1. Introduction
In the last decade, the sharing economy has upended a wide variety of industries, from bike sharing and car sharing to food delivery and hotels. However, despite the rapid rise and success of online sharing platforms, there is much about their design and operation yet to be understood. Our lack of understanding is made clear by the fact that competing sharing platforms in the same space often have fundamentally different approaches to pricing, matching, subsidies, market positioning, etc. The contrast between Uber, Lyft, and Didi with respect to pricing and market positioning in the context of ridesharing is a particularly salient example, e.g., (uber_subsidy, ; uber_lyft_ola, ).
As a result, the design and operation of sharing platforms has emerged as a lively and growing research area, with papers focused on efficiency in matching (stiglic2015benefits, ; alarabi2016demonstration, ), pricing strategies (benjaafar2015peer, ; banerjee2016multi, ), equilibrium analysis (fang2017prices, ; benjaafar2015peer, ), empirical studies (chen2015peeking, ; chen2016dynamic, ), and more.
In this paper, our focus is on the task of designing subsidies, a.k.a., loyalty programs within sharing platforms. Subsidies play a crucial role in sharing platforms because they help to ensure that there is enough shared capacity in order to meet the demand. For example, in the context of ridesharing, various forms of loyalty programs for drivers have been in place since the beginning of the industry, e.g., (didi_subsidy, ; uber_subsidy, ). In addition to ensuring enough shared capacity is present in the system, loyalty programs play another important role in sharing platforms too. They are crucial tools when competing for and retaining sharing capacity. For example, in the case of ridesharing, loyalty programs allow Uber and Lyft to lockin drivers on one platform; thus both increasing the platform capacity and decreasing the capacity of competing platforms (uber_incentives, ).
The importance of these loyalty programs cannot be overstated – maintaining capacity is the fundamental challenge of a sharing platform. Thus, it is not surprising that they have become increasingly sophisticated as competition has heated up in the industry. In ridesharing, Uber and Lyft both began with simply offering a onetime reward, called a signup bonus, to new drivers. However, loyalty programs have become considerably more complex and expensive as the industry has evolved. Uber and Lyft both recently introduced loyalty programs to reward drivers after they reach certain trip milestones, i.e., drivers earn a “power driver” bonus after a minimum service requirement is reached. Indeed, platforms depend so much on these loyalty programs that they are willing to invest heavily in them. The cost of paying subsidies to boost supply and win market share is huge, e.g., Didi spends in excess of $ billion annually on subsidizing drivers (didi_subsidy, ).
Motivated by the importance of loyalty programs for sharing platforms, the goal of this paper is to study the optimal design of loyalty programs for sharing platforms and to understand the competitive advantage provided by moving toward more sophisticated loyalty programs.
While we are motivated by ridesharing, our focus is on sharing platforms more broadly. Thus, we build our work on classical models of twosided platforms in economics (rysman2009economics, ; weyl2010price, ; rochet2006two, ). Specifically, we first consider the case when the platform has monopoly in a market with homogeneous product owners. We show that the optimal revenue can be achieved with a linear loyalty program (LLP) with a form similar to the “power driver” program used in practice. We then consider the heterogeneous market. In this case, however, finding the optimal loyalty program can be intractable as it requires searching over complex subsidy functions. Moreover, the resulting program may be impractical due to its complex structure. To tackle this problem, we prove that the optimal revenue can be achieved by a much simpler class of multithreshold linear loyalty programs (MTLP). Moreover, compared to a general loyalty program which may be impractical, the form of MTLP facilitates implementation in practice. This significantly reduces the required search space. We further design a special MTLP program called a Hyperbolic Bonus (HB), which admits a simple hyperbolic bonus structure and is costefficient, in that it pays the necessary opportunity cost plus a constant regardless of the size of the market.
Our results highlight that the design of the optimal linear loyalty program has the property that it no longer extracts any commission fee when product owners have shared more than some required amount. This property mimics the design of Lyft’s power driver program (lyft_introduce_loyal, ).
Further, we consider loyalty program design under competition. Specifically, we study a setting in which two platforms compete for the scarce supply (shared capacity) with the objective of maximizing revenue. Within this context, we conduct a comparative study of onetime signup bonuses and linear loyalty programs, in order to investigate the impact of the relative sophistication of loyalty programs on competition between the firms. Our results show that sophisticated loyalty programs that reward suppliers via stepwise linear functions provide a meaningful competitive advantage over simple signup bonuses, which give them a one time reward for participating.
Moreover, we show that the heterogeneity of suppliers plays a critical role in deciding the optimal loyalty program used by platforms. This is important because suppliers in the sharing economy tend to exhibit significant heterogeneity in engagement. For instance, it has been reported that there exist two different clusters of active drivers on Uber platform (kooti2017analyzing, ), where the highly active group members have nearly times longer driving time than low activity group members four months after their registration. This heterogeneity has a fundamental impact on platforms’ strategies, both in a monopolistic or competitive market. When a platform has monopoly, introducing a more complicated ladderlike progressive bonus can extract more revenue from heterogeneous suppliers.
While in a competitive heterogeneous market, platforms compete through the design of loyalty programs and can counter the “winner take all” phenomenon, which is known to be prominent in competition between platforms from classic twosided market theory, e.g., (rysman2009economics, ). Intuitively, if users are homogeneous, platforms may try to attract all users and obtain higher revenue. This results in a classic “winner take all” situation since, e.g., the more users a platform has the more desirable it is for drivers, and vice versa. On the other hand, if users are highly differentiated, the aggressive strategy of capturing the whole market can become too expensive. In this case, it is wiser for platforms to focus only on distinct user groups. This creates an opportunity for firms to be strategic about market positioning and may yield situations where multiple platforms can survive, avoiding a “winner take all” scenario.
Contributions of this paper. In summary, this paper makes the following contributions.
First, this paper introduces a novel model to study subsidy program of a sharing platform, which is widely adopted in real world to overcome supply shortage. Our model is built on classical twosided market models, but it captures the fact that suppliers can now benefit from using the resources for their own purposes. It also captures supplier heterogeneity in selfusage values. Further, the model allows the study of competition between platforms.
Second, we prove that a linear loyalty program (LLP) maximizes revenue over all possible loyalty programs in a market with homogeneous suppliers, and we characterize the optimal linear loyalty program design. For markets with heterogeneous suppliers, we establish that optimal revenue can be achieved using a set of multithreshold linear loyalty programs (MTLP) that have a simple bonus structure and are implementationfriendly. We further exploit the structure of MTLP and design an MTLP called Hyperbolic Bonus (HB), which admits a very simple bonus form and is costefficient.
Third, we further consider loyalty program design under competition. We conduct a comparative study of onetime signup bonuses and linear loyalty programs, in order to investigate the impact of the relative sophistication of loyalty programs on competition between the firms. Our findings can be used to explain the fact that Uber and Lyft initially introduced signup bonuses to attract driver registration, but have later shifted to using ladderlike linear loyalty programs (lyft_introduce_loyal, ).
Fourth, we show that supplier heterogeneity has a dramatic impact on the market, e.g., on both the platform’s pricing strategy and market positioning strategy. Our findings can be used to explain why Uber and Lyft adopt different market positioning strategies to attract drivers and riders (uber_lyft_ola, ; uber_lyft_compare, ).
Finally, to validate analytic results, we conduct a simulation using utility functions derived from Didi’s transaction data, which is reported in (fang2017prices, ). Our results show that multithreshold loyalty program achieves nearoptimal revenue in heterogeneous markets, and that linear loyalty programs achieve higher revenue than signup bonuses, and they are more robust to supplier heterogeneity.
Related literature. A large and growing literature studying the sharing economy has emerged in recent years. These works focus on a wide variety of issues from efficiency in matching (stiglic2015benefits, ; alarabi2016demonstration, ), pricing strategies (benjaafar2015peer, ; banerjee2016multi, ), equilibrium analysis (fang2017prices, ; benjaafar2015peer, ), empirical studies (chen2015peeking, ; chen2016dynamic, ), and more.
At this point, existing empirical studies have been able to shed considerable light on the behavior of consumers in sharing platforms. These studies inform the modeling choices we make in this paper. For example, (cohen2016using, )estimates user surplus from demand data under different surge prices, (chen2015peeking, ) studies Uber’s dynamic pricing strategies by collected API data from San Francisco and Manhattan, and shows that demand is much more sensitive to transaction prices compared to supply capacity. Similar studies exist for other sharing platforms. In the case of Airbnb, (lecuyer2017improving, ; quattrone2016benefits, ) study Airbnb’s revenue and occupancy by crawling listing data from the website, and (zervas2014rise, ) estimates the impact of Airbnb on the hotel industry.
The current paper fits into the growing literature that uses analytic models to study and optimize the design of sharing platforms, e.g., see (rysman2009economics, ; weyl2010price, ; rochet2006two, ) and related references for an overview. Such papers focus on twosided market models where both sides in the market benefit from the participation of the other side, and platforms set prices to maximize transaction volumes, defined as the product of population sizes on both sides. Within this literature, some papers focus specifically on the pricing and subsidy strategies for sharing platforms. These are the most related to this paper.
Most such papers focus on optimizing the pricing strategy of a monopolistic platform. For example, (fang2017prices, ) shows that subsidizing product owners can substantially increase supply and, hence, increase social welfare. Similarly, (benjaafar2015peer, ) studies the pricing strategies to achieve optimal revenue and social welfare. Other highlights from the literature on pricing in a single platform are (Banerjee:2015, ), which shows that dynamic pricing is more robust than static pricing in the context of ride sharing; (cachon2017role, ), which shows that fixed ratio commission is nearly optimal for platforms and dynamic pricing helps increase social welfare; and (banerjee2017pricing, )
, which provides an approach to approximately optimize revenue, throughput and welfare for a sharing platform based on a Markov model.
There is far less work studying competition between platforms. The most prominent examples are (rochet2003platform, ; armstrong2006competition, ), which show that it is in the best interest of platforms to compete for users who are singlehoming in order to attract loyal users. These papers highlight the importance of considering supplier heterogeneity when studying competition between platforms, but leave open the question of how to design subsidies to attract and retain suppliers.
The importance of capturing user heterogeneity is also emphasized by (banerjee2017segmenting, ), which considers a model where buyers and sellers are segmented by their characteristics and proposes an algorithm for platforms to control the visibility among segmented groups to achieve approximated optimal revenue. The degree of heterogeneity is also an important feature in empirical studies, e.g., (kooti2017analyzing, ) shows that Uber users are highly differentiated in various aspects.
In summary, there is a growing literature of analytic work studying pricing and subsidization in platforms. This work has highlighted the importance of modeling consumer heterogeneity, and proposed explicit optimal subsidy strategy for practical use. It has also showed the performance of subsidy programs in the competition between two sharing platforms, but has only scratched the surface in understanding how subsidies and competition impact revenue of a sharing platform. Our paper is the first to characterize optimal subsidization and to look at competition that is driven by subsidies.
2. Model and Preliminaries
The goal of this paper is to study the loyalty programs in sharing platforms, focusing on the most common case when supply is insufficient and loyalty programs are necessary to attract and retain supply. To facilitate understanding, we first describe our model for a monopolistic market in this section. Then, we analyze this market in Section 3. After that, we extend the results to study competition between two platforms in Section 4. All the proofs of our results in this paper are in appendix.
2.1. Demand and supply
In our model, demand comes from product renters and supply comes from product owners. The total demand is denoted by , where is the renting price that the platform charges for per unit demand. For example, demand can be ride requests or apartment rental orders. Assume that is decreasing in , and . In our model, demand is considered to be a function of price. Quantitative case studies (mohlmann2015collaborative, ; lamberton2012ours, ) have been conducted to show that economic benefit (maximize economic gain and save money) is the determinant of user’s decision and behavior in the sharing economy, under different services such as bike sharing, car sharing and accommodation sharing. Take the car hailing market for example, empirical studies show that renters are primarily sensitive to price (chen2015peeking, ), and wait time does not have major impact on passenger’s behavior given real world ETA data (fang2017prices, ).
In our paper, we will assume that is fixed, and focus on the optimal price and subsidy program on owners. The reason is that we consider the longterm average effect of pricing and user participation. This style of model has become popular in recent years since it separates design issues associated with dynamics from the underlying economic competition between platforms, e.g., (jiang2016collaborative, ; benjaafar2015peer, ). This assumption captures a platform’s limitation in pricing in reality. Though renter price fluctuates in real time (dynamic pricing), the longterm average is relatively stable. For instance, in the context of ride sharing, passengers are highly price sensitive (fang2017prices, ), so platform has to maintain a stable and reasonable ride price to attract and retain customers. Further, the price of substitute services also limits the pricing power of the sharing platform, e.g., Uber tries to beat taxi fare a little bit in many cities (uber_beat_taxi, ). In fact, the passenger prices for UberX in London have stayed the same for the last two years, while prices in New York have only been changed once in the last two years.^{1}^{1}1See http://uberestimate.com/prices/NewYorkCity/all/ for price change log in New York City.
We denote the set of suppliers by . Since we focus on the case when supply is insufficient, we assume that for . For each user , we normalize his total resource to be unit, and assume that he can divide the allocation of his resources into two parts: selfusage and sharing on platform . Note that . Thus, the total supply on the platforms is:
To determine each owner’s sharing level , we model each product owner’s utility as the sum of two different components: (i) utility derived from selfusage of the product, and (ii) income earned from sharing the product through the platform.
We denote the selfusage utility. Note that each owner’s is not revealed to others, and it is defined on . In this paper, we assume that each owner’s utility function is concave increasing and differentiable on . This assumption guarantees full utilization of resources, i.e., . Denote the per unit sharing price the platform offers to owners. We also assume that and . This ensures that selfusage is not a trivial option for product owners. Therefore, the overall utility of owner is given by:
(1) 
Each owner chooses his and values to maximize his utility (1).
In the twosided market literature, owner utility in (1) often takes the form , to take into account product utilization, e.g., (cachon2017role, ; Evendar:2009, ; fang2017prices, ). In our case, since optimal loyalty programs are primarily needed when supply is insufficient, we assume without loss of generality that in this paper.^{2}^{2}2For the situation when supply is sufficient, there is no need for loyalty programs or other promotions for suppliers. Platforms can achieve maximum revenue by choosing optimal prices (fang2017prices, ; benjaafar2015peer, ).
2.2. Platform price and revenue
In practice, platforms usually extract a commission fee, i.e., a fixed percentage of each ride’s value. We define the commission fee and conveniently write
(2) 
Recall that is the service price to renters and is the payment to owners for per unit resource sharing. Revenue is the profit from the gap of the income from renters and payment to owners.
However, supply shortage has been shown to prevent a platform from obtaining the maximum revenue since many renter orders are not fulfilled (fang2017prices, ). Thus, in practice, a common strategy used by sharing platforms to overcome this challenge is to subsidize resource providers. Yet, despite the wide adoption of subsidy programs, it still remains to understand how to design an optimal subsidy program.
To carry out our study on optimal subsidies, we first introduce a general definition for a subsidy program. A subsidy program gives owner a bonus according to his sharing level . Note that is a nondecreasing function. It is clear that a subsidy program impacts a platform’s revenue, since the platform has to pay its suppliers an additional amount for their sharing. To take this into account when analyzing platform revenue, we define platform’s revenue as:
(3) 
where is the total subsidies paid to owners. For notational convenience, we define to be the marginal bonus, i.e.,
(4) 
2.3. Heterogeneous suppliers
Heterogeneity is a fundamental issue for sharing platforms, as we have highlighted in the introduction. To model heterogeneous suppliers, we consider different classes of owners in the market with selfusage utility functions , . To simplify the analysis, we represent each class as one owner, and use owner to refer to owners in class for simplicity. This is for convenience and is without loss of generality. All of our results can be extended to the case when there are arbitrary number of owners in each class. We assume that any two classes of users are highly differentiated. Specifically, we assume that , for . This assumption is natural given that empirical studies have shown marked differentiation between users (kooti2017analyzing, ).
3. Subsidies in a Monopolistic Market
We begin our analysis with the study of a monopolistic platform. Our main result in this section is that maximal revenue can be achieved in both homogeneous and heterogeneous markets. In the homogeneous market we show that linear loyalty programs are optimal while in heterogeneous markets we show that multilevel threshold programs are optimal. Finally, we also introduce a novel form of subsidies, hyperbolic bonuses, that are costefficient and easy to determine.
3.1. Homogeneous markets
In this subsection we show that linear loyalty programs achieve maximal revenue across any general forms of loyalty program when suppliers are homogeneous. Linear loyalty programs are defined as follows.
Definition 3.1 ().
A linear loyalty program (LLP) is a subsidy program under which the platform pays owners an extra bonus for per unit sharing, in addition to the base pay , to those who share higher than some threshold . Under a linear loyalty program, the utility of owner is:
(5) 
where .
The following theorem summarizes the optimal bonus volume, service threshold and revenue obtained under linear loyalty programs.
Theorem 3.2 ().
Given homogeneous owners, i.e., for all , and platform’s prices on supply and demand, i.e., and , the following loyalty program achieves the maximum revenue over all subsidy programs in a monopoly market:
(6) 
Here and satisfying and are the owner sharing levels with and without the loyalty program, respectively. Moreover, the maximum revenue that the platform obtains is:
(7) 
The meaning of above and is that the platform should boost supply by returning all revenue to owners who meet the minimum sharing requirement, e.g., extract no commission fee for drivers who have reached a certain trip milestone on the platform. This parallels the current design of Lyft’s power driver program (lyft_introduce_loyal, ).
The theorem is proved in two steps. The first step is to show the optimality of the linear loyalty program among all possible forms of loyalty program. We prove that linear loyalty program at least achieves the same revenue as any general loyalty program. The second step is to compute the optimal bonus, which turns out to be exactly the price gap between and . Interestingly, the optimal bonus is independent of the form of function .
The intuition behind this surprisingly simple form of the optimal subsidy program is that, this linear form of loyalty program can exactly cover owner’s opportunity cost for one to increase sharing amount, i.e., barely compensate for the lost of selfusage benefit, in order to maximize platform revenue.
With Theorem 3.2, we can further compute the optimal supply price and the corresponding platform revenue below.
Corollary 3.3 ().
Given homogeneous owners and fixed demand price , the maximum platform revenue is achieved by letting
(8) 
where owner’s sharing level satisfies .
Interestingly, this Corollary 3.3 shows that the optimal supply price is zero. That is to say, it is the platform’s optimal strategy to pay owners nothing unless they share more than some given threshold. From the above, we see that the platform should pay owners exactly the amount to cover their opportunity cost of using the resources themselves (users will start to share only when the marginal benefit of selfusage is lower than ). Therefore, setting gives the minimum opportunity cost of owners. Figure 2 shows an optimal linear loyalty program.
Note that this result extends to the case when owners have linear usage/sharing cost, e.g., gas, taxation (fang2016www, ). In this case, the optimal owner price in Corollary 3.3 should be the price that exactly equals the cost per unit sharing, leaving no extra revenue for owners who share less than the threshold .
3.2. Heterogeneous markets
We now turn to the case when owners are heterogeneous. Recall that for the owners, we have , for . We first introduce a result describing the owners’ participation in loyalty program, given the marginal bonus nondecreasing.
Proposition 3.4 ().
Under a loyalty program with nondecreasing marginal bonus , if owner benefits from participating in the loyalty program, i.e., he increases his sharing level compared to the amount without loyalty program, all owners will also participate in the loyalty program and boost sharing.
The intuition of the above result is that if owner thinks that it is more profitable to share more, other owners will also increase sharing since they have even lower selfusage benefits and hence lower opportunity costs. Thus, the platform can choose a loyalty program such that all owners participate. Without loss of generality, we only consider loyalty programs that all owners will participate in this paper to simplify the presentation.^{3}^{3}3Our results can easily be extended to the case when only owners in participate in the program.
We are interested in finding the optimal loyalty program for this heterogeneous case. It is natural to extend the single threshold in the linear loyalty program to a threshold version with a progressive bonus, as follows.
Definition 3.5 ().
A multithreshold loyalty program (MTLP) is an threshold linear loyalty program, where the corresponding progressive bonus for the sharing amount , is , for and thresholds .
Under MTLP, owner with sharing level has utility:
(9) 
Here is the aggregate bonus for sharing up to , and is additional bonus at rate .
As in the homogeneous market case, it can be shown that the optimal owner price is , since any is essentially given to users for free, when their sharing levels are below the point where . Thus, by using , the platform’s revenue is:
(10) 
where . The first term of (10) is the income from renters, and the second term is the bonus paid to owners.
Theorem 3.6 ().
MTLP achieves optimal revenue among all loyalty programs with nondecreasing marginal bonus .
Theorem 3.6 is proven by showing that for any loyalty program with a nondecreasing B(s), there exists an MTLP counterpart that can achieve the same revenue. Therefore, it is sufficient for a platform to optimize over ladderlike bonus programs with multiple thresholds for achieving optimal revenue. This significantly reduces the search space, and the resulting subsidy programs are also highly practical.
Note that given and the owner utility function in (9), owner ’s optimal strategy should guarantee that the marginal incomes of sharing and selfusage are equal. In other words, a reasonable sharing choice for owner should belong to the intersections of the two curves, i.e., where . We denote the sharing point of owner at the intersection of and (not necessarily his final choice), i.e., we have .
Below, we construct a particular MTLP and show that it is costefficient. To do so, we first derive a necessary condition that any multithreshold loyalty program must satisfy.
Theorem 3.7 ().
For any MTLP, we have for all that:
(11) 
Condition in (11) essentially means that any MTLP guarantees that the bonus owner receives over the interval should at least cover his selfusage opportunity cost in the same interval.
Now we are ready to present a simple MTLP called Hyperbolic Bonus (HB). The advantage of HB is that it admits an exact characterization of the bonus values. The intuition beyond HB is to tighten the subsidies to owners, by making the left hand side and the right hand side of (11) equal for all owners. HB actually exploits the power of differential pricing, so that owner ’s best strategy satisfies , i.e., each owner’s sharing level resides in one corresponding interval. The exact form of HB is as follows.
Definition 3.8 ().
The Hyperbolic bonus (HB) program chooses as
(12) 
The corresponding thresholds satisfy (11) with equality, i.e., for all ,
(13) 
The existence of such in HB is shown in the proof of Theorem 3.6. Figure 3 shows an example of a multithreshold loyalty program and owners’ derivatives in the case of .
Essentially the hyperbolic bonus at each stage is inversely proportional to the number of owners who earn bonus in this stage. Under such bonus strategy, platform pays little amount of reward for sharing levels that many owners are capable to reach. The following theorem shows explicitly that HB is costefficient.
Theorem 3.9 ().
The amount of oversubsidization under the HB program is upper bounded by , i.e.,
(14) 
Here is owner ’s sharing level under HB, is the overall subsidization given by HB and denotes the opportunity cost of owner .
Since any subsidy program has to at least cover the opportunity costs of owners, (14) shows that HB almost does not provide any additional subsidization, i.e., the extra payment is at most , which is independent of the market size . As a platform’s profit typically increases with , a fixed revenue gap also implies that our algorithm is costefficient. Moreover, as we will show in Section 5.1, in addition to its efficiency, HB also achieves very good revenue performance.
4. Subsidies in a Competitive Market
In this section, we consider a market with two platforms, and conduct a comparative study on competition through loyalty programs. In this case, let and denote the renter price and owner reward on platform and , respectively. The following results discuss only the singlethreshold linear loyalty program, showing that when facing a typical heterogeneous market with owners from groups, even a singlethreshold program helps company compete.
Specifically, consider two groups of owners () in the market, with for . Note that is a practical situation, e.g., users can often be categorized as active or inactive (kooti2017analyzing, ). We assume that demand on a platform is a function of renter prices from both platforms, and supply is insufficient, i.e., for . Then, the utility of owner is:
(15) 
where is owner ’s sharing level on platform , and is his sharing on platform . In the case when , we assume that owners will share equally on both platforms.
To demonstrate the performance of linear loyalty program, we also introduce another widely adopted loyalty program, signup bonus, for comparison. To implement a signup bonus, the platform pays a onetime reward for owners who share exclusively on the platform, i.e., if an owner decides to share on this platform and receives a signup bonus, he cannot share on other platform simultaneously. For example, if platform introduces signup bonus and owner joins this program, his utility will be
(16) 
where is the signup bonus from platform . In short, signup bonus is a one time payment to lockin owners in competition.
Therefore, each platform can have three possible situations in the following discussion: adopting no loyalty program, adopting a linear loyalty program and adopting a signup bonus.
In this section, we assume that both platforms adopt the same commission rate, i.e., . The reasons why we consider identical are two fold. The first is that competition in real world shows that platforms usually adopt the same commission rate, e.g., Uber and Lyft may vary their prices in different time and places, but they both extract a fixed commission rate. The second is that an identical commission rate can help define the ”stronger” and ”weaker” platforms in our comparison. With identical , the platform that charges a higher renter price will also have a higher owner price ( recall that ). We say that this platform is ”stronger” and more attractive to users than the other platform in competition. For example, this platform may be better in service, reputation, and efficiency in matching demand and supply, etc., so that customers want to pay for its higher price. Note that the asymmetry in customer attractions and prices are common in real world: a less competitive manufacturer has to charge a lower price to win customers (fang2016market, ), sacrificing its revenue. Naturally, we will show how a loyalty program can help the ”weaker” platform to win market share.
4.1. Competition without loyalty program
We first provide results in a competition without loyalty programs as a baseline. Here, there are two distinct cases to consider: (i) when the market is symmetric () and (ii) when the market is asymmetric ().
Proposition 4.1 ().
When both platforms adopt the same commission rate and neither of two platforms adopt loyalty programs, if , platforms will equally share the supply, i.e., . Otherwise, the platform with a lower supplier price has zero revenue.
In the symmetric case, the market is wellbehaved, i.e., both platforms can win some market share. In contrast, in the asymmetric case, things are different. In particular, the asymmetric case highlights that market is quite fragile and, as soon as any asymmetry exists then the market becomes “winnertakeall.” This is a common phenomenon in twosided markets and is a result of the fact that more supply makes the market more desirable for demand, and vice versa.
For our purposes, this result highlights the need for loyalty programs. A “weaker” firm (one with smaller ), must find a way to compete in order to stay viable, and loyalty programs provide such an opportunity. Note that we only consider the situation when supplier price of the “stronger” firm is less than its competitor’s demand price, i.e., if platform is “stronger” (), we must have . Otherwise, platform will never be able to attract owners, since platform is offering an owner reward even higher than platform ’s income from renters. This condition is natural and common in practice, since platforms do not tend to coexist if one is extremely more efficient.
4.2. Loyalty program on one platform
The results in the case of no loyalty programs highlight the need for a “weaker” firm to compete via the introduction of a loyalty program. The results in this section highlight that such an approach to competition can be successful. However, our results also highlight that there is an “arms race” underlying competition via loyalty programs. As the “weaker” firm adopts a loyalty program, the “stronger” firm also has incentives to adopt a loyalty program. However, our results also highlight that consumer heterogeneity eases the “winner take all” nature of this competition.
We denote , as the sharing level of owner when he choose to share on platform or , respectively. Denote . We always have , since and owner will share more on platform .
To begin, we first consider the case where the “weaker” firm adopts a simple loyalty program – a signup bonus – in order to compete. The following theorem highlights that this approach can be successful.
Theorem 4.2 ().
Suppose and . Further, suppose platform does not use a loyalty program. Then, there exists a signup bonus for platform under which platform has a strictly positive revenue.
Our proof in appendix provides a more detailed characterization of the exact form of the signup bonus and the revenue for each platforms. We also observe in the proof that the resulting revenue of the firms depend heavily on whether firm tries to attract both users, or just one user. If the owners are diverse enough, the two firms will settle at an equilibrium where each attracts one owner (hence, both survive in the competition). Otherwise when owners are more alike, firm will try to take away all users (See Fig. 9 and Fig. 9 in Section 5.2 for experimental results).
Importantly, notice that it is not always possible for platform to use a signup bonus to compete. In fact, we see from Theorem 4.2 that if is very close to , a signup bonus cannot ensure a positive revenue for platform (see Figure 6 for experimental result). On the other hand, the theorem also highlights that platform can obtain a positive revenue if is not too much larger than .
Our next result highlights that introducing the more sophisticated linear loyalty program can also allow the “weaker” firm to gain market share.
Theorem 4.3 ().
Suppose , , and platform does not use a loyalty program. Then, there exists a linear loyalty program for platform under which platform can achieve a positive revenue.
Interestingly, our proof highlights that the optimal linear loyalty program to attract a specific class of owner turns out to be the same as in the monopolistic market (see Theorem 3.2). It gives a bonus that is equal to the platform’s commission fee, i.e.,
(17) 
Importantly, the increased sophistication of the linear loyalty program as compared to the signup bonus means that platform can compete even if it is much weaker, as long as (signup bonus requires ). This highlights the value of increased sophistication.
Similar to Theorem 4.2
, under a linear program, firms will also be strategic about which owner group to attract, and the results also depend largely on the heterogeneity of owners (See Fig.
9 and Fig. 9 in Section 5.2 for experimental results).4.3. Loyalty programs on both platforms
In fact the increased value of increased sophistication shows up even more prominently when signup bonus and linear loyalty programs are compared directly.
To see this, consider competition between two identical platforms with different forms of loyalty programs: platform uses a linear loyalty program and platform uses a signup bonus. In this case, the platform using a linear loyalty program can always squeeze out its competitor while maintaining a strictly positive revenue.
Theorem 4.4 ().
Suppose and . If platform adopts a linear loyalty program and platform adopts a signup bonus, then platform can always maintain a strictly positive revenue while making platform gain zero revenue, i.e., the platform with linear loyalty program can force the competitor with signup bonus out of market.
This result highlights that if one platform adopts a sophisticated linear loyalty program, it is crucial for competing platforms to follow suit quickly. In fact, exactly that was observed in the case of ridesharing platforms, where both Uber and Lyft moved from signup bonuses to linear loyalty programs at nearly exactly the same time (uber_copycat_loyalty, ). The intuition behind the fact that linear programs are more powerful is that they boost supply by incentivizing sharing, while signup bonuses only encourage participation, even under the strong assumption of exclusiveness.
5. Case Study
We conclude the paper by illustrating the results using a case study driven by ridesharing data from Didi Chuxing. In particular, we use data from Didi Chuxing (fang2017prices, ) to reverse engineer the utility functions in the model. This yields the following form:
(18) 
where . In our case study we compute the revenue and the optimal loyalty program under different price and heterogeneity settings, based on the user utility in (18). Note that is strictly concave and increasing in , with a derivative of at .
5.1. Monopolistic market
We start by considering a monopolistic market and provide the following results.
How does the supplier base payment impact revenue? We first demonstrate that the optimal pricing strategy for a firm is to set the base payment and extract revenue via the loyalty program.
Specifically, we fix the renter charge and consider a homogeneous market, which adopt the optimal linear loyalty program derived in Theorem 3.2. Then, we vary the value and compute the revenue. From Figure 6, we see that the platform’s revenue is decreasing in the supplier price . Note that this is not obvious, since a larger facilitates sharing and may lead to a larger total supply. Nonetheless, the results in Figure 6 illustrate Corollary 3.3 and shows that the platform should choose for base payment and rely on the loyalty program for extracting optimal revenue.
How effective is HB? We now investigate the effectiveness of the HB program in a heterogeneous market, and show that the revenue under HB is close to the optimal.
Specifically, we choose the selfusage utility of owner to be:
(19) 
We set the base payment to (this is the optimal case), , and vary the number of owners . Figure 6 shows the revenue and platform’s total subsidy under the Hyperbolic bonus program (). It can be observed that HB achieves good revenue performance since is close to the optimal revenue . Further, we can see that the total subsidy paid by platform is a constant beyond the opportunity cost, i.e., the extra payment is independent of . This validates Theorem 3.9. Note that when , the two curves coincide, as the market is homogeneous and the single threshold linear loyalty program is optimal.
5.2. Competitive market
Next, we consider loyalty program under competition. In this case, we capture the user heterogeneity with a parameter by choosing and , i.e., being the difference between the two classes of owners. We set in the experiments, and let .
How does competition impact the optimal signup bonus? Consider platform adopts signup bonus and platform does not adopt loyalty program. We first compute the optimal bonus under different competition profiles with fixed , and show that user heterogeneity plays a role in deciding market partition.
Figure 6 shows the result for , where the optimal signup bonus is computed for three regimes.

The first regime is when . In this regime, the signup bonus is rapidly increasing in , and platform ’s best strategy is to attract both owners.

In the second regime, where the competition is tougher for platform (a larger ), platform ’s optimal strategy is to only attract owner by reducing revenue. Thus, the bonus increases in .

In the last regime, is large enough such that . In this case, platform cannot stop owners from leaving for platform . As a result, it achieves zero revenue.
How does competition impact the form of the optimal linear loyalty program? We now study the form of the optimal linear loyalty program for platform under different values, while platform does not adopt loyalty program. Platform will only attract one owner when user heterogeneity is extensive. Figure 9 shows the case for and . There are two regimes for the optimal linear loyalty program.

If , platform ’s best strategy is to adopt an optimal bonus to attract both owners and obtain higher revenue (exact form can be found in the proof of Theorem 4.3).

If , i.e., when users are highly heterogeneous, platform ’s optimal strategy is to keep , and to target only owner .
The existence of transition point shows the watershed in platform’s subsidy policy, i.e., when users are highly differentiated (), it will be too expensive to attract both owners. In that case, platform chooses to stay with the more profitable owner and substantially increases the minimum sharing requirement to block owner from entering the platform.
Do linear loyalty programs provide a competitive advantage? We now show that linear loyalty programs always lead to a higher revenue than signup bonuses in the competitive setting.
Figure 9 shows the results for and different competition profiles when only platform adopts loyalty program. It can be observed that linear loyalty program always leads to a higher revenue than signup bonus. The reason is that linear bonuses can substantially boost supply, since owners earn more bonus only if they share more. On the other hand, signup bonus does not increase the sharing amount.
How does heterogeneity impact competition in providers?
We now move to studying the impact of supplier heterogeneity on the optimal subsidizing policy, by finding the critical heterogeneity level beyond which platform switches to attract only one product owner. In Figure 9, we plot the critical point under different competition profiles when platform adopts linear loyalty program and signup bonus, respectively.
The blue line shows the critical values of signup bonus. We see that is very high () before , and drops quickly to when . Then, it continues to decrease as increases. This means that signup bonus can win both owners only when competitor’s price or user heterogeneity is small. The intuition behind this behavior is as follows. When is high, owner ’s sharing level at platform will be high. Thus, if platform wants to attract owner (the one with a smaller marginal selfusage utility), it has to compensate owner ’s loss due to switching by giving a bonus that is larger than this loss. However, since is not large and the signup bonus itself does not facilitate sharing, attracting owner is not beneficial for platform . Hence, it will try to instead only focus on attracting owner once is large. The dash line shows the value of , below which using signup bonus can still gain positive revenue.
The red line shows the critical values for linear loyalty bonuses. Interestingly, is increasing in . This shows that linear loyalty program is more likely to win both owners, even when both and heterogeneity increase. The reason that the critical point is increasing is as follows. First, the bonus for platform to attract owner is always . Hence, if is high, the only way platform can attract owner is by lowering the threshold value . Moreover, the value decreases with . Thus, if is high, platform already uses a small . In this case, it only has to pay a smaller additional bonus to also attract owner (the one with higher marginal selfusage utility), and by doing so, it also obtains owner ’s sharing, which brings additional benefit. As a result, platform will stick to attracting both users until is higher. Note that this implies a higher robustness of the linear program, which is a desired feature for platform strategies, especially at the early stage, when a company does not have enough data to accurately profile user behavior and heterogeneity.
6. Conclusion
In this paper, we consider the problem of optimal loyalty program (subsidy) design, which is central to the development of the sharing economy. We first characterize the optimal loyalty program for a monopolistic market with homogeneous product owners, and show that it has a linear form that is close to loyalty programs in practice. After that, we show that the multithreshold loyalty program (MTLP) can achieve optimal revenue when the platform is facing heterogeneous suppliers. Also, we introduce a hyperbolic bonus program (HB) for the heterogeneous market, and show that this program is highly costefficient and achieves a good revenue performance. Further, we consider loyalty programs in a duopoly market with heterogeneous suppliers, showing that the introduction of loyalty programs can help a weaker platform to win customers, and the more sophisticated ladderlike loyalty program can outperform the simple signup bonus. Our results show that heterogeneity in users helps reduce the competition among platforms, since platforms tend to attract different groups of users in this case. We further validate our results with real transaction data from ride sharing platform Didi Chuxing.
Acknowledgements.
The work of Zhixuan Fang and Longbo Huang is supported by the National Natural Science Foundation of China Grant 61672316 and 61303195, the Tsinghua Initiative Research Grant and the China Youth 1000Talent Grant. The work of Adam Wierman is supported by NSF Grant AitF1637598, CNS1518941.References
 [1] L. Alarabi, B. Cao, L. Zhao, M. F. Mokbel, and A. Basalamah. A demonstration of sharek: an efficient matching framework for ride sharing systems. In Proceedings of the 24th ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems, page 95. ACM, 2016.
 [2] M. Armstrong. Competition in twosided markets. The RAND Journal of Economics, 37(3):668–691, 2006.
 [3] S. Banerjee, D. Freund, and T. Lykouris. Multiobjective pricing for shared vehicle systems. arXiv preprint arXiv:1608.06819, 2016.
 [4] S. Banerjee, S. Gollapudi, K. Kollias, and K. Munagala. Segmenting twosided markets. In Proceedings of the 26th International Conference on World Wide Web, pages 63–72. International World Wide Web Conferences Steering Committee, 2017.
 [5] S. Banerjee, R. Johari, and C. Riquelme. Pricing in ridesharing platforms: A queueingtheoretic approach. In Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC ’15, New York, NY, USA, 2015. ACM.
 [6] S. Benjaafar, G. C. Kong, X. Li, and C. Courcoubetis. Peertopeer product sharing: Implications for ownership, usage and social welfare in the sharing economy. Available at SSRN: 2669823, 2015.
 [7] M. BERGEN. This change to how lyft pays driver bonuses could weed out the cheaters. Recode, 2015.
 [8] J. Bogage. Uber’s controversial strategy to finally defeat lyft. The Washington Post, 2016.
 [9] M. Branman. Ride sharing showdown: should you grab an uber, or hail a lyft? Digital Trends, 2017.
 [10] G. P. Cachon, K. M. Daniels, and R. Lobel. The role of surge pricing on a service platform with selfscheduling capacity. Manufacturing & Service Operations Management, 2017.
 [11] L. Chen, A. Mislove, and C. Wilson. Peeking beneath the hood of uber. In Proceedings of the 2015 ACM Conference on Internet Measurement Conference, pages 495–508. ACM, 2015.
 [12] M. K. Chen and M. Sheldon. Dynamic pricing in a labor market: Surge pricing and flexible work on the uber platform. In EC, page 455, 2016.
 [13] P. Cohen, R. Hahn, J. Hall, S. Levitt, and R. Metcalfe. Using big data to estimate consumer surplus: The case of uber. Technical report, National Bureau of Economic Research, 2016.
 [14] E. Evendar, Y. Mansour, and U. Nadav. On the convergence of regret minimization dynamics in concave games. ACM STOC, 2009.
 [15] Z. Fang and L. Huang. Market share analysis with brand effect. In 2016 IEEE 55th Conference on Decision and Control (CDC), pages 7016–7023. IEEE, 2016.
 [16] Z. Fang, L. Huang, and A. Wierman. Prices and subsidies in the sharing economy. arXiv preprint arXiv:1604.01627, 2016.
 [17] Z. Fang, L. Huang, and A. Wierman. Prices and subsidies in the sharing economy. In Proceedings of the 26th International Conference on World Wide Web (WWW), pages 53–62. International World Wide Web Conferences Steering Committee, 2017.
 [18] C. Gaffey. Uber is losing billions in china, ceo admits. Newsweek, 2016.
 [19] B. Jiang and L. Tian. Collaborative consumption: Strategic and economic implications of product sharing. Management Science, 2016.
 [20] F. Kooti, M. Grbovic, L. M. Aiello, N. Djuric, V. Radosavljevic, and K. Lerman. Analyzing uber’s ridesharing economy. In Proceedings of the 26th International Conference on World Wide Web Companion, pages 574–582. International World Wide Web Conferences Steering Committee, 2017.
 [21] C. P. Lamberton and R. L. Rose. When is ours better than mine? a framework for understanding and altering participation in commercial sharing systems. Journal of Marketing, 76(4):109–125, 2012.
 [22] M. Lecuyer, M. Tucker, and A. Chaintreau. Improving the transparency of the sharing economy. In Proceedings of the 26th International Conference on World Wide Web Companion, pages 1043–1051. International World Wide Web Conferences Steering Committee, 2017.
 [23] M. Möhlmann. Collaborative consumption: determinants of satisfaction and the likelihood of using a sharing economy option again. Journal of Consumer Behaviour, 14(3):193–207, 2015.
 [24] C. Perea. The power driver battle in ca: What it means for everybody. http://therideshareguy.com, 2016.
 [25] PYMNTS. Uber, ola, lyft: Ridesharing platforms battle. PYMNTS, 2016.
 [26] G. Quattrone, D. Proserpio, D. Quercia, L. Capra, and M. Musolesi. Who benefits from the sharing economy of airbnb? In Proceedings of the 25th International Conference on World Wide Web, pages 1385–1394. International World Wide Web Conferences Steering Committee, 2016.
 [27] J.C. Rochet and J. Tirole. Platform competition in twosided markets. Journal of the european economic association, 1(4):990–1029, 2003.
 [28] J.C. Rochet and J. Tirole. Twosided markets: a progress report. The RAND journal of economics, 37(3):645–667, 2006.
 [29] M. Rysman. The economics of twosided markets. The Journal of Economic Perspectives, 23(3):125–143, 2009.
 [30] C. Said. Uber incentives aim to lure power drivers. San Francisco Chronicle, 2016.
 [31] T. L. Siddhartha Banerjee, Daniel Freund. Pricing and optimization in shared vehicle systems: An approximation framework. In ACM EC 17’, 2017.
 [32] S. Silverstein. These animated charts tell you everything about uber prices in 21 cities. Business Insider, 2014.
 [33] M. Stiglic, N. Agatz, M. Savelsbergh, and M. Gradisar. The benefits of meeting points in ridesharing systems. Transportation Research Part B: Methodological, 82:36–53, 2015.
 [34] E. G. Weyl. A price theory of multisided platforms. The American Economic Review, 100(4):1642–1672, 2010.
 [35] G. Zervas, D. Proserpio, and J. Byers. The rise of the sharing economy: Estimating the impact of airbnb on the hotel industry. Boston U. School of Management Research Paper, (201316), 2014.
7. Appendix: Proofs
This section provides proofs of the results stated and discussed in the previous sections.
7.1. Proof of Theorem 3.2
We first show that the linear loyalty program is optimal in a monopolistic market with homogeneous owners ( for all ), given and ). Since all owners are homogeneous, we can normalize the number of owners to be without loss of generality. In this case, we have and thus the total supply , where is the owner’s sharing level. We then derive the optimal . The platform’s revenue is the commission fee subtracting subsidies, as defined in (3):
(20) 
(i) Suppose there exists a general subsidy program such that platform revenue is maximized. We want to show that there exists a linear loyalty program with that can achieve the same revenue. Denote the sharing level of owner under to be , let
Since maximizes , the minimum subsidy is to compensate owners such that the total utility would be the same whether or not participating loyalty program . Similarly, we can set such that
(21) 
where is the sharing level when there exists no loyalty program, which satisfies the marginal condition:
Since maximizes revenue, we should at least have . Note that
Here we use the marginal condition . Thus there must exists a which satisfies (21). In conclusion, given above, owner’s sharing level and platform’s revenue are the same as the optimal general subsidy .
(ii) Similar to analysis above, the linear loyalty program with parameter achieves maximum revenue when:
(22) 
Therefore platform’s revenue is:
consider first order condition,
which gives . Here we use the marginal utility condition that . The bonus threshold is given by (22):
7.2. Proof of Corollary 3.3
By analysis in the proof of Theorem 3.2, the maximum revenue of platform under price and loyalty program is:
where is the sharing level when there is no loyalty program. Therefore it is platform’s strategy to choose supply payment and bonus (which will uniquely determine by (22)). Under first order condition:
which gives optimal . Note that is fixed and is independent of here. Similarly,
The last equation is due to the fact that , and derived above. The result shows that revenue is decreasing in , which means optimal supply payment should be , and introduces loyalty program such that :
7.3. Proof of Proposition 3.4
Given owners with self usage benefit , let denote the sharing level of under no loyalty program, i.e., . Consider some owner with sharing level under loyalty bonus with nondecreasing , such that , i.e, one who participates in loyalty program and thus shares more than initial sharing (it will be trivial if none of owners participate). We must have the bonus covers owner ’s opportunity cost:
(23) 
and the marginal condition that .
We show that for any , we must have owner also participates in the loyalty program. Note that . If for all , owner must participate in the program and share . Otherwise, there must exists some intersection point such that in .
Let , we must have for . Therefore, we have:
In this case, owner will choose to share to earn more from the loyalty program.
7.4. Proof of Theorem 3.6
We prove the theorem by showing that for any subsidy program with increasing marginal bonus, multithreshold loyalty program can achieve the same revenue.
Suppose under the nondecreasing marginal bonus , owner ’s sharing level is , we must have , and . By Proposition 3.4, assume owners joins the loyalty program. If there is only one owner who joins the program, i.e., , the proof is the same as in the Theorem 3.2. Otherwise, consider owner such that . There must exists some such that:
(24) 
For the case when , we can choose any . We also have such that:
(25) 
Construct a multithreshold loyalty program such that , and threshold according to (24) and (25), for all . In this case, each owner will share the same . Note the utility for to share stays the same in MTLP, compared to . The reason is that will only choose to share at intersection points of and , i.e., where , if any such exists. For such point such that , and where , we have:
given choose as optimal sharing under . The first inequality is due to the fact that for . This means has no motivation to over share under MTLP . Similarly, if choose a smaller sharing level such that where , we have:
which also guarantees will not share less. The first inequality here is due to the fact that for .
Hence, the MTLP constructed here gives the same sharing level of each owner, and the revenue of platform is also the same:
In conclusion, for any bonus with nondecreasing marginal , we can construct a multithreshold loyalty program to achieve the same revenue and sharing level, i.e., MTLP can achieve the optimal platform revenue.
7.5. Proof of Theorem 3.7
Note that satisfies . If , since owner ’s optimal sharing is , we must have the income of sharing instead of is higher than the lost of self usage benefit from to :
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