{"title": "Balancing Efficiency and Fairness in On-Demand Ridesourcing", "book": "Advances in Neural Information Processing Systems", "page_first": 5309, "page_last": 5319, "abstract": "We investigate the problem of assigning trip requests to available vehicles in on-demand ridesourcing. Much of the literature has focused on maximizing the total value of served requests, achieving efficiency on the passengers\u2019 side. However, such solutions may result in some drivers being assigned to insufficient or undesired trips, therefore losing fairness from the drivers\u2019 perspective.\n\nIn this paper, we focus on both the system efficiency and the fairness among drivers and quantitatively analyze the trade-offs between these two objectives. In particular, we give an explicit answer to the question of whether there always exists an assignment that achieves any target efficiency and fairness. We also propose a simple reassignment algorithm that can achieve any selected trade-off. Finally, we demonstrate the effectiveness of the algorithms through extensive experiments on real-world datasets.", "full_text": "Balancing Ef\ufb01ciency and Fairness in On-Demand\n\nRidesourcing\n\nNixie S. Lesmana\u2217\n\nnixiesap001@e.ntu.edu.sg\n\nXuan Zhang\u2020\n\nxuan6@illinois.edu\n\nXiaohui Bei\u2217\n\nxhbei@ntu.edu.sg\n\nAbstract\n\nWe investigate the problem of assigning trip requests to available vehicles in on-\ndemand ridesourcing. Much of the literature has focused on maximizing the total\nvalue of served requests, achieving ef\ufb01ciency on the passengers\u2019 side. However,\nsuch solutions may result in some drivers being assigned to insuf\ufb01cient or undesired\ntrips, therefore losing fairness from the drivers\u2019 perspective.\nIn this paper, we focus on both the system ef\ufb01ciency and the fairness among\ndrivers and quantitatively analyze the tradeoffs between these two objectives. In\nparticular, we give an explicit answer to the question of whether there always exists\nan assignment that achieves any target ef\ufb01ciency and fairness. We also propose a\nsimple reassignment algorithm that can achieve any selected tradeoff. Finally, we\ndemonstrate the effectiveness of the algorithms through extensive experiments on\nreal-world datasets.\n\n1\n\nIntroduction\n\nRidesourcing refers to a mode of transportation that connects private car drivers with passengers\nvia mobile devices and applications. Recent advances in technology provide the opportunity for\nridesourcing platforms to dynamically match drivers and passengers in real time. This new generation\nof ridesourcing has the potential to signi\ufb01cantly increase the ef\ufb01ciency of urban transportation\nsystems, consequently reducing congestion and pollution [23]. In most on-demand ridesourcing\nplatforms, private-hire car drivers are not allowed to pick up passengers who hail them on the streets,\nbut can only take booking requests assigned by the platform. One key function of these platforms is\nthus to automatically assign potential passengers to active drivers. The development of an ef\ufb01cient\nreal-time demand assignment algorithm is central to the concept and to the success of a ridesourcing\nenterprise.\nResearch into real-time ridesourcing has often focused on developing algorithms for optimal assign-\nment of sets of requests to drivers [1, 33, 24]. In these studies, the common objective is to minimize\nthe total waiting time for passengers and maximize the service rate, achieving ef\ufb01ciency on the\npassengers\u2019 side. Admittedly, customer satisfaction should be the main goal in any service industry.\nHowever, in the ridesourcing domain, the role of drivers is as important as that of passengers in terms\nof sustaining the business. Drivers have preferences that might not align with those of the passengers\nthat are optimized by the algorithm. A centralized algorithm that only focuses on system ef\ufb01ciency\nwill inevitably result in some drivers being assigned to insuf\ufb01cient or undesired trips. Leaving the\nsystem as it is would affect the sustainability of the ridesourcing business model in the long run, as\nunsatis\ufb01ed drivers will not renew their memberships and new drivers will be deterred from signing\nup. Therefore, fairness on the drivers\u2019 side should be assessed more carefully and should receive\nmore attention.\n\n\u2217Department of Mathematical Sciences, Nanyang Technological University, Singapore 637371\n\u2020Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign,\n\nChampaign, IL 61801-3080\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fIn this paper, we study the batch request-vehicle assignment problem with a focus on both ef\ufb01ciency\nand fairness and address the problem of assigning requests to vehicles that account for the natural\ntension between these two objectives. In the basic setting, we consider a \ufb02eet V of available vehicles\nand a set R of ride requests. Assignment constraints are captured by a bipartite graph G = (V,R, E),\nsuch that edge {v, r} \u2208 E iff vehicle v can be assigned to serve request r under the speci\ufb01ed\nconstraints. Our goal is to design an assignment algorithm that matches each vehicle to at most one\nrequest, such that both ef\ufb01ciency and fairness are optimized. For ef\ufb01ciency, we adopt the utilitarian\ncriterion, de\ufb01ned as the sum of values of all served requests. This is one of the most axiomatically\njusti\ufb01ed measures of ef\ufb01ciency; it has been studied extensively in the literature and employed as a\nnatural metric for practical applications. For fairness, we adopt the max-min fairness criterion that\nemphasizes the maximization of the least value that a vehicle obtains. This criterion is built on the\nRawlsian egalitarian justice [29] and is well-recognized in various application domains (see more\ndiscussions in [34]).\nEf\ufb01ciency and fairness are often competing objectives such that in most cases, the optimum of both\ncannot be achieved simultaneously. Thus, we are naturally led to the question of how to reconcile\nsystem ef\ufb01ciency and drivers\u2019 fairness in a ridesourcing assignment. For instance, a central decision\nmaker may choose a mild strategy that balances the tension between ef\ufb01ciency and fairness in normal\ntraf\ufb01c conditions. However, as demand increases and traf\ufb01c conditions worsen, it may be desirable to\nmove to a strategy that puts higher attention to ef\ufb01ciency to quickly serve the waiting passengers.\nNote that such changes are two-way (i.e., ef\ufb01ciency or fairness-oriented), gradual, and dependent on\ndynamically changing demand conditions. Therefore, to provide better managerial \ufb02exibilities to\ndecision makers, we need to provide a full set of candidate allocation solutions and characterize the\ntrade-offs inherent in applying these concepts.\nIn general, through this paper, we aim to address the following question:\nGiven any problem instance and any required fairness threshold, how do we \ufb01nd a request-vehicle\nassignment that meets the fairness threshold while also has suf\ufb01ciently good system ef\ufb01ciency?\n\n1.1 Our Contributions\n\nOur contributions can be summarized as follows.\n\n\u2022 We answer the above generic question with an ef\ufb01cient algorithm REASSIGN. Our algorithm\ntakes any desired fairness threshold as a parameter, and through a surprisingly simple\nprocedure, computes an assignment that satis\ufb01es the fairness threshold and has provably\ngood ef\ufb01ciency.\nOur constructive answer to the question above provides extra managerial \ufb02exibilities to\ndecision makers with various needs. For example, in the aforementioned traf\ufb01c scenario, the\nrequirements for ef\ufb01ciency and fairness vary at different stages. By applying our algorithm,\none can simply set a speci\ufb01c fairness requirement as the input of the algorithm to generate\nan output assignment with the required ef\ufb01ciency and fairness.\n\n\u2022 We further show that the ef\ufb01ciency-fairness tradeoff guaranteed by our algorithm is provably\noptimal. That is, we prove that for any target ef\ufb01ciency and fairness that go beyond our\nguarantee, there exist a problem instance in which no assignment can achieve both targets\nsimultaneously.\n\n\u2022 Finally, we demonstrate the performance of our algorithm in a case study that considers taxi\nassignment with real taxi data from New York City. Our experiment results show that in\npractical scenarios, algorithm REASSIGN is able to signi\ufb01cantly improve the fairness of the\nassignment with almost no loss on system ef\ufb01ciency.\n\n1.2 Related Works\n\nThe problem of vehicle-request assignment in ridesourcing has been studied extensively. Several\nworks have focused on real-time assignment using different approaches, such as greedy match [22],\ncollaborative dispatch [32, 35, 25], planning and learning framework [33], and receding horizon\ncontrol approach [27].\n\n2\n\n\fWhen requests do not arrive in real-time but are given beforehand, the problem is known as the\nDial-a-Ride Problem (DARP) [12, 28]. Many variants of the dial-a-ride problem were proposed\ndepending on the speci\ufb01c applications [11, 20, 31, 15, 13, 3, 8].\nThe idea that a fairness criterion may affect ef\ufb01ciency in resource allocation problems has been\nexplored in numerous contexts and in a variety of models, such as multiportfolio optimization [18],\ncake cutting [10, 4, 7], load balancing in job scheduling [21, 2, 16], and bandwidth allocation [26,\n17, 9]. Bertsimas et al. [5, 6] has speci\ufb01cally studied fairness and the corresponding ef\ufb01ciency loss\nin a general divisible resource allocation framework and applied their results to a case study in the\ncontext of air traf\ufb01c management. However, none of these works can easily incorporate additional\nwaiting time and pick-up distance constraints because these are unique to the ridesourcing problem.\nTo the best of our knowledge, the ef\ufb01ciency-fairness trade-off in the domain of ridesourcing has not\nbeen addressed in the literature.\n\n2 Preliminaries\n\nWe consider the following bipartite matching problem that models the batch assignment of a set of\nrequests to a set of available vehicles in on-demand ridesourcing. Let us de\ufb01ne a bipartite graph\nG = (V,R, E), with V = {v1, v2, . . . , vn} the set of n available vehicles, R = {r1, r2, . . . , rm} the\nset of m requests, and E \u2286 {{v, r} : v \u2208 V, r \u2208 R} the set of weighted edges such that {v, r} \u2208 E\niff the request r can be served by the vehicle v under some speci\ufb01ed assignment constraints3. A\nutility uvr is the weight associated to each edge {v, r} and de\ufb01ned as the sum of trip utility wvr (i.e.\nthe pro\ufb01t vehicle v could obtain by serving request r) and historical utility hv (i.e. the total utility v\nhas obtained in preceding assignment periods).\nWe de\ufb01ne trip value \u03c4r as the length of the trip or the shortest time needed to travel from r\u2019s pickup\nto its dropoff location and trip cost \u03b9vr as the cruising time of v induced by serving r. Then, we set\nour trip utility function wvr = c\u03c4r \u2212 \u03b9vr with c being a constant to balance the value-cost effect.\nNext, we will introduce \u2206, a parameter that proves to be critical for our tradeoff analysis in Section 3.\nFormally,\n\n\u2206 := max\nr\u2208R\n\nmax\n\n{v,r},{v(cid:48),r}\u2208E |wvr \u2212 wv(cid:48)r|\n\n(1)\n\nor the maximum trip utility difference across all pairs of edges corresponding to the same request,\nacross all requests.\nRemarks. It is important to note that by our de\ufb01nition of trip utility wvr, the same request r contributes\nthe same value \u03c4r to this trip utility when matched with any vehicle. Thus, \u2206 directly translates to\nthe maximum difference in vehicle cruising time; it is easy to check that this is bounded above by\nsome assignment constraints that we set, e.g. request waiting time constraint.\nWe now refer to a setting with graph G = (V,R, E), a set of trip utilities {wvr}{v,r}\u2208E and historical\nutilities {hv}v\u2208V as an instance I.\nGiven an instance I, our goal is to \ufb01nd an assignment M that assigns each vehicle v to at most one\nrequest M (v) and each request r to at most one vehicle M (r). That is, M is always a matching in\nthe bipartite graph G.\nWe focus on two main objectives:\n\n(cid:88)\n\u2022 The ef\ufb01ciency of an assignment M,\n\nE(M ) :=\n\nv\u2208V\n\n(cid:88)\n\nv\u2208V\n\nuv,M (v) =\n\nhv + wv,M (v)\n\n\u2022 The fairness of an assignment M,\nF(M ) := min\n\nv\u2208V {uv,M (v)} = min\n\nv\u2208V {hv + wv,M (v)}\n\n3We make no restrictions on the structure of the set of edges E and allow it to encode any physical or\nperformance-related constraints, such as that request waiting time should be within some threshold, or vehicle\ntype (e.g. regular, luxury) should match the request type.\n\n3\n\n\fLet M be the set of all feasible assignments of instance I, we further de\ufb01ne the optimal ef\ufb01ciency\nEopt := max{E(M ) | M \u2208 M} and optimal fairness Fopt := max{F(M ) | M \u2208 M}. We will\nrefer to the assignments that produce optimal ef\ufb01ciency and optimal fairness as ef\ufb01cient assignment\nMeff and fair assignment Mfair, respectively.\nNote that our model above is \ufb02exible with respect to different features that may be of interest to the\nsystem. Below we discuss how to incorporate ridesharing (or carpooling) and how our model can\nhandle real-time assignment with multiple time periods.\n\n2.1 Ridesharing\n\nRidesharing refers to a ridesourcing mode in which a vehicle can serve multiple (usually no more than\n2) requests simultaneously. Such a service has been provided by all major ridesourcing companies\nworldwide and has enormous potential for positive societal impacts in terms of pollution, energy\nconsumption, and traf\ufb01c congestion.\nOur model can be easily adapted to allow ridesharing and following are the speci\ufb01c changes needed\nto be made. First, we de\ufb01ne a passenger to be a past request assigned in any preceding periods\nthat had not been picked up, or had been picked up by some vehicle and is currently en route to\nits destination. At any batch assignment, each vehicle v \u2208 V will have its own set of passengers\nSv. Thus, to determine whether an edge {v, r} \u2208 E, we need to update our assignment constraints\nsuch that, for instance, the total number of occupied seats by r and p, \u2200p \u2208 Sv, does not exceed\nthe vehicle\u2019s capacity or the delay (corresponding to r and each passenger p \u2208 Sv) imposed by\naugmenting r to v\u2019s current route should be within some speci\ufb01ed threshold.\nNext, by allowing ridesharing, our de\ufb01nition of hv implies the inclusion of trip utility wvp of all\npassengers p \u2208 Sv. It is particularly important to note the slight difference between the interpretation\nof trip cost in this setting and its single-ride counterpart.4\nNote that all these changes only affect the structure (density) of graph G and the values in {hv}v\u2208V\nassociated to its edge weights. The overall model remains the same. The de\ufb01nition of ef\ufb01ciency and\nfairness also remains unchanged. Hence, all the results and algorithm in Section 3 directly extend to\nthis setting. We will describe more details on the speci\ufb01c constraints for ridesharing relevant to our\ncase study in Section 4.\n\n2.2 Multi-Period Assignment\n\nThe above model describes a vehicle-request assignment problem in a single-batch setting. In practice,\nrequests are collected and matched in multiple batches in real-time throughout the day. To this end,\nwe can generalize our model to the following multi-period setting.\nWe split the duration of one day into T discrete time periods {1, . . . , T} (e.g., 30s per period).\nRequests are collected and matched during each time period. Consider a vehicle v that is assigned to\nserve request r in the current time period. In the single-ride setting, v will become unavailable for tvr\ntime periods while serving r and reappear at the r\u2019s destination afterward. In the ridesharing setting,\nr\u2019s pickup and destination locations will be appended to the route associated to the passenger set Sv,\nas long as this satis\ufb01es the speci\ufb01ed assignment constraints.\nThe de\ufb01nition of historical utility hv extends naturally in the context of multi-period assignment in\nthat hv is to be updated after each batch assignment, i.e. ht+1\nvr . Therefore, at t = T ,\nthe de\ufb01nition of utility, ef\ufb01ciency, and fairness remain the same.\n\nv + wt+1\n\nv = ht\n\n4Generally, we de\ufb01ne trip cost as follows,\n\n\u03b9vr := max{0, pu(r) \u2212 max{t, do(p)}}\n\nwith t the assignment time of r, pu(r) the pickup time of r, and do(p) the last dropoff time of passenger p that\nhas been on-board during the assignment of r. This de\ufb01nition caters to the case when Sv is not empty and that\nassigning r to v may involve the altering of v\u2019s original route in a non-trivial way. Meanwhile, in a single-ride\nsetting, this de\ufb01nition just implies r\u2019s pickup duration.\n\n4\n\n\f3 Ef\ufb01ciency-Fairness Tradeoff\n\nIn this section we analyze the ef\ufb01ciency and fairness tradeoff in ridesharing. Our main result is the\nfollowing theorem.\nTheorem 3.1. Given any ridesharing problem instance I and any 0 \u2264 \u03bb \u2264 1, there exists an\n2+\u03bb (Eopt \u2212 n\u2206) simultaneously.\nassignment M with fairness F(M ) \u2265 \u03bbFopt and ef\ufb01ciency E(M ) \u2265 2\nOur proof is constructive. In the following we present a simple reassignment algorithm that, starting\nfrom any existing assignment Mold, outputs a new assignment Mnew satisfying any desired fairness\nthreshold with bounded ef\ufb01ciency loss from Mold.\n\nAlgorithm 1: REASSIGN (I, Mold, f )\nInput\n\n:Instance I = {G(V,R, E),{wvr}{v,r}\u2208E,{hv}v\u2208V},\ncurrent assignment Mold,\nfairness threshold f \u2264 Fopt.\n\nOutput :A new vehicle-request assignment Mnew\n\n1 Compute a fair assignment Mfair\n2 Set Mnew = Mold\n3 while there exists v \u2208 V such that hv + wv,Mnew(v) < f do\n\n\u2208 V such that Mnew(v(cid:48)) = Mfair(v) do\n\nr \u2190 Mnew(v)\nMnew(v) \u2190 \u2205\nwhile there exists v(cid:48)\nMnew(v(cid:48)) \u2190 \u2205\nMnew(v) \u2190 Mfair(v)\nv \u2190 v(cid:48)\n\nend\nMnew(v) = Mfair(v)\n\n4\n5\n6\n7\n8\n9\n10\n11\n12 end\n\nIntuitively, the algorithm repeatedly chooses a vehicle v whose total utility uv,Mnew(v) is lower than\nthe fairness threshold f, and swap its assigned request to the one given out by the fair assignment,\ni.e. assign v to Mfair(v). Note that this new request Mfair(v) may be assigned to another vehicle v(cid:48) in\nMnew and thus, the swapping of solution continues until no such v(cid:48) can be found, as described in line\n6-10 of REASSIGN.\n\nCompute a fair assignment Mfair. Line 1 of REASSIGN requires us to compute a fair assignment\nMfair. This can be done ef\ufb01ciently using a simple variation of the standard bipartite matching\nalgorithm: We add n no-serve requests r1, . . . , rn to set R. Each ri has only one vehicle vi\nconnected to it with wvi,ri = 0; accordingly, we have uvi,ri = hvi. This edge represents the option\nof not assigning vehicle vi to any requests. Let the new request set be R+ and the new edge set be\nE+. Then for any value f, we de\ufb01ne Gf := (V,R+, Ef = {{v, r} \u2208 E+ | hv + wv,r \u2265 f}). It is\nnow easy to see that the optimal fairness Fopt is the largest value f such that Gf still has a perfect\nmatching. Such f can be found via a binary search on all possible fairness thresholds. Mfair is then a\nperfect matching in GFopt.\n\nTo prove Theorem 3.1, we show a more general claim about the output of REASSIGN.\nLemma 3.2. Given instance I, current assignment Mold and any fairness threshold f \u2264 Fopt,\nalgorithm REASSIGN(I, Mold, f ) always outputs an assignment Mnew with fairness F(Mnew) \u2265 f\nand ef\ufb01ciency E(Mnew) \u2265 2Fopt\n\n2Fopt+f (E(Mold) \u2212 n\u2206).\n\nThe idea of the proof of Lemma 3.2 is to consider each iteration of chain swapping (line 6-10 of\nREASSIGN). For some request r, its contribution to the decrease in ef\ufb01ciency is at most \u2206 if r is\nmatched in Mnew, otherwise if r is \u2018dropped\u2019, it is bounded above by f. We can then bound above the\nnumber of \u2018dropped\u2019 requests by constructing a lower bound for E(Mnew). The latter can be obtained\nfrom the fact that we swap the \u2018violating\u2019 edges with its fair counterpart (i.e. the weight of this edge\nis bounded below by Fopt) and in a non-trivial case (i.e. Ei(Mnew) \u2264 Ei(Mold)), we have at least 2\nedges swapped in one iteration.\n\n5\n\n\fFinally, Theorem 3.1 can be proved directly by replacing f with \u03bbFopt in Lemma 3.2.\n\nLower Bound. Next we focus on the theoretical lower bound for the ef\ufb01ciency-fairness tradeoff\nthat any algorithm could achieve. In particular, we show that the tradeoff achieved in Theorem 3.1 is\nactually tight in this model.\nTheorem 3.3. For any 0 \u2264 \u03bb \u2264 1 and any \u03b1 strictly larger than\n2+\u03bb , there always exists a\nproblem instance I, such that no assignment can achieve fairness F(M ) \u2265 \u03bbFopt and ef\ufb01ciency\nE(M ) \u2265 \u03b1(Eopt \u2212 n\u2206) simultaneously.\nThe proof uses a simple counter-example construction and is omitted.\nTheorem 3.1 and 3.3 together show that among all possible algorithms that can achieve a certain\nfairness requirement, the ef\ufb01ciency achieved by our algorithm REASSIGN has the best theoretical\nguarantee.\n\n2\n\n4 Experiments\n\nAt a \ufb01rst glance, the theoretical guarantee obtained in Section 3 may not be enough to convince the\ndecision maker of a ridesourcing platform to consider fairer solutions. Because the loss in ef\ufb01ciency,\nwhich directly translates to a revenue loss of the platform, might be too signi\ufb01cant for fairness\nconsiderations. For example, if one wants to adopt the fairest solution, setting \u03bb = 1 in Theorem 3.1\nshows that in the worst case the platform needs to sacri\ufb01ce more than 33% of ef\ufb01ciency. However, as\nwe will demonstrate in this section, in practice such worst case scenario will almost never happen.\nThrough extensive experiments on real-world datasets, we show that when moving towards fairer\nsolutions, the incurred loss in ef\ufb01ciency is much smaller than the theoretical prediction and in many\ncases negligible.\nWe test the performance of our algorithm in two settings: the single-batch setting, in which we\nconsider all requests within a short period of time and assign them to the set of available vehicles;\nand the multi-period setting, in which the requests are collected and assigned in multiple batches in\nreal-time. We also consider both single-ride and ridesharing setting, as described in the Preliminaries\nSection.\n\nDataset. We use the publicly available dataset of taxi trips in New York City [14], which contains\nfor each day the time and location of all of the pickups and drop-offs executed by each of the active\ntaxis. We choose a representative 2-hour horizon, 1700 - 1900, and extract all requests originating\nand \ufb01nishing within Manhattan, happening in May 2013. We consider the recorded pickup time as the\nrequest arrival time and the recorded passenger count as the request size. There are between 31,694\nto 56,743 extracted requests each day. To re\ufb02ect real road conditions and traveling time, we construct\na road network of Manhattan with 3,671 nodes and 7,674 edges. For simplication purposes, we round\nthe original pickup and drop-off location of data-extracted requests to their respective closest nodes.\nTravel time on each road or edge of the network is estimated based on the daily mean travel time\nestimate following the method in [30]. Shortest paths and travel times between all nodes are then\nprecomputed and stored in a look-up table.\n\nConstruction of Bipartite Graph. Following, we describe the speci\ufb01c constraints that we use in\nthe construction of the edge set E of G = (V,R, E)5: a vehicle v \u2208 V and a request r \u2208 R is\nconnected by an edge {v, r} iff there exists a way for v to serve r such that (i) the difference between\nr\u2019s pick-up time and its request time is within a threshold \u2126; (ii) the total travel delay time, de\ufb01ned\nas the difference between r\u2019s actual drop-off time and its earliest possible drop-off time, is within a\nthreshold \u0393; (iii) the trip utility wvr \u2265 0; (iv) if ridesharing is allowed, the total number of passengers\n(inclusive of r) on the vehicle at any time does not exceed the vehicle capacity \u03c7. For simpli\ufb01cation\npurposes, we assume that any vehicles can serve up to two requests at any time.\n\n4.1 Single-Batch Assignment\n\nOur single-batch setting experiment aims to elicit and analyze worst-case circumstances in terms of\nef\ufb01ciency loss. For this purpose, we construct synthetic cross-sectional scenarios (i.e. when vehicles\n\n5These are the same set of rules used in [1].\n\n6\n\n\fhave been on the road for some time and are available to serve new request) by tuning the parameters\nthat control request data extraction, vehicle positioning, and the structure of our bipartite graph (for\nexample, by relaxing assignment contraints or changing the ratio |V |\n|R| ).\n\n4.1.1 Experimental Setup and Data Preprocessing\n\nWe pick several days in which there are more than 200 requests arriving in the \ufb01rst 30 seconds of\nthe 1700-1900 horizon and test our algorithm on each day under different scenarios. Following, we\ndescribe the exact setting that we used to produce our worst-case results. We consider all requests\nwith trip length at least 400s such that we have m = |R| \u2208 [105, 142]. Upon initialization, we locate\nn = 1.2m vehicles within a reasonable time-distance from the requests such that each vehicle is\nconnected to at least 10 different requests in G. We de\ufb01ne two groups of vehicles, VH and VL,\nwith |VH| = m = 5|VL| to introduce some level of discrepancies to vehicle historical utilities\nand randomly generate hv such that \u2200v \u2208 VH , hv \u223c U (200, 400) and \u2200v \u2208 VL, hv \u223c U (50, 100).\nFinally, we set the maximum waiting time constraint \u2126 = 210s, constant c = 1, and vehicle capacity\n\u03c7 = 4.\n\n4.1.2 Results\n\n(a) Ef\ufb01ciency-Fairness Tradeoff\n\n(b) Utilities\n\nFigure 1: (a) The ef\ufb01ciency and fairness of the assignments output by REASSIGN with respect to\ndifferent fairness thresholds, compared with the theoretical lower bound implied by Thm 3.1 (b) The\nutilities of all vehicles in the ef\ufb01cient assignment and the fair assignment, both sorted from smallest\nto largest.\n\nFigure 1(a) demonstrates the tradeoff between ef\ufb01ciency and fairness when applying our algorithm\nREASSIGN with different values of fairness threshold f.\nAs one can see from the \ufb01gure, when ef\ufb01ciency is the only concern (corresponding to the leftmost\npoint6), the resulting assignment may have the lowest utility of all drivers as low as 51. However,\nas we start applying REASSIGN with higher and higher fairness thresholds, this lowest utility value\ngradually improves, until it reaches the highest point Fopt = 328 in the fair solution (corresponding\nto the rightmost point). For the worst-off driver, the utility improvement from ef\ufb01cient assignment to\nthe fair assignment is over 6-fold.\nIt is also important to note that, although Theorem 3.1 and 3.3 claim that the maximum ef\ufb01ciency\nloss may be as high as 33% (as indicated by the dashed curve in Figure 1(a)) in the worst case, this\nloss is much smaller in reality. In this example, the largest ef\ufb01ciency loss is less than 6% from the\noptimal ef\ufb01ciency. This is our worst-case result; the results corresponding to other days and settings\npreserve similar magnitudes of fairness improvement, while demonstrating even smaller ef\ufb01ciency\nlosses, with many under 1%.\nFrom our experiments, we observe three situations in which we could incur more ef\ufb01ciency loss: (i)\nwhen there is a denser bipartite graph, essentially more leeway to permutate between different vehicle-\n\n6Note that the threshold f = 0 corresponding to no fairness constraint is not necessarily binding in the\nreassignment procedure. For instance, in this particular case, we can keep increasing f up to .16Fopt before the\nalgorithm outputs a matching with different (and higher) fairness value.\n\n7\n\n0.20.40.60.81.0F(M)/Fopt0.40.50.60.70.80.91.01.1E(M)/EoptReassignLower-bound(Thm3.1)020406080100120140VehicleID0500100015002000Utility(secs)E\ufb03cientAssignmentFairAssignment\frequest pairs; (ii) when ef\ufb01cient allocation has a signi\ufb01cantly smaller overall trip cost compared to\nthat of a fair allocation; (iii) when more requests are dropped by reassignment to a fairer solution.\n(i), (ii), and (iii) are intricately connected and may have competing effect up to some extent on the\nef\ufb01ciency loss. Therefore, we can conclude that the theoretical worst-case ef\ufb01ciency loss does not\nnecessarily arise even in arti\ufb01cial examples; here, we have seen the problem instances that show\nmuch more benign behaviour.\nFinally, we compare the utilities of all vehicles (after assignment) in the ef\ufb01cient and fair matching.\nIn Figure 1(b), we plot these two sets of utilities, after sorting the elements from smallest to largest.\nIt is evident that our algorithm manages to redistribute the trip utility increments to the vehicles with\nlow historical utilities, without sacri\ufb01cing too much on the ef\ufb01ciency.\n\n4.2 Multi-Period Assignment\n\nWe further assess the performance of our algorithm in the multi-period case, where we compute a\nmatching assignment using REASSIGN in each period with available vehicles and requests. For this\npurpose, we \ufb01x \u03bb and set f = \u03bbFopt in REASSIGN for each period t \u2208 {1, 2, . . . , T}. Note that\ndue to the dependence of the future instances on current assignments and the uncertainty inherent to\nfuture demand distributions, we cannot claim the same theoretical guarantee as shown in Section 3\nat the end of the multi-period assignment horizon. Nevertheless, experiment results show that our\nalgorithm REASSIGN performs even better in the multi-period setting, demonstrating satisfactory\nfairness improvement with almost no loss in ef\ufb01ciency.\n\n4.2.1 Experimental Setup and Data Preprocessing\n\nWe discretize the 1700-1900 horizon into T = 240 time-steps of 30 seconds. At t = 0, we initialize\nn = 2000 empty vehicles with capacity \u03c7 = 4 at reasonable locations based on the frequency and\nlocations at which requests appear in the whole 2-hour horizon. Requests are generated from the\ndataset and collected in time windows of 30s. Each vehicle will continue to pickup and dropoff\npassengers following the routes assigned in batches by the central. Historical utility hv is set to 0 for\nall vehicles at the beginning of the simulation horizon. Then in any particular period, hv represents\nthe accumulated trip utilities wvr from all the requests that vehicle v has been matched to (see Section\n2 for more details on the updating of hv). In the case of ridesharing, hv should include the trip\nutilities wvp of all passengers p \u2208 Sv. We set the maximum waiting time constraint \u2126 = 150s and\nmaximum delay time constraint \u0393 = 300s.\nRemarks. Note that the controlling of vehicle initial locations is in line with what we did in the\nsingle-batch setting; we make sure that there are suf\ufb01ciently many requests each vehicle can serve\nduring the 2-hour horizon. With this, we want to alleviate the adverse impact that exogenous factors,\nsuch as the neighbouring structure of our network, have on fairness. Speci\ufb01c to the multi-period\nsetting, we also need to take care of vehicle\u2019s intermediate locations (after each assignment). Consider\nthe case when vehicle is assigned to a request with dropoff node having very few degrees. Due to our\nassumption that any vehicle stays still until it is assigned a new request, this vehicle may be stuck\nforever in this node. In this respect, we removed the requests whose drop-off node is not close to\nsuf\ufb01ciently many pickup nodes. We keep such removal under 4 percents of all requests in the dataset.\nFollowing, we present the average results of our algorithm tested on 10 different days.\n\n4.2.2 Results\n\nFigure 2 shows again the ef\ufb01ciency and fairness tradeoff for the algorithm REASSIGN given different\nvalues of \u03bb, in both single-ride and ridesharing setting. Compared to the single-batch setting, these\nresults are even more extreme: there is essentially negligible ef\ufb01ciency loss, even for the fairest\nsolution with \u03bb = 1. One explanation of such phenomenon is that as time progresses, the historical\nutilities of all vehicles will increase to larger and larger values relative to the batch-speci\ufb01c trip utility\nincrements that we can control by specifying \u03bb\u2019s. As a result, the analysis of Theorem 3.1 is no longer\ntight and correspondingly, the ef\ufb01ciency loss will be signi\ufb01cantly smaller than what the theorem\nclaims.\nFigure 3 then shows the fairness improvement when we set different \u03bb as the parameter in REASSIGN.\nOne can still observe a signi\ufb01cant increase in fairness value when we shift our \u03bb from 0 to 1.\n\n8\n\n\f(a) Single-Ride\n\n(b) Ridesharing\n\nFigure 2: The ef\ufb01ciency and fairness of the assignments output by REASSIGN with respect to different\nfairness thresholds in the multi-period setting.\n\n(a) Single-Ride\n\n(b) Ridesharing\n\nFigure 3: Relative fairness values of the assignments output by REASSIGN with regard to different \u03bb\nin the multi-period setting. The vertical intervals represent the 95% con\ufb01dence intervals.\n\nOverall, these results suggest that in practical scenarios, it is often possible to signi\ufb01cantly improve\nfairness with negligible loss in system ef\ufb01ciency. This essentially implies that by considering fair\nsolutions, ridesourcing enterprises will have the capacity to do much greater good without sacri\ufb01cing\ntheir pro\ufb01tability.\n\n5 Conclusion\n\nIn this paper, we deal with the problem of balancing ef\ufb01ciency and fairness in the context of\nridesourcing request assignment. We present a simple reassignment algorithm that can compute an\nassignment with any desired fairness and provably good ef\ufb01ciency. We also provide tight upper\nbound on the relative ef\ufb01ciency loss of our solution compared to the ef\ufb01cient-maximizing assignment.\nExperiment results show that in practical scenarios, this algorithm is able to signi\ufb01cantly improve the\nfairness of the assignment to drivers with very little loss on the system ef\ufb01ciency.\nThe theoretical bounds derived in our work are of independent interest and can be applied to a\nbroader family of matching problems. How to \ufb01nd other suitable applications in which similar\ntechniques or results can be applied to is one interesting future working direction. Other future\nresearch directions may include considering strictly passenger-side ef\ufb01ciency or different fairness\ncriteria, such as proportional fairness [19], and measure their tradeoffs with ef\ufb01ciency. Finally, our\ninvestigations lead to the open question of designing a learning framework to obtain endogenously\nthe optimal string of \u03bb\u2019s, that interacts with and adjusts to real-time supply-demand dynamics.\n\n9\n\n0.40.50.60.70.80.91.0F(M)/Fopt0.650.700.750.800.850.900.951.001.05E(M)/EoptReassignLower-bound(Thm3.1)0.20.40.60.81.0F(M)/Fopt0.650.700.750.800.850.900.951.001.05E(M)/EoptReassignLower-bound(Thm3.1)0.00.20.40.60.81.0\u03bb0.20.30.40.50.60.70.80.91.0F(M)/Fopt0.00.20.40.60.81.0\u03bb0.20.40.60.81.0F(M)/Fopt\fReferences\n[1] Javier Alonso-Mora, Samitha Samaranayake, Alex Wallar, Emilio Frazzoli, and Daniela Rus.\nOn-demand high-capacity ride-sharing via dynamic trip-vehicle assignment. Proceedings of the\nNational Academy of Sciences, 114(3):462\u2013467, 2017.\n\n[2] Javed Aslam, April Rasala, Cliff Stein, and Neal Young. 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The International Journal of Robotics Research, 35(1-3):186\u2013203,\n2016.\n\n11\n\n\f", "award": [], "sourceid": 2854, "authors": [{"given_name": "Nixie", "family_name": "Lesmana", "institution": "Nanyang Technological University"}, {"given_name": "Xuan", "family_name": "Zhang", "institution": "University of Illinois at Urbana-Champaign"}, {"given_name": "Xiaohui", "family_name": "Bei", "institution": "Nanyang Technological University"}]}