1 Introduction
The prevalence of smartphones equipped with internet connectivity and GPS systems have enabled the rapid growth of platforms that allow a passenger to make a shortterm request for the service of independent drivers. When the driver’s motivation is to obtain a recurring income from these trips, such as in the platforms Uber and Lyft, this is called ridesourcing (rayle2014app). This scenario can be distinguished from ridesharing, where the driver seeks to share their trip costs by serving passengers that have a route similar to their own, as for example, in Waze Carpool (Waze) — a commercial platform that facilitates ridesharing. Some ridesourcing platforms also seek sharing as a way to reduce costs, for example, UberPool (Uber) and Lyft Line offer a smaller price by allowing the trip to be shared among multiple passengers. The InDriver (InDriver) platform resembles an auction since it allows passengers to advertise rides and prices, to which drivers can then give a counteroffer. These services are part of a new type of economic interaction, called collaborative consumption or sharing economy (hamari2016sharing).
Beyond the benefits for the participants of the trip, ridesharing is advantageous for the society as a whole, as it may reduce the congestion and pollutant emission caused by the use of personal cars in metropolitan areas by increasing the occupancy rate of vehicles and reducing the need for parking (hahn2017ridesharing)
. As an example, the metropolitan area of São Paulo, one of the largest in the world, is in a longlasting mobility crisis that was estimated to cost up to 1% of Brazilian GDP in 2012
(cintra2014custos), with congestion resulting in up to three times more emission of pollutants than the expected emission without congestion (cintra2014custos).In this paper, we propose pricing solutions that apply to both ridesharing and ridesourcing and uses the term ridesharing to generalize both scenarios, except where the difference is explicitly noted. We are particularly interested in dynamic ridesharing with trips that may be shared among multiple passengers, which incurs lower costs for passengers and society. The Dynamic Ridesharing Problem, or realtime ridesharing, is characterized in agatz_optimization_2012 by the following properties: Dynamic, the trips must be formed rapidly as new requests come in continuously; Independent, drivers and passengers are independent agents and therefore will only share a trip if that is beneficial to them; Cost sharing, participants wish to obtain a lower cost than the one incurred by traveling alone; Nonrecurring trips, single shortterm trips rather than recurring prearranged trips; Automatic, the system should require minimal operational effort from the participants.
Dynamic ridesharing involves complex routing and pricing aspects. The DialARide Problem (DARP), as defined in cordeau_branchandcut_2003, is a good routing model for multipassenger ridesharing, where passengers have pickup and dropoff locations that may not coincide with the driver’s initial location and destination while also having constraints on pick up time and travel time. Routing problems that have cost minimization as the objective have received much attention in the operations research literature (pillac2013review), for example, santos2013dynamic
develop heuristics to maximize taxi sharing in a DARP problem. The ridesharing problem with dynamic pricing brings new challenges such as balancing the number of passengers served and the profit obtained.
Trip pricing gains importance in dynamic ridesharing because the independence of the agents makes it necessary to incentivize the participation of passengers and, especially, drivers in the system. However, pricing has received less attention from the ridesharing literature (furuhata_ridesharing:_2013)
. In moments of high demand, when there are few drivers and many passengers, it is necessary to raise prices to attract drivers into the system and balance supply and demand. In this scenario cost minimization is no longer the main goal of the system’s optimization since the valuation of each passenger for the trip must be considered. Some ridesourcing platforms such as Uber and Lyft raise the price per distance in these scenarios, a policy known as
surge pricing. Current studies suggest that surge pricing achieves its goals of improving economic efficiency (hahn2017ridesharing). Still, surge pricing has been viewed negatively by consumers and regulators (cachon_surge). When the system decides for a higher price consumers might perceive that it is acting against them. In an auction however the consumers’ bids would determine the price, so we explore auctions as a form of balancing the market in moments of high demand with a price that is better justifiable to the consumer.1.1 Literature Review
Two desirable properties of auctions that are studied in auctionbased pricing for ridesharing literature are budgetbalance, which means the price paid by users is at least the driver’s cost for serving them, and strategyproofness, which means participants do not gain by misreporting their valuations to manipulate the auction.
kamar_collaboration_2009 propose a multidriver system based on a VickreyClarkGroves (VCG) (vickrey_counterspeculation_1961; clarke1971multipart; groves_incentives_1973) auction. The VCG auction is not budgetbalanced and, due to the use of heuristics for better computational performance, is implemented in a way that also is not strategyproof. kleiner_mechanism_2011 uses a strategyproof auction adapted from the secondprice auction (vickrey_counterspeculation_1961) where each driver may serve a single passenger. Considering each driver in isolation, this auction is strategyproof and budgetbalanced, as long as the driver is not allowed to choose the served passenger. Our work extends this line of research by proposing mechanisms that are multipassenger, strategyproof, and budgetbalanced.
Another line of research involves making prior assumptions on the passenger’s arrival rate and valuation distributions and then calculating prices that are optimal in expectation. sayarshad2015scalable and chen_optimal explore this type of mechanism while masoud2017using assumes a prior distribution on the passengers’ valuations to propose an optimal price for which a passenger may buy a trip from another passenger.
In recent literature, ZhaoVCG propose a multidriver and multipassenger theoretical model that uses a VCG auction with reserve prices, but they achieve budgetbalance only when the drivers make no detours, while the auctions we propose are budgetbalanced even with detours. zhang2016discounted apply a bilateral trade reduction mechanism that is strategyproof, modifying the McAfee mechanism (mcafee1992dominant) for a ridesourcing scenario. A limitation of this mechanism is that a driver may only serve one passenger at a time, and budgetbalance is not guaranteed. shen2016online propose an online ridesharing system where a passenger that enters the system is offered a price estimate to be served by an available driver, assuming a scenario where the driver supply is sufficient for passengers to be immediately served. The authors claim that the system is strategyproof however the final price may be lower than the accepted estimate, so the passenger may strategically accept a higher estimate hoping to get a lower final price. This system is in fact individually rational, that is, it guarantees that a passenger that only accepts estimates lower or equal to their valuation will not suffer a loss.
A comprehensive survey by furuhata_ridesharing:_2013classifies ridesharing as either for a single passenger or multiple passengers. It also describes four spatial patterns of increasing generality, of which the most general is detour ridesharing, where the driver may take detours and passenger pickup locations do not have to coincide with the driver’s start location and neither do passenger delivery locations have to coincide with the driver’s destination. This survey also identifies challenges for auctionbased ridesharing, among them to find a mechanism that is strategyproof, budgetbalanced, and allows multiple passengers to be served. Even in recent literature this challenge still has not been addressed.
1.2 Our contribution
In this paper, we propose auction mechanisms for singledriver, multipassenger, detour ridesharing that are strategyproof and budgetbalanced. The main proposed auction is the Weighted Minimum Surplus (WMS) auction, for which we obtain a lower bound of the maximum social welfare and profit under certain conditions. We also propose , a budgetbalanced version of the VCG auction which remains strategyproof.
We present an experimental analysis of all proposed auctions, utilizing ridesharing instances generated on realworld maps. Both the WMS and the obtain high social welfare and achieve good profit, particularly in instances with a larger number of passengers. Though these mechanisms were analyzed only in singledriver scenarios, they may serve as a basis for generalization to multidriver scenarios, which appears to be a much harder problem.
2 Auctionbased dynamic ridesharing
In the AuctionBased Dynamic Ridesharing Problem with a Single Driver and Multiple Passengers a driver with a specific destination wishes to share their trip with passengers in exchange for monetary gain. Each passenger wishes to be served, that is, transported from their pickup location to their dropoff location. To solve the problem using an auction, the system requests each passenger to bid a nonnegative value for being served. The passengers have maximum pickup time and travel time constraints. In a ridesharing scenario, the driver may have a constraint on the maximum arrival time at the destination. We focus on the singledriver problem as a step towards a generalization for the multidriver problem.
A trip consists of a route that begins at the driver’s current location, serves a set of passengers, and finishes at the driver’s destination, respecting all time constraints. A trip is represented simply by a set of served passengers, that is, if and only if is served by trip . The value is the cost for the driver to serve , representing costs such as fuel and time. We consider that the distances between locations form a metric, and therefore the triangle inequality is valid. The system must determine the winning trip to be served by the driver and the price to be charged from each passenger. Let be the set of all passengers, then the total value collected will be (noting that, in our auctions, if ).
An auction is a suitable way to solve this problem when there is a high demand since the price will rise along with the bids as an incentive for drivers to join the system. Since there are multiple desirable properties for an auction studied in Auction Theory, we focus on four properties to evaluate auction mechanisms for this problem: Being strategyproof, maximizing social welfare, being budgetbalanced and maximizing profit. The remainder of this section defines these properties.
The passenger’s bid may not be equal to their private valuation for the ride. Strategyproofness means that a passenger can maximize their gain by bidding . Therefore, a single passenger cannot manipulate the auction to their advantage by lying their true valuation for being served. Since we restrict our attention to auctions that are strategyproof we refer to or interchangeably.
Let be set of trips, or alternatives, from which the winning trip is chosen. The social welfare of an alternative is the valuation of the served passengers minus the cost for the driver, that is, . An auction maximizes social welfare if . Budgetbalance is achieved when the total price paid is at least the cost for the driver, that is, and the driver never suffers a loss from serving the passengers. Profit is an important goal both for drivers and for the organization running the system and is defined as . To maximize profit is to maximize the surplus payment after paying the driver cost. In the next section, we propose an auction that is strategyproof, budgetbalanced, and achieves good social welfare and profit in experimental results.
3 Weighted Minimum Surplus auction
In this section, we propose the Weighted Minimum Surplus (WMS) auction for a singledriver, multiple passenger setting which models the situation where a driver enters a system where they are the only driver available to multiple waiting passengers. We prove this auction to be budgetbalanced, which is desirable to the driver as they will never lose money from a trip, and strategyproof which is desirable to the passenger who can maximize their gain simply by bidding the true valuation , instead of strategizing to improve their gain.
3.1 Winning trip and price
For the auction to be budgetbalanced it must be ensured that the cost of the winning trip will be covered by the prices paid. To achieve this, from the valuation of each passenger , the passenger cost is reserved to pay part of the cost of the driver if is served. The exact way to determine is a choice to be made by the auction designer. For strategyproofness to hold it is required that does not depend on nor on the choice of winning trip. For example can be the cost of going from passenger’s pickup location to their dropoff location.
The surplus of passenger is the difference between the passenger’s bid and the passenger’s cost. Only passengers with can be served by the auction since, otherwise, their valuation would not be sufficient to cover their cost.
A trip will be considered feasible if for all and the passenger costs are sufficient to cover the cost of the trip, that is . The set of alternatives for the auction (from which the winning trip is chosen) is composed only of feasible trips.
Let the minimum surplus of a trip be defined as , then the weighted minimum surplus of is . The winning trip is given by the allocation function where . Ties are broken by some criteria that does not depend on bids.
To define , first we must define the value and the alternative . Given a passenger let . If , then is simply the trip of maximum cardinality such that and . In general, is defined for any as the set of maximum cardinality among the sets such that , and
(1) 
In case of an equality in Equation (1), must also be such that it wins a tie against any for which . If no such that fulfills all the requirements exists, then if let . Given and , the price paid by is defined as
(2) 
if , and otherwise. By Lemma 1, if , then exists and is welldefined. The term is used to pay the driver’s cost while the value of the second term may be considered entirely as profit to be shared between the driver and the organization that manages the system.
Lemma 1.
If there is such that for some , then exists.
Proof.
We prove the contrapositive: If there is no then for any . By definition, we have that does not exist if and only if and for all
(3) 
We must show that this implies . Considering more elements cannot increase the minimum surplus of a set, so from (3), we have for all with . Since is the winner we know that . Joining the inequalities, we have that for any such that , from where we conclude that . ∎
The price in Equation (2) is defined so that it is a threshold for . If is above the threshold then wins and pays that threshold, otherwise does not win and pays nothing. This is called the critical value which is analogous to the secondprice in a Vickrey auction. This will be a crucial point in the proof that the mechanism is strategyproof, which is given in Section 3.2.
3.2 Example
Here we illustrate an instance of the Weighted Minimum Surplus auction. The input graph is presented in Figure 1 and consists of a driver with a start and end position and four passengers. The trips with highlighted paths are in red, in green, in blue and in yellow with multiple arrows between two nodes indicating an edge being used by multiple highlighted trips. Table 1 shows the input bids, passenger costs and trip costs. We can verify that these four trips are feasible since for all of , , and . Other feasible trips are omitted because they do not affect the outcome of the auction, so without loss of generality, the set of alternatives is .
The winning trip is since for . Thus, passengers and are served, and their price must be determined. First, we calculate which depends on and . Since is the only alternative that does not contain , trivially . By definition is trip of maximum cardinality such that and . Therefore, and . For , we need do compute and . The secondhighest weighted minimum surplus is that of and , therefore . The only trip such that and is itself, therefore . Finally we have . In this example it is interesting to note that even though passenger had a higher bid than passenger and both had the same cost, the price was higher.


3.3 Strategyproofness
In this section, we build theoretical results that are then used in Theorem LABEL:theorem:strategy to show that the WMS auction is strategyproof.
The objective of a participant in the auction is to maximize their utility. For a served passenger the utility is , the private valuation for the trip minus the price paid . The utility of passenger if is since and there is no value in not being served. A rational passenger will choose a bid such that the utility is maximized. An auction is strategyproof when making always maximizes the utility of a passenger.
We see that in the WMS auction the passenger expresses the private valuation in a single value for winning since the valuation for losing is always zero. So we may say the WMS auction has a singleparameter domain. For discussion on mechanisms with a singleparameter domain see nisan_algorithmic_2007.
The critical value
, given a vector of valuations
by other passengers and an allocation function , is such that for any if then and if then . An allocation function is allocation monotone if a passenger that wins when reporting a valuation also wins when reporting any higher valuation, considering that and all other bids are held constant. Therefore a passenger cannot go from winning to losing by raising their bid.To show strategyproofness, we use the characterization stated in Theorem LABEL:characterization. Proof of this characterization can be found in Theorem 9.36 of nisan_algorithmic_2007.
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