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Endogenous market penetration dynamics of automated and connected vehicles: Transport-oriented model and its paradox
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Transportation Research Procedia 27 (2017) 238–245 www.elsevier.com/locate/procedia
20th EURO Working Group on Transportation Meeting, EWGT 2017, 4-6 September 2017, Budapest, Hungary
Endogenous market penetration dynamics of automated and connected vehicles: Transport-oriented model and its paradox Toru Seo1, Yasuo Asakura Tokyo Institute of Technology, 2-12-1-M1-20, O-okayama, Meguro, Tokyo 152-8552, Japan
Abstract Automated and connected vehicles (ACVs) have received a great deal of attention. Indeed, their full market penetration will be desirable in terms of traffic efficiency, as ACVs can efficiently drive by precisely and instantaneously communicating, recognizing, and reacting to other ACVs. However, it is not yet certain whether traffic efficiency is improved in mixed traffic where ratio of manual vehicles is substantially high. This is because, for example, ACVs in mixed traffic may require excessive safety clearance, as they have to rely on relatively imperfect vision/radar-based vehicle recognition. Meanwhile, relative benefit of ACVs compared to manual vehicles would be proportional to travel time (because the most significant merit of ACVs for their driver is comfortable in-vehicle experience) and therefore severity of congestion. Consequently, equilibrium states of a myopic car market may suffer severer congestion and higher social cost than the current state—this is congestion paradox. This kind of phenomena can be considered as a consequence of market penetration of a good with network externality or social interaction, where market penetration of ACVs is endogenously determined based on their cost/benefit which depend on current number of ACVs users. This study analyzes this problem under idealized conditions. Specifically, a theoretical model of endogenous market penetration of ACVs considering changes in value of time, travel time, and transportation fare, which are the most direct impacts of ACVs to the society, is formulated. Then, its market dynamics is analyzed. Finally, strategic policies to avoid congestion paradox and achieve social optimum are proposed. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. Peer-review Group on on Transportation Transportation Meeting. Meeting. Peer-review under under responsibility responsibility of of the the scientific scientific committee committee of of the the 20th 20th EURO EURO Working Working Group Keywords: autonomous vehicle; traffic flow theory; endogenous market penetration; multiple equilibria; network externality; social interaction
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Corresponding author. Tel.: +81 3 5734 2575 E-mail address: [email protected]
2214-241X © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 20th EURO Working Group on Transportation Meeting.
2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 20th EURO Working Group on Transportation Meeting. 10.1016/j.trpro.2017.12.028
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1. Introduction Automated and connected vehicles (ACVs) have received a great deal of attention (Hörl et al., 2016; Milakis et al., 2017). Indeed, their full market penetration will be desirable in terms of traffic efficiency thanks to the connected vehicle technology (CVT), by which ACVs can efficiently drive by precisely and instantaneously communicating, recognizing, and reacting to other ACVs. It is called cooperative driving. However, it is still an open question whether traffic efficiency is improved in mixed traffic where ratio of manual vehicles (MVs) is substantially high. This is mainly due to that ACVs in mixed traffic may require excessive safety clearance (headway). Specifically, ACVs in mixed traffic have to rely on relatively imperfect vision/radar-based vehicle recognition instead of CVT, and relation between efficiency and safety of ACVs involves various technological and ethical issues (Le Vine et al., 2015, 2016; Bonnefon et al., 2016). If long headway is required to ensure safety, then it results in low traffic capacity and traffic efficiency is degraded. New technology penetrates a market gradually when it is beneficial than the conventional counterpart for the consumers. Relative benefit of ACVs compared to MVs would be almost proportional to the in-vehicle travel time; because the most significant merit of ACVs for their driver is comfortable in-vehicle experience (i.e., reduced value of time (VoT) and safety) (Milakis et al., 2017). In other words, the worse congestion gets, the worthier ACVs are. Therefore, if ACVs are inefficient in mixed traffic and cause congestion, equilibrium states of a myopic car market may suffer severer congestion and even higher social cost than the current state. This can be considered as a congestion paradox where ACVs which naively intend eliminating congestion induce congestion due to market mechanism. It is important to know under what conditions such paradox occurs. The aforementioned features of ACVs can be considered as network externality or social interaction (Rohlfs, 1974; Durlauf, 2001), which means that cost/benefit of ACVs depend on current number of ACVs users. In general, market penetration of goods with such network externality is difficult to be estimated by naive questionnaire surveys that simply ask the willingness-to-pay. Therefore, to predict market penetration of ACVs, it would be necessarily to develop an ACVs’ endogenous market penetration model which explicitly considers the network externality based on its mechanism and determine penetration rate of ACVs endogenously. 2 Such model often has multiple market equilibria with different social cost; and policy implication on how to achieve socially desirable equilibrium can be obtained by analyzing the model (Yang, 1998; Fukuda and Morichi, 2007). The congestion paradox of ACVs corresponds to socially undesirable equilibria which may be achieved if there are no strategic policy intervention. The aim of this study is to develop a model of market penetration dynamics of ACVs considering their impact to traffic flow via the network effects, to show when the congestion paradox occurs, and to propose appropriate strategic policy to avoid the paradox. Specifically, a theoretical model of endogenous market penetration of ACVs considering changes in VoT, travel time, and transportation fare, which are the most fundamental and direct impacts of ACVs to the society (Milakis et al., 2017), is formulated in Section 2. The proposed model is analyzed and some notable solutions of the market penetration problem are presented in Section 3. Conditions when the congestion paradox occurs and measures to avoid the paradox are discussed based on the results. 2. Model of Market Penetration Dynamics of ACVs This study proposes a model that represents endogenous market penetration of ACVs considering their effects to transportation system. In this model, each traveler has to use either an MV or ACV for his/her commuting trip; and this decision is myopically3 made considering cost and benefit of using these vehicles for the trip at that moment. The framework of the proposed model is illustrated in Fig. 1. The model consists of three sub-models, namely, traffic flow model (Section 2.1), congestion model (Section 2.2), and market penetration model (Section 2.3). In this
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Few studies investigated ACVs’ endogenous market penetration based on similar ideas. van den Berg and Verhoef (2016) investigated a departure time choice problem combined with ACV market. Chen et al. (2016) investigated an optimal deployment problem of dedicated lanes for ACVs. Miyoshi (2016) proposed a model of market penetration of collision avoidance system. 3 “Myopic” in this context means that users do not know/predict the future; and long-term vehicle ownership decision making process is not considered. One of the interpretations of this situation is carsharing service where user can use either an MV or ACV at each moment.
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framework, the market model determines the equilibrium penetration rate (PR) of ACVs based on the supply and demand which are determined by the traffic flow model and congestion model. The inputs to the model are such as driving performance, price, and VoT reduction of ACVs, and total demand. As results, we can determine PR of ACVs at market equilibria, stability of each equilibrium, and social cost of each equilibrium, by which policy implications can be obtained.
Fig. 1. Framework of the proposed model.
2.1. Sub-model 1: Traffic flow model 2.1.1. Formulation Newell’s simplified car-following model (Newell, 2002) is employed to describe characteristics of the traffic flow consists of ACVs and/or MVs. In this model, driving behavior of a vehicle is determined by three parameters: namely, desired speed 𝑢𝑢, reaction time 𝜏𝜏, and jam spacing (effective vehicle body length) 𝛿𝛿. The traffic capacity 𝑞𝑞 ∗ and critical headway time ℎ∗ is expressed as ℎ∗ = 1/𝑞𝑞∗ = 𝜏𝜏 + 𝛿𝛿/𝑢𝑢. (1) It is well-known that average traffic capacity in the model with heterogeneous vehicles is derived as 𝑞𝑞∗ = |𝑁𝑁|/Σ𝑛𝑛∈𝑁𝑁 ℎ𝑛𝑛∗ , (2) ∗ where ℎ𝑛𝑛 indicates the critical headway time of vehicle 𝑛𝑛 and 𝑁𝑁 is the set of all the vehicles (Newell, 2002). We assume that reaction time τ can differ depending on use of ACV and whether it is in cooperative driving mode or not; whereas other two parameters, namely, desired speed 𝑢𝑢 and jam spacing 𝛿𝛿 , remain constant. This is a reasonable assumption employed by many studies (e.g., Levin et al., 2016; van den Berg and Verhoef, 2016; Chen et al., 2017). Specifically, let 𝜏𝜏𝑀𝑀 , 𝜏𝜏𝐶𝐶 , and 𝜏𝜏𝐴𝐴 be the reaction time of MVs, ACVs under cooperative driving mode (ACV∗ be the critical headway C), and ACVs under non-cooperative driving mode (ACV-NC), respectively. Similarly, let ℎ𝑀𝑀 time of MVs, and so on. As discussed in Section 1, it is reasonable to assume that 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝐴𝐴 and 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝑀𝑀 hold, because ACV-Cs can drive efficiently. Contrary, it is not clear whether 𝜏𝜏𝐴𝐴 ≤ 𝜏𝜏𝑀𝑀 or 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 holds; thus we don’t assume these conditions. The possible leader–follower combination in mixed traffic can be categorized into following cases: [AA] An ACV follows an ACV. Because these two vehicles are “connected” by CVT and cooperating, the follower can obtain precise information from the leader. This is often referred as “platooning”. Therefore, the follower is an ACV-C, and its reaction time is 𝜏𝜏𝐶𝐶 . [MA] An ACV follows an MV. The follower has to obtain information on the leader without CVT. Thus the follower is an ACV-NC, and its reaction time is 𝜏𝜏𝐴𝐴 . [*M] An MV follows a vehicle in arbitrary class. The follower has to be driven manually regardless of the class of the leader. Thus the follower is an MV, and its reaction time is 𝜏𝜏𝑀𝑀 . An example is illustrated in Fig. 2. Fig. 2. Example of leader-follower combinations and vehicle driving modes.
Consequently, the average capacity of traffic during certain time period, where the PR of ACVs is 𝑝𝑝 and vehicles are distributed randomly, can be derived as follows. By the definition of the leader–follower combinations, the probability can be expressed as 𝑃𝑃([∗ 𝑀𝑀]) = 1 − 𝑝𝑝, 𝑃𝑃([𝐴𝐴𝐴𝐴]) = 𝑝𝑝2 , and 𝑃𝑃([𝑀𝑀𝑀𝑀]) = 𝑝𝑝(1 − 𝑝𝑝). Then, according to Eq. (2), the average traffic capacity, 𝑞𝑞∗ (𝑝𝑝), is expressed as
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𝑞𝑞∗ (𝑝𝑝) =
2.1.2. Basic features of the model
1 . ∗ (1 − 𝑝𝑝)ℎ𝑀𝑀 + 𝑝𝑝(1 − 𝑝𝑝)ℎ𝐴𝐴∗ + 𝑝𝑝2 ℎ𝐶𝐶∗
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(3)
Before integrating the traffic flow sub-model into the market penetration model, basic features of the traffic flow sub-model is analyzed here. If ACV-Cs are more efficient than MVs, namely 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝑀𝑀 , then 𝑞𝑞∗ (𝑝𝑝) is maximized with 𝑝𝑝 = 1 in general. It means that fully automated traffic is desirable in terms of traffic efficiency. If ACV-NCs are also more efficient than MVs, namely 𝜏𝜏𝐴𝐴 ≤ 𝜏𝜏𝑀𝑀 , then d𝑞𝑞∗ (𝑝𝑝)/d𝑝𝑝 > 0 holds. It means that traffic efficiency monotonically increases as ACVs penetrates. If ACV-NCs is less efficient than MVs, namely 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 , then the model behaves qualitatively different. For example, following relations hold: 1−1/𝜙𝜙 𝑞𝑞∗ (𝑝𝑝) = 𝑞𝑞∗ (0), if 𝑝𝑝 = 0, 1−𝜃𝜃 , 𝑞𝑞∗ (𝑝𝑝) < 𝑞𝑞∗ (0), if 0 < 𝑝𝑝 < ∗ (𝑝𝑝)
1−1/𝜙𝜙 , 1−𝜃𝜃
(4)
1−1/𝜙𝜙 if 1−𝜃𝜃 < 𝑝𝑝 ≤ 1, ℎ𝐶𝐶∗ = 𝜃𝜃ℎ𝐴𝐴∗ . Notice that
∗ (0),
𝑞𝑞 > 𝑞𝑞 { ∗ ∗ ∗ where 𝜙𝜙 and θ are ratios such that ℎ𝐴𝐴 = 𝜙𝜙ℎ𝑀𝑀 and 𝑞𝑞∗ (0) = 1/ℎ𝑀𝑀 indicates the today’s traffic efficiency without ACVs. It means that traffic efficiency can be degraded due to inefficient automated driving, when PR of ACVs is low. An example of 𝑞𝑞 ∗ (𝑝𝑝) under 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 is shown in Fig. 3. Note that 𝜙𝜙 > 1 and 𝜃𝜃 < 𝜙𝜙𝜙𝜙 < 1 hold under the given conditions on 𝜏𝜏.
Fig. 3: Relation between traffic capacity and PR when ACV-NCs are inefficient.
2.2. Sub-model 2: Congestion model Here we consider a simple point-queue bottleneck congestion model where a given demand arrives from one origin, and goes through one link with one bottleneck to one destination within a given time period and with random arrival sequence. The departure time choice behavior is not considered, and the demand flow-rate (at which rate commuters arrive the bottleneck) is assumed to be constant.4 In general for a point-queue bottleneck model, the average travel time, 𝑇𝑇(𝑝𝑝), is defined as 𝑇𝑇(𝑝𝑝) = 𝑇𝑇𝑓𝑓 + 𝑇𝑇𝑐𝑐 (𝑝𝑝), (5)
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It must be noted that, van den Berg and Verhoef (2016) recently investigated a similar problem, namely, departure time choice problem with mode choice between ACVs and MVs with different VoT and capacity. However, their focus and model are substantially different from those of our study. One of the main differences is, ACVs always improves traffic capacity in their model; because mixed traffic does not occur due to “user grouping/sorting” in departure time choice equilibrium. This is a theoretically interesting result, but is strongly owing to simplified assumptions which are not necessarily realistic. Besides, they have not considered stability of equilibria and the market dynamics.
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where 𝑇𝑇𝑓𝑓 indicates average free-flow travel time, and 𝑇𝑇𝑐𝑐 (𝑝𝑝) indicates average delay due to congestion. The free-flow travel time can be represented as 𝑇𝑇𝑓𝑓 = 𝑙𝑙/𝑢𝑢 where 𝑙𝑙 is the trip length. The average delay under given ACV’s PR 𝑝𝑝 and demand flow-rate 𝑎𝑎 can be derived as 0, if 𝑎𝑎 ≤ 𝑞𝑞∗ (𝑝𝑝), (6a) 𝑇𝑇𝑐𝑐 (𝑝𝑝) = { 𝑁𝑁 1 1 ( ∗ − ) , otherwise, (6b) 2 𝑞𝑞 (𝑝𝑝) 𝑎𝑎 where 𝑁𝑁 is the total demand and 𝑞𝑞 ∗ (𝑝𝑝) is the capacity of the bottleneck given by Eq. (3). Eq. (6) is derived by the definition of total delay in the cumulative plot. 2.3. Sub-model 3: Market penetration model First, in Section 2.3.1, a framework of endogenous market penetration modeling is introduced following the concept of network externality and social interaction (Rohlfs, 1974; Durlauf, 2001). It is a methodology to determine equilibrium market penetration rate of a certain good whose cost/benefit are affected by current market penetration rate of the good itself. Then, in Section 2.3.2, a model for ACV is formulated based on the sub-models. 2.3.1. Framework of endogenous market penetration modeling Let us consider a myopic and dynamic market where each consumer has to purchase either good X or Y at each time moment. It is reasonable to assume that people use X if and only if using X is more beneficial than using Y at that time moment. This can be formalized as follows. Let 𝑉𝑉(𝑝𝑝) be a relative utility (i.e., benefit minus cost) of using X instead of Y when the PR of X is 𝑝𝑝. Then, the dynamics of the market can be described as If 𝑉𝑉(𝑝𝑝) > 0, then the PR 𝑝𝑝 will increase as the time progresses (i.e., d𝑝𝑝/d𝑡𝑡 > 0 where 𝑡𝑡 represents time). If 𝑉𝑉(𝑝𝑝) < 0, then the PR 𝑝𝑝 will decrease as the time progresses (i.e., d𝑝𝑝/d𝑡𝑡 < 0). If 𝑉𝑉(𝑝𝑝) = 0, then the PR 𝑝𝑝 will not change (i.e., d𝑝𝑝/d𝑡𝑡 = 0). (Note that exceptions exist at 𝑝𝑝 = 0 or 1; they will be explained later.) The case 𝑉𝑉(𝑝𝑝) = 0 is called an equilibrium of this market, and the value of PR is called an equilibrium PR. Furthermore, the stability of equilibria can be analyzed by using the model. A stable equilibrium is a state which attracts any sufficiently nearby states as the time progresses. It means that stable equilibria are likely to be realized as a consequence of the dynamical process of the market, whereas unstable equilibria are not. Specifically, following statements hold true for equilibrium 𝑝𝑝: If 𝑉𝑉(𝑝𝑝) = 0 and d𝑉𝑉(𝑝𝑝)/d𝑝𝑝 < 0, then the equilibrium is stable. If 𝑉𝑉(𝑝𝑝) = 0 and d𝑉𝑉(𝑝𝑝)/d𝑝𝑝 > 0, then the equilibrium is unstable. The aforementioned special cases are found when 𝑝𝑝 = 0 or 1, namely, If 𝑉𝑉(0) < 0, then 𝑝𝑝 = 0 is an equilibrium and is stable. If 𝑉𝑉(1) > 0, then 𝑝𝑝 = 1 is an equilibrium and is stable. In transportation research literature, similar models have been utilized to analyze market penetration or social diffusion of new transportation-related technology or behavior in which the effect of inter-traveler interaction is essential, such as advanced traveler information systems (Yang, 1998), illegal bicycle parking (Fukuda and Morichi, 2007), and automated collision avoidance systems (Miyoshi, 2016). 2.3.2. Market penetration model for ACVs The market penetration dynamics of ACVs is developed by specifying 𝑉𝑉(𝑝𝑝) for ACV market considering its properties. For easier model representation, the relative utility function 𝑉𝑉(𝑝𝑝) is decomposed as (7) 𝑉𝑉(𝑝𝑝) = 𝑓𝑓(𝑝𝑝) − 𝑔𝑔(𝑝𝑝), where 𝑓𝑓(𝑝𝑝) and 𝑔𝑔(𝑝𝑝) are benefit and cost, respectively, of using ACVs for one trip instead of MVs when the PR of ACVs is 𝑝𝑝. The benefit function of ACVs can be specified as follows. Basically, the benefit of using ACVs consists of the reduced VoT and the reduced accident risk. In this model, the effect of accident risk is ignored, because it is difficult to develop a precise model at this moment (benefit of safety improvement would monotonically decrease as the PR increases). Without loss of generality, the utility is normalized by the VoT for using MVs; thus, the VoT for using
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MVs is defined as 1, and that for using ACVs is defined as 1 − 𝜂𝜂, where 𝜂𝜂 ∈ [0,1] is the VoT reduction rate for using ACVs. Therefore, the average benefit of using ACVs is defined as (8) 𝑓𝑓(𝑝𝑝) = 𝜂𝜂𝜂𝜂(𝑝𝑝), Regarding the cost function 𝑔𝑔(𝑝𝑝) of ACVs, it is difficult to formulate a precise one at this moment, because it involves future prediction on industry. Therefore, as a first step, we only assume that 𝑔𝑔(𝑝𝑝) is a linearly monotonically decreasing function because of the mass production effect. It can be expressed as (9) 𝑔𝑔(𝑝𝑝) = 𝑐𝑐0 (1 − 𝜆𝜆𝜆𝜆) − 𝑐𝑐𝑀𝑀 , where 𝑐𝑐0 represents per-trip-value of using ACVs when they firstly enter the market, 𝜆𝜆 ∈ [0,1] represents mass production effect, and 𝑐𝑐𝑀𝑀 represents the per-trip cost of using MVs. Each of per-trip-cost 𝑐𝑐0 and 𝑐𝑐𝑀𝑀 represents sum of the initial cost (vehicle production cost divided by the life-time number of total trips performed by each vehicle) and running cost (fuel/maintenance cost for each trip). The total social cost, 𝑆𝑆(𝑝𝑝), which indicates undesirability of a market state in terms of the entire society, is defined as the value of the total travel time: (10) 𝑆𝑆(𝑝𝑝) = (1 − 𝜂𝜂𝜂𝜂)𝑁𝑁𝑁𝑁(𝑝𝑝). 3. Analysis
One can easily obtain the general solution of the proposed model by solving equation 𝑉𝑉(𝑝𝑝) = 0 and checking the sign of d𝑉𝑉(𝑝𝑝)/d𝑝𝑝. For example, the model always has 1 to 5 equilibria, among which 1 to 3 are stable. However, describing the entire solution is tedious because it has too many cases. Therefore, in order to demonstrate features of the proposed model and obtain policy implications, here we present some of the notable cases which are interesting and could be realized under fairly realistic model parameters. A case with the congestion paradox, where stable market equilibria have severer congestion due to inefficient ACVs, is presented in Section 3.1.1; and then countermeasures for such paradox is discussed in Section 3.1.2. Cases under efficient ACVs with qualitatively different equilibria are presented in Section 3.2. 3.1. Congestion paradox 3.1.1. Example A solution where ACV-NCs are inefficient (i.e., 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 ) and the initial society (𝑝𝑝 = 0) does not have congestion is shown in Fig. 4. There are three market equilibria, namely, 𝑝𝑝 ≃ 0%, 33%, and 67%, among which 0% and 67% are stable. According to the benefit curve, congestion occurs when 0 < 𝑝𝑝 ≤ 80% due to the inefficient ACVs. Note that the severity of the congestion is indicated by the benefit curve 𝑓𝑓, since it is proportional to the travel time as shown in Eq. (8). Since 𝑝𝑝 = 0 is a stable equilibrium, ACVs will not penetrate this society without external force such as policy intervention by the government or strategic pricing by the industry. If 𝑝𝑝 exceeds 33% due to such external force, ACVs will automatically penetrate up to the next stable equilibrium, 𝑝𝑝 ≃ 67%. The stable equilibrium at 𝑝𝑝 ≃ 67% has severer congestion than the initial state at 𝑝𝑝 = 0%. This is the congestion paradox, where the society gets “locked-in” a stable equilibrium in which travelers are spending increased travel cost. Specifically, the MV users are annoyed by the traffic jam and the ACV users are purchasing expensive ACVs in order to withstand (i.e., kill time) the traffic jam. Even worse, the social cost is also higher than the initial state, due to the wasted time in the traffic jam. Ironically, the car manufacturer is earning larger income than the initial state, because many consumers are purchasing the expensive ACVs. In general, such congestion paradox occurs if the scenario parameters have following tendencies.5 First, it occurs only if ACV-NCs are inefficient (𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 ). Second, the cost of ACVs should be moderately expensive compared with 5 One can easily obtain the exact conditions in mathematical form by solving 𝑇𝑇(𝑝𝑝∗ ) > 𝑇𝑇(0) or 𝑆𝑆(𝑝𝑝∗ ) > 𝑆𝑆(0) where 𝑝𝑝∗ satisfies 𝑉𝑉(𝑝𝑝∗ ) = 0 and 𝑉𝑉′(𝑝𝑝∗ ) < 0. However, they are not presented here due to the space limitation.
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the benefit, so that 𝑓𝑓 and 𝑔𝑔 have an intersection at the “decreasing part” of 𝑓𝑓. The social cost is not necessarily increased (or social surplus is not necessarily decreased) at a congestion paradox state, because of the income of car manufacturer.
Fig. 4: An example of market penetration dynamics of ACVs where the congestion paradox occurs.
3.1.2. Strategic measures to avoid paradox A simple measure to avoid the paradox is technology regulation which bans ACVs with low efficiency. However, it will obstruct penetration of ACVs completely, meaning that we cannot benefit from ACVs at all. More constructive solution would be subsidy for ACVs in which the authority subsides part of the cost of using ACVs. Such subsidy makes ACVs beneficial for users, and changes the market dynamics. The required amount of subsidy per trip can be derived by the proposed model: namely, the amount has to be greater than 𝑔𝑔(𝑝𝑝) − 𝑓𝑓(𝑝𝑝) when 𝑔𝑔(𝑝𝑝) > 𝑓𝑓(𝑝𝑝) and 𝑆𝑆(𝑝𝑝) is larger than a desired value. However, ill-designed subsidies will cause the congestion paradox; for example, the paradox state in Fig. 4 can be achieved by a subsidy which only promotes PR from 0% to 33%. Besides, subsidy will arise equity issues. Therefore, this policy should be decided carefully. Another approach to solve the problem is extensive introduction of carsharing (c.f., shared autonomous vehicles (Levin et al., 2016)). As carsharing significantly increases number of trips per vehicle during its lifetime (according to Levin et al. (2016), it can increase twofold or more) and diminishes parking cost, it drastically decreases the pertrip-cost of ACVs, 𝑐𝑐0 , meaning that there will be only one stable equilibrium at 𝑝𝑝 =100%. 3.2. Market penetration of efficient ACVs
Solutions where ACV-NCs are efficient are shown in Fig. 5. In both of the solutions, the total travel time decreases monotonically as ACVs penetrate; because the congestion of the initial stages diminishes. However, the properties of market equilibrium differ completely between the solutions. In case 1 (Fig. 5a), unique stable equilibrium is found at 𝑝𝑝 ≃ 67%; therefore, ACVs automatically penetrate the market up to 𝑝𝑝 ≃ 67%. Contrary, in case 2 (Fig. 5b), stable equilibria is found at 𝑝𝑝 = 0% and 100%, and an unstable equilibrium is found at 𝑝𝑝 ≃ 50%. Therefore, ACVs will not penetrate the society without policy intervention; however, if PR exceeds the midpoint equilibrium because of policy intervention, the PR will increase to 100% automatically. It means that the midpoint equilibrium can be considered as a critical mass of the system. 4. Conclusion This study proposed a theoretical model of market penetration dynamics of ACV considering effects of ACVs to VoT, travel time, and transportation fare, which are the most fundamental impacts of ACVs to the society. The notable feature of the model is that it explicitly modeled network externality of the ACVs via the connected vehicle technology in traffic flow. By using the proposed model, the market dynamics, equilibrium, and properties are analyzed and discussed. For example, we show that under particular conditions a stable e quilibrium of a myopic car
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Fig. 5: Examples of market dynamics where ACV-NCs are efficient (for the notation, see Fig. 4).
market has severer congestion—and sometimes higher social cost—than society with 100% manual vehicles. Strategic policies to avoid this pitfall are also proposed. The proposed model is based on quite simple assumptions which might not be very realistic. In fact, the model and presented results might be pessimistic from the view point of ACV utilization, because the assumptions of the model do not consider some of the advanced strategies for ACV utilization, such as use of dedicated-link/lane (Chen et al., 2016; Chen et al., 2017). Therefore, sophistication of the model considering such advanced strategies is desirable. Other factors such as safety improvement (Miyoshi, 2016), consumer preference heterogeneity (e.g., early-adopter, commuting condition), endogenous departure time choice (van den Berg and Verhoef, 2016), induced demand, and technological advancements are also considerable. All these factors can be incorporated into the proposed framework by modifying the sub-models. As an application of the proposed model, the trade-off relation between safety and efficiency of traffic would be worth investigating. Besides, the direct impacts of ACVs which this study considered will deliver some indirect effects, such as land use and human activity transformation, which could be huge impacts to the society. Expanding the model to consider such indirect effects is also important. References Bonnefon, J.-F., Shariff, A., Rahwan, I., 2016. The social dilemma of autonomous vehicles. Science 352 (6293), 1573–1576. Chen, D., Ahn, S., Chitturi, M., Noyce, D., 2017. Towards vehicle automation: roadway capacity formulation for traffic mixed with regular and automated vehicles. Transportation Research Part B: Methodological 100, 196–221. Chen, Z., He, F., Zhang, L., Yin, Y., 2016. Optimal deployment of autonomous vehicle lanes with endogenous market penetration. Transportation Research Part C: Emerging Technologies 72, 143–156. Durlauf, S. N., 2001. A framework for the study of individual behavior and social interactions. Sociological Methodology 31 (1), 47–87. Fukuda, D., Morichi, S., 2007. Incorporating aggregate behavior in an individual’s discrete choice: An application to analyzing illegal bicycle parking behavior. Transportation Research Part A: Policy and Practice 41 (4), 313–325. Hörl, S., Ciari, F., Axhausen, K. W., 2016. Recent perspectives on the impact of autonomous vehicles. Working Paper, Institute for Transport Planning and Systems. Le Vine, S., Kong, Y., Liu, X., Polak, J., 2016. Vehicle automation, legal standards of care, and freeway capacity. Available at SSRN: http://ssrn.com/abstract=2794628. Le Vine, S., Zolfaghari, A., Polak, J., 2015. Autonomous cars: The tension between occupant experience and intersection capacity. Transportation Research Part C: Emerging Technologies 52, 1–14. Levin, M. W., Li, T., Boyles, S. D., Kockelman, K. M., 2016. A general framework for modeling shared autonomous vehicles. In: Transportation Research Board 95th Annual Meeting. Milakis, D., Van Arem, B., Van Wee, G. P., 2017. Policy and society related implications of automated driving: a review of literature and directions for future research. Journal of Intelligent Transportation Systems. Miyoshi, H., 2016. Diffusion policies of automated driving systems: Rear-end collision-prevention systems. ITEC Working Paper Series, Doshisha University. Newell, G. F., 2002. A simplified car-following theory: a lower order model. Transportation Research Part B: Methodological 36 (3), 195–205. Rohlfs, J., 1974. A theory of interdependent demand for a communications service. The Bell Journal of Economics and Management Science 5 (1), 16–37. van den Berg, V. A. C., Verhoef, E. T., 2016. Autonomous cars and dynamic bottleneck congestion: The effects on capacity, value of time and preference heterogeneity. Transportation Research Part B: Methodological 94, 43–60. Yang, H., 1998. Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration. Transportation Research Part B: Methodological 32 (3), 205–218.
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20th EURO Working Group on Transportation Meeting, EWGT 2017, 4-6 September 2017, Budapest, Hungary
Endogenous market penetration dynamics of automated and connected vehicles: Transport-oriented model and its paradox Toru Seo1, Yasuo Asakura Tokyo Institute of Technology, 2-12-1-M1-20, O-okayama, Meguro, Tokyo 152-8552, Japan
Abstract Automated and connected vehicles (ACVs) have received a great deal of attention. Indeed, their full market penetration will be desirable in terms of traffic efficiency, as ACVs can efficiently drive by precisely and instantaneously communicating, recognizing, and reacting to other ACVs. However, it is not yet certain whether traffic efficiency is improved in mixed traffic where ratio of manual vehicles is substantially high. This is because, for example, ACVs in mixed traffic may require excessive safety clearance, as they have to rely on relatively imperfect vision/radar-based vehicle recognition. Meanwhile, relative benefit of ACVs compared to manual vehicles would be proportional to travel time (because the most significant merit of ACVs for their driver is comfortable in-vehicle experience) and therefore severity of congestion. Consequently, equilibrium states of a myopic car market may suffer severer congestion and higher social cost than the current state—this is congestion paradox. This kind of phenomena can be considered as a consequence of market penetration of a good with network externality or social interaction, where market penetration of ACVs is endogenously determined based on their cost/benefit which depend on current number of ACVs users. This study analyzes this problem under idealized conditions. Specifically, a theoretical model of endogenous market penetration of ACVs considering changes in value of time, travel time, and transportation fare, which are the most direct impacts of ACVs to the society, is formulated. Then, its market dynamics is analyzed. Finally, strategic policies to avoid congestion paradox and achieve social optimum are proposed. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. Peer-review Group on on Transportation Transportation Meeting. Meeting. Peer-review under under responsibility responsibility of of the the scientific scientific committee committee of of the the 20th 20th EURO EURO Working Working Group Keywords: autonomous vehicle; traffic flow theory; endogenous market penetration; multiple equilibria; network externality; social interaction
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Corresponding author. Tel.: +81 3 5734 2575 E-mail address: [email protected]
2214-241X © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 20th EURO Working Group on Transportation Meeting.
2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the 20th EURO Working Group on Transportation Meeting. 10.1016/j.trpro.2017.12.028
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1. Introduction Automated and connected vehicles (ACVs) have received a great deal of attention (Hörl et al., 2016; Milakis et al., 2017). Indeed, their full market penetration will be desirable in terms of traffic efficiency thanks to the connected vehicle technology (CVT), by which ACVs can efficiently drive by precisely and instantaneously communicating, recognizing, and reacting to other ACVs. It is called cooperative driving. However, it is still an open question whether traffic efficiency is improved in mixed traffic where ratio of manual vehicles (MVs) is substantially high. This is mainly due to that ACVs in mixed traffic may require excessive safety clearance (headway). Specifically, ACVs in mixed traffic have to rely on relatively imperfect vision/radar-based vehicle recognition instead of CVT, and relation between efficiency and safety of ACVs involves various technological and ethical issues (Le Vine et al., 2015, 2016; Bonnefon et al., 2016). If long headway is required to ensure safety, then it results in low traffic capacity and traffic efficiency is degraded. New technology penetrates a market gradually when it is beneficial than the conventional counterpart for the consumers. Relative benefit of ACVs compared to MVs would be almost proportional to the in-vehicle travel time; because the most significant merit of ACVs for their driver is comfortable in-vehicle experience (i.e., reduced value of time (VoT) and safety) (Milakis et al., 2017). In other words, the worse congestion gets, the worthier ACVs are. Therefore, if ACVs are inefficient in mixed traffic and cause congestion, equilibrium states of a myopic car market may suffer severer congestion and even higher social cost than the current state. This can be considered as a congestion paradox where ACVs which naively intend eliminating congestion induce congestion due to market mechanism. It is important to know under what conditions such paradox occurs. The aforementioned features of ACVs can be considered as network externality or social interaction (Rohlfs, 1974; Durlauf, 2001), which means that cost/benefit of ACVs depend on current number of ACVs users. In general, market penetration of goods with such network externality is difficult to be estimated by naive questionnaire surveys that simply ask the willingness-to-pay. Therefore, to predict market penetration of ACVs, it would be necessarily to develop an ACVs’ endogenous market penetration model which explicitly considers the network externality based on its mechanism and determine penetration rate of ACVs endogenously. 2 Such model often has multiple market equilibria with different social cost; and policy implication on how to achieve socially desirable equilibrium can be obtained by analyzing the model (Yang, 1998; Fukuda and Morichi, 2007). The congestion paradox of ACVs corresponds to socially undesirable equilibria which may be achieved if there are no strategic policy intervention. The aim of this study is to develop a model of market penetration dynamics of ACVs considering their impact to traffic flow via the network effects, to show when the congestion paradox occurs, and to propose appropriate strategic policy to avoid the paradox. Specifically, a theoretical model of endogenous market penetration of ACVs considering changes in VoT, travel time, and transportation fare, which are the most fundamental and direct impacts of ACVs to the society (Milakis et al., 2017), is formulated in Section 2. The proposed model is analyzed and some notable solutions of the market penetration problem are presented in Section 3. Conditions when the congestion paradox occurs and measures to avoid the paradox are discussed based on the results. 2. Model of Market Penetration Dynamics of ACVs This study proposes a model that represents endogenous market penetration of ACVs considering their effects to transportation system. In this model, each traveler has to use either an MV or ACV for his/her commuting trip; and this decision is myopically3 made considering cost and benefit of using these vehicles for the trip at that moment. The framework of the proposed model is illustrated in Fig. 1. The model consists of three sub-models, namely, traffic flow model (Section 2.1), congestion model (Section 2.2), and market penetration model (Section 2.3). In this
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Few studies investigated ACVs’ endogenous market penetration based on similar ideas. van den Berg and Verhoef (2016) investigated a departure time choice problem combined with ACV market. Chen et al. (2016) investigated an optimal deployment problem of dedicated lanes for ACVs. Miyoshi (2016) proposed a model of market penetration of collision avoidance system. 3 “Myopic” in this context means that users do not know/predict the future; and long-term vehicle ownership decision making process is not considered. One of the interpretations of this situation is carsharing service where user can use either an MV or ACV at each moment.
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framework, the market model determines the equilibrium penetration rate (PR) of ACVs based on the supply and demand which are determined by the traffic flow model and congestion model. The inputs to the model are such as driving performance, price, and VoT reduction of ACVs, and total demand. As results, we can determine PR of ACVs at market equilibria, stability of each equilibrium, and social cost of each equilibrium, by which policy implications can be obtained.
Fig. 1. Framework of the proposed model.
2.1. Sub-model 1: Traffic flow model 2.1.1. Formulation Newell’s simplified car-following model (Newell, 2002) is employed to describe characteristics of the traffic flow consists of ACVs and/or MVs. In this model, driving behavior of a vehicle is determined by three parameters: namely, desired speed 𝑢𝑢, reaction time 𝜏𝜏, and jam spacing (effective vehicle body length) 𝛿𝛿. The traffic capacity 𝑞𝑞 ∗ and critical headway time ℎ∗ is expressed as ℎ∗ = 1/𝑞𝑞∗ = 𝜏𝜏 + 𝛿𝛿/𝑢𝑢. (1) It is well-known that average traffic capacity in the model with heterogeneous vehicles is derived as 𝑞𝑞∗ = |𝑁𝑁|/Σ𝑛𝑛∈𝑁𝑁 ℎ𝑛𝑛∗ , (2) ∗ where ℎ𝑛𝑛 indicates the critical headway time of vehicle 𝑛𝑛 and 𝑁𝑁 is the set of all the vehicles (Newell, 2002). We assume that reaction time τ can differ depending on use of ACV and whether it is in cooperative driving mode or not; whereas other two parameters, namely, desired speed 𝑢𝑢 and jam spacing 𝛿𝛿 , remain constant. This is a reasonable assumption employed by many studies (e.g., Levin et al., 2016; van den Berg and Verhoef, 2016; Chen et al., 2017). Specifically, let 𝜏𝜏𝑀𝑀 , 𝜏𝜏𝐶𝐶 , and 𝜏𝜏𝐴𝐴 be the reaction time of MVs, ACVs under cooperative driving mode (ACV∗ be the critical headway C), and ACVs under non-cooperative driving mode (ACV-NC), respectively. Similarly, let ℎ𝑀𝑀 time of MVs, and so on. As discussed in Section 1, it is reasonable to assume that 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝐴𝐴 and 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝑀𝑀 hold, because ACV-Cs can drive efficiently. Contrary, it is not clear whether 𝜏𝜏𝐴𝐴 ≤ 𝜏𝜏𝑀𝑀 or 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 holds; thus we don’t assume these conditions. The possible leader–follower combination in mixed traffic can be categorized into following cases: [AA] An ACV follows an ACV. Because these two vehicles are “connected” by CVT and cooperating, the follower can obtain precise information from the leader. This is often referred as “platooning”. Therefore, the follower is an ACV-C, and its reaction time is 𝜏𝜏𝐶𝐶 . [MA] An ACV follows an MV. The follower has to obtain information on the leader without CVT. Thus the follower is an ACV-NC, and its reaction time is 𝜏𝜏𝐴𝐴 . [*M] An MV follows a vehicle in arbitrary class. The follower has to be driven manually regardless of the class of the leader. Thus the follower is an MV, and its reaction time is 𝜏𝜏𝑀𝑀 . An example is illustrated in Fig. 2. Fig. 2. Example of leader-follower combinations and vehicle driving modes.
Consequently, the average capacity of traffic during certain time period, where the PR of ACVs is 𝑝𝑝 and vehicles are distributed randomly, can be derived as follows. By the definition of the leader–follower combinations, the probability can be expressed as 𝑃𝑃([∗ 𝑀𝑀]) = 1 − 𝑝𝑝, 𝑃𝑃([𝐴𝐴𝐴𝐴]) = 𝑝𝑝2 , and 𝑃𝑃([𝑀𝑀𝑀𝑀]) = 𝑝𝑝(1 − 𝑝𝑝). Then, according to Eq. (2), the average traffic capacity, 𝑞𝑞∗ (𝑝𝑝), is expressed as
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𝑞𝑞∗ (𝑝𝑝) =
2.1.2. Basic features of the model
1 . ∗ (1 − 𝑝𝑝)ℎ𝑀𝑀 + 𝑝𝑝(1 − 𝑝𝑝)ℎ𝐴𝐴∗ + 𝑝𝑝2 ℎ𝐶𝐶∗
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(3)
Before integrating the traffic flow sub-model into the market penetration model, basic features of the traffic flow sub-model is analyzed here. If ACV-Cs are more efficient than MVs, namely 𝜏𝜏𝐶𝐶 < 𝜏𝜏𝑀𝑀 , then 𝑞𝑞∗ (𝑝𝑝) is maximized with 𝑝𝑝 = 1 in general. It means that fully automated traffic is desirable in terms of traffic efficiency. If ACV-NCs are also more efficient than MVs, namely 𝜏𝜏𝐴𝐴 ≤ 𝜏𝜏𝑀𝑀 , then d𝑞𝑞∗ (𝑝𝑝)/d𝑝𝑝 > 0 holds. It means that traffic efficiency monotonically increases as ACVs penetrates. If ACV-NCs is less efficient than MVs, namely 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 , then the model behaves qualitatively different. For example, following relations hold: 1−1/𝜙𝜙 𝑞𝑞∗ (𝑝𝑝) = 𝑞𝑞∗ (0), if 𝑝𝑝 = 0, 1−𝜃𝜃 , 𝑞𝑞∗ (𝑝𝑝) < 𝑞𝑞∗ (0), if 0 < 𝑝𝑝 < ∗ (𝑝𝑝)
1−1/𝜙𝜙 , 1−𝜃𝜃
(4)
1−1/𝜙𝜙 if 1−𝜃𝜃 < 𝑝𝑝 ≤ 1, ℎ𝐶𝐶∗ = 𝜃𝜃ℎ𝐴𝐴∗ . Notice that
∗ (0),
𝑞𝑞 > 𝑞𝑞 { ∗ ∗ ∗ where 𝜙𝜙 and θ are ratios such that ℎ𝐴𝐴 = 𝜙𝜙ℎ𝑀𝑀 and 𝑞𝑞∗ (0) = 1/ℎ𝑀𝑀 indicates the today’s traffic efficiency without ACVs. It means that traffic efficiency can be degraded due to inefficient automated driving, when PR of ACVs is low. An example of 𝑞𝑞 ∗ (𝑝𝑝) under 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 is shown in Fig. 3. Note that 𝜙𝜙 > 1 and 𝜃𝜃 < 𝜙𝜙𝜙𝜙 < 1 hold under the given conditions on 𝜏𝜏.
Fig. 3: Relation between traffic capacity and PR when ACV-NCs are inefficient.
2.2. Sub-model 2: Congestion model Here we consider a simple point-queue bottleneck congestion model where a given demand arrives from one origin, and goes through one link with one bottleneck to one destination within a given time period and with random arrival sequence. The departure time choice behavior is not considered, and the demand flow-rate (at which rate commuters arrive the bottleneck) is assumed to be constant.4 In general for a point-queue bottleneck model, the average travel time, 𝑇𝑇(𝑝𝑝), is defined as 𝑇𝑇(𝑝𝑝) = 𝑇𝑇𝑓𝑓 + 𝑇𝑇𝑐𝑐 (𝑝𝑝), (5)
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It must be noted that, van den Berg and Verhoef (2016) recently investigated a similar problem, namely, departure time choice problem with mode choice between ACVs and MVs with different VoT and capacity. However, their focus and model are substantially different from those of our study. One of the main differences is, ACVs always improves traffic capacity in their model; because mixed traffic does not occur due to “user grouping/sorting” in departure time choice equilibrium. This is a theoretically interesting result, but is strongly owing to simplified assumptions which are not necessarily realistic. Besides, they have not considered stability of equilibria and the market dynamics.
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where 𝑇𝑇𝑓𝑓 indicates average free-flow travel time, and 𝑇𝑇𝑐𝑐 (𝑝𝑝) indicates average delay due to congestion. The free-flow travel time can be represented as 𝑇𝑇𝑓𝑓 = 𝑙𝑙/𝑢𝑢 where 𝑙𝑙 is the trip length. The average delay under given ACV’s PR 𝑝𝑝 and demand flow-rate 𝑎𝑎 can be derived as 0, if 𝑎𝑎 ≤ 𝑞𝑞∗ (𝑝𝑝), (6a) 𝑇𝑇𝑐𝑐 (𝑝𝑝) = { 𝑁𝑁 1 1 ( ∗ − ) , otherwise, (6b) 2 𝑞𝑞 (𝑝𝑝) 𝑎𝑎 where 𝑁𝑁 is the total demand and 𝑞𝑞 ∗ (𝑝𝑝) is the capacity of the bottleneck given by Eq. (3). Eq. (6) is derived by the definition of total delay in the cumulative plot. 2.3. Sub-model 3: Market penetration model First, in Section 2.3.1, a framework of endogenous market penetration modeling is introduced following the concept of network externality and social interaction (Rohlfs, 1974; Durlauf, 2001). It is a methodology to determine equilibrium market penetration rate of a certain good whose cost/benefit are affected by current market penetration rate of the good itself. Then, in Section 2.3.2, a model for ACV is formulated based on the sub-models. 2.3.1. Framework of endogenous market penetration modeling Let us consider a myopic and dynamic market where each consumer has to purchase either good X or Y at each time moment. It is reasonable to assume that people use X if and only if using X is more beneficial than using Y at that time moment. This can be formalized as follows. Let 𝑉𝑉(𝑝𝑝) be a relative utility (i.e., benefit minus cost) of using X instead of Y when the PR of X is 𝑝𝑝. Then, the dynamics of the market can be described as If 𝑉𝑉(𝑝𝑝) > 0, then the PR 𝑝𝑝 will increase as the time progresses (i.e., d𝑝𝑝/d𝑡𝑡 > 0 where 𝑡𝑡 represents time). If 𝑉𝑉(𝑝𝑝) < 0, then the PR 𝑝𝑝 will decrease as the time progresses (i.e., d𝑝𝑝/d𝑡𝑡 < 0). If 𝑉𝑉(𝑝𝑝) = 0, then the PR 𝑝𝑝 will not change (i.e., d𝑝𝑝/d𝑡𝑡 = 0). (Note that exceptions exist at 𝑝𝑝 = 0 or 1; they will be explained later.) The case 𝑉𝑉(𝑝𝑝) = 0 is called an equilibrium of this market, and the value of PR is called an equilibrium PR. Furthermore, the stability of equilibria can be analyzed by using the model. A stable equilibrium is a state which attracts any sufficiently nearby states as the time progresses. It means that stable equilibria are likely to be realized as a consequence of the dynamical process of the market, whereas unstable equilibria are not. Specifically, following statements hold true for equilibrium 𝑝𝑝: If 𝑉𝑉(𝑝𝑝) = 0 and d𝑉𝑉(𝑝𝑝)/d𝑝𝑝 < 0, then the equilibrium is stable. If 𝑉𝑉(𝑝𝑝) = 0 and d𝑉𝑉(𝑝𝑝)/d𝑝𝑝 > 0, then the equilibrium is unstable. The aforementioned special cases are found when 𝑝𝑝 = 0 or 1, namely, If 𝑉𝑉(0) < 0, then 𝑝𝑝 = 0 is an equilibrium and is stable. If 𝑉𝑉(1) > 0, then 𝑝𝑝 = 1 is an equilibrium and is stable. In transportation research literature, similar models have been utilized to analyze market penetration or social diffusion of new transportation-related technology or behavior in which the effect of inter-traveler interaction is essential, such as advanced traveler information systems (Yang, 1998), illegal bicycle parking (Fukuda and Morichi, 2007), and automated collision avoidance systems (Miyoshi, 2016). 2.3.2. Market penetration model for ACVs The market penetration dynamics of ACVs is developed by specifying 𝑉𝑉(𝑝𝑝) for ACV market considering its properties. For easier model representation, the relative utility function 𝑉𝑉(𝑝𝑝) is decomposed as (7) 𝑉𝑉(𝑝𝑝) = 𝑓𝑓(𝑝𝑝) − 𝑔𝑔(𝑝𝑝), where 𝑓𝑓(𝑝𝑝) and 𝑔𝑔(𝑝𝑝) are benefit and cost, respectively, of using ACVs for one trip instead of MVs when the PR of ACVs is 𝑝𝑝. The benefit function of ACVs can be specified as follows. Basically, the benefit of using ACVs consists of the reduced VoT and the reduced accident risk. In this model, the effect of accident risk is ignored, because it is difficult to develop a precise model at this moment (benefit of safety improvement would monotonically decrease as the PR increases). Without loss of generality, the utility is normalized by the VoT for using MVs; thus, the VoT for using
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MVs is defined as 1, and that for using ACVs is defined as 1 − 𝜂𝜂, where 𝜂𝜂 ∈ [0,1] is the VoT reduction rate for using ACVs. Therefore, the average benefit of using ACVs is defined as (8) 𝑓𝑓(𝑝𝑝) = 𝜂𝜂𝜂𝜂(𝑝𝑝), Regarding the cost function 𝑔𝑔(𝑝𝑝) of ACVs, it is difficult to formulate a precise one at this moment, because it involves future prediction on industry. Therefore, as a first step, we only assume that 𝑔𝑔(𝑝𝑝) is a linearly monotonically decreasing function because of the mass production effect. It can be expressed as (9) 𝑔𝑔(𝑝𝑝) = 𝑐𝑐0 (1 − 𝜆𝜆𝜆𝜆) − 𝑐𝑐𝑀𝑀 , where 𝑐𝑐0 represents per-trip-value of using ACVs when they firstly enter the market, 𝜆𝜆 ∈ [0,1] represents mass production effect, and 𝑐𝑐𝑀𝑀 represents the per-trip cost of using MVs. Each of per-trip-cost 𝑐𝑐0 and 𝑐𝑐𝑀𝑀 represents sum of the initial cost (vehicle production cost divided by the life-time number of total trips performed by each vehicle) and running cost (fuel/maintenance cost for each trip). The total social cost, 𝑆𝑆(𝑝𝑝), which indicates undesirability of a market state in terms of the entire society, is defined as the value of the total travel time: (10) 𝑆𝑆(𝑝𝑝) = (1 − 𝜂𝜂𝜂𝜂)𝑁𝑁𝑁𝑁(𝑝𝑝). 3. Analysis
One can easily obtain the general solution of the proposed model by solving equation 𝑉𝑉(𝑝𝑝) = 0 and checking the sign of d𝑉𝑉(𝑝𝑝)/d𝑝𝑝. For example, the model always has 1 to 5 equilibria, among which 1 to 3 are stable. However, describing the entire solution is tedious because it has too many cases. Therefore, in order to demonstrate features of the proposed model and obtain policy implications, here we present some of the notable cases which are interesting and could be realized under fairly realistic model parameters. A case with the congestion paradox, where stable market equilibria have severer congestion due to inefficient ACVs, is presented in Section 3.1.1; and then countermeasures for such paradox is discussed in Section 3.1.2. Cases under efficient ACVs with qualitatively different equilibria are presented in Section 3.2. 3.1. Congestion paradox 3.1.1. Example A solution where ACV-NCs are inefficient (i.e., 𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 ) and the initial society (𝑝𝑝 = 0) does not have congestion is shown in Fig. 4. There are three market equilibria, namely, 𝑝𝑝 ≃ 0%, 33%, and 67%, among which 0% and 67% are stable. According to the benefit curve, congestion occurs when 0 < 𝑝𝑝 ≤ 80% due to the inefficient ACVs. Note that the severity of the congestion is indicated by the benefit curve 𝑓𝑓, since it is proportional to the travel time as shown in Eq. (8). Since 𝑝𝑝 = 0 is a stable equilibrium, ACVs will not penetrate this society without external force such as policy intervention by the government or strategic pricing by the industry. If 𝑝𝑝 exceeds 33% due to such external force, ACVs will automatically penetrate up to the next stable equilibrium, 𝑝𝑝 ≃ 67%. The stable equilibrium at 𝑝𝑝 ≃ 67% has severer congestion than the initial state at 𝑝𝑝 = 0%. This is the congestion paradox, where the society gets “locked-in” a stable equilibrium in which travelers are spending increased travel cost. Specifically, the MV users are annoyed by the traffic jam and the ACV users are purchasing expensive ACVs in order to withstand (i.e., kill time) the traffic jam. Even worse, the social cost is also higher than the initial state, due to the wasted time in the traffic jam. Ironically, the car manufacturer is earning larger income than the initial state, because many consumers are purchasing the expensive ACVs. In general, such congestion paradox occurs if the scenario parameters have following tendencies.5 First, it occurs only if ACV-NCs are inefficient (𝜏𝜏𝐴𝐴 > 𝜏𝜏𝑀𝑀 ). Second, the cost of ACVs should be moderately expensive compared with 5 One can easily obtain the exact conditions in mathematical form by solving 𝑇𝑇(𝑝𝑝∗ ) > 𝑇𝑇(0) or 𝑆𝑆(𝑝𝑝∗ ) > 𝑆𝑆(0) where 𝑝𝑝∗ satisfies 𝑉𝑉(𝑝𝑝∗ ) = 0 and 𝑉𝑉′(𝑝𝑝∗ ) < 0. However, they are not presented here due to the space limitation.
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the benefit, so that 𝑓𝑓 and 𝑔𝑔 have an intersection at the “decreasing part” of 𝑓𝑓. The social cost is not necessarily increased (or social surplus is not necessarily decreased) at a congestion paradox state, because of the income of car manufacturer.
Fig. 4: An example of market penetration dynamics of ACVs where the congestion paradox occurs.
3.1.2. Strategic measures to avoid paradox A simple measure to avoid the paradox is technology regulation which bans ACVs with low efficiency. However, it will obstruct penetration of ACVs completely, meaning that we cannot benefit from ACVs at all. More constructive solution would be subsidy for ACVs in which the authority subsides part of the cost of using ACVs. Such subsidy makes ACVs beneficial for users, and changes the market dynamics. The required amount of subsidy per trip can be derived by the proposed model: namely, the amount has to be greater than 𝑔𝑔(𝑝𝑝) − 𝑓𝑓(𝑝𝑝) when 𝑔𝑔(𝑝𝑝) > 𝑓𝑓(𝑝𝑝) and 𝑆𝑆(𝑝𝑝) is larger than a desired value. However, ill-designed subsidies will cause the congestion paradox; for example, the paradox state in Fig. 4 can be achieved by a subsidy which only promotes PR from 0% to 33%. Besides, subsidy will arise equity issues. Therefore, this policy should be decided carefully. Another approach to solve the problem is extensive introduction of carsharing (c.f., shared autonomous vehicles (Levin et al., 2016)). As carsharing significantly increases number of trips per vehicle during its lifetime (according to Levin et al. (2016), it can increase twofold or more) and diminishes parking cost, it drastically decreases the pertrip-cost of ACVs, 𝑐𝑐0 , meaning that there will be only one stable equilibrium at 𝑝𝑝 =100%. 3.2. Market penetration of efficient ACVs
Solutions where ACV-NCs are efficient are shown in Fig. 5. In both of the solutions, the total travel time decreases monotonically as ACVs penetrate; because the congestion of the initial stages diminishes. However, the properties of market equilibrium differ completely between the solutions. In case 1 (Fig. 5a), unique stable equilibrium is found at 𝑝𝑝 ≃ 67%; therefore, ACVs automatically penetrate the market up to 𝑝𝑝 ≃ 67%. Contrary, in case 2 (Fig. 5b), stable equilibria is found at 𝑝𝑝 = 0% and 100%, and an unstable equilibrium is found at 𝑝𝑝 ≃ 50%. Therefore, ACVs will not penetrate the society without policy intervention; however, if PR exceeds the midpoint equilibrium because of policy intervention, the PR will increase to 100% automatically. It means that the midpoint equilibrium can be considered as a critical mass of the system. 4. Conclusion This study proposed a theoretical model of market penetration dynamics of ACV considering effects of ACVs to VoT, travel time, and transportation fare, which are the most fundamental impacts of ACVs to the society. The notable feature of the model is that it explicitly modeled network externality of the ACVs via the connected vehicle technology in traffic flow. By using the proposed model, the market dynamics, equilibrium, and properties are analyzed and discussed. For example, we show that under particular conditions a stable e quilibrium of a myopic car
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Fig. 5: Examples of market dynamics where ACV-NCs are efficient (for the notation, see Fig. 4).
market has severer congestion—and sometimes higher social cost—than society with 100% manual vehicles. Strategic policies to avoid this pitfall are also proposed. The proposed model is based on quite simple assumptions which might not be very realistic. In fact, the model and presented results might be pessimistic from the view point of ACV utilization, because the assumptions of the model do not consider some of the advanced strategies for ACV utilization, such as use of dedicated-link/lane (Chen et al., 2016; Chen et al., 2017). Therefore, sophistication of the model considering such advanced strategies is desirable. Other factors such as safety improvement (Miyoshi, 2016), consumer preference heterogeneity (e.g., early-adopter, commuting condition), endogenous departure time choice (van den Berg and Verhoef, 2016), induced demand, and technological advancements are also considerable. All these factors can be incorporated into the proposed framework by modifying the sub-models. As an application of the proposed model, the trade-off relation between safety and efficiency of traffic would be worth investigating. Besides, the direct impacts of ACVs which this study considered will deliver some indirect effects, such as land use and human activity transformation, which could be huge impacts to the society. Expanding the model to consider such indirect effects is also important. References Bonnefon, J.-F., Shariff, A., Rahwan, I., 2016. The social dilemma of autonomous vehicles. Science 352 (6293), 1573–1576. Chen, D., Ahn, S., Chitturi, M., Noyce, D., 2017. Towards vehicle automation: roadway capacity formulation for traffic mixed with regular and automated vehicles. Transportation Research Part B: Methodological 100, 196–221. Chen, Z., He, F., Zhang, L., Yin, Y., 2016. Optimal deployment of autonomous vehicle lanes with endogenous market penetration. Transportation Research Part C: Emerging Technologies 72, 143–156. Durlauf, S. N., 2001. A framework for the study of individual behavior and social interactions. Sociological Methodology 31 (1), 47–87. Fukuda, D., Morichi, S., 2007. Incorporating aggregate behavior in an individual’s discrete choice: An application to analyzing illegal bicycle parking behavior. Transportation Research Part A: Policy and Practice 41 (4), 313–325. Hörl, S., Ciari, F., Axhausen, K. W., 2016. Recent perspectives on the impact of autonomous vehicles. Working Paper, Institute for Transport Planning and Systems. Le Vine, S., Kong, Y., Liu, X., Polak, J., 2016. Vehicle automation, legal standards of care, and freeway capacity. Available at SSRN: http://ssrn.com/abstract=2794628. Le Vine, S., Zolfaghari, A., Polak, J., 2015. Autonomous cars: The tension between occupant experience and intersection capacity. Transportation Research Part C: Emerging Technologies 52, 1–14. Levin, M. W., Li, T., Boyles, S. D., Kockelman, K. M., 2016. A general framework for modeling shared autonomous vehicles. In: Transportation Research Board 95th Annual Meeting. Milakis, D., Van Arem, B., Van Wee, G. P., 2017. Policy and society related implications of automated driving: a review of literature and directions for future research. Journal of Intelligent Transportation Systems. Miyoshi, H., 2016. Diffusion policies of automated driving systems: Rear-end collision-prevention systems. ITEC Working Paper Series, Doshisha University. Newell, G. F., 2002. A simplified car-following theory: a lower order model. Transportation Research Part B: Methodological 36 (3), 195–205. Rohlfs, J., 1974. A theory of interdependent demand for a communications service. The Bell Journal of Economics and Management Science 5 (1), 16–37. van den Berg, V. A. C., Verhoef, E. T., 2016. Autonomous cars and dynamic bottleneck congestion: The effects on capacity, value of time and preference heterogeneity. Transportation Research Part B: Methodological 94, 43–60. Yang, H., 1998. Multiple equilibrium behaviors and advanced traveler information systems with endogenous market penetration. Transportation Research Part B: Methodological 32 (3), 205–218.