Managing bottleneck congestion with incentives

Highlights

• Queues can be eliminated completely if incentive budget is sufficient.

• Optimal incentive profile is "U-shape" during morning peak with limited budget.

• A smaller budget can provide a higher unit benefit with respect to welfare gain.

• Participants are on sides of the peak, while non-participants travel in the middle.

• Both rich and poor commuters are participants with limited budget in ideal case.

Abstract

Incentive-Based Traffic Demand Management (IBTDM) is a strategy that adopts incentives to demotivate driving trips, or to redistribute demand across space and time. In this paper, we demonstrate the effectiveness of an IBTDM strategy that provides incentives to shift the commuting public's departure times so that the queueing delay is reduced. Based on Vickrey's bottleneck model, this paper considers the impact of incentive budget and market penetration rate on the optimal incentive profile for both homogeneous and heterogeneous commuters. The resulting departure pattern created by the optimal incentive profile achieves Pareto Optimality. The results indicate that an optimal incentive profile is “U-shape” during the morning peak with a limited budget. Additionally, we find that the marginal benefit of incentive is diminishing. Lastly, although Pareto improvement is achieved, commuters with higher values of time are found to benefit more under the optimal incentive design. It is also discovered that the incentive provider should promote IBTDM to the two ends of the income level of the commuters to achieve the lowest total system travel time under an insufficient marketing budget.

Introduction

Commuters in mega-cities usually experience traffic congestion on their way to workplaces from residential areas during the morning rush hours. Aside from expanding the capacity of bottlenecks within the road network, Traffic Demand Management (TDM) strategies have been found to be a cost-effective alternative for alleviating such congestion. Economic instruments are often adopted in TDM strategies as either punishments or rewards to either push travelers out of bad traveling behavior or to pull them towards adopting good behavior, where the pushing strategies are often mandatory. Pigou (1920) and Knight (1924) creatively proposed the concept of congestion pricing, which has become one of the most widely-studied examples of pushing strategies in TDM. On the other hand, an Incentive-Based TDM (IBTDM) approach (a pulling strategy) addresses the potential inequity issue raised by pushing strategies. It has received growing attention from both researchers and practitioners in recent years.

The conventional Vickrey's bottleneck model (Vickrey, 1969) is often adopted to describe congestion propagation or evaluate the effectiveness of various traffic management strategies. In the bottleneck model, it is assumed that a group of homogeneous commuters must pass a bottleneck in order to reach their workplaces. In equilibrium, commuters who want to get closer to their target arrival time will face longer queues. If the commuter advances or delays his/her departure, s/he would then face a shorter queueing delay. Ever since Vickrey's seminal work, there have been many improvements made to the original bottleneck model. For instance, Daganzo (1985) proved that the equilibrium state is unique with a more general distribution of target arrival times. Arnott et al. (1987) derived the user equilibrium and system optimum with heterogeneous commuters. Arnott et al. (1993) considered the impact of elastic demand on the model and linked the static model to the dynamic model. A comprehensive review of the bottleneck model can be found in Small (2015).

There is a rich body of literature on the evaluation of tolling strategies regarding the bottleneck model. It can be found that the social optimum with homogeneous commuters can be reached with a toll that increases linearly from zero at the time when the queue would begin if there were no toll to a maximum at the target arrival time, and subsequently decreases linearly to zero at the end of the queue (Cohen, 1987). Moreover, Cohen (1987) used numerical example to analyze the equilibrium state of heterogeneous commuters and proposed that there is a degree of relative gain for higher-income motorists, while losses may accrue for lower income ones. Arnott et al. (1990) proposed a coarse toll aimed at finding an optimal fee and time interval.

In the case of heterogeneous commuters, Arnott et al. (1994) considered three groups of commuters with different travel time costs and schedule early delay costs. In a system optimum state, the shape of the time-varying tolling profile was the same as the shape of the cumulative departure one with no toll, and one can see that tolls without rebates, on average, tend to benefit the rich and hurt the poor. Smith (1984a) proved that there is a time-dependent equilibrium distribution of arrivals at a single bottleneck, and Smith (1984b) proposed a proportional swap system and tried to find the optimal solution in an iterative way. However, Guo et al. (2018) later proved that these iterative methods could not reach convergence. In recent years, different from the discrete value of time, Van Den Berg and Verhoef (2011) studied heterogeneous commuters with a continuously distributed value of time.

These punishment-based tolling strategies may easily introduce mental and behavioral burdens to the traveling public, as they are required to pay out-of-pocket and the policies might increase the travel costs of some commuters (Xiao et al., 2011). In the meantime, mandatory punishment-based strategies are often questioned since the impacts of charging a toll on different income groups are different and may thus introduce unfairness. Theoretically speaking, commuters with high Value of Time (VOT) will fare better with tolling than those with low VOT (Small, 1983; Hau, 1992). In addition, although congestion charges may reduce system congestion, commuters with lower VOT tend to choose other free road but experienced travel time might be even longer (Hau, 1998; Raux and Souche, 2004).

Various ideas other than a toll have been explored in order to design a fairer congestion management strategy. In recent years, researchers have discovered that a quantity constraint may be an alternative to the pricing toll (Tian et al., 2019). Akamatsu et al. (2006) put forward the concept of a tradable bottleneck permit and they found that queue congestion can be completely eliminated as long as the number of permits issued per unit time equals the capacity at the bottleneck and the permits are tradable in auction markets. Yang and Wang (2011) first applied the tradable credit scheme to optimizing the congestion in a transportation network with homogeneous commuters and proved that it can emulate traditional congestion pricing in a revenue neutral manner. Tradable credit schemes have also been theoretically validated for the bottleneck model. For instance, Xiao et al. (2013) considered the distribution and equity of the tradable credit scheme based on the bottleneck model. Xiao et al. (2019) proposed a link-based cyclic tradable credit scheme (CTCS) in which the compensatory credits could be charged from (i.e. positive credit rate) or subsidized to (i.e. negative credit rate) the travelers.

Incentives in IBTDM refer to positive monetary gains, such as money-like gift cards or other rewards (Bauer et al., 2018). The positive incentives provided by IBTDM contrast with road tolling which creates disincentives for motorists to make socially undesirable choices such as those that contribute to congestion.

In recent years, researchers have been working on the empirical validation of the feasibility and effectiveness of IBTDM on various behaviors. Pioneering research studied the effectiveness of IBTDM strategies with a focus on travel mode choice, personal car usage, and departure time shift of transit commuters. Fujii and Kitamura (2003) found that providing drivers with a one-month free bus ticket increased the frequency of bus usage. Zhang et al. (2014) found that offering low-cost tickets to passengers during off-peak hours effectively alleviated congestion during the peak hours. A pilot study has also shown that a lottery-based revenue-neutral incentive mechanism can reduce congestion in urban transportation systems (Rey et al., 2016).

With the proliferation of mobile technology nowadays, new IBTDM strategies that are mobile device-based, personalized, and have a precise impact on the traveling public's behavior have emerged in the market. IBTDM strategies that target travelers’ route choice and departure time choice have also emerged. Examples include the FAIR lanes concept in the US (Decorla-Souza, 2000), which dictated that if the commuter chose regular lanes instead of fast lanes, she would be compensated with credits, as well as the INSTANT project (Merugu et al., 2009), which provided incentives to off-peak commuters. Another instance was the famous Dutch ‘Spitsmijden’ (‘Peak Avoidance’) project in the Netherlands, where travelers received positive incentives if they avoided traveling during peak hours with their personal car, and cut the amount of peak traffic for all participants by 60% (Ettema et al., 2010).

IBTDM strategies can improve traffic conditions based on empirical observations. However, it remains unclear how to design the optimal incentive scheme that can achieve the most congestion alleviation, especially in situations with a limited budget and a restricted penetration rate.

Rouwendal et al. (2012) compared the relative efficiency between different management strategies including tolls, feebates and rewards in the context of elastic demand and concluded that rewards can be a useful second-best measure to relieve congestion. In their setting there is a trade-off between reducing queueing delay and limiting the number of travelers (i.e., elastic demand case). Our study aims to determine the optimal time-dependent incentive profile to minimize the total system travel time (TSTT) with a given budget in the context of fixed demand. In this setting, the trade-off is between limiting the incentive budget and reducing queueing delay.

Another work on the theoretical modeling of IBTDM in the context of time-varying fine reward is Chakraborty et al. (2018) who formulated IBTDM as a linear program considering a given incentive budget. A link transmission model was adopted as the network loading model. Chakraborty et al. presented a framework to estimate the benefits of a given incentive profile. However, this framework did not try to derive the optimal incentive profile. Currently there exists a considerable knowledge gap between the theoretical development and the practices of IBTDM, especially when designing the optimal time-varying fine incentive profile. The gaps lie in the following three aspects.

Firstly, unlike congestion pricing and a tradable credit scheme, IBTDM is usually managed by small private mobility solution companies (e.g. Metropia, IncenTrip, RideAmigos, etc.) with a limited incentive budget (Hu et al., 2015). The optimal incentive scheme therefore remains restricted by a total incentive budget, as the company would otherwise easily end up bankrupt.

Furthermore, IBTDM is usually a voluntary project with a penetration rate less than 100%. Therefore, it is necessary to consider the impact of penetration rate on an optimal incentive scheme. In other words, it remains unclear how to determine the optimal penetration rate with a given incentive budget.

Finally, individuals’ VOT towards either queueing delay time or schedule delay time varies and highly depends on factors such as income level, trip purpose, etc. Observations reveal that ignoring VOT variance can lead to overestimation of toll road usage when the toll rate is low and vice versa (Jiang et al., 2011). The prediction of commuters’ reactions in homogeneous cases would be off for IBTDM as well. It is also of great research interest to investigate how to select target participants and to perform marketing campaigns accordingly in a heterogeneous case.

In conclusion, the bottleneck model has been explicitly utilized to evaluate the impact of various traffic demand management strategies including congestion tolling and tradable mobility credits. However, the concerns of budget constraints and penetration rate have never been discussed in evaluating either approach, and the success of IBTDM hinges on the consideration of those two concerns. Thus, this paper incorporates budget constraints and penetration rate into Vickrey's bottleneck model in order to reveal valuable insights into the optimal incentive design and optimal operation strategy of IBTDM to deliver persuasive outcomes, and to justify the advantages of IBTDM from a theoretical perspective.

Personal cars are still the primary commuting mode in society today (Federal Highway Administration, 2017). Therefore, this research aims to determine the optimal time-dependent incentive profile for an IBTDM strategy to minimize the total system travel time (TSTT) with a given budget, thereby alleviating the congestion of bottleneck. Players in such an IBTDM program engage in a Stackelberg game, where an incentive provider determines a time-dependent incentive profile and travelers then choose their departure times. In the Nash equilibrium, no traveler can reduce her travel cost by unilaterally choosing another departure time. Unlike Rouwendal et al. (2012) and Chakraborty et al. (2018), we consider the penetration of commuters and user heterogeneity with different values of travel time.

We take the following steps to investigate the optimal incentive profile in various situations. First, we derive the optimal incentive profile in the most basic case (homogeneous commuters without consideration of penetration rate). Intuitively, there exists a minimum incentive budget at which queueing delay is eliminated. We derive this threshold budget, and determine the optimal incentive profile when the incentive budget is below the threshold.

Second, IBTDM program is usually not mandatory but voluntary, and some participants may know the program while some others may not. Thus, we will then explore the pattern of the optimal incentive profile if market penetration is incomplete.

Furthermore, in real life, commuters differ from each other with regards to value of travel cost. We then assume that commuters fall into two bins with discrete values of travel time and derive the optimal incentive profile and the optimal penetration rate in the context of heterogeneous case. Penetration and user heterogeneity are then jointly considered.

The rest of this paper is organized as follows: Section 2 reviews Vickrey's bottleneck model's usage when evaluating various traffic demand management strategies and presents the user equilibrium with no incentives in the bottleneck model. Section 3 presents the solution of the optimal incentive profile with homogeneous commuters under the incentive budget constraint. The model is then expanded to consider the impact of penetration rate in Section 4. Section 5 further considers the joint impact of both penetration rate and user heterogeneity, and Section 6 concludes the paper.