr ). Routes have line capacities defined as w a, a A r. An indicator δ asr is set to 1 if a match between user s and route r requires using link a, and 0 otherwise. In fact, this capacity limits the route not to have more than a certain amount of users at any time. In our proposed model, users may use just some part of route and so the line capacity constraint is imposed on the links of that route. As a result of the parameters, the output of the model is a set of matches x sr and the region of user profits u sr and operator profits v rs that are stable. In this model, a single buyer s can represent a homogeneous group of travelers (e.g. for a single OD pair), to allow for population assignment. The q s to a demand for homogeneous travelers of commodity s. For a given set of user and operator profits, there is a cost allocation p sr charged to the user by the matching route. Figure 1 illustrates the model setting; based only on travel costs (as typically done in traffic assignment models) Feasible Output seems to be the preferred choice, but based on the choices of other passengers, the operating cost C r, and the savings from other travel options, Feasible Output 1 might be the chosen one. Model inputs Feasible Output 1 Feasible Output Each output Flow? (x sr ) 3 Price? (p sr ) User performance? (u sr ) Operator performance? (v rs ) Legend i i j Origin of user i Destination of user i Node j on a route Figure 1. Illustration of explaining different assignment configurations using the proposed model. 7

8 The proposed model s primal formulation is shown in Eq. (5) (9). max a sr x sr s S r R s.t. (5) x sr r R q s s S/{k} (6) s S/{k} δ asr x sr w a a A r, r R (7) x sr M(1 x kr ) s S/{k} r R (8) x sr Z + s S/{k}, r R (9-a) x kr {0,1} r R (9-b) Eq. (5) is the standard assignment game objective to maximize payoffs. Eq. (6) and (7) correspond to the matching quotas for a many-to-one assignment game. The line capacity for the route operators is met on their links. Eq. (8) is a new constraint that ensures the routes are only matched if the sum of all the matched payoff allocations exceeds the operating cost. By setting a kr = C r, r R, a route would only be matched if the total payoff (in generalized user utility converted to operating dollars) exceeds the operating cost. Eq. (9) are the integral constraints. The payoff value used in the objective function in Eq. (5) is defined as a sr = max {0, U sr t sr }. The payoff value that classically defined in Shapley and Shubick (1971) as a net payoff value which translates to: a sr = max {0, U sr t sr c sr }. The issue is that based on our proposed model we don t know c sr in advance because it depends on the value of x sr. We claim that our model is equivalent to the case where Shapley and Shubick (1971) defined net payoff. We will show this in the proposition 1. Assumption 1. Operating cost C r can be divided into different c sr for each user s that is matched to r (C r = r R s S/{k} c sr x sr ). Proposition 1. Model (5) (9) is equivalent to the classic assignment game with net payoff value of a sr = max {0, U sr t sr c sr }. Proof. First we will prove this proposition with demand value of one (q s = 1 s S/{k}) and then we will generalize this proof to the general case. So for now, we assume all the x sr are binary instead of nonnegative integers. Objective function of model (5) - (9) can be written as Eq. (10). 8

9 max a sr x sr s S r R = max ( (U sr t sr )x sr + C r x kr ) (10) s S/{k} r R r R The right hand side of Eq. (10) consists of two parts. The second is the total route cost of unused routes. Let s define R R as the set of used routes. Then the second part of Eq. (10) can be written as Eq. (11). C r x kr r R = C r r R C r r R (11) Now the objective function of the main model (Eq. (5)) can be rewritten again as Eq. (1). max a sr x sr s S r R = max ( (U sr t sr )x sr + C r s S/{k} r R r R C r ) r R (1) The term r R C r is constant and can be eliminated from objective function (1). Based on what is defined, r R C r can be written as r R s S{k} c sr x sr. Eq. (1) becomes Eq. (13). max ( s S/{k} r R (U sr t sr )x sr + C r c sr x sr ) r R r R s S/{k} (13) = max ( (U sr t sr c sr )x sr + C r ) s S/{k} r R This net payoff is equivalent to Shapley and Shubik (1971). In the case of demand greater than one (q s > 1), let s assume set H as set of these q s agents ( H = q s ). We can consider these agents as separate users with the quota of 1 for their demand (q f H = 1). With this assumption with a similar proof this proposition holds. Another issue to consider is the route enumeration. Our model assumes a choice set of routes to begin with and assumes that there is only one user path associated with each operator route-user pair. In conventional traffic assignment, route enumeration is not desired because practical networks may have extremely large route sets. However, the proposed model deals with MaaS which focus less on the user route choice and more on the incentives and strategies of users and operators to facilitate the interaction. In practice, a modeler will likely not be analyzing a network with tens of thousands of routes or more; instead, they will analyze a set of several hundred existing operator routes. There still may be tens of thousands of user groups (e.g. 100 zones lead to 10,000 OD pairs), but the user paths for each group is searched within each operator route, which is a r R 9

10 trivial computation. In the computational experiments, the routes are generated simply with different combinations of origins and destinations. Future research may consider endogenous equilibration of the route set as well (e.g. as studied by Watling et al., 015, and Rasmussen et al., 015). Lastly, the utility parameter U sr needs to be accurately calibrated in advance. For user groups, this parameter would have to be representative of a population. The parameter will vary over time, as the benefit of a trip will change by time of day. The value will also depend on the presence of other transport options outside of the set of R being considered, and on the types of trip purposes at the destination. For example, a non-compulsory trip may relate primarily to not making a trip at all, whereas a compulsory work trip may have a utility based on comparing against another travel mode even if only walking is available. Parameter estimation for mathematical programming models can be done using inverse optimization (Ahuja and Orlin, 001; Xu et al., 017). This involves establishing a prior estimate of the parameters and adjusting those parameters over multiple observations such that matched flows are optimal in the model. Since the model is a straightforward mixed integer program, there should not be any complications with solving an inverse IP (see Wang, 009). 3.. Stable matching properties The optimal solution to the assignment game determines the assignment of users and operators to each other. Properties of the cost allocation space can be defined. Let s consider a cost transfer (e.g. fare price, among others shown in Table 1) from user s to operator route r, p sr and v rs be the profit that operator r earns from serving user s. By definition, p sr = c sr + v rs. Let u sr be the value gained by user s from matching with operator route r. It is important to note that cost allocation between the users of a route, should be done in a way that that allocation stays stable (equations (14), (15)). Before starting to define stability criteria, let us define the notation used. We define D(r, x) as set of users that under x assignment are currently unmatched to r, C(r, x) as the set of users that are matched to r (except the unmatched quota we assumed that are matched to null user) (D(r, x) C(r, x) = S/{k}). τ D(r, x) represents a user that is not currently matched to r, and E(τ, r, x) are the set of combinations of s C(r, x) that should break their coalition with r in advance to make τ D(r, x) eligible to match. μ rτ is the profit that route r loses if it makes a coalition with user τ. As we defined earlier R is the set of routes that are matched to at least one user. Definition 1. (Sotomayor, 199) outcomes u sr, c sr and v rs are feasible and denoted by ((u, v, c); x) if: (i) u sr + v rs = a sr c sr if x sr = 1 and u sr 0, v rs 0, c sr 0 s S, r R (ii) u sr = v rs = c sr = 0 if x sr = 0 (iii) s C(r,x) c sr = C r for all r R Proposition. A feasible outcome ((u, v, c), x) of the assignment game in Eq. (5) (9) is stable if it meets the following criteria in Eq.(14) - (15). u s + μ rs a sr s D(r, x), r R (14) u s a sr c r s S, r R R (15) 10

11 Proof. Note that E(τ, r, x) consists of set of combinations (which we name them obstacles of τ D(r, x)) that with removal of one, τ D(r, x) can be matched to r. If τ D(r, x) has no obstacle, E(τ, r, x) =. γ E(τ, r, x) is a set of one or more users. The aggregated values for each user and operator are defined as shown in Eq. (16) (17). μ rτ = {min π rγ π rγ = (v rs + c sr ) γ E(τ, r, x),} τ D(r, x), r R (16) s γ u s = {min r C(s,x) u sr } s S\{k} (17) Eq. (14) shows the stability for the routes that are already matched to at least one user (r R ). To have a stable match in this case, the user that is not currently matched to such route (s D(r, x)) and the route should not have incentive to encourage the user to break its current coalition to be matched to the new route. For this to happen, the summation of their current outcomes should be greater than what they can generate together. Eq. (14) shows that the value user s currently gets plus the value route r may lose if it wants to make coalition with s should be greater than their payoff. Since the route already has operated (r R ), the new user doesn t have to pay any share for the cost of operation for the route. Eq. (15) shows the stability for the case of routes that are unmatched currently (r R R ). Obviously S = D(r, x) of such routes. Also, μ rτ = 0 τ S, r R R because there is no obstacle for any of the users to be matched to such routes. The left hand side of Eq. (15) will be only users current payoffs. But the route cost is also considered in the right hand side of Eq. (15). The reason is that because such route is not currently operated and if a user wants to switch to it, they should pay all the route s operating cost by themselves. Looking at line capacity makes the stability condition of matches more complex than conventional ones. A small example can be helpful to illuminate the defined concepts in the previous paragraphs. As shown in Figure, suppose there is a set of three routes {r 1 = ( ), r = (3 6), r 3 = (5 4)} where each has a quota of on each of its links, and a set of four users S = {(1,6), (3,), (5,4), (3,6)}. Under assignment x, s 1, s, s 3 and s 4 are respectively matched to routes r 1, r 1, r 1 and r. Then we have: C(r 1, x) = {s 1, s, s 3 }, C(r, x) = {s 4 }, C(r 3, x) = and D(r 1, x) = {s 4 }, D(r, x) = {s 1, s, s 3 }, D(r 3, x) = {s 1, s, s 3, s 4 } and E(s 4, r 1, x) = {{s 1 }, {s, s 3 }}. r 1 revenue from serving users s 1, s and s 3 is v 11, v 1 and v 13 plus their share of operating cost c 11, c 1 and c 31 respectively. Stability (Eqs. (14) and (15)) between s 4 and r 1 implies u 4 + min{(v 11 + c 11 ), ((v 1 + c 1 ) + (v 13 + c 31 ))} a

12 Figure. Users and routes interaction in stability. Proposition 3. In the case with q s > 1, the benefits (u sr ) that users gather from matching to different operators are equal. Proof. For a user s S/{k} that has q s > 1, let s define the set of this q s as H. Since the user group s is matched to more than one route, user s can be considered as q s users with q s H = 1. Let s say {s, s"} H and they are matched to routes r and r" respectively. The feasibility and stability of outcomes (Eqs (14), (15) and (3)) implies Eqs. (18) to (1). u s r + v r s = a s r c s r (18) u s " r " + v r " s " = a s " r " c s " r " (19) u s r + v r " s " + c s " r " a s r " (0) u s " r " + v r s + c s r a s " r (1) By knowing a s r " = a s " r " or a s " r = a s r (because agents s and s" are completely identical) and using Eq. (18), Eq. (0) can be rewritten as u s r + v r " s " a s r " c s " r " = a s " r " c s " r " = u s " r " + v r " s " u s r u s " r ". In similar way and by using Eq. (19), Eq. (1) can be rewritten as u s " r " + v r s a s " r c s r = a s r c s r = u s r + v r s u s " r " u s r. From u s r u s " r " and u s " r " u s r, we can easily conclude that u s r = u s " r ". From Proposition 3, it can be concluded that u s = u sr = u sρ r ρ & r, ρ c(s, x). Proposition 4. The set of stable outcomes (Eqs. (14), (15)) is a convex space. 1

13 Proof. Definition of stability implies that set of feasible outcomes are stable if the Eq. (14) or (15) hold. Since Eq. (15) is a linear inequality, it builds a convex space. Eq. (14) can be rewritten in the form of Eq. (). u s + min{ π rγ π rγ = (v rs + c sr ) s γ γ E(s, r, x)} a sr s D(r, x), r R () Since the left hand side of Eq. () is a minimization which is greater than the right hand side, it can be rewritten in separate equations like Eq. (3). u s + (v rτ + c τr ) τ γ a sr s D(r, x), r R γ E(s, r, x) Eq. (14) is rewritten in the form of Eq. (3) which is a set of linear equations and forms a convex space. Therefor it can be said that set of stable outcomes is a convex space 3.3. Solution method The proposed model is conveniently cast as an integer programming problem with quota constraints. Any conventional IP solution algorithm can be applied to solve this model. There are several approaches which can be divided into three main categories of methods: cutting planes (e.g. reducing feasible area by adding extra constraints), heuristic methods (e.g. search methods), and enumeration techniques (e.g. Branch and Bound). Commercial optimization packages use one or combine some of these methods to solve the integer programming problems. For convenience we used Matlab s intlinprog solver for the numerical examples in this study, as it uses a variant of the branch and bound algorithm to obtain a solution. Having found a set of stable matching outcomes, an objective function is defined to find the optimal stable cost sharing for the policy. Based on the outcomes of Proposition and 4, the stable cost sharing is expressed as a mathematical programming model shown in Eq. (4) to (30). The decision variables are u s, v rs and c rs. The constraints (5) and (30) are related to definition of stability in Proposition but in linear form as shown in Proposition 4. The objective function max H can be any desired objective function for design a cost allocation mechanism, such as maximizing revenue, social welfare, fairness, etc. When Z is set as either max s S u s and max s S r R v rs, the solutions obtain the user optimal and operator optimal outcomes, respectively. These two objective functions are the two opposite vertices of the space of the set of stable outcomes. Any other objective function gets the optimal solution for outcome allocation that lies between these two boundaries. The resulting model is a simple LP for obtaining the cost allocations under the stable matching. For convenience, we use a commercial solver in Matlab, linprog, for the computational experiments below, although other commercial solvers can also be used. In the computational experiments we included just u s and v rs in the objective function of Eq. (4) to (30) and not c rs. The reason is that we mostly were interested in looking at interactions between users and operators. By considering the c rs in the objective function, we would control for the interactions between users as well which is an interesting point for future studies. (3) max Z (4) 13

14 s.t. u s + (v rs + c sr ) s γ a sr s D(r, x), r R γ E(τ, r, x) (5) u s a sr c r s S, r R R (6) u sr + v rs = a sr c sr s S, r R x sr = 1 (7) u sr = v rs = c sr = 0 s S, r R x sr = 0 (8) s C(r,x) c sr = C r r R (9) c rs 0, v rs 0, u s 0 s S, r R (30) 3.4. Illustration Network parameters To illustrate how this model works, we present a simple 4-node network shown in Figure 3. The number over each link in the figure represents the travel time between the nodes. In this network, we allow for all possible routes to be enumerated, of which there are 5 candidate routes (R = {r 1,, r 5 }). We consider a cost structure based on the number of links in the route (a route with more links has higher operating cost), which is set equal to C r = A r Figure 3. 4-node network example. Two different problem settings are considered. The first illustrates individuals with binary decisions, where no capacity is assumed (w a Ar ). In the second case, each user represents 5 individuals that want to make trip. The number of users on a route should not exceed a value of (line capacity of ) at any time. These examples can be interpreted as transit routes that serve a population of travelers Binary, non-capacitated case A set of demand OD pairs between some of the OD pairs are randomly chosen (S = {(1,), (1,3), (,3), (3,), (4,1), (4,)}) for this experiment. The utility (U rs ) for conducting each trip for all the users is set to a constant value equal to 0 (U sr = U = 0 s S/{k}, r R). 14

15 After solving the assignment game model of Eq. (5) (9), three different optimal assignments emerge as shown in Table 3a and Figure 4. Solving the LP in Eq. (4) to (30) with Z equal to s S u s and s S r R v rs results in the payoffs for user-optimal and operator-optimal cases as shown in Table 3b and 3c. Table 3a. Result of assignment game for four node network Optimal r solution X X X number of route Links of route Cost of route Users (1,) (1,3) (,3) (3,) (4,1) (4,) Table 3b. Ticket prices in operator-optimal allocation Cost User Optimal Route Links of route of Solution route (1,) (1,3) (,3) (3,) (4,1) (4,) X X X Operator revenue Table 3c. Ticket prices in user-optimal allocation mechanism Cost User Optimal Route Links of route of Solution route (1,) (1,3) (,3) (3,) (4,1) (4,) X X X Operator revenue In this example each user s share from the operating cost of the routes is assumed to follow C an equal splitting mechanism; the user s cost is c rs = r. It is obvious that the s C(w x r,x) sr denominator of the right hand side will never be zero because the equation is for routes that have been matched to at least one user. Although this strategy gives us a stable cost sharing in this example, it is not guaranteed and should be checked to verify its stability. Under the X matching solution in Table 3a, c 9,1 = c 9, = c 9,3 = C 9 3 = = 1.83 and c 4,4 = c 4,5 = c 4,6 = C 4 3 = 6 3 =. 15

16 Route 9 X 1 3 Route Route 1 Route 51 X Route 6 X 4 3 Route s User s indicator n Node n of network Figure 4. 4-node example user route matching in different optimal solutions. Based on feasibility of outcomes, matched partners have the following payoffs: u 1 + p 19 = 17; u + p 9 = 15; u 3 + p 39 = 18; u 5 + p 4,4 = 16.5; u 5 + p 5,4 = 15; u 6 + p 6,4 = 18 or u 1 + v 9,1 = 15.17; u + v 9, = 13.17; u 3 + v 9,3 = 16.17; u 4 + v 4,4 = 14.5; u 5 + v 4,5 = 13; u 6 + v 4,6 = 16. Both b rs and g sr are assumed to be zero. The result for payoff splitting under the three different optimal assignment solutions between the players is shown in Table 3b and 3c. In the user-optimal (operator-optimal) allocation, the ticket prices are as small (big) as possible and the operators (users) will gain no profit in this matching. Since the set of stable outcomes is convex, every other ticket price between these two ticket prices are also stable ticket price. In summary, this test demonstrates how a given network of service routes would be matched to a population of users and the corresponding unique price allocation space in which no user or operator would unilaterally switch partner without being worse off. Two allocation policies 16

17 representing extreme ends of the stable outcome space are shown: a user-optimal allocation and an operator-optimal allocation Continuous, capacitated case In this scenario, the users from the prior setting are modified into user groups, each representing five agents, q s = 5 s S/{k}. A line capacity is added to each route as well, where w a = a A r, r R. The result of the assignment game is shown in Table 4 and Figure 5. Due to capacity, some users switch to other operator routes. The user-optimal and operator-optimal allocations from Eq. (4) to (30) are shown in Table 4b and 4c, respectively, illustrating the range of stable prices that exist for this scenario. In Figure 5 the number of passengers is written on the edge from user to network node. Table 4a. Result of assignment game for 4-node network with demand and route link capacity r User group route Cost Links of route (1,) (1,3) (,3) (3,) (4,1) (4,) number of route Total Table 4b. Ticket prices in user-optimal allocation mechanism Cost User group Operator Route Links of route of revenue route (1,) (1,3) (,3) (3,) (4,1) (4,)

18 Table 4c. Ticket prices in operator-optimal allocation mechanism Cost Ticket price Operator Route Links of route of revenue route (1,) (1,3) (,3) (3,) (4,1) (4,) This test demonstrates how the stable matching assignment game can be applied to OD-level population groups in a network of routes that exhibit link capacities. The results show that a stable matching can be found along with a corresponding unique stable outcome space for determining stable cost allocations. Route Route Route Route

19 Route Route Route Route s User s indicator n Node n of network Figure 5. matching of users and network nodes in 4-node network example with capacity 4. Case study 4.1. Experimental design Having demonstrated how the model works, we now apply it to a case study calibrated to real NYC taxi data. While we could have analyzed a transit route case study instead, we chose a taxi case study for several reasons: 1) Taxi data is readily available through the NYC Open Data portal. ) Taxi trips represent individual trips, and recent news about allowing shared rides (Hu, 017) suggests the question of demand for shared rides remains an open question that is of interest to policymakers. 3) Tools to address this question have looked at either simulations that assume inelastic demand (Santi et al., 014) or considered demand only for very specific trip purposes like airport access (Ma et al., 017). We choose to study this problem as follows. A random sample of taxi riders are selected based on pickups and drop-offs during a particular time period in lower Manhattan. We consider their assignment game with a taxi market. Two scenarios are defined; one in which only single rider taxi routes are used, and one in which ridesharing is an option. The single rider taxi routes are used to calibrate the payoff values. We assume utilities are equal to the cost of the trips made in that base scenario. The assignment game model is expected to inform on: Percent of users who choose to share rides 19

20 Price allocations under two schemes: user-optimal and taxi-optimal allocation Decision support for designing a policy to request travelers meet the vehicles at a common distance The advantage of our model is that we should be able to forecast the mode split and fare prices that would be stable such that no user or taxi would choose to switch partner without being worse off. While the sample is small, it can allow us to draw comparisons against the optimistic conclusions from Santi et al. (014) that having shared rides leads to 80% taxi sharing in which user and operator incentives are not jointly captured. 4.. Data NYC Taxi and Limousine Commission releases data every month on taxi trips in NYC. The data provides valuable information about each trip, such as pick up and drop off time and locations, trip distance, number of passengers, payment type and also detailed fare amount (fare + tax + tip). For this example, taxi data from Wednesday October 5 th 016 from 8AM to 8:30AM was used. There are 19,97 taxi trips made in NYC during this time. We consider the lower Manhattan region below 3 rd Street as our study area, as shown in Figure 6. We create 1 nodes to represent zone centroids. Distance and travel time matrices between the nodes are extracted from Google API. Figure 6. Study area of NYC taxi case study. Within these 1 zones, 755 taxi trips were conducted during the study period. We are going to calibrate the payoffs of the travelers based on the realized trips and then use the proposed model to evaluate ride sharing option. The model allows us to see the modal split of users between solo 0

21 trip versus shared ride and to design their cost allocations based on stability criteria defined in previous sections. For the taxi market, a set of potential routes are generated: single rider pickups and drop-offs as well as all possible combinations of two rider pickups and drop-offs. For each pair combination, the best route is generated. For example, two users f and g have origins and destinations (O f, D f ) and (O g, D g ) respectively. The shortest travel route from the following set is added to the route set: {(O f O g D f D g ), (O f O g D g D f ), (O g O f D f D g ), (O g O f D g D f )}. To model the dynamics of the trips, we break the study period into multiple time intervals ΔT. In each interval, all the generated trips are pooled together and all the possible routes are generated to run the assignment game model of Eq. (5) - (9). For larger ΔT, the number of trips in the pool will increase so they will have better options in sharing their ride. On the other hand, their potential waiting time will increase. As a result, there is a tradeoff for choosing the value of ΔT. For this study we have chosen ΔT equal to 1 minute (the taxi app would notify whether a shared ride is possible for a new passenger within 1 minute). This means that every minute we pool the trips and solve the assignment game for them. It is assumed that each person s utility is equal to the travel time that they had in their observed single taxi ride plus the amount of fare they paid (which is known from the data). We consider the line capacity of route links equal to two passengers (w a = 3 a A r, r R). Each taxi user is assumed to be a single passenger so q s = 1. All the cost and gain values are converted to monetary values. The value of time for travelling (TVOT) is assumed to be $0.40/min. Value of waiting time (WVOT) assumed that is more than traveling value of time and is equal to twice that amount, $0.80/min. These values are conservative estimates based on numbers reported in Balcombe et al. (004). Operating cost of routes are assumed to be $0.90/mile (based on an estimate of annual cost of $36,000 (including fixed and variable costs) and average annual mileage of 40,000 miles) Results Comparison of total shared taxi rides versus single taxi ride All the calculation are performed with a desktop computer with core GHz processor and 16.0 GB RAM. All the codes are written in Matlab 016a. The full set of results for all 755 users will be uploaded to GitHub ( upon publication of this study. We run the assignment game model on this data for the shared ride policy with a run time of seconds. The first major result is that the total mileage of taxis for serving 755 users is miles, compared to a total mileage for single-ride taxi in the original data of miles. This shows a 31.4% decrease in vehicle miles traveled. Second, the assignment shows that 54 of 755 users (71.8%) decide to rideshare. This is similar to the 80% estimate from Santi et al. (014), although our value is more pessimistic because it jointly incorporates incentives for both users and operators. In Figure 7, the matches before (single taxi ride) and after (shared taxi ride) matching taxi riders are shown. 1

22 a) Single ride taxi b) Shared ride taxi Figure 7. NYC taxi assignment results for single and shared taxi riding. In Figure 8, the generalized ticket price in three conditions of single riding taxi, user optimal shared taxi and operator optimal shared taxi are shown. The blue line represents the single ride taxi ticket price observed from the data. Two red and green lines are ticket prices in the condition of operator and user optimal ticket pricing respectively (without minimum acceptable profit for operator and user). The space between these two lines is the space that a decision maker can find ticket price of system based on a desired policy. With considering minimum acceptable profit for users, the green and red lines will shift to right. The difference between red/green line with blue line illustrates the savings due to sharing a ride for taxis.

23 Percent Ticket Price ($) Single Riding Ticket Price Operator Optimal Ticket Price user optimal ticket price Figure 8. Ticket price percentage of users pay in three different scenario 4.3. Illustration of policy design using the stable cost allocation space The benefit of the proposed model is that the set of stable outcomes is a convex polyhedron, which means that any weighted average price between the user-optimal and operator-optimal ticket price for each user and operator is also a stable price that can be used. In fact, using this method provides a tool for explicitly pricing or setting design constraints for policies rather than blindly finding prices. This tool is a powerful method for pricing any matching situation in transportation systems, of which the NYC taxi is one example. Consider the case of what happens if the shared ride taxi policy was incorporated with a fare price set at the user-optimal fare pricing level with the additional policy of requiring users to move up to a distance D to be picked up. From the stable price space we know how much we can equivalently transfer operator cost to passenger cost without causing some users to break from the coalition. The total cost of the two minus the average operating cost savings per person should not exceed the operator optimal cost allocation. This model allows us to determine a good threshold for D. If D = 1 block, assuming 1 mile is approximately 0 blocks, pedestrian walking speed is 4 ft/s, and value of time of walking is approximately 1.5 of in-vehicle time (Balcombe et al., 004), then the policy can impose up to $0.44 on a traveler. Let us look at the distribution of gaps between the operator- and user-optimal pricing in Figure 9. It shows if the operator can perfectly recover the $0.44/traveler cost in operating cost savings, there are 484 out of 755 travelers (64.1%) who would have enough gap to stably absorb the additional walking cost without having to reduce operating cost to compensate. For the remaining 35.9%, however, the average gap is $0.08. If we want to ensure that no passenger is turned away by the walking policy, the cost savings from rerouting the vehicles need to be approximately an additional ( ) 71 = $0.13 per user on top of the $0.44/traveler 755 transfer. 3

24 GAP BETWEEN USER- AND OPERATOR- OPTIMAL ($) USERS Figure 9. Sorted gap between user- and operator-optimal pricing under shared taxi policy Detailed breakdown for select trips Lastly, we look closer at what happens to single users. As shown in Figure 10, three users with OD pairs (3,10), (10,17) and (10,17) originally conducted their trips as solo taxi rides based on the data. Two of them (10,17) have the same OD pair. Because of the availability of shared rides, they are now incentivized to share their rides together with the route ( ). For each of these users there were other available options such as sharing their ride with other users or even riding alone. The results for pricing of users (3,10), (10,17) and (10,17) are shown in Table 5. In the single taxi riding condition, users (3,10), (10,17) and (10,17) are observed to pay $7.55, $7.50 and $7.85 respectively for their trips. The total profit of the operators from running these three trips is $ $5.5 + $5.87 = $ In the ridesharing condition, one operator serves these three users by a single route of ( ). Stability conditions from Section 3. lead to the values shown in the lower portion of Table 5. The value b is assumed to be a constant minimum acceptable profit for these two operators (b < $8.77) that can serve these two users. For example, assuming b = $5, the operator profit from operating one route will vary from $15 in user-optimal ticket pricing to $16.98 in operator optimal ticket pricing. These values compare to $5.57, $5.5 and $5.87 for each route in single riding taxi. 4

25 Ridesharing taxi riding Single taxi riding Figure 10. Illustration of single taxi riding (dashed) versus ridesharing for two select users (solid). Table 5. Pricing of users (1,) and (1,7) in single and ridesharing taxi riding Path Results Users (3,10) (10,17) (10,17) Travel time (min) 8 Ticket price ($) 7.55 User profit ($) 0 Operator profit ($) (ticket price operation cost) 5.57 Travel time (min) 11 Ticket price ($) 7.50 User profit ($) 0 Operator profit ($) (ticket price operation cost) 5.5 Travel time (min) 11 Ticket price ($) 7.85 User profit ($) Operator profit ($) (ticket price operation cost) Travel time (min) Ticket price ($) User optimal b b+1.3 b Operator optimal b b b User profit ($) User optimal 9.43 b b b Operator optimal 8.77 b 9.9 b 10.7 b Operator profit ($) User optimal 3b (ticket price operation cost) Operator optimal b 5