A Differentially Private Incentive Design for Traffic Offload to Public Transportation

06/04/2019
by   Luyao Niu, et al.
Worcester Polytechnic Institute
0

Increasingly large trip demands have strained urban transportation capacity, which consequently leads to traffic congestion. In this work, we focus on mitigating traffic congestion by incentivizing passengers to switch from private to public transit services. We address the following challenges. First, the passengers incur inconvenience costs when participating in traffic offload due to delay and discomfort, and thus need to be reimbursed. The inconvenience cost, however, is unknown to the government when choosing the incentives. Furthermore, participating in traffic offload raises privacy concerns from passengers. An adversary could infer personal information, (e.g., daily routine, region of interest, and wealth), by observing the decisions made by the government, which are known to the public. We adopt the concept of differential privacy and propose privacy-preserving incentive designs under two settings, denoted as two-way communication and one-way communication. Under two-way communication, we focus on how the government should reveal passengers' inconvenience costs to properly incentivize them while preserving differential privacy. We formulate the problem as a mixed integer linear program, and propose a polynomial-time approximation algorithm. We show the proposed approach achieves truthfulness, individual rationality, social optimality, and differential privacy. Under one-way communication, we focus on how the government should design the incentives without revealing passengers' inconvenience costs while still preserving differential privacy. We formulate the problem as a convex program, and propose a differentially private and near-optimal solution algorithm. A numerical case study using Caltrans Performance Measurement System (PeMS) data source is presented as evaluation.

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I Introduction

The trip demands of the increasing urban population have created a shortage in transportation capacity due to relatively slower development of transportation infrastructure [1]. Several strategies have been proposed to overcome this challenge [2]. A straightforward approach is to increase the capacity of transportation infrastructure by adding more lanes to existing highways and creating new highways in identified congestion areas. However, this approach is expensive and time consuming. Other than physically changing the existing transportation infrastructures, operational improvements aim at utilizing existing transportation infrastructures in a more efficient way. Transportation management centers (TMC) [2, 3] improve highway operations by ramp metering [4], managing traffic flow when incidents happen [5], and reporting real-time information about weather and route conditions [6]. The last category focuses on demand side management. The intuition of the approaches belonging to this category is to serve more passengers using less vehicles so that the congestion can be mitigated. Typical strategies include ridesharing [7, 8], alternative travel options (e.g., bicycle/pedestrian) [9], and pricing scheme design [10, 11, 12].

In this work, we focus on demand side management. In particular, we investigate how to incentivize passengers to switch from private transit service (PVTS) to public transit service (PBTS). Before analyzing how to incentivize the passengers, we first characterize PVTS (e.g., taxis and ride-hailing service) and PBTS (e.g., buses and subways). PVTS provides passengers high quality of service (QoS) with few or no stop, and high flexibility in terms of route selection and travel time. However, PVTS normally charges high fares. PBTS provides a group of passengers a shared ride following a fixed route and schedule time table. Although PBTS sacrifices QoS, passengers incur less fares.

By the nature of PVTS and PBTS, we observe that if more passengers are willing to switch from PVTS to PBTS to satisfy their trip demands, the number of operating vehicles decreases and therefore congestion can be mitigated. However, there are several difficulties to incentivize passengers to switch from PVTS to PBTS. First, passengers incur inconvenience costs while changing their transit behavior by switching from PVTS to PBTS. The inconvenience cost is due to several factors including reduced QoS and delay of arrival time. Thus the passengers need to be reimbursed. Moreover, the inconvenience cost, which varies from passenger to passenger depending on the preference of each passenger, is unknown to the government and the passengers are not willing to reveal their inconvenience cost functions since the inconvenience cost functions contain personal information such as region of interest and daily routine. Finally, the passengers have privacy concerns when participating in traffic offload. A privacy sensitive passenger might lie on its inconvenience cost when revealing it due to privacy concerns to achieve privacy guarantees at the expense of sub-optimal utility. Therefore, the government needs to design a mechanism which not only incentivizes the passengers to switch from PVTS to PBTS, but also addresses their privacy concerns.

In this paper, we focus on the mechanism design to incentivize the passengers to switch from PVTS to PBTS. We make the following contributions.

  • We formulate the problem of mechanism design to incentivize passengers to switch from PVST to PBST under two settings, including two-way communication between the government and passengers and one-way communication from the government to the passengers. We present an adversary model under each setting that characterizes passengers’ privacy concerns.

  • We model the interaction between the government and passenger under two-way communication using a reverse auction model. We formulate the problem as a mixed integer linear program. We propose an efficient approximation algorithm to reduce the computation complexity.

  • We prove that the proposed mechanism design under two-way communication achieves approximate optimal social welfare, truthfulness, individual rationality, and differential privacy.

  • For the one-way communication, we formulate the problem as an online convex program. We give a polynomial-time algorithm to solve for the mechanism design. We prove that the proposed mechanism is differentially private and Hannan consistent.

  • We present a numerical case study with real-world trace data as evaluation. The results show that the proposed approach achieves individual rationality and non-negative social welfare, and is privacy preserving.

The remainder of this paper is organized as follows. We discuss the related works in Section II. In Section III, we present the problem formulation under two-way and one-way communication settings, respectively. We present the proposed incentive mechanism design in Section IV for the two-way communication setting. Section V gives the proposed solution for one-way communication setting. The proposed approaches are demonstrated using a numerical case study in Section VI. We conclude the paper in Section VII.

Ii Related Work

In this section, we present literature review on intelligent transportation system and differential privacy. Significant research effort has been devoted to achieving intelligent and sustainable transportation system. Planning and routing navigation problems have been investigated by transportation and control communities [13, 14, 15, 16, 17, 18]. These works focus on finding the optimal path for each vehicle, and are not sufficient to address the imbalance between trip demands and transportation capacity. Various approaches have been proposed to improve operation efficiency of existing transportation infrastructure, among which vehicle balancing [19] has been extensively studied for bike sharing [20] and taxis [21]. Metering strategy has also been investigated [4]. Different from works mentioned above, this paper focuses on the demand side management.

In the following, we discuss related works on demand side management. Alternative travel facilities have been implemented all over the world, e.g., bike sharing system [22]. Moreover, ridesharing system and the associated ridesharing match system has been investigated [23, 24], which grouped passengers with similar itineraries and time schedules together to reduce the number of operating vehicles. Most of these works focus on taxis and ride-hailing services such as Uber and Lyft, and ignore the potential from PBTS. Pricing schemes have been proposed to reduce the number of operating vehicles at peak hour [10, 11, 12]. These works focus on private cars and ignore the PBTS. This paper fills the gap in existing literature by considering the switch from PVTS to PBTS. Researchers have identified the factors that prevent passengers from PBTS [25, 26]. However, to the best of our knowledge, it has received little research attention on how to incentivize passengers to switch from private to public transit services.

Mechanism design has recently been used in engineering applications such as cloud computing. In particular, Vickrey-Clarke-Groves (VCG) mechanism [27] is widely used to preserve truthfulness. However, truthful communication raises the privacy concerns. To address the privacy issue, we adopt the concept of differential privacy [28, 29, 30]. Mechanism design with differential privacy, such as exponential mechanism, has been proposed [31, 32, 30, 33]. However, they are not readily applicable to the problem investigated in this paper because the presence of inconvenience cost functions leads to violation on individual rationality. Moreover, the computation complexity in exponential mechanism is addressed in this paper.

Trial-and-error implementation for toll pricing has been proposed in [34]. Different from [34], we consider a closed-loop Stackelberg information pattern, and compute the optimal incentive price. To solve the problem under one-way communication setting, we adopt the Laplace mechanism to preserve differential privacy [30]. This paper extends our preliminary conference version [35], in which two-way communication setting is studied. We extend the preliminary work by investigating the one-way communication setting.

Iii Problem Formulation

In this section, we first give the problem overview. Then we present the problem formulations under two settings, denoted as two-way communication and one-way communication. We finally discuss the privacy model.

Iii-a Problem Overview

Let denote the set of origin-destination (OD) pairs that will require traffic offload in the near future time horizon , with each OD pair requiring amount of traffic offload at time . Let be the set of passengers. At each time , any passenger that participates in traffic offload for any OD pair receives revenue issued by the government, where is the amount of traffic offload that passenger can provide for OD pair at time . Passenger also incurs inconvenience cost if it switches from private to public transit service due to discomfort and time of arrival delays. We remark that each passenger is physically located close to some OD pair at each time . Hence each passenger is only willing to participate in traffic offload for one OD pair that is physically close to its current location. For other OD pairs , we can regard the associated inconvenience cost approaches infinity. We assume that the inconvenience cost function is continuously differentiable, strictly increasing with respect to for all , and convex with for all and . The utility of the passenger at each time step is given by

(1)

In this work, we assume the passengers are selfish and rational, i.e., the passengers selfishly maximize their utilities and never accept negative utilities.

Iii-B Case 1: Interaction with Two-way Communication

In this subsection, we present the problem formulation under two-way communication setting. In this case, the interaction between the government and the set of passengers is captured by a reverse auction model.

The passengers act as the bidders. Each passenger can submit a bid to the government at each time , where element contains the amount of traffic offload that passenger can provide and the associated inconvenience cost. Note that is the inconvenience cost claimed by passenger , which does not necessarily equal the true cost .

The government is the auctioneer. It collects the bids from all passengers, and then selects a set of passengers that should participate in traffic offload. In particular, the government computes a selection profile , with each element if passenger is selected and otherwise. If a passenger is selected by the government for OD pair , an associated incentive is issued to passenger .

The utility (1) of each passenger at time is rewritten as

(2)

The social welfare can be represented as

(3)

where contains for all and . The government aims at maximizing social welfare . This social welfare maximization problem is given as

(4a)
s.t. (4b)
(4c)
(4d)

Constraint (4b) implies that a passenger can only be selected for one OD pair at each time . Constraint (4c) requires the desired traffic offload must be satisfied for all and . Constraint (4d

) defines binary variable

.

Under the two-way communication setting, a malicious adversary aims at inferring the inconvenience cost function of each passenger by observing the selection profile . The adversary can observe be eavesdropping on communication channel. Let be the selection profile at time . Then the information perceived by the adversary up to time is . In this case, the government needs to compute a privacy preserving incentive mechanism such that the passengers truthfully report their inconvenience cost functions so that the social welfare is (approximately) optimal.

Besides the privacy guarantees, we state some additional desired properties that the government needs to achieve under this two-way communication setting. First, individual rationality for each passenger should be guaranteed, i.e., each passenger must obtain non-negative utility when contributing to traffic offload. Second, the incentive design is required to be social welfare maximizing. Third, the government wishes to reveal the true inconvenience cost functions from the passengers to seek the optimal solution to (4). Therefore the government needs to ensure that the passengers bid truthfully. Truthfulness is defined as follows.

Definition 1.

(Truthfulness). An auction is truthful if and only if bidding the true inconvenience cost function, i.e., for all , is the dominant strategy for any passenger regardless of the bids from the other passengers. In other words, bidding maximizes the utility (2) of passenger for all .

Iii-C Case 2: Interaction with One-way Communication

In this subsection, we present a problem formulation when two-way communication is infeasible, while one-way communication from the government to the passengers is enabled. Under this setting, the passengers cannot report any information to the government. The government hence broadcasts an incentive price for each OD pair at each time step , and then observes the responses from the passengers to design the incentive price for next time step . Different from two-way communication, the passengers respond to the incentive price rather than bidding a fixed amount of traffic offload. Hence, we define the amount of traffic offload provided by each passenger for OD pair at time is defined as a function of incentive price , and denote it as . We assume that the the traffic offload provided by each passenger is strictly increasing with respect to .

The government predicts the traffic condition for the set of OD pairs in the near future time horizon based on the historical traffic information (e.g., traffic conditions during rush hours). Suppose the government requires amount of traffic offload on OD pair at each time index . To satisfy amount of traffic offload, the government designs a unit incentive price for each time index to incentivize individual passengers to participate in the traffic offload program. The information perceived by the government up to time includes the following: (i) the historical incentives , (ii) the historical traffic offload offered by the passengers . Thus the government’s decision on for each time and OD pair can be interpreted as a policy mapping from the information set to the set of non-negative real numbers .

At each time step , the passengers observe the incentives , and then decide whether to participate in traffic offload and earn the incentive based on their own utility functions. Passengers that participate in traffic offload incur inconvenience cost . The inconvenience cost function is private to each passenger . The information available to passenger up to time includes the following: (i) the historical incentives , (ii) the traffic offload function , and (iii) its inconvenience cost function .

Let be the incentive prices for all OD pairs at time . The utility of each passenger at time step can be represented as

(5)

The social cost is given by

(6)

where contains the incentive prices for all and , represents , and represents the penalty due to deficit of traffic offload. The social cost minimization problem (or equivalently the social cost minimization problem) is formulated as

Under the one-way communication setting, the malicious party could not observe the participation of each passenger directly as in two-way communication setting. We focus on a malicious party that can observe the incentive prices issued by the government up to time and then infer the amount of traffic offload offered by each passenger , which might be further used to infer the inconvenience cost functions of the passengers. Denote the information obtained by the government up to time as . Then we have . The objective of a malicious party is to compute given . In this case, the government’s objective is to compute a privacy preserving incentive design such that the social welfare is (approximately) maximized.

Besides the privacy guarantee, we briefly discuss the game-theoretic properties under one-way communication setting. Since the government broadcasts the incentive price while the passengers decide if they will participate or not, individual rationality is automatically guaranteed for rational passengers. Truthfulness is not required under one-way communication setting since the passengers cannot send messages to the government under this setting.

Iii-D Notion of Privacy

In this subsection, we give the notion of privacy adopted in this paper. We focus on differential privacy [28, 29], which is defined as follows.

Definition 2.

(-Differential Privacy.) Given , a computation procedure is said to be -differentially private if for any two inputs and that differ in a single element and for any set of outcomes , the relationship holds, where is the set of all outcomes of .

Definition 2 requires computation procedure to behave similarly given similar inputs, where parameter models how similarly the procedure should behave. A more relaxed and general definition of differential privacy is as follows.

Definition 3.

(-Differential Privacy.) Given and , a computation procedure is said to be -differentially private if for any two inputs and that differ in a single element and for any set of outcomes , inequality holds.

To quantify the privacy leakage using the proposed incentive designs, we adopt the concept of min-entropy leakage [36]. We first introduce the concepts of min-entropy and conditional min-entropy [37], and then define the min-entropy leakage. Let and

be random variables. The min-entropy of

is defined as , where

represents the probability of

. The conditional min-entropy is defined as , where is the probability that given that . Then the min-entropy leakage [36] is defined as

Under two-way communication setting, the min-entropy leakage is computed as

where is the probability that a bidding profile is submitted, and is the probability that the bidding profile is submitted given the selection profile is observed. Under one-way communication setting, the min-entropy leakage is computed as

where is the probability that the collection of passengers’ inconvenience cost functions is , and is the probability that the collection of inconvenience costs is given the historical incentives is observed.

Iv Solution for Two-way Communication Setting

Motivated by exponential mechanism [32, 31], we present an incentive design for the two-way communication setting in this section. We propose a payment scheme that achieves individual rationality. We mitigate the computation complexity incurred in exponential mechanism using an iterative algorithm. We prove that the desired properties are achieved using the proposed incentive design.

Iv-a Solution Approach

In this subsection, we give an exact solution under two-way communication. We formally prove that truthfulness, approximate social welfare maximizing, and differential privacy are achieved using the proposed mechanism.

The mechanism is presented in Algorithm 1. The algorithm takes the bid profile from the passengers as input, and gives the selection profile and the incentives issued to each selected passenger. The algorithm works as follows. At each time , the government selects a feasible solution to social welfare maximization problem (4). The probability of selecting each feasible is proportional to the exponential function evaluated at the associated social welfare with scale , where is the difference between the upper and lower bound of social welfare . Although the computation of selection profile is motivated by exponential mechanism [32, 31], the Vickrey-Clarke-Groves (VCG)-like payment scheme adopted by exponential mechanism is not applicable to the problem investigated in this work. The reason is that the VCG-like payment scheme violated individual rationality and truthfulness in our case, due to the fact that the passengers do not only have valuations over the incentives, but also inconvenience costs during traffic offload. To this end, the payment scheme (8) is proposed for the problem of interest, in which the incentive issued to each passenger is determined by the social cost introduced by each passenger. In the following, we characterize the mechanism presented in Algorithm 1.

Theorem 1.

The mechanism described in Algorithm 1 achieves truthfulness, individual rationality, near optimal social welfare, and -differential privacy.

Proof.

We omit the proof due to space limit. See [35] for a detailed proof. ∎

The mechanism proposed in Algorithm 1 is computationally expensive. The payment scheme (8) is intractable when the passenger set is large since (8) needs to compute the social welfare associated with and for all . Therefore, a computationally efficient algorithm is desired.

1:procedure Mechanism()
2:     Input: Bid profile
3:     Output: Selection profile , incentives
4:     while  do
5:         Choose a selection profile that is feasible for social welfare maximization problem (4) with probability
(7)
6:         For each passenger that is selected, issue incentive as
(8)
where is the difference between the upper and lower bound of social welfare , is the Shannon entropy,

is the probability distribution over selection profile

, and and are the matrix obtained by removing the -th row and -th column in selection profile and bid profile, respectively.
7:         
8:     end while
9:end procedure
Algorithm 1 Mechanism design for the government.

Iv-B Efficient Algorithm

Algorithm 1 is computationally intensive and hence we need an efficient algorithm. In this subsection, we give a mechanism that achieves the desired game-theoretic properties and privacy guarantees and runs in polynomial time.

In real world implementation, since the passengers are geographically distributed, the government can decompose the social welfare maximization problem (4) with respect to OD pair . Then problem (4) becomes a set of optimization problems associated with each OD pair as follows:

(9)
s.t.

Given the set of decomposed problems, if we can achieve the optimal solution to each decomposed problem using an incentive design, then we reach social optimal solution. Thus our objective is design a mechanism that achieves the (approximate) optimal solution of each decomposed problem, individual rationality, truthfulness, and differential privacy.

The proposed efficient algorithm for each decomposed problem is presented in Algorithm 2. The algorithm iteratively computes the set of passengers selected by the government for OD pair at time . First, the set is initialized as an empty set. Then at each iteration , the probability that selecting a passenger that has not been selected at time is proportional to the exponential function , i.e.,

(10)

where . Then the set of selected passengers are removed from the passenger set . For each , the government issues incentive computed as

(11)

where . We characterize the solution presented in Algorithm 2 as follows.

Lemma 1.

Algorithm 2 achieves truthfulness, individual rationality, and -differential privacy. Moreover, Algorithm 2 achieves near optimal social welfare with probability at least , where is the maximum social welfare for OD pair .

Proof.

We omit the proof due to space limit. See [35] for detailed proof. ∎

Given Algorithm 2 for each decomposed problem, we present Algorithm 3, which utilizes Algorithm 2 as subroutine, to solve for the selection profile for problem (4). Algorithm 3 works as follows. It first makes copies of the passenger set , with each denoted as for all . Then Algorithm 2 is invoked iteratively to compute the selected passengers for each OD pair . The selection profile for time is finally returned as the union .

1:procedure Decompose()
2:     Input: Bid profile , current time
3:     Output: Selection profile
4:     Initialization: Selected passenger set ,
5:     while  do
6:         for  do
7:              Compute the probability of selecting passenger as (10).
8:         end for
9:         if passenger is chosen then
10:              
11:         end if
12:     end while
13:     return
14:end procedure
Algorithm 2 Solution algorithm for decomposed problem (9).
1:procedure Social_Max()
2:     Input: Bid profile
3:     Output: Selection profile
4:     while  do
5:         Initialization: for all
6:         Remove all passengers that provide negative social welfare
7:         for  do
8:              Decompose()
9:              
10:         end for
11:         return
12:         
13:     end while
14:end procedure
Algorithm 3 Solution algorithm for problem (4)

We conclude this section by characterizing the properties achieved by Algorithm 3.

Theorem 2.

Algorithm 3 achieves truthfulness, individual rationality, and -differential privacy. Moreover, Algorithm 2 achieves near optimal social welfare with at least probability , where is the maximum social welfare and .

Proof.

We omit the proof due to space limit. See [35] for detailed proof. ∎

V Solution for One-way Communication Setting

In this section, we analyze the problem formulated in Section III-C. We first present an incentive mechanism design without privacy guarantee. Then we give the incentive mechanism design that satisfies differential privacy.

V-a Incentive Mechanism Design without Privacy Guarantee

Different from the two-way communication scenario, the passengers observe the incentive price signal sent by the government and respond to it by maximizing their own utility. In the following, we first analyze passengers’ best responses to price signal. Then we analyze how the government should design the incentive price to achieve optimal social welfare.

Lemma 2.

Given an incentive price , a selfish and rational passenger would contribute amount of traffic offload to maximize its utility .

Proof.

Let be the OD pair that passenger can contribute to. Then for any , we have . Given an incentive price , the maximizer of can be computed as the solution to due to the convexity of . Therefore, we have that if the incentive price is no less than the marginal cost of contributing for each passenger , then passenger participates in traffic offload. By solving for , we have , where the operator is due to the fact that , and the existence of the solution follows by the convexity of . ∎

We have the following two observations by Lemma 2. First, a selfish and rational passenger that optimizes its utility will contribute the amount of traffic offload if and only if it can obtain non-negative utility. Moreover, by observing the participation of each passenger, the government can infer the gradients of inconvenience cost functions.

Taking the gradient of inconvenience cost function of each participating passenger as feedback, the government can then use the gradient descent algorithm [38] to approximately minimize the social cost. In Algorithm 4, the government first initializes a set of learning rates that adjusts the step size between two time instants. In the meanwhile, Algorithm 4 initializes of small value for time . Then for each time step , the government iteratively updates the incentive price as

1:Initialize the sequence of learning rates
2:while  do
3:     Initialize incentive price for time step arbitrarily
4:     Update incentive price as for all
5:end while
Algorithm 4 Computation of incentive price

In the following, we characterize Algorithm 4 by analyzing the social cost incurred using the incentive price returned by Algorithm 4. Analogous to online convex algorithm [38], we define the regret of the government. The regret over time horizon is defined as

(12)

where is the social cost when when selecting a sequence of incentive prices as defined in (6), and

(13)

is the optimal social cost when using a fixed price. Then the regret (12) models the difference between the social cost when selecting a sequence of incentive prices and optimal social cost from using a fixed price for each .

In the following, we characterize the mechanism design proposed for one-way communication by analyzing the regret (12). In particular, we analyze the regret (12) by showing that it satisfies Hannan consistency, i.e.,

(14)

The Hannan consistency implies that the average regret (14

) vanishes when time horizon approaches infinity. We define the following notations. Define row vectors

and as :

(15)
(16)

We denote the vectors and that are associated with as and , respectively. Let and . Denote the maximum incentive price the government would issue as . We also define column vectors for all and as . Similarly, vector represents the vector associated with . We finally define , where is the dot product of and . Let be the maximum for all and . Next we show that regret (12) is upper bounded.

Lemma 3.

The regret of Algorithm 4 is bounded as

(17)
Proof.

The proof is motivated by [38]. Denote the optimal incentive price associated with optimal social cost as for each OD pair . Due to convexity of inconvenience cost functions , for any and we have

By definition of (15), we have that the optimal social cost satisfies the following inequalities:

(18)
(19)
(20)

where with dimension , inequality (18) follows by the convexity of , inequality (19) follows by concavity of , and inequality (20) holds by the fact that . Rearranging the inequality above, we have that

where , and represents the dot product of and .

At time step , we have

(21)
(22)

where inequality (21) holds by the updating rule of and inequality (22) holds due to . Then we obtain

By definition (12), we have

which completes our proof. ∎

Leveraging Lemma 3, we are ready to show Hannan consistency holds for the proposed incentive mechanism design.

Proposition 1.

Let