I Introduction
There is an increasing notion of shareability in transportation systems. The popular trend of ondemand ridehailing services (e.g., Uber, Lyft and Didi) allows commuters to arrange chauffeured vehicle services conveniently on online platforms. Certain ridehailing platforms also provide sharing services among multiple commuters (e.g., UberPool, Lyft Line, Didi Hitch). Sharing rides is a prominent example of sharing economy [12], which promotes economic sharing activities in a peertopeer manner. Furthermore, worldwide governments are introducing policies to encourage vehiclepooling [9]. Private vehicles are often occupied by single passengers. Vehiclepooling is an effective solution to improve traffic congestion, air quality and parking availability.
Despite the promising benefits, it is not clear whether commuters will be motivated to adopt ridesharing and vehiclepooling themselves. While there have been extensive studies [21] suggesting a significant reduction in the total transportation cost by centralized ridesharing arrangement, it is unlikely that commuters will conform to centralized arrangement without considering their individual payments. In particular, ridesharing is a decentralized decisionmaking paradigm. Commuters are often selfinterested and only motivated to team up with each other based on individual objectives. Also, many existing commercial ridehailing service platforms are restricted to matching commuters with similar trips. Matching noncollocated commuters is significantly more challenging. This paper sheds light on how decentralized mechanisms should be designed to support effective ridesharing and vehicle pooling among noncollocated commuters. We aim to provide the theoretical foundation for decentralized mechanisms, as a departure from the centralized and restrictive mechanisms provided in nowadays ridehailing service platforms.
To illustrate the concept of decentralized ridesharing arrangement process, we provide an example^{1}^{1}1This ridesharing arrangement process can be operated as a standalone services, along with any ridehailing platforms (like Uber or taxis). in Fig. 1. First, the commuters will post their trip information and time constraints on an open data repository (e.g., a social network or an open ledger like blockchain). Then, the commuters will identify potential ridesharing partners and plan the possible shared rides with transportation costs. The commuters will also compute their individual payments by splitting the transportation costs in a certain fair manner. Next, the commuters will propose to potential ridesharing partners according to the ranking order of individual payments. By a proper matching mechanism, they will reach a mutual agreement, such that no better rides can be arranged otherwise. Note that this process is not dictated by a centralized operator. Each commuter is free to accept or reject any ridesharing proposals.
In particular, we highlight three key elements in decentralized ridesharing arrangement:
Ia Fair CostSharing Mechanisms
Central to decentralized ridesharing arrangement is a commonly agreed costsharing mechanism for splitting the transportation costs among the parties of ridesharing. The choice of costsharing mechanisms should take into consideration of fair contribution of each party, which provides a rationale on how to split the cost of a sharable ride in a fair manner. In particular, commuters may not share the same destinations or sources. There are a variety of possible fair costsharing mechanisms. For example, one simple costsharing mechanism is to split the the transportation cost equally between two parties. Another way is to split proportionally according to the original costs of standalone rides. Also, one may consider to split in a way to induce equal savings from standalone rides. Note that the choices of costsharing mechanisms will affect individual commuters’ preferential orders of possible ridesharing options and the outcome of an agreement. Given multiple choices of costsharing mechanisms, it is important to understand their ramifications on ridesharing agreement.
IB Stable Matching
Based on individual commuters’ preferential orders of possible ridesharing options, a decentralized matching mechanism is needed to arrange ridesharing. In practice, commuters usually form a coalition as a pair to share a ride, as this reduces the complexity of reaching an agreement. Matching mechanisms have been studied in various applications, such as college admissions and dating [10]. One useful concept is stable matching, which is particularly desirable in decentralized decisionmaking mechanisms, because no participants would be better off to deviate from a stable matching outcome. Hence, stable matching captures the likely outcome of an agreement in a decentralized matching process. But different costsharing mechanisms will induce different stable matching outcomes. In this paper, we compare stable matching outcomes under different fair costsharing mechanisms in ridesharing arrangement.
IC Social Optimality
To compare different costsharing mechanisms for ridesharing arrangement, a natural approach is to benchmark against a social optimal outcome that minimizes the total transportation cost of all commuters. Decentralized ridesharing arrangement will not reach a social optimal outcome, because everyone is selfinterested to minimize his individual cost, rather than the total cost. However, a good costsharing mechanism should achieve high social optimality. We measure the ratio between the cost of a stable matching outcome over the one of a social optimal outcome. In this paper, we present theoretical bounds on the social optimality ratios for various fair costsharing mechanisms, which then show high social optimality in these costsharing mechanisms. To corroborate our theoretical study, we also present a data analysis on practical taxi sharing in New York City. We compare the empirical social optimality of various fair costsharing mechanisms used in taxi sharing by an extensive study based on NYC taxi trip dataset [16].
Outline: This paper presents an extensive study of how decentralized mechanisms can support effective ridesharing. We first present a brief survey on the related literature in Sec. II. The model and notations of costsharing mechanisms and stable matching are formulated in Secs. IIIIV. We compare the social optimality of various fair costsharing mechanisms by theoretical bounds in Sec. V. We also present a modified stable matching algorithm for finding a stable ridesharing assignment in Sec. VI. To corroborate our theoretical study, we present an empirical data analysis on practical taxi sharing in New York City in Sec. VII.
Ii Related Work
Ridesharing research belongs to a large body of literature (e.g., see a survey in [9]). There are various paradigms of ridesharing among commuters, or between drivers and passengers. For example, see a survey in [15]. One of the critical components in ridesharing is the matching process that pairs commuters to share a ride, or finding suitable drivers for the requested passengers [26, 17, 23, 1, 24, 19, 18].
In particular, the studies in [21, 2] investigated how ridesharing and vehicle pooling can reduce transportation delay. These papers generally assume that ridesharing and vehicle pooling is arranged by centralized entities to achieve the desirable benefits. There is no consideration of the selfinterested nature of commuters who will not always follow centralized arrangement. On the other hand, there are recent papers considering stable matching in ridesharing [26, 17, 23]. However, the motivation of stable matching in these papers is related to arbitrary passengers’ or drivers’ preferences about each other, which is not necessarily related to costsharing of transportation costs. [1, 24, 19, 18] consider stable matching between drivers and passengers, but do not consider sharing transportation cost among the passengers. Also, these papers did not compare stable matching with social optimal ridesharing arrangements in the context of sharing transportation cost among commuters.
Ridesharing belongs to the general topic of transportation scheduling. Scheduling algorithms for ridesharing have been investigated in other studies [27, 14], where a scheduler optimizes a ride to pick up and drop off multiple passengers at different locations. While optimization of ride is outside the scope of the paper, the scheduling algorithms can be incorporated in the matching algorithm to arrange the best shared rides among a given set of commuters.
Our study of fair costsharing mechanisms for ridesharing belongs to the broad topic of network costsharing and coalition formation problems [8, 11, 20, 13]. A study related to our results is the strong price of anarchy for stable matching [3]. Our ridesharing matching problem is a subclass of coalition formation games [4] that allows arbitrary coalitions with at most two participants per coalition. However, our results are based on social optimality ratio, which is different than the previous studies. The results of social optimality ratio in this work can be derived in a general setting in [5, 6, 7, 25]. But we simplify the proofs by considering a coalition as pairs in the context of ridesharing. Furthermore, we corroborate our theoretical study by an empirical data analysis of practical taxi sharing in New York City.
The New York City taxi trip dataset is a large publicly available dataset [16], which provides detailed records of pickup and dropoff locations and times in New York City. The dataset can enable a wide range of empirical studies of taxi service strategy optimization [22]. There have been a number of studies about taxi sharing in the literature based on New York City taxi trip dataset [21, 2]. But none of the previous studies considered decentralized stable matching for taxi sharing using New York City taxi trip dataset.
Symbol  Definition 

Graph representing road network  
Set of commuters participating in ridesharing  
Source and destination locations, and earliest departure and latest arrival times of  
Set of sharable rides for satisfying feasibility constraints  
Standalone ride of not shared with others  
Matching graph, where has nonempty and for all  
Minimum cost sharable ride in  
Cost of shared ride for  
Cost of standalone ride for  
Payment from for shared ride based on a costsharing mechanism  
Utility of for of shared ride  
Feasible ridesharing assignment, a subset of satisfying the feasibility properties  
Social cost of ridesharing assignment  
Social optimality ratio between a stable ridesharing assignment () and a social optimal ridesharing assignment () 
Iii Model and Notations
This section presents a general model of ridesharing by matching commuters for sharing hired vehicles^{2}^{2}2In practice, the matching process may be carried out on an online platform automated by computer agents representing the users.. Table I lists some key notations. Consider a road network represented by a directed graph , where is a set of road junctions and is a set of road segments. There is a set of commuters . Each commuter is associated with a tuple of parameters , where is the source location, is the destination location, is the earliest departure time, and is the latest arrival time. Our goal is to pair up the commuters for potential ridesharing.
Iiia Sharable Rides
Given road network , a ride is defined by a sequence of locations , where each , and a sequence of arrival times , where is the arrival time at location . For each , we denote as the arrival time of ride at location .
Definition 1
(Sharable Ride) A ride is sharable by a pair , if the following feasibility constraints are satisfied:

(Location Constraint): are in the sequence of locations of ride . Namely, .

(Temporal Constraint for ): .

(Temporal Constraint for ): .
Given , there are two types of sharable rides:

Hitchhiking Ride: A ride is called an hitchhiking ride, if .

Combined Ride: A ride is called an combined ride, if .
A sharable ride for can either be hitchhiking, hitchhiking, combined, or combined. For example, in Fig. 2 (b), shares hitchhiking ride. In Fig. 2 (c), shares combined ride.
In this paper, we consider the matching of pairs of commuters, who have declared their requests in advance, barring cancellation. Also, we consider the cost of transportation as the primary factor to ridesharing decisions. However, the model can be extended by incorporating additional constraints in the matching process. Given a ride , let be the associated transportation cost, which will be the fare of a taxi or hired vehicle. Let be the set of sharable rides in road network for a pair of distinct commuters , and be the minimum cost sharable ride in , which is the ride of minimum cost among all sharable rides between . Let be the standalone ride for commuter if does not share with another commuter. Note that . Otherwise, the commuter ( or ) can always choose another lower cost standalone ride.
IiiB Matching for RideSharing
Matching for ridesharing can be attained by an undirected matching graph where . Namely, includes two types of edges: (1) represents a sharable ride with a pair of distinct commuters, and (2) represents a standalone ride. Let and be the edge costs of edges and , respectively.
Definition 2
(Feasible RideSharing) Given matching graph , we define a feasible ridesharing assignment as a subset of edges satisfying the following feasibility properties:

Every is covered by an edge in . Namely, there exists or for every .

No pair of edges in share any nodes. Namely, there do not exist and for .
Definition 3
(Socially Optimal RideSharing) Given a feasible ridesharing assignment , let the social cost be . A feasible ridesharing assignment is called a social optimum (denoted by ), if it minimizes the total transportation cost:
A ridesharing assignment
can also be equivalently represented by a binary vector
, where each binary variable
indicates whether the pair of commuters will share a ride. Note that finding a social optimal ridesharing assignment is equivalent to solving the following minimum weight edge covering problem:subject to  (1)  
(2) 
Constraints (1)(2) ensure the feasibility of ridesharing assignment.
Example: We consider an example of road network in Fig. 2 (a). There are four commuters . For commuter , his sources location is denoted by and destination location by . The number on each edge represents the transportation cost of the respective segment. In this example, the social optimal ridesharing assignment is with social cost , as illustrated in Fig. 2 (b).
Iv Decentralized RideSharing Mechanisms
Note that ridesharing is a decentralized decisionmaking paradigm. It is unlikely that every commuter will follow a ridesharing assignment according to the social cost. In reality, commuters are selfinterested and only motivated to team up with each other based on individual savings. It is natural to consider how individual commuter will decide in arranging ridesharing among themselves.
Iva Fair CostSharing Mechanisms
Consider a sharable ride for a pair of commuters . There are many ways to split to cost among in a fair manner. A costsharing mechanism is defined by payment function , which is the payment by commuter for ride . A budgetbalanced costsharing mechanism requires . We also let the utility (i.e., saving) of commuter for ride be , where is the standalone ride for commuter .
A fair costsharing mechanism should provide a rationale on how to split the cost of a sharable ride in a fair manner. There are several fair and budgetbalanced costsharing mechanisms (denoted by different superscripts in ) as follows:

(Equal CostSharing): Each commuter should split equally.
(3) Note that equal costsharing mechanism may produce negative utility ().

(Egalitarian CostSharing): Each commuter should split in a way to attain the same utility.
(4) Namely,

(Proportional CostSharing): Each commuter should contribute proportionally to the cost of standalone ride.
(5) 
(Segmentbased CostSharing): Each commuter should only contribute to the participated segments of ride. Let the transportation cost of the segment from to in ride be .

If is an hitchhiking ride, then
(6) (7) 
If is an combined ride, then
(8) (9)

Each of these fair costsharing mechanisms captures a notion of fair contribution from the involved commuters. Different fair costsharing mechanisms will induce different outcomes in decentralized ridesharing arrangement. We next investigate the impacts of these fair costsharing mechanisms. In the following, we consider the sharable ride for each pair pf commuters to be the minimum cost sharable ride .
IvB Stable RideSharing Assignment
Given a costsharing mechanism, any pair of commuters may join to share a ride based on individual payments. Every commuter aims to minimize his individual payment. We consider any unilateral switch that allows any pair of commuters abandon their current rides to create another ride. As a consequence, a stable assignment is likely to emerge, such that no one would be better off to deviate from the current assignment.
Definition 4
(Stable RideSharing) Given payment function , a pair of commuters is called a blocking pair with respect to ridesharing assignment if and can strictly reduce their payments when they form a pair to share a ride instead of the respective rides in . A feasible ridesharing assignment is called stable ridesharing assignment, if there exists no blocking pair with respect to
. Stable ridesharing assignment is based on the concept of stable coalition in cooperative game theory
[5].A stable ridesharing assignment is equivalent to a stable matching outcome (with the inclusion of possibly singleton groups for standalone commuters). Note that a stable ridesharing assignment is also a strong Nash equilibrium in game theory [5]. A strong Nash equilibrium is a Nash equilibrium, in which no group of players can cooperatively deviate in an allowable way that benefits all of its members, whereas a Nash equilibrium only allows a player to make a unilateral action. Some previous papers (e.g., [1]) studied Nash equilibrium the matching in ridesharing, but not strong Nash equilibrium.
Definition 5
(Social Optimality Ratio) Define the social optimality ratio as the ratio between the cost of a stable ridesharing assignment () and that of a social optimal ridesharing assignment () for a particular instance of ridesharing problem by
(10) 
If the social optimality ratio is small for a particular costsharing mechanism, then such a costsharing mechanism can achieve high social optimality by inducing a stable ridesharing assignment close to a social optimum.
eq  ega  pp  sb  
3.25  3.25  3.25  3.25  
3.25  3.25  3.25  3.25  
3.5  3  3.11  2  
3.5  4  3.89  5  
3.5  3  3.11  2  
3.5  4  3.89  5  
stable ridesharing 
4  4  4.9  4.9 

Example: We consider the road network in Fig. 2 (a). Table II shows the individual payments based on equal (eq), egalitarian (ega), proportional (pp), and segmentbased (sb) costsharing mechanisms for commuters , and the costs of standalone rides. The stable ridesharing assignments for different costsharing mechanisms are derived as follows:

(Equal CostSharing): We obtain
(11) (12) Hence, will be motivated to share a ride instead of other options. Then, the stable ridesharing assignment is and the social optimality ratio is 1.16.

(Egalitarian CostSharing): We obtain
(13) (14) (15) (16) Hence, and will be motivated to share rides respectively. The stable ridesharing assignment is and the social optimality ratio is 1.

(Proportional CostSharing): Similar to egalitarian costsharing, the stable ridesharing assignment is and the social optimality ratio is 1.

(Segmentbased CostSharing): We obtain
(17) (18) (19) (20) Hence, and will be motivated to remain standalone rides, and then will share a ride. The stable ridesharing assignment is and the social optimality ratio is 1.16.
Remark: In the preceding example, egalitarian and proportional costsharing mechanisms can induce a social optimal stable ridesharing assignment. But in general, different stable ridesharing assignments will be induced by different costsharing mechanisms. It is crucial to understand the social optimality ratios of different costsharing mechanisms. In Sec. V, we provide theoretical upper bounds on the social optimality ratios, which shows that several fair costsharing mechanisms can achieve high social optimality.
V Bounds on Social Optimality Ratio
Given any feasible ridesharing assignment , let the set of commuters present in be . For , let the payment of with respect to be , where . In the following, we provide general theories to upper bound the social optimality ratio () between the cost of a stable ridesharing assignment () and that of a social optimal ridesharing assignment () for any instances of problem. In Sec. VII, we will corroborate our theoretical results with an empirical data analysis on taxi sharing in New York City.
Theorem 1
For equal costsharing mechanism, let be a stable ridesharing assignment. We show that the social optimality ratio is upper bounded by .
Proof:
First, we assume that . Otherwise, . Suppose . Then there must exist and , such that , because all commuters must belong to some sharable rides in and , and both and are feasible.
We assume that and . Note that the cases of or can be proven straightforwardly. Recall and . Since , we obtain
(21) 
On the other hand, since is a stable ridesharing assignment, we obtain
(22)  
(23) 
Hence, it follows that
(24) 
Because is a stable ridesharing assignment and noting that and , we obtain
(25)  
(26) 
Together, by noting that and , we obtain
(27)  
(28)  
(29)  
(30) 
When (or ), Eqn. (30) can also be proven straightforwardly, by omitting (or , respectively).
Since equal costsharing mechanism is budgetbalanced, , summing over can obtain
(31) 
Theorem 2
For egalitarian costsharing mechanism, let be a stable ridesharing assignment. We show that the social optimality ratio is upper bounded by .
Theorem 3
For proportional costsharing mechanism, let be a stable ridesharing assignment. We show that the social optimality ratio is upper bounded by .
Theorem 4
For segmentbased costsharing mechanism, let be a stable ridesharing assignment. The corresponding social optimality ratio is upper bounded by .
Remark: Theorems 14 show that the social optimality ratios under equal, egalitarian, proportional and segmentbased costsharing mechanisms are at most in any instances of any number of commuters, which is a small constant. Therefore, these fair costsharing mechanisms can achieve high social optimality. Note that in practice, the social optimality ratios are much smaller than the theoretical bounds, and hence, can achieve even higher social optimality.
Vi Stable Matching Algorithm
To complement our analysis of stable matching on ridesharing, we present a modified stable matching algorithm for finding a stable ridesharing assignment. This algorithm is based on the classical GaleShapley algorithm for stable marriage problem and Irving’s algorithm for stable roommates problem [10]. Here, we extend the classical algorithms to allow the possibility of standalone rides (namely, with no ridesharing partner). The modified stable matching algorithm will be used in Sec. VII for an empirical data analysis on taxi sharing in New York City.
We first define some notations. For each , define a preferential order () over all possible options of ridesharing pairs with (including standalone ride). For example, “” means that is the most preferred by , then followed by and standalone ride , and so on. Each commuter’s preferential order is formulated according to the ranking order of individual payments of a given costsharing mechanism (e.g., ). For equal payments, we will enforce deterministic tiebreaking in a consistent manner across all commuters. Note that the preferential orders also include the option of standalone ride . Each commuter will remove the options of ridesharing below the standalone ride, as they will not be selected at the end.
The stable matching algorithm is consisted of several rounds. Initially, all commuters are set to be unsuspended, which allows them to propose to any partners. In each round, first each unsuspended commuter will propose to a ridesharing partner he prefers most in his preferential order to whom has not proposed in the previous round, and is better than the partner with whom is currently provisionally matched, if any. Note that for a standalone ride, will propose to himself. Next, each commuter will collect a number of proposals at each round. He will select the most preferred one from the received proposals. If commuter is currently provisionally matched with, say , and prefers another new proposer to , then will be unmatched and will be provisionally matched, instead. This process will continue until every commuter is provisionally matched with another ridesharing partner.
The pseudocode of stable matching algorithm is described in StableMatching.
Next, we define a cyclic preference as a sequence of commuters , such that
(32) 
Theorem 5
If there exists no cyclic preference, then StableMatching will converge to a stable matching outcome.
The full proof of existence of stable matching outcome under the condition of no cyclic preference can be found in [5]. The basic idea is that the absence of cyclic preference rules out the possibility of oscillation, where commuters keep switching matched pairs without termination. Note that there is no cyclic preference under equal, egalitarian and proportional costsharing mechanisms, as shown in [5]. Although segmentbased costsharing mechanism may induce cyclic preference, this is uncommon in practice.
Although StableMatching is an extension of the classical matching algorithm from Irving [10], there are some subtle differences in our algorithm. First, our algorithm allows any commuter to be unmatched (and hence taking a standalone ride by himself). Second, Theorem 5
shows that the common fair costsharing mechanisms will induce no cyclic preferences. Hence, there is no need to deal with cyclic preferences by considering odd rotations in the original Irving’s algorithm.
Vii Empirical Data Analysis of Taxi Sharing
To corroborate our theoretical results of social optimality ratios in Sec. VII, we present an empirical big data analysis on practical taxi sharing in New York City. We compare various properties of different fair costsharing mechanisms used for taxi sharing in an empirical study based on New York City taxi trip dataset. We provide useful insights on effective costsharing mechanisms for ridesharing in practice.
In our data analysis, we used the taxi trip dataset of New York City (NYC) of 2013 [16]. The dataset contains over 450K taxi trips per day with the average distance per trip is around 4.2 km. Each data record of a trip includes the information of Taxi ID, trip distance and duration times of pickups and dropoffs of commuters as well as the GPS locations of pickups and dropoffs of commuters.
Viia Density Maps of Matched Commuters
Setting: We first present a case study of the effectiveness of equal, proportional, egalitarian and segmentbased costsharing mechanisms, as compared to the social optimal outcome. We consider the taxi trips during 12pm1pm on 23th Feb 2013 (weekend) with over 5000 taxi trips. We employed the modified stable matching algorithm in Sec. VI to find stable ridesharing assignments. We consider a pair of commuters who can be potentially matched, if their pickup times in the NYC dataset are within 3 minutes with each other. We examine the properties of matched and unmatched commuters in stable matching outcomes in Fig. 2(a). We compare the outcomes with the social optimal outcome, which minimizes the total cost of all commuters. We also examine the distributions of total separation distance in pickup locations between pairs of matched commuters in Fig. 2(b).
Observations: To visualize the outcomes of ridesharing, we plot five density maps of pickup locations of commuters in Fig. 2(a). Not all commuters can be matched for ridesharing. Some of them have standalone rides. The green dots indicate the pickup locations of matched commuters, whereas the red dots indicate those of unmatched commuters.
We observe that the density maps of different costsharing mechanisms are rather similar, and most of the pickup locations of matched commuters are located in similar places. However, the portions of matched commuters are different. Over 70% of commuters can be matched based on any of the four costsharing mechanisms. However, equal and segmentbased costsharing mechanisms can match a fewer number of commuters (27%30%) than proportional and egalitarian costsharing mechanisms (22%23%), which are fewer than the social optimal outcome (14%). Since equal costsharing mechanism can induce negative utility, it may discourage sharing between commuters, and lead to lower percentage of matched pairs. On the other hand, social optimal matching ignores stability in matching, and hence can match the maximum number of pairs of commuters.
We also plot the distributions of total separation distance in pickup locations between pairs of matched commuters in Fig. 2(b). We observe that most matched commuters are separated by less than 2km in their pickup locations. We also notice that over 40% of matched commuters are separated by over 500m in their pickup locations and segmentbased costsharing mechanism induce the highest percentage of separated matched commuters (50% over 500m).
ViiB Stable Matching Structures
Setting: We next study the structures of stable matching outcomes under different costsharing mechanisms. We first sorted the matched commuters according to the distances of their standalone rides. In Fig. 4, each commuter is represented by a point on the perimeter of a circle, following the order of distances of standalone rides. An edge is drawn between a pair of matched commuters. We also color the commuters with long standalone rides by yellow, and the commuters with short standalone rides by blue.
Observations: In Fig. 4, we visualize the stable matching structures. We observe that different costsharing mechanisms induce different stable matching structures. For equal costsharing mechanism, commuters with long standalone rides are more likely to be matched among themselves, and so are those with short standalone rides. On the other hand, for proportional, egalitarian and segmentbased costsharing mechanisms, commuters with long standalone rides are more likely to be matched with those of short standalone rides. Hence, proportional, egalitarian and segmentbased costsharing mechanisms can bolster diversity in stable matching with heterogeneous commuters of different pickup and dropoff locations.
ViiC Social Optimality
Setting: We next examine the social optimality of different costsharing mechanisms for practical taxi sharing. The study is based on the data of January 4, 2013 (i.e., Weekday) and February 23, 2013 (i.e., Weekend) in the NYC taxi trip dataset. We selected five representative onehour periods (12 am, 89am, 121pm, 56pm, 910pm) on each day. In Fig. 4(a), we compare the social optimality ratios of the social costs over the costs of social optimal outcomes in the respective periods. In Fig. 4(b), we also compare the social utilities of all matched commuters (which is the total savings from standalone rides of all commuters: ).
Observations: We observe that the empirical social optimality ratios of four costsharing mechanisms ( 1.2) are well below the theoretical upper bound in Sec. VII. Indeed, the social optimality ratios are much in practice smaller than the theoretical bounds, and hence, can achieve higher social optimality. In particular, equal costsharing mechanism induce less social utilities and larger social optimality ratios, whereas the other three costsharing mechanisms produce comparable results. In terms of social utilities, we observe that all costsharing mechanisms can achieve higher social utilities, when the social optimal outcome has a higher social utility.
ViiD Distributions of Normalized Utilities
Setting: To shed light on the individual benefits of matched commuters, we define the normalized utility of commuter by the ratio , which is the normalized utility over the standalone ride cost. In Fig. 5(a), we compare the cumulative distributions of normalized utilities among the matched commuters under different costsharing mechanisms.
Observations: We observe that the cumulative distributions of normalized utilities under different costsharing mechanism on weekday and weekend are similar. Comparing the outcomes of different costsharing mechanisms, more commuters have lower normalized utilities under equal costsharing mechanism. The number of commuters who have the highest normalized utilities (0.5) under proportional mechanism is the largest among the four costsharing mechanisms, which is approximately 15% of total commuters. Thus, proportional costsharing mechanism can benefit commuters with higher normalized savings. Furthermore, proportional and egalitarian costsharing mechanisms induce similar cumulative distributions of normalized utilities of less than 0.35, whereas proportional costsharing mechanism induces more commuters having the normalized utilities around 0.4 than egalitarian costsharing mechanism.
ViiE Distributions of Standalone Cost Ratios
Setting: Next, we study the similarity between commuters in a matched pair. We define the standalone cost ratio of a matched pair of commuters by , which is the maximum over different standalone ride costs for a pair . If , the matched pair are relatively similar. In Fig. 5(b), we compare the cumulative distributions of standalone cost ratios among the matched commuters under different costsharing mechanisms.
Observations: The cumulative distributions of standalone cost ratios under different costsharing mechanism on weekday and weekend commuters are similar. Overall, commuters are more likely matched with those of similar standalone costs since more commuters have high standalone cost ratios ( 0.8) in any one of the costsharing mechanisms. It’s also worth to notice that approximately 70% of commuters have high standalone cost ratios ( 0.8) under equal costsharing mechanism, whose percentage is more than that the other three costsharing mechanisms. Egalitarian and segmentbased costsharing mechanisms produce the similar cumulative distributions of standalone cost ratios, whereas proportional costsharing mechanism produces more commuters who have lower standalone cost ratios than the other costsharing mechanisms, which indicates that commuters are more likely matched with those of different standalone costs under proportional costsharing mechanism.
ViiF Distributions of Delay Ratios
Setting: Ridesharing may incur additional delay to a commuter because of a detour to pickup another commuter. Particularly, we consider the incurred delay in terms of additional geographical distance in ridesharing, in the presence of similar traffic condition. We denote the geographical distances traveled of a matched pair of commuters and in a shared ride by and respectively. We denote the geographical distance by commuter in his standalone ride by . Define the delay ratio of a matched pair of commuters by , which is a natural metric of delay in a shared ride. In Fig. 7, we compare the cumulative distributions of delay ratios among the matched commuters under different costsharing mechanisms.
Observations: The cumulative distributions of delay ratios under different costsharing mechanisms on weekday and weekend are similar. We also notice that the average delay ratios are around 1.1, and over 90% of matched commuters have delay ratios less than 1.4. Hence, small delays are incurred among the matched commuters. Segmentbased costsharing mechanism induces more matched commuters having lower delay ratio ( 1.1) than other three costsharing mechanisms. In addition, equal and egalitarian costsharing mechanisms produce similar cumulative distributions of delay ratios, whereas proportional costsharing mechanism produce more matched commuters who have slightly lower delay ratios than equal and egalitarian costsharing mechanisms.
ViiG Insights and Ramifications
Overall, we draw the following insights and ramifications based on our empirical study of practical ridesharing with New York City taxi trip dataset:

The four fair (i.e., equal, egalitarian, proportional and segmentbased) costsharing mechanisms can enable effective decentralized ridesharing arrangement in practice by achieving high social optimality, such that the induced social costs are as comparably low as the social optimal arrangement by a centralized planner whose objective is to minimize the total cost.

Egalitarian and proportional costsharing mechanisms can induce more matched commuters than equal and segmentbased costsharing mechanisms.

Egalitarian, proportional and segmentbased costsharing mechanisms can bolster diversity among the matched commuters with heterogeneous locations.

Proportional costsharing mechanism can benefit commuters with higher normalized savings.

All four costsharing mechanisms incur small delays among the matched commuters.
Viii Conclusion
Ridesharing is a popular paradigm in intelligent transportation systems, requiring decentralized decisionmaking processes among commuters. This paper offers a thorough study of decentralized ridesharing arrangements based on the principle of fair costsharing of transportation costs. We define several fair costsharing mechanisms, including equal, egalitarian, proportional and segmentbased costsharing mechanisms. We compare the stable matching outcomes induced by these costsharing mechanisms with a social optimal outcome by deriving the theoretical bounds of social optimality ratios. Our results show that these fair costsharing mechanisms can achieve high social optimality. We also corroborate our results with an empirical study of taxi sharing under fair costsharing mechanisms by a data analysis on New York City taxi trip dataset. In particular, we observe from our empirical study that egalitarian and proportional costsharing mechanisms can approach a social optimal outcome, although all equal, egalitarian, proportional and segmentbased costsharing mechanisms share the same theoretical bound of social optimality ratio.
Apart from the study of social optimality ratio, there are other potential areas to explore in future work. First, our theoretical analysis does not consider the shareability network structure of feasible coalitions. Some network structures may favor a particular type of costsharing mechanism. Second, this work so far considers transportation cost as the primary factor of coalition formation. Another possible factor is delay, which can lead to different coalition structures. Third, we consider stable matching for ridesharing with a pair of commuters. There are other possibilities, for example, with more than two commuters in each group in shuttle sharing, and between drivers and passengers, who have asymmetric roles. Also, we will consider uncertain information (e.g., traffic information) and unknown arrivals of future commuters in an online fashion, which will require different matching mechanisms.
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