Ridesharing with Driver Location Preferences

05/30/2019 ∙ by Duncan Rheingans-Yoo, et al. ∙ 0

We study revenue-optimal pricing and driver compensation in ridesharing platforms when drivers have heterogeneous preferences over locations. If a platform ignores drivers' location preferences, it may make inefficient trip dispatches; moreover, drivers may strategize so as to route towards their preferred locations. In a model with stationary and continuous demand and supply, we present a mechanism that incentivizes drivers to both (i) report their location preferences truthfully and (ii) always provide service. In settings with unconstrained driver supply or symmetric demand patterns, our mechanism achieves (full-information) first-best revenue. Under supply constraints and unbalanced demand, we show via simulation that our mechanism improves over existing mechanisms and has performance close to the first-best.

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1 Introduction

Uber connected its first rider to a driver in the summer of 2009,111https://www.uber.com/newsroom/history/, visited 02/25/2019. and since then, ridesharing platforms have dramatically changed the way people get around in urban areas. Ridesharing platforms allow a wide array of people to become drivers and—in contrast to traditional taxi systems—use dynamic “surge pricing” at times when demand exceeds supply. Properly designed, dynamic pricing improves system efficiency [Castillo et al.2017], increases driver supply [Chen and Sheldon2015], and makes the system reliable for riders [Hall et al.2015].

A growing literature studies how to structure prices for riders and compensation for drivers so as to optimally account for variation in supply and demand [Banerjee et al.2015, Bimpikis et al.2016, Castillo et al.2017, Ma et al.2019]. However, existing models leave aside driver heterogeneity. In practice, some drivers may prefer to drive in the city and others in the suburbs, and many prefer to start or end their days in a particular location. A matching system that treats drivers as homogeneous can make inefficient matches, with drivers preferring to fulfill each others’ dispatch instructions instead of their own.

The problem, however, goes beyond simple efficiency loss. A core feature of platforms is that drivers retain the flexibility to choose when and where to provide service. Every ride the platform proposes needs to be accepted voluntarily, forming an optimal response for the driver [Ma et al.2019]. This kind of incentive alignment simplifies participation for drivers and also makes behavior more predictable. But without accounting for heterogeneity, a platform cannot fully understand a driver’s preferences or achieve full incentive alignment.

Indeed, platforms have experimented with methods to incorporate driver heterogeneity. As of Summer 2019, Uber allows drivers to indicate—twice a day—that they would like to take trips in the direction of a particular location.222https://help.uber.com/partners/article/set-a-driver-destination?nodeId=f3df375b-5bd4-4460-a5e9-afd84ba439b9, visited 2/25/19. However, mechanisms that account for driver preferences can also have unintended consequences if not designed properly. By saying “I want to drive South,” a driver biases her dispatches in a way that could also promote more profitable trips.333https://therideshareguy.com/uber-drops-destination-filters-back-to-2-trips-per-day/, visited 2/25/19.

In this paper, we introduce the study of driver location preference in a mechanism design framework. In Section 2, we adapt a model originally conceived by Bimpikis et al. Bimpikis2016 to an economy where drivers prefer a particular location. In Section 3, we present the Preference-Attentive Ridesharing Mechanism (PARM), which elicits driver preferences and sets a revenue-optimal pricing policy. We show that PARM is incentive-compatible, and that it achieves the full-information, first-best revenue when supply is unconstrained or when demand is symmetric. In Section 4, we study settings with constrained supply and asymmetric demand, using simulations to compare the revenue and welfare performance of PARM to existing ridesharing mechanisms. We show that PARM achieves close to first-best revenue and typically outperforms even the best case for preference-oblivious pricing (where strategic behavior hurts efficiency). Proofs not presented in the text are deferred to Appendix A.

1.1 Related Work

Existing work on pricing and dispatching in ridesharing platforms does not account for the heterogeneity of drivers.

Our work builds on work of Bimpikis et al. Bimpikis2016, who show that under a continuum model, and with stationary demand and unlimited driver supply, a ridesharing platform’s revenue is maximized when the demand pattern across different locations is balanced. They show via simulation that in comparison to setting a uniform price for all locations, pricing trips differently depending on trip origins improves revenue. Relative to the Bimpikis2016 [Bimpikis2016] model, we allow limited driver supply; moreover, each driver in our model has a preferred location. We thus introduce a reporting phase, in which drivers report their preferred locations. We then modify the matching and pricing formulations in order to align incentives.

ma2018spatio [ma2018spatio] study the incentive alignment of drivers in the presence of spatial imbalance and temporal variation of supply and demand. castillo2017surge [castillo2017surge] show that dynamic pricing mitigates inefficient “wild goose chase” phenomena for platforms that employ myopic dispatching strategies. Modeling a shared vehicle system as a continuous-time Markov chain, banerjee2017pricing [banerjee2017pricing] establish approximation guarantees for a static, state-independent pricing policy. ostrovsky2018carpooling [ostrovsky2018carpooling] study the economy of self-driving cars, focusing on car-pooling and market equilibrium. Queuing-theoretic approaches have also been adopted: banerjee2015pricing [banerjee2015pricing] show the robustness of dynamic pricing, afeche2018ride [afeche2018ride] study the impact of driver autonomy and platform control, and Besbes2018Spatial [Besbes2018Spatial] analyze the relationship between capacity and performance.

There are various empirical studies, analyzing the impact of dynamic pricing [Hall et al.2015, Chen and Sheldon2015], the labor market for Uber drivers [Hall and Krueger2016, Hall et al.2017], consumer surplus [Cohen et al.2016], the value of flexible work [Chen et al.2017], the gender earnings gap [Cook et al.2018], and the commission vs. medallion lease based compensation models [Angrist et al.2017].

2 Model

We consider a discrete time, infinite horizon model of a ridesharing network with discrete locations, . Following the baseline model of Bimpikis et al. Bimpikis2016, we assume unit distances, i.e., it takes one period of time to travel in between any pair of locations. At the beginning of each time period, for each location , there is a continuous mass of riders requesting trips from . The fraction of riders at with destination is given by (thus ). We assume that the components of rider demand and

are stationary and do not change over time. Riders’ willingness to pay for trips are i.i.d. random variables with CDF

. Thus, for any , the number of trips demanded from to at price would be . (Riders who are unwilling to pay the stated prices for their rides leave the market.)

Each driver has a preferred location . Drivers receive additional utility whenever they start a period in their preferred locations (irrespective of whether they have a rider). This preferred location is private information, and represents a driver’s type. For each location , the total mass of available drivers of type is given by . Drivers have a discount factor of , and an outside option that delivers utility .444Throughout the paper, we consider to be very close to — this is natural, since an annual interest rate of 4% implies an exponential discount factor of over the course of ten minutes. We assume , meaning that the utility from being in one’s favorite location at all times does not outweigh the outside opportunity.

A ridesharing mechanism matches drivers and riders to trips, sets riders’ trip prices and drivers’ compensation, and (potentially) imposes drivers’ penalties for strategic behavior. At the beginning of each period, a driver whose previous trip ended at location chooses whether or not to provide service at location . If a driver provides service, the mechanism may dispatch that driver to (i) pick up some rider with trip origin , (ii) relocate to some location, or (iii) stay in the same location. If a rider going from to is picked up by some driver, then the rider pays the platform the trip price . If a driver of (reported) type is dispatched from to , the platform pays them , regardless of if her dispatch was to pick up a rider or relocate.555It bears mentioning that need not be a fixed proportion of . In fact, Bimpikis et al. Bimpikis2016 find that for certain types of networks, making driver compensation a fixed proportion of trip price drastically reduces platform revenue. Drivers who choose to not provide service in a period can relocate to any location in the network, are not compensated by the mechanism in this period, and may be charged a penalty .666Drivers only choose whether or not to provide service at a location and cannot decline dispatches based on the trip destination. This is consistent with current ridesharing platforms, which hide rider destinations because of drivers “cherry-picking” rides. Denote , and .

2.1 Ridesharing Mechanisms

We design a ridesharing mechanism that attends to drivers’ heterogeneous preference over locations. Before the beginning of the first time period, the mechanism elicits the preferred locations from potential drivers. Accounting for rider demand , and the supply of drivers of each type, the mechanism determines rider and driver flow, setting trip prices , driver compensation , and driver penalties . With the consideration of the pricing, penalties, and their outside options, drivers decide whether or not to participate, and then the platform starts to dispatch drivers to trips and processes payments accordingly in each period.

2.2 Steady-State Equilibrium

Consider a ridesharing mechanism with pricing and compensation and no penalties.777 Under an incentive-compatible ridesharing mechanism, drivers will report truthfully and always provide service, so they will never be charged any penalty. At the beginning of each period, let be the number of drivers of (reported) type at location , and let denote the total number of drivers of (reported) type on the platform. Let the trip flow be , where is the number of riders from to assigned to drivers of type . Let be the mass of drivers of type at who are dispatched to relocate to without a rider, and set and . No driver or rider can be matched multiple times in the same period, thus assuming drivers always provide service, we have for all and all , and and for all .

For a trip with origin and destination , if the total rider demand exceeds driver supply (i.e., ), the mechanism may increase the trip price and achieve higher revenue. Therefore for revenue optimization, we can assume without loss that . When , meaning that some type- drivers are at location

, the probability that a given driver of type

is dispatched to destination is . Assuming a driver of type has truthfully reported her type and will provide service in all periods, her lifetime expected utility for starting from location is of the form

(1)

where is the indicator function. The first term in (1) is the expected compensation and future utility a driver gets when dispatched to one of the possible destinations. The second term corresponds to the idiosyncratic utility drivers get from starting trips in their favorite locations.

Definition 1 (Steady-State Equilibrium).

A steady-state equilibrium under pricing policy is a tuple s.t.:

  1. (Driver best-response) Drivers providing service always maximizes their payoff, i.e. , .

  2. (Flow balance) For all locations and driver types , .

  3. (Market-clearing) .

  4. (Individually rational driver entry) Participating drivers get at least their outside option ; with excess supply of drivers with type , all participating type- drivers get exactly their outside option .

  5. (Feasibility) Rider and driver flows are non-negative, i.e., , , , ; the supply constraints are satisfied, i.e., , .

With full knowledge of driver types, the first best revenue is achieved by setting rider prices and driver compensations s.t. the revenue is optimized in steady state equilibrium:

(2)

The design problem is to compute rider prices , driver compensations , and driver penalties to optimize platform revenue in the steady state equilibrium, in a way that drivers will truthfully report their location preferences and will choose to always provide service.

3 The Preference-Attentive Ridesharing Mechanism (PARM)

We now introduce our Preference-Attentive Ridesharing Mechanism (PARM) and show that this mechanism (i) truthfully elicits drivers’ location preferences, (ii) incentivizes drivers to provide service, and (iii) achieves first-best revenue when supply is unconstrained or when demand is symmetric.

3.1 Alternate Form of the Optimization

The optimization problem (2) need not be convex, and moreover, even when an optimal solution can be found, it may not incentivize drivers to report their types truthfully. Denoting , we present an alternate problem (3), which guarantees that any optimal solution can be converted into an optimal solution for (2) using compensation scheme (4)—while preserving the objective. Specifically, we consider:

(3)

Our approach is analogous to a similar move by Bimpikis et al. Bimpikis2016— assuming that is distributed , the solution space is convex and the optimization problem is quadratic. We also go a step further by accounting for driver heterogeneity and the possibility of zero demand at a location, the latter by paying drivers for relocation dispatches.

Consider the following compensation scheme:

(4)
Lemma 1.

Consider an optimal solution to problem (3), and let be the compensation scheme (4). Then:

  1. is an optimal solution to optimization problem (2) with steady-state equilibrium ; and

  2. lifetime utility (payment and location value) of a truthful driver is exactly at every location, i.e.  for all .

Briefly, feasible solutions to (3) satisfy conditions (C2), (C3) and (C5). Moreover, with , the compensation as in (4) is non-negative.

Given (4), drivers receive utility in expectation per period (so over their lifetimes); this implies (C1) and (C4). Furthermore, the solution is optimal, since no compensation scheme can lower the total payment to drivers while fulfilling the same rider trip flow .

3.2 Constructing PARM

Definition 2.

Given rider demand , the Preference-Attentive Ridesharing Mechanism (PARM):

  1. Elicits the location preferences from drivers.

  2. Solves (3) with an additional constraint

    (5)

    for dispatching , and pricing determined by (4).

  3. If a type driver did not provide service and relocated to , the platform treats her as a type driver from then on. If this is the first such deviation for this driver, the driver pays penalty for as solved for in the following system of linear equations:

    (6)

The system (7) has linear equations and unknowns ( of the and of the ). Intuitively, describes the utility of a driver of type pretending to be type and providing service everywhere except , where she instead relocates to . By construction, is the minimum penalty needed to equalize driver earnings between this deviation and truthtelling plus providing service. We take the maximum over such penalties so no driver wants to pretend to be type and employ this strategy, then max with in case the calculated penalty is negative.888A negative penalty might arise if the deviation is itself very bad for drivers, in which case the only way to make the deviating drivers’ utility is to pay those drivers.

If a driver declines to provide service but relocates to her reported preferred location, she faces no penalty or type re-assignment. A driver might have a legitimate (idiosyncratic) reason for not being able to provide service in a period, but if she relocates to a location she did not report as preferred, that is taken as an indication that her original report was not truthful.

We now prove, under the assumption that drivers always provide service and as a result are never charged any penalty, that imposing (5) is sufficient to guarantee truthful reporting.

Theorem 1.

Assuming all drivers always provide service, it is a dominant strategy for drivers to report their location preferences truthfully under PARM.

Proof.

First observe that by being truthful, each driver gets utility per period—getting paid at preferred locations, and at every other location. As a result, for all and all . Suppose an infinitesimal driver of type reports that she is of type . At all she gains utility per period. At , the driver makes because the platform, treating her as a type driver, is still paying her . At , she is paid , and does not get the extra utility .

With , misreporting in place of leads to an increase in the expected payoff in static steady-state equilibrium if and only if in equilibrium, the driver with reported type spends more time in location than in location . Considering the location of a driver treated as type as a Markov chain, then is the stationary distribution, meaning that a driver with type spends a plurality of her time at location , and a driver does not benefit from misreporting her type if all drivers follow the platform’s dispatches. ∎

We now consider drivers who may strategically decline to provide service and show that such deviations are not useful under PARM, which updates its belief about a driver’s type after deviations and imposes a penalty on the first such deviation (step 3 of Definition 2).

Theorem 2.

Under PARM, it is an ex post Nash equilibrium for drivers to report their types truthfully and to always provide service.

Briefly, Theorem 1 and the following Lemma 2 imply that (i) a profitable misreport must be paired with post-reporting deviation, and (ii) the most profitable deviation must be the driver providing service everywhere except the location that she signaled. The penalties ensure the driver does not get a utility higher than from this deviation (or any other), so there does not exist a profitable deviation.

Drivers are never charged any penalty under the equilibrium outcome, but the threat of a penalty is necessary to ensure truthful reporting. In certain special economies, a misreporting driver might spend many periods at her true preferred location before being sent to her reported preferred location. Without penalties, she may simply decline service and relocate back to her actual preferred location, thereby sacrificing one period of income for the possibility of many periods of extra idiosyncratic utility. See Appendix C for an example and discussions.

Lemma 2.

Consider any set of driver reports (possibly untruthful), and assume that the rest of the drivers provide service in each period. Consider a driver of true type and reported type . If (truthful), always providing service is a best response. If , one of the following is a best-response: (i) always providing service, or (ii) providing service at every location except , where the driver instead drives to .

As an outline of the proof, first note that a truthful driver makes at every location, and so it always optimal to provide service. What if at some point it is optimal for a driver with to not provide service and drive to some location ? If , the driver has given up a period of income and relocated to her least profitable location, which is sub-optimal. So the driver drives to , paying a penalty. In subsequent periods before another deviation, the platform considers her as a type driver and pays her less for trips starting at . If , this decrease in income is exactly offset by her idiosyncratic utility . If , the stationary distribution of her location (as a Markov chain) is given by . Therefore, the expected number of times she visits (and gets extra utility ) before returning to (and again making less) is exactly , thus the loss in income is greater than the gain in idiosyncratic utility. By deviating again before returning to , the driver gains less idiosyncratic utility in expectation. Furthermore, symmetry dictates that it is still optimal to relocate to location . This implies that in comparison to deviating and relocating to location , a deviation to location results in lower continuation payoff and therefore cannot be a best response. Because the driver will relocate to her true preferred location, she will make in subsequent periods. Before deviating, she makes at , at , and elsewehere. So the only place she will deviate from is possibly , and in that case will relocate to .

3.3 Cases with First-Best Revenue and No Penalty

Although the IC constraint (5) may reduce revenue, we can characterize some settings where imposing IC does not lead to a revenue loss: when supply is unconstrained, or when rider demand is symmetric, PARM achieves the full-information first-best revenue. Furthermore, no penalty is necessary to ensure incentive compatibility.

Theorem 3.

Suppose for all . Then PARM achieves full-information first-best revenue, and no penalty is necessary to ensure incentive compatibility.

Briefly, the IC constraint (5) does not bind because at location , drivers of type cost less than drivers of other types. If drivers of type fulfill many rides at location , the platform can improve its revenue by dispatching drivers of type to fill those rides instead, as long as there are enough drivers of type . An individual driver will visit her signaled location before visiting any other too many times (in expecation), so she cannot profitably use a misreport-plus-deviation to sacrifice one period of income for many periods of idiosyncratic utility (as described following Theorem 2). This makes the threat of penalties unnecessary for ensuring incentive compatibility. Note that the preceding argument makes no assumption on the demand pattern, and requires only the availability of supply.

Definition 3.

Rider demand is symmetric if we have and .

Theorem 4.

Suppose that rider demand is symmetric. Then we can construct a solution to optimization problem (3) with incentive compatibility constraint (5) such that PARM achieves full-information first-best revenue, and no penalty is necessary to ensure incentive compatibility.

To understand Theorem 4, we prove two additional lemmas.

Lemma 3.

With symmetric demand, any optimal solution to (3) satisfies for all .

Intuitively, drivers of type cost the platform less when they complete trips at location , so it is optimal for the marginal ride they give at location to have a lower price than at other locations.If demand is symmetric, this means drivers of type provide more rides at location than at any other location.

Lemma 4.

If the demand pattern is symmetric, there exists an optimal solution to (3) such that for all and all , and .

Intuitively, there is no need for drivers to relocate when demand is fully symmetric. Moreover, given any optimal solution to (3), we can construct an alternative optimal solution, where the flow of drivers of each type can be decomposed as cycles with length , i.e., .

We can now sketch the proof of Theorem 4. Given symmetric demand, Lemma 3 implies driver flow for within-location trips satisfies the IC constraint (5). For all inter-location trips, Lemma 4 lets us focus only on bilateral driver flow between pairs of locations and , which should be served by drivers of type and . Even if there are not enough of those drivers, type drivers cost less giving rides to and from , so they will naturally fill more rides in and out of and fewer between and . Combining the two cases, drivers of each type do not spend more time at another location than they do at , and imposing the IC constraint (5) is without loss of revenue. As in Theorem 2, drivers visit their reported favorite locations before visiting any other location too many times (in expecation), so they cannot profitably use misreport-plus-deviations to sacrifice one period of income for many periods of idiosyncratic utility (as described following Theorem 2). This means that penalties not necessary for ensuring incentive compatibility here.

4 Simulation Results

In this section, we use simulations to analyze the revenue and social welfare under PARM for settings outside the cases covered by Theorems 3 and 4—i.e., settings with limited supply and unbalanced demand.

Social welfare is defined as the total rider value plus drivers’ utilities from being in their preferred locations, minus the total opportunity costs incurred by drivers. We compare PARM with the full-information first-best, and also a Preference-Oblivious Ridesharing Mechanism (PORM) which sets prices as in Bimpikis2016 [Bimpikis2016] without considering drivers’ location preferences, while assuming that drivers always follow dispatches. In Section 4.2, we also study the equilibrium outcome under PORM, allowing driver autonomy. Additional simulation results are presented in Appendix B. For ease of illustration, we consider two locations throughout the analysis.

4.1 Varying Demand Patterns

Suppose that there are an equal number of drivers favoring each location: . Drivers have outside option , discount factor , and gain utility per period from being in their preferred locations. Each rider has value independently drawn .

Varying Total Demand.

We first assume an unbalanced trip flow and (i.e., three quarters of riders from each location would like to go to location ). Fixing the total demand at location at , and varying from to , the revenue and welfare under PARM and benchmarks are as in Figure 1. Although PARM only necessarily achieves first best revenue when (symmetric demand), we see that PARM achieves the first best and outperforms PORM unless is very small, such that demand from the two locations is highly asymmetric.

(a) Revenue
(b) Welfare
Figure 1: Revenue and welfare varying demand at location .

When , almost all rides originate and terminate at location , thus the first best and PORM dispatch most drivers of both types to provide service at location . Figure 2 illustrates the rider trip flows fulfilled by drivers of each type under different mechanisms, when . To satisfy PARM’s incentive compatibility (IC) constraint, however, drivers of type must spend a plurality of their time at location 0. Therefore, PARM completes fewer trips at location , dispatches more type  drivers to fulfill (the less profitable) between-location trips, and asks many type  drivers to relocate back to  once they arrive at location (the numbers after the “+” sign represent driver relocation flow), resulting in lower revenue and social welfare.

(a) Type 0: First Best

(b) Type 0: PARM

(c) Type 0: PORM

(d) Type 1: First Best

(e) Type 1: PARM

(f) Type 1: PORM
Figure 2: Total rider trips fulfilled by drivers of each type, with , , and .

Varying Imbalance in Demand.

(a) Revenue
(b) Welfare
Figure 3: Revenue and welfare varying for .

Fixing and varying for (i.e., changing the proportion of rides with destination ), the revenue and welfare achieved by different mechanisms are shown in Figure 3. Similar to Figure 1, PARM achieves first best revenue and outperforms PORM for a wide range of (though demand is only symmetric when ). For similar reasons as in the above scenario, we see a decline of revenue and welfare under PARM when demand becomes highly imbalanced: in this case, when approaches or and almost all riders have the same destination.

4.2 PORM in Equilibrium

In this section, we analyze a scenario for which we are able to compute the equilibrium outcome given the pricing under PORM, and under the setting where drivers are given the flexibility to decide how to drive. Consider two locations and drivers of type  only: . All trips start and end in the same location, i.e., . Being oblivious to drivers’ preferences, PORM sets the same trip price for the two locations and expects the spatial distribution of drivers to be proportional to the distribution of demand. In equilibrium, however, more drivers decide to drive in location  (the preferred location), such that in each period drivers in  are dispatched with probability less than and achieve the same expected utility as drivers in .

Varying Location Preference .

In Figure 4, we fix demand and plot revenue and welfare as , the idiosyncratic driver utility, varies from to . As increases, welfare and revenue under PARM coincide with the first best and increase as expected. However, revenue under PORM (assuming driver compliance) remains constant since the mechanism is oblivious to drivers’ preferences. We also see a decrease in welfare and revenue achieved in equilibrium under PORM, since more drivers decide to supply in location , instead of in location  as dispatched, resulting in unfulfilled rides in 0 and idle drivers in 1. Beyond , revenue and welfare remain constant, since all drivers are already supplying location .

(a) Revenue
(b) Welfare
Figure 4: Equilibrium revenue and welfare varying .

Figure 5 illustrates rider trip flow fulfilled by the type  drivers when . PARM assigns more drivers to location 1 than location 0, but PORM does not. However, in equilibrium more drivers end up at location 1 anyway, leading to 25 units of drivers idling at location .

(a) Type 1: PARM

(b) Type 1: PORM

(c) Type 1: PORM Eq.
Figure 5: Rider trips fulfilled by type drivers, with , , , , and .

Varying Demand Ratio .

In Figure 6, we fix , , and vary from to . We see that PARM revenue coincides with the first-best and significantly exceeds the revenue of PORM. The revenue and welfare of the equilibrium outcome under PORM is much lower, however, because drivers over-supply the preferred location , leaving rider trips in unfulfilled. It is curious that with highly imbalanced demand, an increase in initially leads to reduced equilibrium revenue and welfare—this is because with higher demand at location , PORM sets a higher price at location  and accepts fewer location  trips in order to complete more trips in . The drivers, however, are only willing to drive in when is high enough that the low probability of getting a ride in offsets the extra utility . That said, PARM does not always outperform the equilibrium outcome under PORM. (See Appendix B for an example.)

(a) Revenue
(b) Welfare
Figure 6: Equilibrium revenue and welfare varying

5 Discussion

We have proposed the Preference-Attentive Ridesharing Mechanism (PARM) for pricing and dispatch in the presence of driver location preferences. It is an equilibrium under PARM for drivers to report their preferred locations truthfully and follow the mechanism’s dispatches. PARM achieves first-best revenue in settings with unconstrained driver supply or symmetric rider demand, and we show via simulations that even outside those scenarios, PARM achieves close to first-best welfare and revenue and outperforms a mechanism that is oblivious to location preferences.

Our analysis suggests that incorporating drivers’ location preferences is compatible with other aspects of ridesharing pricing and marketplace design—even though drivers could in principle game the system by expressing preferences for locations associated with more highly compensated rides. There are two key elements to our approach that both seem likely to provide practical insight beyond the specific framework and mechanism considered here: First, we recognize that respecting drivers’ location preferences creates value, which can at least partially substitute for cash compensation. Then, we incentivize truthful location preference revelation through a variation on a revealed preference approach. Indeed, PARM uses drivers’ deviations from proposed dispatches to learn about their preference types—a driver who chooses to drive to instead of her assigned location is inferred to prefer location and subsequently faces the compensation profile of other drivers with that preference .

We note that for our approach to work, it is important that drivers’ preferences do not change frequently over the course of the day. If drivers’ underlying types were moving targets, it would be much harder to enforce incentive compatibility by tracking endogenous responses to dispatch assignments.

Acknowledgements

Kominers gratefully acknowledges the support of National Science Foundation grant SES-1459912 and the Ng Fund and the Mathematics in Economics Research Fund of the Harvard Center of Mathematical Sciences and Applications. Rheingans-Yoo gratefully acknowledges the support of the Economic Design Fellowship of the Harvard Center of Mathematical Sciences and Applications, the Harvard College PRISE Fellowship, and the Harvard College Research Program.

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Appendix A Proofs

a.1 Proof of Lemma 1

See 1

Proof.

The first term of the objective of (3) is the same as the income portion of the objective of (2). The constraints on this optimization are exactly the equilibrium conditions (C2), (C3) and (C5). Thus, it suffices to devise a compensation scheme where driver income everywhere is exactly equal to outside option, i.e.  (this will necessarily satisfy (C1) and (C4)). This will be revenue-optimal because by (C4) drivers cannot be making less than . Consider compensation scheme (4):

The second term means that any idiosyncratic utility a driver gets is extracted by the platform, so any dispatched driver makes exactly in that period. Thus, exactly when probability of dispatch at every location is 1, i.e.

This is equivalent to the second-to-last constraint of (3), so any optimal solution to (3) will have , so (C1) and (C4) are satisfied. Thus an optimal solution to (3) corresponds to an optimal solution to (2) under compensation scheme (4). ∎

a.2 Proof of Theorem 2

See 2

Proof.

By Theorem 1, if a driver is going to always provide service, she cannot profitably misreport. By Lemma 2, if a driver reports her type truthfully, it is not profitable for her to strategically decline to provide service. So for a misreport to be profitable, it must be paired with post-reporting deviation (strategically declining to provide service). Lemma 2 characterizes what this deviation must be: providing service everywhere except her signaled location. There she instead drives to her true preferred location.

Thus to show incentive compatibility without penalty, it suffices to show this strategy is not more profitable than truthfully reporting and always providing service. Recall how we calculate penalties:

(7)

Where . If we had set penalties for switching types from to to be , a driver of type pretending to be type following the strategy from Lemma 2 would get utility as defined in (7). The first term is the compensation from providing service at locations that are not . The second term is idiosyncratic utility from being at her favorite location. The third term is the utility from declining to provide service at and driving to , after which she pays the penalty, the platform updates her type, and she makes afterwards. Then with random initialization, by (7) the driver’s expected utility is exactly . Because we set , The equality in (7) is an inequality, but still the driver’s expected utility is no more than , which is what she would make by reporting truthfully and always providing service. Therefore, it is a best response for each driver to report truthfully and always provide service. ∎

a.3 Proof of Lemma 2

See 2

Proof.

Suppose we have a driver of type who has signaled she was of type . If , then the driver makes every period she provides service. If , the driver makes every period she provides service at (extra utility but not paid less), makes every period they provide service at (paid less but not extra utility), and makes every period elsewhere.

First, we will show that the driver will never choose to not provide service and drive to . This move will not change how the platform treats the driver in the future. So in either case ( and ) the driver has lost in income and then relocated to the location where she makes weakly least in the network. This is not profitable, so the driver will not choose to not provide service and drive to .

Next, we will show that if a driver chooses to not provide service, she will drive to her true preferred location. Suppose this driver of type declines to provide service. As established in the previous paragraph, she will not relocate to her previously signaled location, so she may pay a penalty, but the penalty is the same no matter where she relocates to. she has the choice of where to drive in the network and will choose the location that maximizes her expected lifetime earnings given that she will be treated as type in the future. If location is such a best-response choice, then for every future deviation, the symmetry of the situation implies that will be a best-response choice then too. So we can assume that for all future deviations, the driver drives to .

In the case that , the driver expects to make in every period after arriving at location . Now suppose that . The driver gets utility every period she is at location , and every period she is at , and everywhere else. With , this means choice of can only be better than if she spends more time at than before her next deviation, at which point she faces the same choice. However, we know that , which by the Ergodic Theorem implies that a driver treated as type will spend on average at least as much time at location than location . This implies that if the deviant driver starts at location , the expected number of times she visits location before returning to is at most 1 (otherwise she would spend more time on average at than ). So for any , , where is the number of time periods and is the number of periods in which the driver is in location . So there is no time in the future by which point the driver expects to have been at more than if she follows platform instructions. The driver can deviate from platform instructions, but as established previously, without loss of best response she will drive to location , which puts in the same position as before.

Finally, we will show that a driver will choose to provide service everywhere if and will choose to provide service everywhere except possibly if . We have already established that a driver’s best relocation is her true preferred location but that a driver will not decline to provide service and drive to her previously signaled location. It follows immediately that truthful drivers (those whose previously signaled location is their true preferred location) will always provide service. So we assume . Because a driver’s post-deviation relocation is her preferred relocation, she will make in every period thereafter, after paying the platform a penalty. Before deviation, the driver makes at , at , and everywhere else. So the only location she might not want to provide service at is . Everywhere else her earnings are the same as post-deviation, and the only time it will be less is when at . So the driver will always choose to provide service at locations other than . ∎

a.4 Proof of Theorem 3

See 3

Proof.

We show full-information first-best revenue by showing the IC constraint does not bind. Consider an optimal solution to (3) without IC constraint (5). The flow constraint allows us to decompose the flow of type drivers into cycles with various mass. Suppose one of these cycles does not go through location . Then it is optimal to replace that driver flow with drivers of type , for location in the cycle. (this can necessarily happen because there is no supply constraint). The new drivers do exactly what the old drivers did, so the demand met is exactly the same and the flow constraints are still satisfied. However, they end up in their preferred location strictly more, so the last term of the objective strictly increases. So the previous solution was not optimal. So all the cycles of type driver flow go through location . This means because all flow of drivers through also goes through . So any optimal solution to (3) naturally satisfies the IC constraint (5). So imposing the IC constraint does not lead to an objective loss.

We show no penalty is necessary to achieve incentive compatibility by showing that the strategy described in Lemma 2 is not profitable. Consider and suppose a driver of type reports he is of type . Every time he visits location , he will visit location before returning to . Otherwise, there would be some non-negative flow in a cycle through but not through , which the platform could more optimally fill with (reported) type drivers. So the driver will always eventually visit and will never visit location more than once before doing so. The driver makes at location , then every period before giving a ride to , then at location , before relocating to and being thereafter treated as , making in every subsequent period. So compared to truthful behavior, the driver makes a maximum of extra at and loses a minimum of at location before revealing his true type and making every period thereafter. With , this strategy is not more profitable than truthful reporting and always providing service. So even without a penalty, PARM is incentive compatible. ∎

a.5 Proof of Lemma 3

See 3

Proof.

Consider an arbitrary . First notice that in any optimal solution, because these are the drivers are not fulfilling rides and staying in the same location, which adds cost to the system without fulfilling demand or helping to satisfy any of the constraints. Next, if and for , then it is more optimal to switch drivers of type going with drivers of type going . This fulfills the same demand, but with less cost because you have drivers of type at location more. This violates the optimality of the original solution, so we can assume that and that s.t. . So we find ourselves in one of two cases:

Case 1. : If this is the case, then because .

Case 2. : Then . Suppose . Then because the demand pattern is symmetric. I will show it is more optimal for the platform to raise and lower by infinitesimal amounts. Making substitutions and differentiating, we get the derivative of the first term of the objective with respect to is given as follows:

We know that and that . So

So the platform can make a marginal increase in the first term of the objective by raising and lowering by infinitesimal amounts, shifting an infinitesimal amount of to to make the market clear. The same number of drivers are in the the system, so it does not change the second term of the objective. And it increases the third term in the objective because we just shifted drivers of type to only be at location . So this shift brings us to a more optimal solution, which is a contradiction. So without the IC constraint imposed,

a.6 Proof of Lemma 4

See 4

Proof.

Consider an optimal solution to the platform’s optimization problem. Now set and . The same number of drivers of each type is used, so the supply constraint is still satisfied. Flow into and out of a location before was the same, so it will be the same now too, so the flow constraint is satisfied. And the symmetry of the demand pattern implies that and so if the market cleared before, setting prices and flows to be the same will still clear the market.

So all we need to show is that revenue is weakly better when , holding number of drivers used (and thus ) constant. Let . Then isolating the part of the objective that changes under this switch yields:

So holding constant, equal prices is optimal. So any solution to the optimization problem can be made without loss of optimality into a solution with . Having drivers flowing back and forth without giving rides is sub-optimal, so this implies . ∎

a.7 Proof of Theorem 4

See 4

Proof.

We will show full-information first-best revenue by showing the IC constraint does not bind. By Lemma 3, the symmetry of the demand pattern means that . By Lemma 4, we can restrict our attention to solutions where all flow of drivers is bilateral and there are no floating drivers. Suppose in an optimal solution . This implies , which in turn implies that there exists a location s.t. . We find ourselves in one of two cases:

  1. Exists s.t.. Then we can switch some type drivers going with some type k drivers going . Formally, we make smaller by and make larger by . All the same demand is filled, the same number of drivers of each type are used, and we’ve swapped drivers in a way that preserves the flow constraint, so all the constraints are satisfied. We’ve just strictly increased while weakly increasing (increase in the case that ), which decreases our cost. This improvement is a contradiction with the optimality of the original solution.

  2. DNE s.t. . Then . The symmetry of the demand pattern then implies that , and so it is an objective improvement to raise by and lower by . This violates the optimality of the original solution.

So there exists an optimal bilateral-flow solution, and this solution must have .

We will show no penalty is necessary to achieve incentive compatibility by showing that the deviation described in Lemma 2 is not profitable for the solution constructed in Lemma 4. Consider . We know that for all that , so a driver of (reported) type is more likely to be sent to than no matter what location she is at. So, with random initialization of drivers, a driver of type pretending to be type is more likely to be sent to before than than the other way around. Even if sent to first, she is more likely than not to visit before visiting again. When the driver visits , she declines to provide service, giving up in income and then relocating to , making in every subsequent period. When the driver visits , she gets in extra idiosyncratic utility. However, the probability she visits (and lose before is at least . So from the beginning of the game to the end of her first period in location or , she loses more in expectation than she gains. Even if she is sent to first, the probability she visits (losing and then revealing her true preference) before returning to and making another is less than , so she is still making less than if she had truthfully reported and always provided service. ∎

Appendix B Additional Simulation Results

We present in this section additional simulation results omitted from the body of the paper. We consider the same setting as in Section 4.2, with two locations , drivers of type  only: , and trips start and end in the same location: .

Varying Demand Ratio .

In Figure 7, we set , and vary from 0 to 2000. We see a similar trend as in Figure 1, with PARM doing worse than even equilibrium PORM for very small values of . When is small compared to , but all the drivers are of type 1, PARM employs some drivers to do idle in location 1, just to satisfy the IC constraint, so that more drivers can be employed to provide service in location 0. This is very costly, and leads to a larger reduction in revenue and welfare than just setting prices obliviously and having drivers strategize.

(a) Revenue
(b) Welfare
Figure 7: Equilibrium revenue and welfare varying

(a) Type 1: PORM

(b) Type 1: PORM Eq.

(c) Type 1: PARM
Figure 8: Rider trip and idle driver flows, with , , , and .

Figure 8 illustrates driver flow for . PORM and equilibrium PORM are similar because there are not that many many rides to fill at location 1, therefore having a few units of idle drivers is sufficient to reduce the earnings at location , so that the majority of the drivers are sill willing to supply in location 0. PARM, in contrast, employs many drivers to idle around location 1 to satisfy IC. This is very costly, and leads to worse performance. It is worth noting that as increases, the social welfare achieved by the equilibrium outcome under PORM in fact does not increase, due to the increased amount of idle drivers at location .

Varying Driver Supply .

We now examine the effect of varying the supply of drivers of type (while still keeping the supply of driver of type at zero). In Figure 9, we set , , and vary from to . Revenue and welfare under PARM coincide with first-best and outperform PORM. All the mechanisms improve in profit and welfare as supply increases, but PARM is better able to use the additional drivers. Under PORM in equilibrium, drivers again over-supply the preferred location , causing rides at location to get dropped. Eventually, there are so many drivers that they can fill all the demand, even with drivers idling at location At this point, equilibrium PORM revenue coincides with PORM revenue, though the welfare is still lower.

(a) Revenue
(b) Welfare
Figure 9: Equilibrium revenue and welfare varying

Appendix C Incentive Problem Without Penalty

Figure 10 illustrates driver flow for , , , . Importantly, there is no demand between locations, so once a driver is assigned to a location, they will continue giving rides there and never visit another location, unless they decline to provide service in a period and relocate. Then a type 1 driver has a useful deviation. She reports that she is type 0, then if she is sent to location 1, she provides service in every period. Thus she prefers location 1 but is being paid as if she does not, and is never sent to location 0 (where she would be paid less) because there is no demand between locations. If this happens, she gets utility over her lifetime. If instead she is sent to location 0, she does not provide service the first period and relocates to location 1, after which she provides service and the platform treats her as type 1. If this happens, she gets utility over her lifetime. So her expected lifetime utility from this deviation is , which is greater than her utility from reporting truthfully and providing service, which is . So this deviation is useful. The key thing going on here is that although more type 0 drivers are at location 0 than 1, an individual type 0 driver might spend their entire lifetime at location 1, which incentivizes type 1 drivers to misreport, taking the risk of one lost period of income in order to get a lifetime of idiosyncratic utility. In this case, the markov chain describing the movement of a type 0 driver is disconnected, but this issue also occurs when the markov chain is only very weakly connected and the driver is balancing one period of income versus many periods of idiosyncratic utility. The penalty is calculated such that if a driver does the deviation described here, the extra loss when they decline to provide service makes the deviation not useful. And in our proof of Theorem 2, we show that this deviation is the best of all non-truthful strategies, so the penalty ensures incentive compatibility.