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Boosting Sharing Economy: Social Welfare or Revenue Driven?

Product sharing over online platforms, or sharing economy, has become a growing trend and has the potential to mitigate many social problems such as wasted products, idle resources, road congestions, and even greenhouse gas emissions.Despite its quick and successful development so far, there has been a lack of clear understanding about who is a better candidate for boosting sharing economy: government-like organizations who care about social welfare, or profit-driven entities who mainly focus on revenue? To investigate this problem, we propose a game-theoretic model among users participating in a sharing platform and analyze the social welfare under different platform pricing strategies. Specifically, we derive tight bounds of social welfare loss of the revenue maximization policy compared to social welfare maximization, and show that revenue maximization leads to more sharing supply and ensures a better quality-of-service to users. We further conduct case studies and show the advantage of sharing over a common platform compared to separated sharing groups. Our numerical results based on real data also show that, when the sharing demand/supply ratio is large, the revenue maximizing policy also achieves optimal social welfare. To the best of our knowledge, we are the first to study social welfare with general user utility functions in sharing economy, and to compare the performance of different pricing policies.


INTRODUCTION
Along with the rapid recent developments of internet and mobile technologies, sharing economy [1] (also known as collaborative consumption) has emerged as an efficient way to utilize available resources. For example, Airbnb [30] for sharing rooms, Uber [3] and Lyft [2] for car sharing, and many bike sharing systems around the world served as part of city public transportation [29]. In fact, sharing economy attracts great amount of capital, which supports dozens of startups to chase the trend, as the revenue in the industry may reach ACM ISBN 978-1-4503-2138-9. DOI: 10.1145/1235 $3.5 billion this year according to Forbes [13]. Meanwhile, governments and many other non-profit organizations are eager to help develop sharing economy since it can "address social and economic challenges" [18], quoted from the British government's response to the sharing economy.
Among many factors that facilitates the great success of sharing economy, a very important one is that it enables efficient usage of many resources that are essentially wasted: According to [27], the average time in use for cars in US and Europe is 8% and the average used time of an electric drill over its lifetime is 6-13 minutes by [6]. Data from European Environment Agency [11] also shows that the average number of passengers per car is 1.58 in UK and 1.42 in Germany. One can enumerate many other kinds of resources remain idle in daily life: empty rooms, idle computing resources, etc. Due to the large quantity of these under-utilized resources, even a small improvement in utilization can result in great societal benefits. Indeed, UK parliament [26] estimates that increasing the average utilization of cars to 2 passengers per car would save 17 billion tonnes of CO2. This is exactly the great potential of sharing economy.
Due to its importance, there have been many works discussing the definition, motivation, and development of sharing economy, e.g., [6], [17]. There are also concrete case studies, e.g., [23] on car sharing, phone minute sharing and bike sharing, and [30] on the impact of Airbnb to the hotel industry. Different from these prior works that mainly focus on qualitative assessments, in this paper, we try to provide an analytical framework for modeling sharing economy and for understanding how it can be effectively boosted. Specifically, we consider a general product-sharing system with a set of product owners and renters. The sharing platform regulates sharing supply and demand by choosing an appropriate market price. Product owners and renters then decide on their own whether to share or rent, and determine their usage level. Product owners receive monetary payments sharing their products, but spend maintenance costs, and all users obtain utility from using the products. Our goal is to understand the overall social welfare of the system and the quality-of-service (QoS) in sharing under two most common operation modes: social-welfare maximization, e.g., run by non-profit organizations, or revenue maximization, e.g., run by companies like Uber. This general framework captures various important aspects of sharing economy including user incentive, system efficiency, and QoS, a metrics that reflects both users' satisfaction and resources utilization.
Most prior works on sharing have focused on peer-to-peer (P2P) networks. For instance, [24] discusses the social wel-fare under assumption of linear supply and demand, and quadratic utility. [15] and [22] study the equilibrium either under quasi linear utility or simple discrete strategies set. Outside the peer-to-peer area, a recent work [16] considers user's network usage and social welfare under prices set by a platform and focuses on the network congestion problem. [7] studies social welfare and a sharing platform's strategy with linear user utility, and focuses on how users decide to purchase products. However, our work is different from the aforementioned works in the following: (i) we consider a sharing platform that regulates sharing and demand via pricing, while most previous works in network resources sharing do not consider pricing, (ii) in our model, product owners can also benefit from personal usage of the product, whereas existing works often neglect this part, (iii) we consider general concave user utility functions, while prior works often adopt a specific form of utility functions, and (iv) we propose a novel QoS metric for product sharing, which captures the "wasted supply" of owners, e.g., Uber cars running without customers, while existing works have not discussed this aspect.
The main contributions of the paper are summarized as follows.
• We propose a general modeling framework for studying the sharing economy. Our model captures various important aspects including user incentive, QoS and system efficiency.
• Focusing on general concave user utility functions, we establish the existence of market equilibrium and provide tight bounds of social welfare loss under the revenue maximizing policy compared to social welfare maximization. We also show that the revenue maximization policy guarantees more sharing supply, and ensures a better QoS compared to social welfare maximization.
• We study two classic cases where utility functions are linear and quadratic. We show that under linear utility, sharing over a common sharing platform can achieve a higher social welfare compared to sharing inside multiple isolated groups.
• We conduct numerical simulations based on real data from DiDi [5], the biggest car sharing platform in China.
Our results show that the social welfare loss under revenue maximization quickly decreases to zero as the renter-owner ratio increases (scarce resource). We also show that a moderate product usage cost can lead to both high QoS and low resource idle rate. This result implies that a government-like organization can also improve sharing by choosing a proper usage cost, e.g., choosing tax or gasoline price for car usage.
The rest of the paper is organized as follows. In Section 2, we provide two examples of sharing and discuss common features of them. Section 3 introduces our system model. Section 4 provides main results of this paper, including analysis of social welfare, supply and demand properties under different pricing policies. Section 5 studies two typical cases in sharing based on the results in the previous section. Section 6 provides simulation results and discussions. Conclusions come at last in section 7.

EXAMPLES
Among the thriving sharing industries, we take two common cases as examples. The first one is car sharing. Companies like Uber, Zipcar, RelayRides provide car sharing services. Zipcar provides their own cars for sharing, while the other two match car owners and renters via online platform. In the case of Uber, car owners put their cars online for share when they want to, while renters applied for cars through Uber's platform, and rents are charged by distances or duration time of use. The other example is house sharing: hosts put their available rooms online and guests can book them via online platform, e.g., Airbnb.
The same pattern of these online sharing is that, product owner shares his products by putting some portion of them (part of the product or part of the usage time) available online, and he can use what is left of the product by himself. Owner's problem is to decide how much amount of product to put online for share, e.g., a driver may share 100% usage time of his car when the renting price is really high, or 0% if the profit is lower than his cost. He earns income by doing so, but also pays a maintenance cost, and can face competitions in getting renters due to other owners' sharing actions. As for renters, they pay as they rent the products.

Owners and Renters
Consider a product sharing system that consists of two groups of users: product owners denoted by the set O, and renters denoted by R, with NO = |O|, NR = |R| denoting the numbers of owners and renters.
For each user i ∈ O ∪R, we normalize his maximum usage of the product to be 1, e.g., product usage frequency. Then, we use ui ∈ [0, 1] to denote one's usage of the product, e.g., fraction of time one uses his own/rented car. If user i is a product owner, we also use si ∈ [0, 1] to denote the level at which he shares the product, e.g., drives it as a Uber car. Note that we always have ui + si ∈ [0, 1].
Note that with ui, si ∈ [0, 1], our model can be viewed as considering the average usage over a period of time. This is different from using a realtime supply and demand model, and facilitates the quantification of long-term sharing behavior of users in the market.

Sharing Demand and Supply
We assume that sharing takes place over a platform, e.g., Uber. In this platform, renters pay the market renting price for using cars from owners. The market price P is set by the sharing platform. Depending on their own usage preferences, desire to share products, and the renting price, users choose their own ui and si (si for product owners only) values, which result in different demand and supply conditions in the system.
We denote the total sharing supply S(P ) and demand D(P ) (renting demand) of the market under price P as the sum of owners' sharing levels and the sum of renters' usage levels, respectively. That is, (1)

User Utility
For each user i ∈ O ∪ R, we denote Ui(ui) his self-use utility of using the product, e.g., personal satisfaction. The Ui(ui) is unrevealed to others. In this paper, we make the following assumption about Ui(ui). Assumption 1. For each i, Ui is a continuous and strictly concave function in ui with Ui(0) = 0. Moreover, Ui's derivative at ui = 0 exists and is bounded.
Note that the concavity and continuity assumption is general and is commonly adopted when studying system utilities, e.g., [28] and [21], while bounding the derivative at zero is for the convenience of proofs without loss of generality.
In addition to usage utilities, monetary transactions on the platform also affect users' overall utility as follows, e.g., product owners can profit from sharing their product while renters will pay for their usage.
To be specific, product owner will gain utility from both his self-use and the income of sharing his product, meanwhile he will suffer from product's maintenance cost, i.e., the overall utility obtained by an owner is: Here Ui(ui) is i's self-use utility and cui +csi is maintenance cost, where c is a universal constant representing the usage cost per unit, e.g., gasoline, tolls, or house keeping. P min{ D(P ) S(P ) , 1}si in (2) is owner i's sharing income. The term min{ D(P ) S(P ) , 1} is introduced to captured the fact that in practice, it is typical that a certain percentage of owner's sharing levels will not generate any income, especially when the total demand is less than the supply. For instance, the time a Uber driver spends driving around waiting for customer's order, or the time an Airbnb room provider spends keeping his apartment empty. In other word, min{ D(P ) S(P ) , 1} can be treated as the probability that an owner meets a customer.
Interestingly, the term min{ D(P ) S(P ) , 1} in (2) is a also nature indicator of quality-of-service (QoS) in sharing: a lower D(P ) S(P ) value implies a higher service quality to customers, as it reflects a lower difficulty level for renters to rent a product. Indeed, if we intuitively imagine that the system operates according to an M/M/1 queueing model, with D(P ) being the arrival rate (demand) and S(P ) being the service rate (supply), then the expected delay for renters is equal to 1 S(P )−D(P ) . As a result, a lower D(P )/S(P ) value indicates a smaller service delay. 1 That is to say, the item P min{ D(P ) S(P ) , 1} can also be seem as the QoS-weighted market price, i.e., when the service quality is low ( D(P ) S(P ) is close to 1), the "effective price" P min{ D(P ) S(P ) , 1} seen by owners becomes relatively high, and this attracts more supplies.
Note that the denominator of S(P ) includes every owner's contribution in sharing. Due to this coupling among sharing owners, the problem becomes indeed a game among users (after the sharing price P is set).
As for renters, they have to pay the renting price P per unit usage for using the product. Hence, the overall utility a renter obtains with product usage ui is given by 1 The fraction of time the server is busy in the M/M/1 queue is exactly D(P )/S(P ).
Therefore renter i will choose his demand independently according to his own overall utility function. When total demand exceeds total supply, some renters may not be able to get access to products and thus obtain zero utility. There are three reasons that we use independent demand decision instead of putting a similar "discount factor" such as min{S(P )/D(P ), 1} in renter's utility functions to capture the fraction of time a renter actually gets served: • Renters' independence. Product owner can choose their self-use amount and sharing amount based on the expected payoff when he put his product online for share, while renters in real world usually apply for shared product on demand, which is independently decided, similar independent setting can be found in [8].
• Short term behavior. Product owner can be seen as long term participators in the sharing platform who value long term expected payoff. By contrast, renters can be seen as "one shot" comers who only care about whether they can get access to the product "this time", e.g., one may immediately try taxi or subway if he can not get served through Uber, therefore setting his utility gained from sharing platform to be zero is more meaningful to be a long term average.
• most importantly, using (3) does not lose any generality, as we will prove that in the cases of interests, i.e., revenue maximization and social welfare maximization, supply always exceeds demand. That is, even we add a factor min{S(P )/D(P ), 1} in renters' utility function, we will still have min{S(P )/D(P ), 1} = 1 in the two cases we discuss in this paper.

User's Response and Platform's Policy
Given the utility functions (3) and (2), under a market price P , each owner chooses his usage and amount of share by maximizing his utility, i.e., choosing u * i (P ) and s * i (P ) by solving the following optimization problem: and each renter choose u * i (P ) similarly by maximizing his utility (3).

System Objective
Under a given market price P , we say that the market reaches a Nash equilibrium when each owner is executing his best response action, i.e., the solution of problem (4). The platform's objective is to find an optimal market price to maximize certain given objective function when the market is at a Nash equilibrium (if it exists). For example, a maximum social welfare policy maximizes user's social welfare at the Nash equilibrium.
In this paper, we focus on two important objectives that are commonly studied, revenue maximization and social welfare maximization. They represent two dominant modes under which sharing economy can be operated, i.e., government or non-profit organization driven, and revenue-driven. We are interested in understanding how the system behaves under these two objectives.

Remarks
A few remarks are in place for our model. First of all, the setting of fixed identities for owners and renters is based on the assumption that users' identities do not change frequently. 2 Second, only assuming concavity of the utility functions makes our framework and results very general. In fact, most of the main results in this paper, e.g., Theorem 3, are derived based only on this assumption and are independent of the form and parameters of each individual's utility function.

SHARING ANALYSIS
In this section, we present our results. To facilitate reading, all detailed proofs are presented in the appendices and additional technical report [4].

Market Equilibrium
We first establish the existence of market equilibrium under any fixed sharing price. This result is critical, as the existence allows subsequent studies about the market performance.
Theorem 1. Nash equilibrium exists among all users in O for any given market price P .
The theorem is proved by showing that each owner's utility function is continuous in everyone's strategy and is concave in his own strategy. Note that according to (3), renters decide their levels of demand to the platform independently, considering only their own utilities and the market price. That is, demand request is fixed given price P , and the overall payoff is (3) if they get served. However, it can happen that under certain equilibria (determined by P ), the resulting total supply is less than the total demand, i.e., S(P ) < D(P ). In this case the actual utility a renter gets in practice is different from that determined by (3), as he is not even served. Fortunately, in two cases of interest in this paper, we show that this will never happen, i.e., for both revenue and social welfare maximization, we have S(P ) ≥ D(P ) at the equilibria.

Social Welfare and Revenue Maximization
In this section, we look at the aggregate social welfare under two different market price mechanisms that correspond to two types of platform providers, i.e., government-like organizations who care about maximizing total social welfare, and companies who are after maximum revenue. Our goal is to understand the differences of user behavior and system performance under these two modes.
We start with social welfare maximization. To this end, we define the aggregate social welfare as follows: Here R denotes the set of renters who are actually served (when D(P ) > S(P ), some renters may not be able to meet an owner). Notice here that the welfare is defined over all possible uniform price policies. We adopt this definition to avoid unfair comparisons with welfare maximization policies with non-uniform prices, i.e., different charges can be set to different users in the system, e.g., [25]. Maximum social welfare policy is to maximize W (P ) by choosing optimal P . The revenue maximization case is to maximize the volume of trade defined as follows; The adoption of this objective is motivated by the fact that in many sharing systems, the platform obtains a transaction fee from each successful business event, e.g., Uber charges its drivers 20% commission fee [19]. Thus, maximizing the trading volume is equivalent to maximizing the company revenue.
We first present a few propositions to describe some basic features of demand and supply in the market. They also serve as preliminaries for later theorems.
Proposition 1. The total demand D(P ) is non-increasing in the market renting price P and total demand eventually becomes 0 when P is high enough.
Proposition 1 is intuitive, since renters' decisions are independent and are guided by objective (3). Thus, as the price increases, each renter will cut down his demand.
Some intuition about Pupper and Pc in Proposition 2 is in place. First, if we increase P from zero, supply will continue to increase as long as supply is less than demand. However, as supply increases, it eventually exceeds demand and competition among owners becomes more and more intense. Then when the price is higher than some threshold price Pupper, resulting in = c, no owner will share any more since the profit can not even cover the cost. Second, the price Pc is the lowest market clearing price: Pc = min{P | D(P ) = S(P )}. However, establishing its existence requires showing the continuity of S(P ) at the demand-supply intersection point and is nontrivial.
From Proposition 2, we see that when the market renting price is low such that D(P ) > S(P ), some renters may not be able to rent a product, complicating the analysis of the aggregate social welfare. Fortunately, we prove in the following proposition that both the social welfare maximizer and revenue maximizer guarantee that the supply exceeds demand, i.e., S(P ) ≥ D(P ).
Proposition 3. Let Psw denote the market price which maximizes social welfare (5) and Pr denote the market price which maximizes revenue (6). We have S(Psw) ≥ D(Psw) and S(Pr) ≥ D(Pr).

Proposition 3 guarantees that renters have enough supply at
Pr and Psw. Therefore, all renters will get served. Hence, R = R in (5), and that V (P ) = P · D(P ) at Pr and Psw.
After the above propositions, we now have our first main theorem.
Theorem 2. We have following results with respect to Psw and Pr: • lowest market clearing price P maximizes aggregate social welfare, i.e., Psw = P , and • maximum revenue price is higher than maximum social welfare prie: Pr ≥ Psw; • The total sharing supply under revenue maximization is no less than that under social welfare maximization, i.e., S(Pr) ≥ S(Psw), where Pr and Psw are the revenue maximizing price and the welfare maximizing price, respectively.
It is interesting to note that revenue maximization oriented sharing actually encourages more supply from owners. The reason is that the pursuit of trading volume is consistent with owners' interests. Thus, this objective increases their enthusiasm on sharing. To be specific, the "effective price" P D(P ) S(P ) seen by an owner has the numerator exactly the form of trading volume, therefore higher trading volume magnifies "effective price", which stimulates supply. Another interesting point is that a higher sharing amount does not imply a higher social welfare. Take car sharing as an example, a higher sharing amount at higher price may lead to more empty running cars and result in lower utility for everyone.
Despite different prices and supply under these two cases, the following theorem bounds the social welfare loss under the maximum revenue policy.
Theorem 3. The social welfare gap between the maximum social welfare policy and a maximum revenue policy is bounded by: Moreover, the bounds are tight.
Here W (Pr) denotes the aggregate social welfare (5) under the revenue-maximizing price Pr. Theorem 3 states that maximum revenue policies are "good enough," in the sense that their social welfare losses are bounded. One important feature of the gap is that it can be easily evaluated by third-party organizations who are interested, e.g., government, without the necessity to know user's private utility functions, i.e., it only depends on the total demand and supply under prices Pr and Psw. The tightness result will be demonstrated by examples in section 5. Our next main result is an immediate corollary of Theorem 3. We decide to make it a theorem due to its importance. Theorem 4 shows an interesting result that, in addition to providing a bounded loss in social welfare, the revenue maximizing approach also guarantees a better QoS for customers.

Robustness
In this subsection, we investigate the robustness of our results to the heterogeneity of users. Specifically, we assume that each owner i may have different costs for sharing and personal usage, denoted by ci,s and ci,u, respectively, and that users can have different preferences on sharing, e.g., the effective price seen by an altruist maybe P ·( D(P ) S(P ) + i), > 0, which is always higher than P · D(P ) S(P ) . Similarly, i < 0 means that a user is not very willing to share. 3 Therefore, an owner's utility function becomes: −ci,uui − ci,ssi.
In this case, we have the following result indicating model's robustness: Theorem 5. Under the owner's utility of (7), though Pr, Psw and supply may be different, all theorems above still hold.
Theorem 5 shows that our results indeed hold for a more general class of scenarios, where (i) owners can have different usage costs (captured by ci,u), due to product diversity, e.g., SUVs may cost more gasoline than sedans, (ii) owners can have different endurance levels to the wear and tear costs due to renter's use, or the moral hazard because of renter's potential abusive use (captured by ci,s), as considered in [20] and [7], (iii) owners have different preferences towards sharing (captured by i), i.e., being revenue-driven or altruistic.
A positive for product owners can help lower Pr since supply altruists provide more supply. Note that this is crucial for platform to obtain higher revenue, as shown in Fig.1 (with 200 renters and 120 owners, see 6.3 for detail settings), since Pr will only be chosen in the interval where supply is no less than demand, Pr( = 0) = Pc( = 0) = 13 and Pr( = 0.5) = Pc( = 0) = 12, here Pc is the market clearing price. That is, according to the revenue curve in this example, a positive can increase supply and help platform gain more revenue.

CASE STUDY
In this section, we present two concrete cases to demonstrate our results. Specifically, we consider the linear usage utility and quadratic usage utility. These two scenarios are commonly adopted in social welfare analysis, see [7] [16]. Though linear function is not strictly concave, we find that our results also fit well here. Moreover, we show that in these two cases, the bound in Theorem 3 is tight.

Linear Usage Payoff
Consider the case when each user's payoff is linear, where αi is the private value of user i.
In this case, owners i's best response is choosing either to share his car all the time (si = 1) or to use his car all the time (ui = 1), depending on whether his private value αi is lower or higher than the market price. Similarly, renter i will choose to request either full use of a car (ui = 1) or zero demand (ui = 0), depending on whether his private value αi is higher or lower than the market price. Without loss of generality, suppose no users have the same private value. We label all users, including owners and renters, ranked and labeled by their private value:

Algorithm of Allocation
We now give a concrete maximum social welfare policy.
Proposition 4. The maximum social welfare price is set such that users (including owners and renters) of the first NO highest private value αi get full use of the product (ui = 1).
Note that this is the best possible arrangement for maximizing social welfare, since all users with higher private values get access to products. In this case, demand equals supply and none of the products are wasted, e.g., no cars are running empty.
Below, we also check the performance of the bound we give in Theorem 3. Specifically, we show that the bound given in Theorem 3 is tight between maximum social welfare and maximum revenue policies under linear usage payoff.
We consider the following concrete example in car sharing. Suppose NR = NO = 3. Each owner's and renter's private value is showed in Table 1. From Proposition 4, let c = 0.01,the maximum social welfare price is Psw ∈ (1.5, 2), letting renter r1 use owner o3's car and owners o1, o2 use their own cars. The trade volume is 1 · Psw ∈ (1.5, 2) and the social welfare is 11 + − 3c. Meanwhile, the maximum revenue price Pr will be 4+ such that only renter r1 can rent a car and owners o2, o3 share their cars all the time. The trade volume is 4 + and the social welfare is 9 + − 3c.
Thus, the social welfare gap between these two policies is W (Psw) − W (Pr) = 2, and the bound given by Since is infinitely small, the bound is tight.

Understanding the Social Welfare Loss Bound
We further interpret the welfare loss bound in the linear utility case. First, we rewrite the welfare loss as follows.

Power of Aggregation
Here we also discuss the platform's contribution, by comparing the system to the one without the platform. In that case, users are separated into many groups, e.g., people live nearby group together to share with each other. Such sharing groups are difficult to grow since they are mainly based on location, or relationship etc. Let J denote the set of all isolated sharing groups.
We optimistically suppose each small sharing group can reach a contract price which maximizes the social welfare inside the group. We will show that even in this case, maximizing social welfare among all users over the platform achieves higher social welfare. To do so, let the maximum social welfare price of small group J ∈ J be PJ . Then, the total social welfare can be computed by the sum of each isolated sharing group's utility, each of which is at their maximum social welfare price, i.e., Wiso = J∈J i∈J WJ (PJ ).
The contribution of a sharing platform is to bring different people into one whole group so that all renters can access to all owner's products. The following theorem guarantees that a platform can achieve a higher social welfare compared to having multiple small isolated sharing groups. Theorem 6. When users' utility functions are linear, sharing over a common platform achieves a higher social welfare than having isolated sharing groups, i.e., W (Psw) ≥ Wiso This theorem is straightforward since the policy showed in Proposition 4 gives the best possible social welfare, and separated groups can only degenerate the performance. For example, consider users showed in Table 1 again. If users are divided into two groups, group 1 includes {o1, o2, r1} and group 2 includes {o3, r2, r3}, maximum social welfare of group 1 and group 2 will be 9 + − 2c and 1 − c, which is less than the social welfare showed in 5.1.1.

Quadratic Usage Payoff
In this subsection, we study the case when each user's usage utility is a quadratic function: Under quadratic usage payoff, each renter' best response demand is b i −P 2a i , which means the total demand is a descending piecewise linear function.
Similar to the linear utility case, we present an example where the bound given in Theorem 3 is tight. Consider the following example. Let NO = NR = 1 and their ai, bi parameters are given in Table 2.
As proved in Proposition 3, we only consider the case when S(P ) ≥ D(P ). Under this setting, renter's demand o + 2uo renter r Ur(ur) = −u 2 r + ur will be 1−P 2 . As for the owner, he will have no desire to provide supply more than the demand since there is no competition. Thus, demand equals supply. Let c = 0, the owner's best responses will be to fully utilize the product, i.e., uo + so = 1. Hence, the owner's utility function is Wo = −u 2 o + 2uo + P min{D, so} and uo + so = 1. Therefore owner's best response will always be choosing so = D and uo = 1 − so.
A revenue maximizer will maximize the volume of trade, i.e., P D(P ) = P 1−P 2 , which yields the optimal revenue price Pr = 1 2 . A social welfare maximizer, instead, will maximize the total social welfare: which also gives Psw = 1 2 . As a result, both policies use the same price, which leads to the same demand and supply. The bound given in The- S(Pr ) ] = 0, which is tight in this example.

EXPERIMENT
In this section, we conduct numerical experiments to validate the results in this paper, on both generated data and real data. In addition, we also discuss some insights delivered by simulations.

Simulation Setup
In 6.2, we conduct simulation on generated data with quadratic usage payoff. A few different parameter settings are examined. All user usage utility functions Ui(ui) are set to be quadratic functions here, i.e., Ui(ui) = −aiu 2 i + biui with ai, bi > 0. This is because quadratic utilities provide both tractability and representativeness, since quadratic functions can be seen as second order approximations for concave functions under Taylor expansion. Specifically, each user's ai, bi are uniformly chosen from (0.1, 1.2) and (0, 1). In 6.3, we use real data from DiDi [5], the biggest car sharing platform in China. Data includes transaction records on the first three weeks in January, 2016, inside a Chinese city, including driver ID, passenger ID, starting district ID, destination district ID and fee on each ride. We choose to use data on Jan 8th randomly, which includes 395938 order records.
In searching Nash Equilibrium, we use best response iteration, which turns out to be a fast converging algorithm under our setting.

Supply-Demand Relationship
Here we consider the tightness of social welfare loss bound (Theorem 3) under different supply-demand relationship. That is, we set number of owner to be 100 and compare It is clearly showed in Fig.2 that when the number of renters increases, the social welfare gap decreases to zero. There are two important implications that we want to point out: (i) with the growth of renter population, the theoretical bound not only provides an upper bound of actual social welfare loss, but also an accurate estimation, (ii) when the renter population is a few times more than owners, which is common in real world scenarios, maximum revenue policy also achieves maximum social welfare.
Intuitively, this is because when the resources become scarce, a maximum social welfare policy will first satisfy the needs of renters with higher utilities, which is the same as what a maximum revenue policy will do. In other words, when resources are scarce, a revenue maximization platform run by company simultaneously guarantees maximum social welfare.

Psw and Pr
Here we compare Psw and Pr under different settings, e.g., different cost or different degree of scarcity of the products. From Fig.3(a), Psw and Pr are stable when c is low, but along with the cost increment, Psw increases quickly towards Psw and eventually becomes the same when c is high enough. Compare the results in 6.3.2 and Fig.3(a), if the cost can be chosen properly, say c = 0.3, one can both keep a low price and low wastage rate.
Different degree of product scarcity also influences Psw and Pr. When resources become scarce, as show in Fig.3(b), Psw rapidly increases towards Pr, that is because when there is insufficient supply, a maximum social welfare policy also needs to guarantee the usage of renters with higher utilities, which is the same as what a maximum revenue policy will do. Therefore, it is reasonable that when supply is abundant (Fig.3(c)), Psw becomes much lower than Pr: a maximum social welfare policy wants to let more people get access to idle resources while a maximum revenue policy still focuses on high value clients.

Results on Real Data
We also conduct experiment based on real data from DiDi. Since data are transaction records on Jan 8, 2016, we use the statistical volume of transaction at different prices to represent renters' demand under different prices. Total demand curve is fitted to exponential function D(P ) = αe −βP with and plugging it into (3), we will have exactly the exponential form of total demand mentioned above. Since (12) are based on statistical results from real data, with out loss of generality, we also assume all product owners have the same form of usage utility as (12). To balance the computation time and credibility, we scale α and Nr to α = Nr = 1919 in computation, then each renter's demand will be e −βP ∈ [0, 1] and total demand will be Nre −βP = αe −βP . Since DiDi has a minimum charge on each ride (at 10 RMB), we can reset the zero price point at the minimum charge fee.

Social Welfare Loss Under Exponential Demand
Here we want to see the social welfare loss when DiDi is maximizing revenue under exponential form of demand function listed above. We examine some practical cases: when supply are less or slightly more than demand (No = 100, 500, 1000, 2000, 2500, 3000 while Nr = 1919). Fig.5 shows that when supply is not abundant, DiDi is actually achieving maximum social welfare simultaneously when it sets the price to maximize revenue.

The Role of Cost
We study the role of cost c here. Fig 6(a) shows demand and supply under different cost, and Fig.6(b) is drawn under

Price
Owners P P r P sw P r Figure 5: Under exponential form of real demand, when supply is not abundant, maximizing revenue is actually maximizing social welfare at the same time.
quadratic usage payoff given in 6.2 for comparison. These two settings actually shows similar results. As showed in both two figures, cost c is the dominant factor in determining the lowest price to enable sharing, i.e., there will be no sharing when price is lower than c. In Fig.6(a), if we compare the supply curves of c = 5 and c = 10 (similar situation happens if we compare c = 0.1 and c = 0.3 in Fig 6(b) ), we also find that higher cost significantly suppresses sharing, e.g., drivers will choose their sharing level more conservatively if they have to pay more cost when their cars are running empty waiting for customers' order. This shows that a properly chosen cost c can help reduce redundant supply while maintaining an acceptable QoS. This is crucial to many social issues such as greenhouse gas emission, road congestion, because it is feasible to implement such a scheme: government can carefully choose a proper c, e.g., gasoline price, instead of designing very complex mechanisms.

How can DiDi do Better?
After showing the experiment results, we can also derive the possible optimal revenue that the platform can get. Since trade volume is defined as (6) and by Proposition 3, P = 1/β = 12 is the price which maximizes P D(P ), therefore maximizes revenue. The platform obtains maximum revenue when they set the price such that average charge for each ride is P plus minimum charge, under the condi-  tion that S(P ) ≥ D(P ). Meanwhile, since all data are from actual transaction, we can calculate the average transaction price by: which is exactly the optimal price mentioned above. That is, DiDi has already accurately chosen their price which could generate best possible revenue. Nevertheless, the platform still need to make sure there are enough supply at P , otherwise P will not be the correct maximum revenue price according to Proposition 3. Given 1919 renter and 1500 owners, we know from Fig. 5 that the revenue maximization price will be 12.9, which is not at the best optimal revenue price P = 12 due to the lack of supply. Actually, among all the 395938 records of order in the data we study, there are only 43648 car driver IDs, while number of customer ID is 237800. In another word, shortage of supply is now the biggest obstacle for sharing platform to obtain best possible revenue. There are many ways to augment supply, e.g., Uber adopt "surge price" [9] to raises the price at the area where supply is scarce in the hope of attracting more drivers.
However, under some circumstance where price is fixed, recruiting more altruistic product owners will be a better choice, as illustrated in 4.3. Fig 6(c) shows the impact of recruiting some actively sharing product owners, under the setting of 1919 renter and 1500 owners: with only a slightly increase of product owner's enthusiasm ( = 0.05), actual Pr can be set to the best possible revenue maximization price P = 12 so that platform will gain optimal revenue. Meanwhile, from social welfare maximizer's perspective, altruistic product owners' participation will also lower Psw and therefore increase total social welfare.

CONCLUSION
In this paper we consider the fundamental problem about how to efficiently boost sharing economy. Based on a gametheoretic model, we derive various tight bounds of social welfare loss of the maximum revenue policy compared to maximum social welfare pricing policy. We also show that the revenue maximization ensures higher sharing supply level and ensures better quality-of-service to users in the sharing system. Our numerical results also demonstrate that as the renter/owner ratio increases, revenue optimal policy is also welfare optimal. Our results provide novel insights and guidances on how to boost sharing economy towards maximizing social welfare by commercial power who chases maximum revenue.

APPENDIX
Proofs of Theorem 2 and Proposition 4 are in additional technical report [4] due the space limitation.
First we notice that an owner's strategy space is (u, s) ∈ {u ≥ 0, s ≥ 0, u + s ≤ 1} and a renter's action space is u ∈ [0, 1], both of which are compact convex sets.
Then, for renter i, it is clear that his utility function shown in (3) is continuous in S and U, and concave in ui. As for the owners, consider Equation (2). The total demand D is fixed as long as P is set, as it only involves renters. The total supply is the sum of si. Thus, owner's utility function is continuous in {s1, s2, ...si−1, si+1, ...sN o }, u.
Depending on D(P ) and others owners' supplies j =i sj, there are three cases: • D(P ) < j =i sj: the utility function for i becomes Wi(ui) = Ui(ui) + P D(P ) S(P ) si − cui − csi and we know that P D(P ) S(P ) si = P D(P ) j =i s j +s i si is concave in si. • D(P ) > j =i sj + 1: the utility function for i becomes Wi(ui) = Ui(ui) + P si − cui − csi, which is also concave in si. • j =i sj < D(P ) < j =i sj + 1, the utility function for i becomes and it is continuous. It is concave in si since each piece is concave in si and at the intersection point of these two pieces, the left derivative P − c is greater than the right derivative P D(P )(S(P )−s i ) Hence, owner i's utility function in above three cases are always continuous and concave to its own strategy (ui, si).
According to [14][10] [12], a pure strategy Nash equilibrium exists since each user's strategy space is convex and compact, and the utility function of each user is concave in his strategy, and continuous in S and U.

Proof of Proposition 1
Proof. Under Assumption 1, for concave and continuous Ui, and fixed P , renter has unique optimal demand u * i (P ) ∈ [0, 1] which maximizes his overall utility Wi(P ) given by (3). Suppose the market price is set to be some P0, we must have: . Suppose for any market price P1 > P0, we must have: . This implies u * i (P1) ≤ u * i (P0), i.e., each renter's demand is non-increasing in P , for otherwise by setting ui = u * i (P1) > u * i (P0), we get the following contradiction: For any renter i, by Assumption 1, there exists mi such that Ui(ui) < miui, ui ∈ [0, 1]. Define P no demand = maxi{mi}. Then, every renter has zero demand when P ≥ P no demand .

Proof of Proposition 2
Proof. Due to Assumption 1, each renter's demand u * i (P ) is continuous in P , so D(P ) is continuous in P . Let Pupper be the price satisfying the equation PupperD(Pupper) = c, which means even only one owner provides production sharing with no competition, he can not make positive profit. It is clear that no owner will share products at price beyond Pupper. We start by proving (ii) of the proposition, and then (i) and (iii) (ii) We first prove that there exists P < Pupper such that D(P ) = S(P ). By Proposition 1, total demand decreases to 0 when market price exceeds Pupper. Thus, there must exists some price P such that D(P − ) > S(P − ) for P − = P − δ and D(P+) < S(P + ) for P + = P + δ, here δ > 0 is infinite small. We only need to show that D(P ) = S(P ), i.e., S(P ) is continuous at the intersection point P . Suppose the opposite that S(P + ) − S(P − ) = ∆ > 0, where ∆ is some positive constant. Then, there must exists some owner i, such that s * i (P + ) − s * i (P − ) = ∆i > 0, and the effective price at P + is P + D(P + ) S(P − )+∆ . Since δ → 0, i share more at P + while the effective price is lower, which leads to contradiction. We can also show that P < Pupper. That is because since S(P ) = D(P ) > 0, we must have P < Pupper. Now we show that D(P ) ≤ S(P ) for P ≥ P . Consider owner i's utility function at any P0 ≥ P satisfying D(P0) ≤ S(P0), e.g., P0 = P (we have D(P0) = S(P0)): and the best response is denoted by (u * i (P0), s * i (P0)). Assume for contradiction that for some P1 > P0 we have D(P1) > S(P1). Then, owner i's utility function becomes: where the best response is (u * i (P1), s * i (P1)). Compare equation (14) and (15), since P1 > P0 D(P 0 ) S(P 0 ) , we must have: and from (17) we have: , plugging (16) into it we have: since P1 > P0 D(P 0 ) S(P 0 ) > 0, we have: Since i is arbitrary, we have S(P1) ≥ S(P0), and S(P1) ≥ S(P0) ≥ D(P0) ≥ D(P1) by Proposition 1, which leads to contradiction.
Therefore let P be the smallest market clearing price, P = min{Pc | D(Pc) = S(Pc)}, we have S(P ) ≥ D(P ), ∀P ≥ P .
(i)Similarly we know S(P ) is nondecreasing when P ≤ P , since the utility function at P < P for each owner is: and each one's si is nondecreasing when P increases.
(iii)It has been proved at the beginning.

Proof of Proposition 3
Proof. (i) Consider the case when the price is P0 and D(P0) > S(P0). There exists some renters who are not able to rent products since available products are not enough. Let R0 denote the set of renters who have successfully rented a product and R/R0 denotes the other renters. We know that Wi(ui) ≥ 0, ∀i ∈ R0 and Wi(ui) = 0, ui = 0, ∀i / ∈ R0, i ∈ R. According to Proposition 2, there exists some market clearing price P > P0 such that D(P ) = S(P ).
For owners, we must have each owner's social welfare Wi(ui) = Ui(ui) + P si − cui − csi increases compared to the case when price is P0.
For renters in R/R0, each renter gains non-negative social welfare increment. For renters in R0, each renter i suffers utility lost not exceeding (P −P0)u * i (P0), with u * i (P0) being his best response usage at price P0. Therefore total social welfare lost for group R0 will be less than (P − P0)S(P0). However for the owners who have served users in R0, they have at least (P −P0)S(P0) utility increment at price P due to the price increase even if they do not augments supplies.
To sum up, the social welfare at lowest market clearing price P , is higher than the social welfare at any price satisfying D(P ) > S(P ), i.e., Psw ≥ P . According to Proposition 2, we always have S(Psw) ≥ D(Psw).

Proof of Theorem 2
(1) We prove the first bullet in Theorem 2 here. Suppose for some P1 > P , we must have Σi∈RWi(P1)−Σi∈RWi(P ) < −(P1 − P )D(P1), which is the additional payment paid by renters for demand D(P1) after price changed to P1. As for the renters, they have earned additional payment A from renters: where the first term in (20) is exactly the lower bound of renters' loss, and we only need to prove that owners can not completely retrieve the loss in the second term by choosing their own usage. For owner i, we have: U (u * i (P )) + P D(P ) S(P ) s * i (P ) − cu * i (P ) − cs * i (P ) ≥ U (u * i (P1)) + P D(P ) S(P ) + si s * i (P1) − cu * i (P1) − cs * i (P1) where si = s * i (P1) − s * i (P ). Thus, U (u * i (P1)) − cu * i (P1) − cs * i (P1) − U (u * i (P )) + cu * i (P ) + cs * i (P ) ≤ P D(P ) S(P ) s * i (P ) − P1D(P1) S(P ) s * i (P1) (22) summarize (22) If S(P1) > S(P ), we show that for any owner i, si ≥ 0. Suppose the contradiction that there exists some i such that si > S(P1) − S(P ), while some j with sj < 0. Therefore, the effective price seen by j will be lower at P1, i.e., P 1 D(P 1 ) S(P 1 ) ≤ P D(P ) S(P ) , otherwise j can always increase its sharing by arbitrarily small amount beyond s * i (P ) to gain higher utility. Hence, we know that i can also see the decreased effective price and decrease his sharing supply, which leads to contradiction. Thus, we have: Similarly, if S(P1) < S(P ), we will have si ≤ 0 for all owner i. Since both the supply and effective price decrease at P1, we have P D(P )

S(P )
> P 1 D(P 1 ) S(P 1 ) . So the owner will bear the payment loss P D(P ) − P1D(P1), which can be showed similarly as above to be irretrievable by changing self use: (2)By Proposition 2, 3 and results above, we must have Pr ≥ Psw.
(3)The volume of trade can be represented as PrD(Pr) and PswD(Psw) by Proposition 3. Since Pr maximizes the volume of trade, we know that PrD(Pr) ≥ PswD(Psw).
We prove by contradiction, suppose that S(Pr) < S(Psw). Let the best response of owner i at Pr and Psw is s * i (Pr) and s * i (Psw). Since S(Pr) < S(Psw), without loss of generality, we can assume s * i (Pr) < s * i (Psw) for some i. Then, as far as i can see, the "effective" price Pr D(Pr )

S(Pr )
is greater than the "effective" price Psw D(Psw ) S(Psw ) . Under such circumstances, i will find that he can always increase its sharing supply, at least by positive δ → 0, to increase his utility, as long as the total effective price be still higher than Psw D(Psw ) S(Psw ) , since i's best response at Psw D(Psw )