Bike sharing is a new transportation mode with many benefits in offering convenience and flexibility as well as lowering economic cost. By 2015, more than 7000 bike sharing systems have been deployed around the world . However, the flexibility of bike sharing systems, in particular the “anywhere-parking” convenience, brings forth a serious issue of imbalance between the distribution of bike supply and demand. This leads to many users being unable to find a bicycle nearby when they need it, and ultimately affects company revenue adversely. Hence there is an urgent need to rebalance the supply and demand by repositioning the bicycles, which we refer to as a bike rebalancing problem.
There are two approaches to solving this problem. One is to relocate bicycles by the staff of the bike-sharing company or platform, for example using trucks. This involves route planning and typically uses linear programming techniques, as has been extensively studied with static models[14, 17] and dynamic models [7, 3]. However, this approach is costly and not eco-friendly in terms of carbon footprint.
Another approach is to design incentive mechanisms to motivate users to reposition bicycles at platform-desired locations for rebalancing supply and demand. This falls under the research area of crowdsourcing  which takes advantage of the power of crowd to complete tasks that are otherwise difficult . With this idea, ghosh2017incentivizing ghosh2017incentivizing proposed a solution that generates repositioning tasks together with the use of bike trailers, and pays users using the Vickrey-Clarke-Groves (VCG) mechanism. singla2015incentivizing singla2015incentivizing introduced a crowdsourcing method that offers users monetary incentive for parking their bikes at recommended locations.
However, most of existing mechanisms for this problem have not considered the predicted revenue (or value) of a repositioning task. Such predicted values are made available in recent years, due to the prosperity of deep learning technique which can well predict the bike demand in such systems[21, 10, 23]. With this predictive power, the expected revenue or value of a repositioning task can be easily obtained. Therefore, the platform would assign a task to a user only if the payment to the user does not exceed the predicted revenue. This constraint is akin to the “reserve price” in forward auctions which are used to promote revenue. In our problem, it is used to better control the payments and thereby promote the profit of the platform. This constraint is generally overlooked in prior work on bike sharing or bipartite matching [2, 5, 19], making them no longer applicable. For instance with classic budget feasible mechanisms such as , the payment to a winning user is determined by the bids of other users and/or the budget. But in our case, this payment is invalid if it is above the value of the task assigned to the winning user.
In this paper, we propose a truthFul and budgEt feasIble incentive mechanism with predicted Task rEvenue (FEITE) as the solution to the bike rebalancing problem, which we model as a reverse auction. In this auction, users bid for repositioning tasks and the platform determines task allocation and user payments. While we show that commonly used auctions and pricing mechanisms do not apply to our case, we specifically show that FEITE solves the problem and satisfies a number of desired economic and computational properties. We also prove two important theoretical guarantees backed by our mechanism. The performance of FEITE is then evaluated via simulations using a real industrial dataset obtained from a large bike-sharing company.
Our main contributions are summarized as follows:
We model the bike rebalancing problem under a crowdsourcing framework using an reverse auction model. Importantly, we incorporate a bipartite graph into the auction - with a payment constraint - to determine the allocation rule and the payment rule for the auction mechanism. This also implies that a bridge is set up between bike demand prediction and incentive mechanism design.
We propose a mechanism FEITE which satisfies desired properties including incentive compatibility, budget feasibility, individual rationality, and computational efficiency. Notably, as is novel, we achieve this by combining Myerson’s Lemma and a greedy weighted maximum matching technique.
We prove two important theoretical guarantees: When the budget is tight, our mechanism ensures that the platform revenue is no less than its budget, under a practical large-market assumption. When the budget is sufficient, FEITE achieves a 2-approximation ratio as compared to the optimal solution which maximizes revenue; in addition, we show that the lower bound of the approximation ratio is which is rather close to our ratio. Putting in practice perspectives, we provide a guideline of how to use this result to set company budget in real dynamic systems.
We evaluate the effectiveness of our mechanism via extensive experiments using a real dataset from a large bike-sharing company. The results show that our mechanism outperforms other benchmark mechanisms in terms of revenue and profit.
Consider a dynamic bike sharing system in which there is a batch of users who have hired their bicycles and have not parked them (at the end of the hire) yet. There is also a set of discrete locations111We do not consider continuous locations because, nowadays, many countries such as Singapore and China have stipulated municipal regulations that shared bikes must be parked at designated locations rather than arbitrarily. as bicycle parking lots, where each location can accommodate multiple bicycles but is conceptually considered a point on a map. We assume that the destinations of each user is known to the platform when she hires a bicycle, since it can be either reported by users as is a practice adopted by some companies (e.g., Hellobike in China), or predicted using historic information . The platform aims to incentivize the users to park their bicycles at system-desired locations for rebalancing the supply and demand of bicycles. Each user has a maximum relocation range , out of which they would not accept a repositioning task; in other words, is the maximum “extra mile” they are willing to relocate.
Similar to the concept of first come first served flow in 
, we assume that a bicycle that is parked earlier will have a higher probability to be hired than a bicycle parked later. As such, each bicycle has a different probability of being hired even at the same location. Accordingly, we define a repositioning taskas “park at location as the -th bicycle”, which is associated with an expected revenue . As mentioned above, can be derived from the prediction of bike demand at this location [21, 10].
To address the task assignment problem, we construct a bipartite graph , where the left nodes are the set of users , and the right nodes are the set of tasks . Since there are maximum possible tasks at each location, there are totally tasks in . The set is the edges connecting users and tasks and there is an edge between and if location is within the maximum relocation range of user . For notation simplicity, henceforth we use to denote a task when there is no ambiguity.
Each user has a relocation cost which is a private value only known to user . She bids for a task with a claimed cost which is not necessarily equal to . Each task has an expected revenue (or “value” as we use interchangeably) for the platform. We aim to design an incentive mechanism that consists of an allocation rule and a payment rule, where the allocation rule specifies which task is allocated to which user, i.e., the matched pairs , and the payment rule specifies a payment for each matched user . The utility of a user is defined as . In addition, the platform has a budget for the rebalancing process, and the overall payment should not exceed the budget. As are desired, we want to design a mechanism that satisfies the following properties:
Incentive Compatibility: a user can only maximize her utility by bidding truthfully, i.e., . This property is also known as truthfulness or strategy-proofness.
Individual Rationality for both users and platform: the payment to each winning (i.e., matched) user should be no less than her cost, i.e., ; the payment for each matched task should also be no more than its value, i.e., , which is similar to the term reserve price in the theory of mechanism design.
Budget Feasibility: the overall payment should be no more than the budget, i.e., .
Computational Efficiency: the mechanism should terminate in polynomial time.
Our objective is to maximize the platform revenue , where is denoted as the set of matched user-task pairs. Thus, the problem of revenue maximization from task allocation can be formulated as
where the first two constraints indicate the individual rationality for platform and users, and the third one indicates the budget feasibility.
We also evaluate platform profit which is defined as
and will be compared with revenue in our experiments.
Infeasibility of Existing Mechanisms
In this section, we show that some widely used mechanisms are not feasible for the bike rebalancing problem.
This is a classical mechnism that is strategy-proof and maximizes social welfare. However, it does not guarantee budget feasibility as required in our case. Proof by counter-example: see Figure 1 (a), where the platform has budget and there are two users both with a small cost ; the two tasks both have value and there are two edges . The VCG mechanism will output matching and the payment for each user is 1. Thus, the overall payment exceeds the budget and hence the mechanism is not budget feasible.
Proposed in , this is another well-known mechanism yet is budget feasible. It greedily allocates user-task pairs with the highest ratio of , but does not consider the individual rationality of the platform (i.e., ). If we adapt the mechanism by adding the constraint into the payment rule, it will no longer guarantee incentive compatibility. To illustrate this, see Figure 1 (b), the platform has sufficient budget, and there are two users with costs , two tasks with values , and three edges . The adapted mechanism outputs match , and . However, if user misreports her cost to be a small number , she will be allocated task and her payment becomes , making her better off.
The third method is to choose a subgraph with and then use the optimal (i.e., maximum) matching (that maximizes platform revenue) to allocate tasks, where the payments to winners are set to the values of their matched tasks. Not only does this mechanism has zero profit for the platform, but more importantly, it is also not truthful. To see this, consider Figure 1 (c), where the budget is sufficient, and there are two users with costs , three tasks with values , and four edges . The optimal matching will output match since it achieves maximized revenue and the payments are and . However, if user untruthfully bids a cost of 3 instead of 1, then she will be assigned task 2 instead of 3 because task value must be no less than the cost. Therefore, she will receive a higher payment than bidding truthfully. Therefore, incentive compatibility is violated.
In fact, we will prove in Theorem Theorem that there is no optimal mechanism that can satisfy the four properties simultaneously. Therefore, we propose an approximate mechanism (i.e., FEITE) in this paper.
Mechanism Design of FEITE
Inspired by , in this section, we present our proposed mechanism FEITE for the bike rebalancing problem. We first introduce a notion called right-perfect matching in a bipartite graph.
A right-perfect matching in a bipartite graph is a matching with size , where is the set of right nodes in graph .
In other words, we say a bipartite graph has a right-perfect matching if all tasks in the graph on the right can be matched to a user on the left. It is easy to see that a right-perfect matching is also a maximum matching of a bipartite graph.
The key idea of FEITE is to maintain a subgraph that always has a right-perfect matching. FEITE sorts all the tasks and users together in decreasing order of their values (costs), and then iterates over this sorted list. The mechanism tries to include more tasks with high values in and delete more users with high costs from it until the budget is exhausted or all elements are processed.
The complete pseudo-code of the mechanism FEITE is presented in Algorithm 1. In the mechanism, we maintain a variable of the remaining budget which is initially set as . Once a user is matched with payment , we update as . In addition, we update a decreasing global price during the algorithm process. Intuitively, for each element in the iteration of the sorted list, if it is a task , the task and its affiliated edges (the connected user should be unmatched) will be added to if two conditions are satisfied after adding them: 1) still has a right-perfect matching and 2) the upper bound of the payment for all tasks in (i.e., ) is below the remaining budget. If task is added to , we update the global price as . If the element is a user , then if can maintain a right-perfect matching after discarding (i.e., is not critical for the right-perfect matching of ), we then remove her from , in this case, we also update the global price as . Once the subgraph is changed (either a task is added or a user is removed), we check if there are critical users for the right-perfect matching of , if so, for a critical user , we allocate task in to if has a right-perfect matching. The payment is set as the global price at this step, and then we update the remaining budget .
A walk-through example of FEITE is given in Figure 2.
Analysis of FEITE
The mechanism FEITE satisfies incentive compatibility.
Since each user has only one private value (i.e., cost), this is a single-parameter problem and hence we can use the Myerson’s lemma:
In single parameter auctions, for a normalized mechanism , where is the allocation rule and is the payment rule, is incentive compatible iff it satisfies:
Monotone allocation rule: , if , then implies for every ;
Threshold payment rule: payment to each winning bidder is .
First, we prove the monotone allocation rule, i.e., once user is matched by bidding , she must be matched by bidding . Let be the element that updates the global price as . When bidding , we use to denote the graph after is processed (i.e., the step that is updated as ), and has same definition while in the case of bidding . It’s obvious that the algorithm process before is updated as is not affected by the bid of . Therefore, we have , and user is also critical for the right-perfect matching of , hence the monotone allocation rule is satisfied.
Next, we prove the threshold payment rule, that is, if user bids any cost larger than the payment , she will not be matched, otherwise, she will be matched with a task. If , it can be observed that the value of all tasks in is higher than , thus the edges of will not be deleted because . Therefore, similar to the proof of the monotone allocation rule, we have that , and user is still critical, so she will be matched. If , user will be considered before the element . However, we can observe that, in the steps before element is processed, user is either not added to or is not critical for the right-perfect matching of , otherwise she will be matched before element . As a result, we have that user will be discarded if , and the threshold payment rule is satisfied.
The mechanism FEITE is budget feasible.
We can observe that once a task with value is added to , the payment to any user in is no more than , since the global price is non-increasing. Let be the last task added to , the remaining budget before is added, and the number of tasks in before adding . We have that
which concludes the proof.
The mechanism FEITE is individually rational for both users and platform.
We need to prove that when matching a user with a task , the payment to user should be larger than and smaller than . It can be observed that, at any step during the algorithm process, the global price is always smaller than the values of all tasks in , and any tasks with lower value should not be considered until this step. Similarly, is also always larger than the costs of all users in , otherwise the user with higher cost should be either discarded or matched before. Therefore, we have that for any matched pair , , which concludes our proof.
The mechanism FEITE satisfies computational efficiency.
Our proposed mechanism FEITE is an incentive compatible, budget feasible, individually rational, and computational efficient mechanism.
Theoretical Guarantee of Revenue
To show the theoretical guarantee of FEITE on revenue, we first introduce the large market assumption.
Assumption (Large Market Assumption)
We assume and for each user and each task .
Intuitively, it’s assumed that each individual user or task is negligible compared with the budget. This assumption is widely adopted in previous work [19, 1] and it is practical in real world as the revenue of a single ride is indeed very small.
Next, we prove the theoretical guarantee under tight budget.
Under the large market assumption, we have if the budget is tight.
Let be the first task that is discarded because of budget constraint, the last task added to before , and () the number of tasks in before considering (). Assume that the set of tasks allocated between considering and is , and the total payment for them is . We have and Since the payments to users in are all between and , we can get that
Combining the above inequations, we have
and further we obtain
where the first inequation is because of the individual rationality of both platform and users and the last approximate equation is due to the large market assumption.
Before we prove the theoretical guarantee under sufficient budget, we first demonstrate that a greedy algorithm as shown in Algorithm 2 has an approximation ratio of 2, i.e., it can achieve at least half of the optimal revenue under sufficient budget.
Algorithm 2 is a 2-approximation algorithm if the budget is sufficient.
Let be the optimal matching of maximum revenue, and be the matching of the greedy algorithm. For each task , let be its matching user in . Given that is not matched in Algorithm 2, we have has been matched with a task with higher revenue. Note that all are included in , so we have that
Then we can get
Next, we prove the following lemma by showing that the allocation of our mechanism coincides with a particular run of the greedy algorithm.
The mechanism FEITE is a 2-approximation mechanism if the budget is sufficient.
Since the budget is sufficient, we know that no task is discarded due to the budget limitation. We denote the matching in our mechanism as . It’s assumed that this matching is produced as following: For each task in decreasing order of , if , we allocate task to its matched user in , otherwise the task is skipped. Now we prove that, in the greedy algorithm, this process can also happen.
It’s obvious that for task , we can assign task to its matched user in in the greedy algorithm, thus we only need to prove that for each task , when we consider it in the greedy algorithm, there is no unassigned user that has an edge to .
Next, for contradiction, assume that we can find such an unmatched user that has an edge to when processing in the greedy algorithm. Then in our mechanism, when considering , there will be a right-perfect matching for graph , i.e., . Thus, the task will be added to . Note that in our mechanism, any task added to will end up being matched. This contradicts with our assumption and hence the lemma is proved.
To understand how “good” the approximation ratio of 2 is, next we prove that the lower bound is at least .
There is no mechanism that satisfies incentive compatibility and individual rationality can achieve better than -approximation when the budget is sufficient.
We prove the lemma with a concrete counter example. Assume there exists a mechanism that can achieve an approximation ratio better than . Consider case 1 where there are two users , three tasks and 4 edges . In addition, we have that , where is a small positive number. We can observe that the optimal matching should be which achieves revenue of , we now prove that any mechanism that satisfies the above properties can achieve total revenue of at most .
First we consider case 2 where the only difference with case 1 is that and hence the edge has to be deleted. In case 2, mechanism can only output the matching , otherwise, the approximation can be at least. Moreover, due to individual rationality, .
Then we consider case 1, in the output matching of , if user is matched with task , the payment is at most and the utility of user is at most 0. If user misreports cost of , it becomes case 2, and as stated above, the utility of user can be at least . Thus, for incentive compatibility, mechanism has to allocate task to user and pay at least . Hence the output matching of can achieve revenue of at most , and the approximation ratio limit is when .
Practical Implication: We now show how those results can be used when setting budget in a real dynamic system. The platform can set sufficient budget in the first time and obtain a revenue , then based on Theorem Theorem, we know the optimal revenue is at most . After that, the platform is able to set a tight budget , and Theorem Theorem guarantees that it achieves at least of the revenue of optimal solution. This way, the platform can control the budget while there are not a lot of revenue loss.
We conduct simulation using a real-world dataset obtained from a large bike-sharing company in China called Mobike. The bike riding data cover regions of Beijing with each region being 0.6km 0.6km, and are dated from May 10th to 14th, 2017. With the same distribution of destinations in this dataset, we build a simulator which can randomly generate users’ destinations. In the experiments, we set the number of users , and test different location numbers . The cost of each user
is drawn from uniform distribution overwhere . The value of a task is calculated as the difference between the Kullback-Leibler (KL) divergences  before and after fulfilling the task, similarly to previous work [16, 13]. Because of the space limitation, we refer the authors to  for concrete calculation of the task value. The acceptable range is set as 300m and 600m, respectively. We also test the budget of 50 and 500 where 500 is sufficient while 50 is not. In addition, we conduct each experiment 10 times and take the average.
We compare FEITE with the following mechanisms:
APP-OPT: As the budgeted matching problem is an NP-hard problem (it can be reduced to the knapsack problem), we cannot give an optimal allocation as benchmark. However, if the budget is tight and the large market assumption holds simultaneously, the following strategy is approximately an optimal mechanism: consider edges in decreasing order of , match user and task if they are not ever matched before, and pay user exactly her bid. The process stops until the budget is exhausted or there are no more edges. This mechanism achieves the maximum revenue under the above two assumptions but it’s not truthful.
Greedy: The greedy mechanism considers users in increasing order of their bids and allocates the task with the highest value as long as it is higher than the bid of the user. Once a user is matched, the price for all winning users is updated as the bid of her next user (whose bid is higher). The mechanism stops once a user cannot find a feasible task or the overall payment is above the budget, and pays all the winning users the bid of this unmatched user. If all users can be matched, we don’t match the last one, and set her bid as the price for all other users. This greedy mechanism is incentive compatible but cannot provide any theoretical guarantee.
Surge: Surge pricing is widely used in practice and it’s an effective way to promote the revenue of platform . We adopt a simple version of surge pricing here. Consider users in increasing order of their bids and allocate the task with the highest value to them once is higher than the bid of the user, where is a surge factor and is the value of the task. The user is paid . The algorithm also terminates until the budget is exhausted or there are no feasible users. In the experiments, we set . It can be easily shown that the Surge mechanism is not truthful.
The experiment results for different acceptable extra distance and are given in Figure 3 and Figure 4, respectively. In Figure 3, we show the revenue and profit of platform under different parking location number and different budget . In general, as increases, revenue and profit both increase, because of more right nodes in the bipartite graph. When the budget is tight (), we make the following specific observations. The revenue and profit of our mechanism are constantly higher than Surge and Greedy. When is less than 30, as the bipartite graph is small, the budget is enough and our mechanism achieves nearly the same revenue as APP-OPT. When is larger than 30, the budget will be exhausted and the revenue (and hence profit) of Surge mechanism stops increasing because it always has a constant ratio between payment and revenue. However, FEITE can output a better matching when the bipartite graph is larger and it shows much better performance over Surge and Greedy on both revenue and profit. When the budget is sufficient, our mechanism is not as good as APP-OPT but outperforms the others. As APP-OPT pays users exactly their costs, the mechanism is budget-saving but not practical in application because users will bid higher costs.
In Figure 4, is larger so more edges appear in the bipartite graph. In this case, the performance of APP-OPT, FEITE and Surge are similar to Figure 3, but Greedy shows both much better revenue and profit. This is due to the fragility of Greedy mechanism as a single user can determine the termination of the mechanism. Thus, when , the mechanism stops quickly because of the lack of edges, and when , the termination is delayed. To show this, we test another 100 rounds of
for both Greedy and FEITE, and the maximum revenue of Greedy is 161.3, the minimum revenue is 19.7, and the variance is 648.1, while the data for FEITE is 127.2, 98.8 and 32.2 respectively. The profit shows similar results. Therefore, although Greedy can sometimes be slightly better than our mechanism, it is fragile and has much larger fluctuation.
In summary, our mechanism FEITE performs well on both revenue and profit, on top of its several desirable theoretical properties.
Conclusion and Future Work
In this paper, we have proposed an incentive mechanism to solve the bike rebalancing problem with predicted task value for bike sharing systems. It has desired theoretical properties as it satisfies incentive compatibility, budget feasibility, individual rationality and computational efficiency. In particular, we gave two theoretical guarantees under different budget constraints, and we showed how it can be applied in real dynamic systems. Its practical performance was evaluated using simulations based on real-world data, and the results demonstrate its superiority in terms of both revenue and profit. In our future work, we will explore tighter lower bounds and upper bounds for the problem. In addition, we plan to extend our algorithm into a real-time decision-making mechanism while maintaining the theoretical properties, and conduct pilot studies in actual cities.
This work was supported in part by Science and Technology Innovation 2030 –“New Generation Artificial Intelligence” Major Project No. 2018AAA0100905, in part by China NSF grant No. 61972252, 61972254, 61672348, and 61672353, in part by Joint Scientific Research Foundation of the State Education Ministry No. 6141A02033702, in part by the Open Project Program of the State Key Laboratory of Mathematical Engineering and Advanced Computing No. 2018A09, and in part by Alibaba Group through Alibaba Innovation Research Program. The opinions, findings, conclusions, and recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the funding agencies or the government.
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