Balancing the Tradeoff between Profit and Fairness in Rideshare Platforms During HighDemand Hours
Abstract
Rideshare platforms, when assigning requests to drivers, tend to maximize profit for the system and/or minimize waiting time for riders. Such platforms can exacerbate biases that drivers may have over certain types of requests. We consider the case of peak hours when the demand for rides is more than the supply of drivers. Drivers are well aware of their advantage during the peak hours and can choose to be selective about which rides to accept. Moreover, if in such a scenario, the assignment of requests to drivers (by the platform) is made only to maximize profit and/or minimize wait time for riders, requests of a certain type (e.g., from a nonpopular pickup location, or to a nonpopular dropoff location) might never be assigned to a driver. Such a system can be highly unfair to riders. However, increasing fairness might come at a cost of the overall profit made by the rideshare platform. To balance these conflicting goals, we present a flexible, nonadaptive algorithm, , that allows the platform designer to control the profit and fairness of the system via parameters and respectively. We model the matching problem as an online bipartite matching where the set of drivers is offline and requests arrive online. Upon the arrival of a request, we use to assign it to a driver (the driver might then choose to accept or reject it) or reject the request. We formalize the measures of profit and fairness in our setting and show that by using , the competitive ratios for profit and fairness measures would be no worse than and respectively. Extensive experimental results on both realworld and synthetic datasets confirm the validity of our theoretical lower bounds. Additionally, they show that under some choice of can beat two natural heuristics, Greedy and Uniform, on both fairness and profit. Code is available at: https://github.com/nvedant07/ridesharefairnesspeak/.
1 Introduction
Rideshare platforms have received significant attention in both the computer science and operations research communities. The following are the three main categories of research. The first studies matching policy design in the rideshare setting (i.e., matching riders and drivers), e.g., [30, 1, 18, 25, 24, 26, 5, 9, 8]. The second considers the spatialtemporal pricing aspects of rideshare, e.g., [19, 6, 13, 2, 3]. The third focuses on applying reinforcementlearning approaches to planning and matching problems in rideshare see, e.g., [28, 17].
In all the aforementioned prior work, the objective is either to maximize the total profit of the system or minimize the waiting time of riders (or a combination thereof). However, both of these objectives are global in that they do not ensure subgroup level fairness (e.g., riders of a particular protected class being systematically underserved). Consider a scenario during peak hours when there is increased demand for rides, thus, giving the drivers a bargaining advantage to drop rides. Typical criteria used by drivers to reject riders include riders’ starting/ending location, trip length, gender, race and age. Recently, it has been reported that drivers also reject riders based on attributes such as gender, race and disability either intentionally or unintentionally, see, e.g., [15, 7, 12]. \citeauthorwebrace \shortcitewebrace reported that “black passengers using ridehailing apps have to wait an average of minute and seconds longer than white counterparts and are 4% more likely to have drivers cancel on them.”
Thus, current rideshare platforms can enable and amplify prejudices in society. To counter this, current rideshare apps implement the following measures:
Ensuring group level fairness and optimizing global profit tends to be somewhat conflicting goals in general; particularly so in rideshare platforms during peak hours. The rationale is that the driver to request ratio is small and thus, drivers can afford to be more choosy. This may lead to unfair practices such as rejecting trips to unpopular destinations and rejecting riders with disabilities. To promote group level fairness among the riders, systems should aggressively match requests which typically incur high cancellation rates. This action, however, would increase the number of cancellations by drivers. The current penalty in popular rideshare platforms is that drivers are deactivated if they cancel too many requests. Thus, prioritizing high risk (in terms of cancellation) trips could lead to more drivers being deactivated and/or leaving the system. In summary, on one hand, prioritizing group level fairness leads to limited drivers and thus, limited total trips. On the other hand, letting the market find its equilibrium leads to amplifying societal biases. Thus, the central question is the following: Can we design policies that can smoothly tradeoff between the two conflicting objectives?
In this paper, we answer the above question in the affirmative by providing provably efficient policies. As is common (see, e.g., \citeauthorxuAAAI18 \shortcitexuAAAI18 for additional motivation) we model the dynamics of the rideshare platform as an onlinematching model as follows. We have a bipartite graph , where and represent the set of available drivers (static or offline) and the request types (dynamic or online arrival) respectively.
We have the following key assumptions that we use to show provably guarantees. In the experimental section, we work with real data and show that the algorithms are robust even when some of these assumptions do not necessarily hold.
Arrival of Requests. We consider a finite set of requests that are drawn from a known identical distribution independently; this is commonly [8] called the known identical independent distributions (KIID). The motivation for this assumption stems from the fact that we can often learn the arrival distribution from historical logs [29, 16, 27]. KIID is widely used in many practical applications of online matching markets including rideshare and crowdsourcing [30, 9, 22, 23]. Further, we call the expected number of times any request is sampled from this distribution in the rounds as the arrival rate, denoted by . Thus, it is easy to see that . We further assume that the total number of arrivals of online requests is far larger than that of drivers in the system, i.e., .
Edge existence probabilities. Each edge is associated with an existence probability : once we assign to , we observe an immediate random outcome of the existence, which is present (i.e., accepts ) with probability and not ( cancels ) otherwise. The probability captures the statistical chance that a driver of type would accept to serve the request of type . We assume that (1) the randomness driving the edge existence is independent across all the edges; (2) the values are provided as part of the input. The first assumption is motivated by individual choice and the second from the fact that historical logs can be used to compute such statistics with high precision.
Cancellation budget. Each driver is associated with a given budget of cancellation, . In other words, driver will be removed from the graph if they cancel more than requests in the rounds (in which case we assume that is temporarily deactivated by the system as a penalty). Once a request gets rejected, we assume that the system will not try any reassignment of to other available drivers. This is without loss of generality since any reassignment can be modeled as resampling in the succeeding timesteps.
We assume that the system gains a profit from if driver completes (i.e., is assigned and the driver accepts) the trip (in this case, we call it a successful assignment or a match). For a given policy , let (possibly random) be the set of successful assignments; we interchangeably use the term matching to denote this set . We define two objectives, namely profit and fairness, as follows.
 Profit:

The expected total profit over all matches, which is defined as .
 Fairness:

Let be the subset of edges incident to . Note that can be larger than due to multiple arrivals of type in the timesteps. We define the fairness achieved by over all request types as , which refers to the minimum ratio of the expected number of matches of type to that of arrivals.
^{4} Thus, maximizing fairness corresponds to maximizing this minimum ratio.
We aim to design an online matching policy that balances the tradeoff between the two objectives of maximizing profit and fairness.
1.1 Preliminaries and Main Contributions
Competitive ratio. Competitive ratio is a commonlyused metric to evaluate the performance of online algorithms. Consider an online maximization problem for example. Let denote the expected performance of on a given distribution , where the expectation is taken over the random arrival sequence . Let denote the expected offline optimal, where refers to the optimal value after we observe the full arrival sequence . Then, the competitive ratio is defined as . It is a common technique to use a Linear Program () to upper bound (called the benchmark ) and hence get a valid lower bound on the target competitive ratio. In our paper, we conduct competitiveratio analysis on both objectives.
Main contributions. This paper provides threefold contributions.
First, we formalize the metric of fairness in rideshare. More specifically, we consider the onlinematching based model with multiple objectives.
Second, we present a provably efficient algorithm and provide formal mathematical guarantees. To do so, we first propose a biobjective linear program ((1) and (2)), whose optimal value is at least as large as that of any online algorithm that maximizes either of these objectives (or a combination thereof). Our main algorithm uses this biobjective LP to guide the online decisionmaking process. In particular, we prove the following main theorems.
Theorem 1.
achieves a competitive ratio at least simultaneously on the profit and fairness for any given with .
Theorem 2.
Third, we provide an extensive evaluation of the algorithm and modeling assumption on a realworld dataset collected from a large ondemand taxi dispatching platform. The experiments have many novel insights; among others, we show that even when some of the assumptions that were used to prove mathematical guarantees do not hold, performs well in practice.
2 Additional Related Work
There is a large body of work which studies fairness issues in resource allocation problems (divisible or indivisible goods), see, e.g., [11, 4, 20, 14, 10]. Most of these works require the allocation policy to satisfy certain properties in fair mechanism design such as strategyproofness and envyfreeness. These properties do not apply here, however. Recent work by \citeauthorsuhr2019 \shortcitesuhr2019 proposes a matching policy to balance fairness and profit over time; however, they do not provide any optimality guarantess for their proposed policy. \citeauthorlesmana2019neurips \shortcitelesmana2019neurips has a setting similar to ours, however they consider the case when there are more drivers than riders. Our focus is on the case when the number of drivers in the system are less than the demand for rides (i.e., peak hours).
Our model belongs to a more general optimization paradigm, called MultiObjective Optimization. \citeauthorravi1993many \shortciteravi1993many presented approximation algorithms for a variety of networkdesign problems. \citeauthorgrandoni2009 \shortcitegrandoni2009 designed several iterativerounding based approximation algorithms for multiobjective optimization problems. More recently, \citeauthoraggarwal2014 \shortciteaggarwal2014 studies the Biobjective Online Bipartite Matching where there is essentially one single objective: the minimum matching ratio over two disjoint sets of edges. \citeauthoresfandiari2016bi \shortciteesfandiari2016bi considers the Biobjective Online Submodular Optimization problem, where the two objectives are two monotone submodular functions. Our objectives are two linear functions, but our model assumes a more complicated setting (i.e., edge existence probabilities and cancellation quotas on the offlineside vertices). The models studied by \citeauthorbansal2012lp \shortcitebansal2012lp and \citeauthorBSSX17 \shortciteBSSX17 have the closest setting to us: each edge has an independent existence probability and each vertex from the offline and/or online side has a patience constraint (similar to cancellation quota for each driver in our setting) on it. However, both those works investigated only one single objective: maximization of the total profit.
3 Valid Benchmarks for Profit and Fairness
We first present our benchmark LPs and then an LPbased parameterized algorithm. For each edge , let be the expected number of probes on edge (i.e., assignments of to but not necessarily matches) in the offline optimal. For each (), let () be the set of neighboring edges incident to (). Consider the following biobjective LP.
(1)  
(2)  
s.t.  (3)  
(4)  
(5)  
(6) 
Let (1) and (2) denote the two LPs with the respective Objective (1) and (2), each with constraints (3), (4), (5), (6). Note that we can rewrite Objective (2) as a linear one like with additional linear constraints as for all . For presentation convenience, we keep the current compact version. The validity of (1) and (2) as benchmarks for our two objectives can be seen in the following lemma.
Lemma 1.
Proof.
We can verify that objective functions (1) and (2) each captures the exact expected profit and fairness achieved by the offline optimal, according to our definition in Section 1. To prove the validity of the benchmark for each objective, it suffices to show the feasibility of all constraints for any given offline optimal.
Constraint (3) is valid since each driver has a unit capacity; Constraint (4) is valid since each can be probed at most times according to our assumption; Constraint (5) is valid since the expected number of probes related to each should be no more than that of online arrivals (recall that we can try at most one assignment upon the arrival of ). Thus we justify the feasibility of all constraints for any given offline optimal. ∎
4 An LPbased Parameterized Algorithm
Now we present an LPbased parameterized algorithm. Let and be an optimal solution to (1) and (2) respectively. Consider a given pair of parameters with . will sample an assignment guided by the two LP solutions and , with probabilities and respectively. Note that constraint (4) for the LP models the expectation of number of probes. While this number might exceed , it’s important to note that this solution is for the offline case and does not correspond to the actual assignments. During the execution of our algorithm, we explicitly check if a driver is available before making the assignment, thus ensuring that no driver is assigned more than rides. The details are as follows.
The key part to prove Theorem 1 is the computation of the probability that each driver is available at each time. Focus on a given driver and a time . Let be the event that is available at (the beginning of) . Note that the occurrence of can be guaranteed by these two events: (1) never received and simultaneously accepted any assignment prior to ; (2) received no more than assignments prior to . For each and , let indicate if comes (or comes) at time , indicate if gets sampled in at and indicate if accepts after the assignment. Set and . From our analysis, we see that .
Lemma 2.
For any given and , we have
Proof.
The last inequality follows from and due to Constraint (3), and .
Lemma 3.
Lemma 3 says that the two events and are positively correlated. Proof of Lemma 3 can be found in the full version of the paper (on Arxiv).
Now we have all ingredients to prove the main Theorem 1.
Proof.
For each , let be the expected number of successful assignments of in . Here an assignment is successful iff is available when we assign to (but no necessarily means accepts ).
The last term is obtained after taking .
Let be the expected total profit obtained by . By linearity of expectation, we have . From Lemma 1, we know that the expected profit in offline optimal is upper bounded by . Thus we claim that achieves a ratio at least on the profit. Similarly, we can argue that achieves a ratio at least on the fairness. ∎
5 Hardness Results
The model in our paper has two objectives which complicate the hardness analysis. To simplify it, we focus only on those nonadaptive algorithms. We characterize a nonadaptive algorithm as where each ( is the size of , the set of edges incident to ) such that . A nonadaptive algorithm parameterized with , denoted by will sample an assignment with probability upon the arrival of , and assign to if is available. Note that our LPbased parameterized algorithm can be viewed as a specific nonadaptive with for each .
Example 1.
Consider a star graph where with , and . We use to denote and edge when the context is clear. Let and for all . In other words, our star graph is unweighted and the arrival distributions are uniform. Set for and for each . Let and denote the optimal LP values of (1) and (2) on this example respectively. We can verify that: (1) , where there is a unique optimal solution and for all ; (2) , where there is a unique optimal solution and for all .
Consider a given nonadaptive algorithm , with on the above example. Let and be the expected total profit and fairness achieved by . Set . Note that .
Lemma 4.
The sum of competitive ratios achieved by for the profit and fairness on Example 1 is no larger than .
Proof.
From the above computation, we see that the expected sum of acceptances of assignments over all during the online phase is . Thus the minimum ratio should be no larger than the average, from which we have
Note that and . Thus the sum of competitive ratios on the profit and fairness should be
∎
6 Experiments
Realworld dataset processing.
We now provide experimental validation of our algorithms. We use a realworld dataset: the New York City yellow cabs dataset
We consider the rush hour of 7–8PM; in our dataset, this period typically has the largest number of trips per hour of the day. On Jan. 31st 2013, a total of trips were completed by a total of drivers. One driver might make multiple trips during this period. However, here we assume that once a driver starts the trip, they are out of the system. So, we assume that all drivers are present at the start and need to serve a total of trips (i.e., ). This is consistent with peak hour assumptions (). Similar to \citeauthorxuAAAI18 \shortcitexuAAAI18, we focus on longitudes from to and latitudes from to , both with a step size of . We then assign an index to each grid and every location with latitudes and/or longitude in a particular grid is represented by the index of that grid.
We construct our graph as follows. Each represents a driver type which has attributes of the starting location and race. Each represents a request type which has attributes of the starting location, ending location, and race. We downsample from all driver and request types such that and . For each , we set a uniform cancellation quota ; across three experiments, we vary . For each , we generate a random value (normal distribution) and assign the value as its arrival rate. Set . We assume that the driver’s starting coordinates belong to the same bin as that of the start coordinates of the trip. An edge exists between a driver type and a request type , iff the starting coordinates of is in the same bin as that of . We set the existing probabilities mainly based on the combination of driver and rider’s races. When both driver and rider belong to the advantaged class, we assign ; if driver and rider are both members of the disadvantaged class then we assign and in all other cases . These probabilities are then scaled up by a factor , i.e., = + . In our experiments we set . The profit associated with each edge () is defined as the normalized distance of the request type (so ).
Synthetic dataset. To generate a synthetic dataset, we fix , and sample arrival rates randomly from a multinomial distribution to ensure they sum up to . Between each pair and , an edge exists with probability and does not exist with probability . The probabilities are randomly sampled such that and the profit values () associated with each edge are randomly sampled such that . We assign each the same cancellation quota , and vary across three experiments.
Algorithms. We run for iterations and take the average values over these runs to be the expectations. We use profit computed via (1) as the benchmark and use it to calculate the competitive ratio of profit for . Similarly, we use (2) as the benchmark for fairness and use it to calculate the competitive ratio of fairness for . For and , say a request of type arrives at time and let be the set of available assignments at and let be all possible assignments for . In , we sample an assignment uniformly from and check if it exists in . If it exists, then we make the assignment; otherwise, we reject the request. In , we select the assignment with the highest value among .
Results on the real and synthetic datasets. Figures 1 and 2 show results on the real dataset. We see that the solid lines (performance of ) always stay above the dotted ones (theoretical lower bounds). Note that a higher means that is guided more heavily by and a higher means is guided more heavily by . For the cases when this trend is not apparent (e.g., for in Figure 1), we posit that the optimal solutions ( and ) are highly correlated. Figure 2 shows that the effectiveness and flexibility of on both objectives compared to and . Specifically, is very weak and dominated by almost all variants of on both objectives. This is expected since as a heuristic is shortsighted and does not make assignments based on the what kind of requests the system expects to get in the future. seems to achieve relatively high fairness, which makes sense due to the nature of . However, provides a wide range of options and some of them can beat on both objectives as well. Figures 3 and 4 show results on the synthetic dataset. The trends are more consistent since all values here are randomly picked. We see that even in this case is able to do better than and on both the objectives.
7 Conclusion
In this work, we present a flexible approach for matching requests to drivers during peak hours that balances fairness and profit. Our proposed approach allows the system designer to specify how fair and how profitable they want the system to be via two separate parameters. We take a nuanced view of the ridesharing market and model the problem as an online bipartite matching problem with stochastic rewards. One highlight is the introduction of existence probability on each edge, which captures the potential acceptance rate for each pair of driver and rider types. We present an LPbased algorithm that dynamically assigns requests to drivers. Extensive experimental results on both of the real and synthetic datasets show that our proposed approach is not only above the theoretical lower bounds but also can beat natural approaches such as and on both objectives. Our work presents many interesting directions for future research. For example, we propose a nonadaptive algorithm in which once a request is rejected (either by the system or the driver), it will not be assigned again. A possible direction for future work could be to reassign a rejected request. Finally, we assume that drivers do not appear again in the system for the peak hour, however, in reality, that might not always be the case. We hope that our work will encourage the community to look at such problems.
8 Acknowledgements
We would like to thank anonymous reviewers for their helpful comments. Nanda and Dickerson were supported by NSF CAREER Award IIS1846237 and DARPA SI3CMD Award S4761. Srinivasan was supported in part by NSF CNS1010789, CCF1422569, and CCF1749864, and by research awards from Adobe, Amazon, and Google. Dickerson and Srinivasan were both supported by a gift from Google and a seed grant from the Maryland Transportation Institute. Work and the corresponding funding was done when Sankararaman was affiliated with University of Maryland, College Park.
Footnotes
 See, for example, §4A “Sharing Between Users” in Lyft’s privacy policy found at https://www.lyft.com/privacy.
 We consider a small time window during the peak hours, thus assuming that the set of drivers are static.
 We can create multiple copies of to address the case when each can be matched multiple times. Note that each copy corresponds to the same driver and shares same underlying variables.
 Since we are in the context of peak hours, we consider the group level fairness of riders.
 http://www.andresmh.com/nyctaxitrips/
 https://tinyurl.com/y56gonvs
 https://tinyurl.com/y7q8dppr
 https://tinyurl.com/y27anseh
 For simplicity, we ignore other potential discrimination factors such as gender and disability.
References
 (2019) Edge weighted online windowed matching. In Proceedings of EC, pp. 729–742. Cited by: §1.
 (2017) Pricing and optimization in shared vehicle systems: an approximation framework. In , EC ’17, pp. 517–517. External Links: ISBN 9781450345279, Link, Document Cited by: §1.
 (201609) Dynamic pricing in ridesharing platforms. SIGecom Exch. 15 (1), pp. 65–70. External Links: ISSN 15519031, Link, Document Cited by: §1.
 (2016) Fair resource allocation in a volatile marketplace. EC ’16, pp. 819–819. External Links: ISBN 9781450339360, Link, Document Cited by: §2.
 (2018) Algorithms for tripvehicle assignment in ridesharing. In , AAAI ’18, pp. 3–9. Cited by: §1.
 (2017) Spatial pricing in ridesharing networks. NetEcon ’17, pp. 5:1–5:1. External Links: Link, Document Cited by: §1.
 (2019) Woman says uber driver denied her ride because of her wheelchair. Note: \urlhttps://tinyurl.com/yxea2r97Accessed: 20190612 Cited by: §1.
 (2018) Allocation problems in ridesharing platforms: online matching with offline reusable resources. In , AAAI ’18, pp. 1007–1014. Cited by: §1, §1.
 (2018) Assigning tasks to workers based on historical data: online task assignment with twosided arrivals. In Proceedings of the 17th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’18, pp. 318–326. Cited by: §1, §1.
 (2018) Fair allocation of indivisible public goods. In Proceedings of EC, pp. 575–592. Cited by: §2.
 (2011) Dominant resource fairness: fair allocation of multiple resource types.. In NSDI, Vol. 11, pp. 24–24. Cited by: §2.
 (2019) Ont. woman says uber driver rejected her guide dog. Note: \urlhttps://tinyurl.com/y7sd268yAccessed: 20190612 Cited by: §1.
 (2019) Near optimal control of a ridehailing platform via mirror backpressure. arXiv preprint arXiv:1903.02764. Cited by: §1.
 (2014) No agent left behind: dynamic fair division of multiple resources. JAIR 51, pp. 579–603. Cited by: §2.
 (2019) Uber driver fired for refusing to drive woman to get abortion. Note: \urlhttps://tinyurl.com/y2cadftkAccessed: 20190612 Cited by: §1.
 (2018) Multitask representation learning for travel time estimation. In , KDD ’18, pp. 1695–1704. Cited by: §1.
 (2018) Efficient largescale fleet management via multiagent deep reinforcement learning. In , pp. 1774–1783. Cited by: §1.
 (2018) Online spatiotemporal matching in stochastic and dynamic domains. Artificial Intelligence 261, pp. 71 – 112. External Links: ISSN 00043702, Document, Link Cited by: §1.
 (2019) Spatiotemporal pricing for ridesharing platforms. In , EC ’19, pp. 583–583. Cited by: §1.
 (2015) Beyond dominant resource fairness: extensions, limitations, and indivisibilities. ACM TEAC 3 (1), pp. 3. Cited by: §2.
 (2018) There’s a quiet battle of wills between uber drivers and customers over who cancels first. Note: \urlhttps://tinyurl.com/y8jx6dolAccessed: 20190612 Cited by: §1.
 (2013) Pricing mechanisms for crowdsourcing markets. In , WWW ’13, pp. 1157–1166. Cited by: §1.
 (2013) Truthful incentives in crowdsourcing tasks using regret minimization mechanisms. In , WWW ’13, pp. 1167–1178. Cited by: §1.
 (2016) Online minimum matching in realtime spatial data: experiments and analysis. Proceedings of the VLDB Endowment 9 (12), pp. 1053–1064. Cited by: §1.
 (2016) Online mobile microtask allocation in spatial crowdsourcing. In , ICDE ’16, pp. 49–60. Cited by: §1.
 (2017) Flexible online task assignment in realtime spatial data. Proceedings of the VLDB Endowment 10 (11), pp. 1334–1345. Cited by: §1.
 (2018) Learning to estimate the travel time. In , KDD ’18, pp. 858–866. Cited by: §1.
 (2018) Largescale order dispatch in ondemand ridehailing platforms: a learning and planning approach. In , KDD ’18, pp. 905–913. Cited by: §1.
 (2018) Deep multiview spatialtemporal network for taxi demand prediction. In , AAAI ’18, pp. 2588–2595. Cited by: §1.
 (2019) Preferenceaware task assignment in ondemand taxi dispatching: an online stable matching approach. In , AAAI ’19, pp. 2245–2252. Cited by: §1, §1.