Departure Time Choice Models in Urban Transportation Systems Based on Mean Field Games
Mostafa Ameli, Mohamad Sadegh Shirani Faradonbeh, Jean-Patrick Lebacque, Hossein Abouee-Mehrizi, Ludovic Leclercq
SSubmitted
Departure Time Choice Models in CongestedTransportation Systems Based on Mean Field Games
M. Ameli a , M.S. Shirani F. b , J.P. Lebacque a , H. Abouee-Mehrizi b , L. Leclercq c a Univ. Gustave Eiffel, COSYS, GRETTIA14-20 Boulevard Newton, 77420 Champs-sur-Marne, France,mostafa.ameli@univ-eiffel.fr, jean-patrick.lebacque@univ-eiffel.fr b Department of Management sciences, University of Waterloo200 University Avenue W, Waterloo, Ontario, Canada N2L 3G1,[email protected], [email protected] c Univ. Gustave Eiffel, Univ. Lyon, ENTPE, LICIT3 Rue Maurice Audin, 69518 Vaulx-en-Velin cedex, France,ludovic.leclercq@univ-eiffel.fr
Departure time choice models play a crucial role in determining the traffic load in transportation systems.This paper introduces a new framework to model and analyze the departure time user equilibrium (DTUE)problem based on the so-called Mean Field Games (MFGs) theory. The proposed framework is the combi-nation of two main components including (i) the reaction of travelers to the traffic congestion by choosingtheir departure times to optimize their travel cost; and (ii) the aggregation of the actions of the travelers,which determines the congestion of the system. The first component corresponds to a classic game theorymodel while the second one captures the travelers’ interactions at the macroscopic level and describes thesystem dynamics. In this paper, we first present a continuous departure time choice model and investigatethe equilibria of the system. Specifically, we demonstrate the existence of the equilibrium and characterizethe DTUE. Then, a discrete approximation of the system is provided based on deterministic differentialgame models to numerically obtain the equilibrium of the system. To examine the efficiency of the proposedmodel, we compare it with the departure time choice models in the literature. We observe that the solutionsobtained based on our model is 5.6% closer to the optimal ones compared to the solutions determined basedon models in the literature. Moreover, our proposed model converges much faster with 87% less number ofiterations required to converge. Finally, the model is applied to the real test case of Lyon Metropolis. Theresults show that the proposed framework is capable of not only considering a large number of players butalso including multiple preferred travel times and heterogeneous trip lengths more accurately than existingmodels in the literature.
Key words : Departure time choice models, Departure time user equilibrium, Deterministic differentialgames, Mean Field Games, Bathtub model a r X i v : . [ m a t h . O C ] J a n meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
1. Introduction
In urban transportation systems, representing trip-making behavior requires a deep understandingof the interactions and interrelation between travelers’ decisions and the systems’ performance(Mahmassani, Chang et al. 1985). In this context, studying the dynamics of travelers’ departuretime choice behavior, particularly in congested systems, is of fundamental importance (Ben-Akivaand Bierlaire 2003). Departure time choice models represent how travelers choose their departuretime considering their own desired arrival time (Hendrickson and Kocur 1981). The goal of eachtraveler in the system is to optimize its own travel cost. This means that all travelers are assumedto be fully rational decision-makers. They anticipate other travelers’ behavior in order to make anoptimal decision, i.e., minimizing the additional travel cost due to the traffic congestion (Perakisand Roels 2006) and narrowing the final arrival time to the desired one (Smith 1984, Guo, Yang,and Huang 2018). Thus, the system behaves as a game in which the winners experience minimumtravel cost (Cominetti, Correa, and Larr´e 2015) considering not only the travel cost but also thedeparture time. Based on the first principle of Wardrop (Wardrop 1952), the game may have anequilibrium state called Departure Time User Equilibrium (DTUE) where no traveler can improvehis individual travel cost by changing his departure time (Mahmassani and Herman 1984, Ran,Boyce, and LeBlanc 1993).Most of the previous studies on the DTUE problem in the literature are based on solving aclassic Nash equilibrium problem coupled with a single point-queue (bottleneck) model based onthe pioneering paper of Vickrey (1969). The idea behind the point-queue models is to assume thatthe travel cost on the transportation system consists of a free-flow travel cost plus a congestioncost represented by a queueing cost (Daganzo 1985). Therefore, DTUE arises because the queuecapacity is limited, and travelers should consider the trade-off between the travel time losses andthe costs corresponding to arriving later or earlier than the preferred arrival time (Ata and Peng2018). A comprehensive literature review on the bottleneck models has been recently conducted byLi, Huang, and Yang (2020), which highlights the developments and the applications of bottleneckmodels to transportation systems in the past half-century.Representing the urban transportation network by an origin, a destination, and a single bottle-neck is not realistic (Lamotte and Geroliminis 2018, Nagel, Wagner, and Woesler 2003). Indeedcongestion depends on the detailed topology of the transportation network (e.g., the spatial dis-tribution of origins, destinations, routes, and roads). In addition, congestion is impacted by thedistribution of trips and vehicle densities (Jin 2020a, Ji, Luo, and Geroliminis 2014). To integratesuch features while keeping the macroscopic scale, a common approach in urban transportationeconomics and transportation science is to model the transportation network using the bathtubmodels (Arnott, Kokoza, and Naji 2016, Mariotte, Leclercq, and Laval 2017, Lamotte and Geroli-minis 2016). meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted The first bathtub model was introduced by Vickrey (1991, 2020). The travel demand is described by(i) the total number of trips, (ii) the distribution of trips’ departure time, and (iii) the distributionof trips’ length. The single bathtub model considers an undifferentiated movement area to representa dense network of congested links. The motion of travellers is assumed to take place at a speedwhich is considered to be uniform over the network but varies over time depending on the overallnetwork loading (Bao, Verhoef, and Koster 2020). Therefore, the model does not need the locationinformation of the origin and destination of travelers. A trip is defined by its length and departuretime in the dynamic setting. When a trip starts, its remaining distance to travel decreases followingthe evolution of the network mean speed. Note that the network speed depends on the networkcharacteristics (e.g., network size and road capacities) as well as the load on the network (networkdensity) (Fosgerau 2015). In order to capture the dynamics of the system, Vickrey (1991) definedan ordinary differential equation to describe the evolution of the number of active trips (users inthe network). Such a model resort to a strong assumption that the average remaining distance ofactive trips is constant. Another option is to assume that the remaining trip distance of activetrips follows a time-independent negative exponential distribution (Vickrey 1994, 2019). However,based on the empirical studies of Liu et al. (2012), Thomas and Tutert (2013), travelers’ trip lengthdistribution is neither time-independent nor exponential.Recently, Jin (2020a) reviewed and analyzed several studies that relaxed the assumption onthe trip length distribution (see, e.g., Leclercq, S´en´ecat, and Mariotte 2017, Mariotte, Leclercq,and Laval 2017, Lamotte and Geroliminis 2018) and then proposed the generalized bathtub modelwhich captures any distribution of the trip length. From a mathematical point of view, the totalnumber of active trips is the primary variable for most bathtub models. The generalized bathtubmodel focuses on the number of active trips with remaining distances greater than or equal to athreshold. By this definition, Jin (2020a) derived a set of partial differential equations to trackthe distribution of the remaining trip lengths. Further properties of the model are discussed inJin (2020a). In this work, we use the generalized bathtub model to capture the state of the urbantransportation network for the departure time equilibrium problem.
The departure time choice (also known as “morning commute”) problem at the network level iswell-reviewed by Lamotte (2018). One of the main questions that have not been well studied in theliterature is how a departure time choice model can take into account the heterogeneity of the triplengths with multiple preferred arrival time (Lamotte 2018). The most complex equilibrium problemthat has been addressed in the literature is modeling and numerically solving the DTUE problem for meli et al.:
Departure Time Choice Models Based on Mean Field Games Submitted a group of travelers with a single distributed preferred arrival time and heterogeneous trip lengths(Lamotte and Geroliminis 2018). This model is supported by empirical data and simulation. Notethat even when a simpler bottleneck formulation is used for solving the DTUE problem (reviewedby Jin (2020b)), few studies in the literature consider multiple preferred arrival time for commuters(Akamatsu et al. 2020, Lindsey 2004, Doan, Ukkusuri, and Han 2011, Ramadurai et al. 2010,Takayama and Kuwahara 2017, Akamatsu et al. 2018, Lindsey, De Palma, and Silva 2019). Recallthat the heterogeneity of the travelers’ trip distance is not considered by single bottleneck modelsbecause they consider a single origin-destination (Akamatsu et al. 2020). The goal of this paper isto develop a more general mathematical framework to address departure time choice equilibriumwith heterogeneous trip lengths and many desired arrival times in an urban transportation network.The concept of DTUE, originally, comes from game theory and Nash-equilibrium principles (Sunet al. 2017). In general, with rational travelers, the user equilibrium problems represent fixed points(Wang et al. 2018, Bortolomiol, Lurkin, and Bierlaire 2019). The equilibration process of DTUEmodels is always addressed by population game theory at the network level (Arnott 2013, Yang2005, Arnott and Buli 2018). Population games have one strong assumption, which is called Myopia.Myopia in our problem means that travelers only take into account the current utilities of eachalternative when choosing the departure time, without predicting other users’ reactions (i.e., thedeparture time adjustment) (Sandholm 2015). User interactions create new system states, whichmodify in return perceived utilities. This evolution process pushes the system at each iteration orday-to-day process toward equilibria. Iryo (2019) proved that when an evolution dynamics playsthe role of the replicator dynamics, no stable equilibrium solution can be determined in the DTUEproblem even when the demand profile is homogeneous, i.e., all users have the same travel distance.This study aims to overcome this limitation by employing a mean-field approximation and derivinga macroscopic framework.
To propose a new perspective on the DTUE, we resort to the Mean Field Games (MFGs) frame-work. The mathematical foundations of this theory were introduced in the seminal papers of Lasryand Lions (2006, 2007). The theory and methodology of MFGs have rapidly developed in differentengineering fields (Djehiche, Tcheukam, and Tembine 2016). The theory of MFGs studies decision-making problems with an infinite number of interacting players (Adlakha and Johari 2013). TheMFGs theory restates the classical game theory model as a micro-macro model (Cardaliaguet2013). It allows defining players at the microscopic level similar to classical game theory modelswhile translating the effect of players’ decision to macroscopic models (Caines, Huang, and Mal-ham´e 2015). Therefore, instead of solving a large set of highly coupled equations that represent meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted the interactions among players on a microscopic level, the core idea of MFGs is to exploit the“smoothing” effect of large numbers of interacting players. The MFGs’ main assumption (calledmean field approximation) states that each player only reacts to a “mass”, which is defined byaggregating the effect of all the players. This approach simplifies the complex multi-agent dynamicsystems at a macroscopic level (Degond, Liu, and Ringhofer 2014).There are few studies in the literature that apply MFGs to analyze transportation systems andmost of them apply MFGs theory in the context of control theory (Chevalier, Le Ny, and Malham´e2015, Huang et al. 2019), vehicle routing problem (Tanaka et al. 2020, Salhab, Le Ny, and Malham´e2018) or pedestrian moving models (Aurell and Djehiche 2019). This paper, for the first time,develops a MFGs-based framework for the departure time equilibrium problem. In our framework,each traveler looks for the optimal departure time by predicting the other travelers’ departuretime choices, given the current information of the traffic network congestion (mean-field), whichis extracted from the generalized bathtub model. Then, the mean field is updated based on theoptimal departure time choice of the travelers. The Nash equilibrium state occurs when the initialmean field approximation of the system is equal to the final mean field derived from the travelers’optimal departure time distribution. This process is equivalent to solving a fixed-point problem(Friesz et al. 1993).To numerically solve the DTUE model and determine the equilibrium of the system, many studiesin the literature limit the feasible space of the DTUE problem by making strong assumptions onthe trip-length distribution of the demand profile. For instance, recent studies on the morningcommute problem assume that the optimal solution fulfills some sorting property relating to thetrip length and departure time, e.g., First-In, First-Out (FIFO) by (Daganzo and Lehe 2015),partial FIFO by Lamotte and Geroliminis (2018), and Last-In, First-Out (LIFO) by Fosgerau(2015). Such assumptions restrict the exploration of the solution space. Moreover, most approachesin the literature have a common drawback: they do not guarantee the solution’s optimality andstability while they are costly computationally at large-scale (Huang et al. 2020). In this paper,we relax all the assumptions concerning sorting properties in the solution method (i.e., departuretime rescheduling process) to better explore the solution space. We also propose a new heuristicmethod to speed-up the calculation process while converging to a solution which is closer to theDTUE equilibrium compare to the existing methods in the literature.In this study, we first express the dynamic departure time choice problem at the network levelbased on the mean field games theory and generalized bathtub model (Section 2). Then, we discussthe properties of the DTUE in the continuous and discrete settings and prove that the model canrepresent the morning commute problem without any strong assumptions of homogeneity on thedemand profile (Sections 3 and 4). Finally, in Section 5, we evaluate the performance of the model meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted against one of the recently proposed approaches in the literature to solve the DTUE (Section 5.1)and apply the proposed model to the real test case of Lyon Metropolis network (Section 5.2).We numerically demonstrate that the model can not only consider heterogeneous demand profilefor the morning commute problem but also a large transportation system with a high number oftravellers. In Section 6, we provide concluding remarks.
2. Problem Definition
Table 1 presents the list of notations used in this paper. Consider a system with n independenttrips indexed by i ∈ [ n ] := { , , . . . , n } in a time horizon T := [0 , T max ]. The trip length of the i -thtrip is denoted by x i ∈ X := [ X min , X max ]. The goal of the player i is to choose his departure time t id ∈ T d to arrive at the desired arrival time t ia ∈ T a , where T d and T a are two compact subsets of T . We assume that the joint distribution of the desired arrival times and trip lengths is given as ademand profile which is represented by m . If we define m n := 1 n n (cid:88) i =1 δ ( x i ,t ia ) , (1) TABLE 1 List of notations T Time horizon. n The total number of trips. i Index of trips, i ∈ N . X max Maximum trip length. X min Minimum trip length. x i Trip length of trip i . t id Departure time of trip i . T ( t id , x i ) Travel time of a trip started at t id with trip length x i . t ia Desired arrival time of trip i .¯ t ia Actual arrival time of trip i . v t Velocity of the system at time t . c t Fraction of the total demand that traveling in the system at time t . z ( t ) Characteristic travel distance. o t Outflow fraction of the system at time t . ϕ ( t, · ) Probability density function of the active trips’ remaining distances at t .Φ( t, x ) Fractions of active trips with trip lengths more than x at time t . F In-flow measure, the empirical distribution of the departures. V Speed function, which maps the fraction of active travellers to the velocity.∆ t Small time interval.∆ x Small space interval. meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted where δ denotes the Dirac delta function, then m n → m as n → ∞ . Note that m is a probabilitymeasure in the space of all probability measures defined on X × T a , i.e. m ∈ P ( X × T a ). Therefore,it fully describes the demand characteristics. We make the following regularity assumption on m . Assumption 1.
There exists a constant M m such that m ( B, T a ) ≤ M m λ ( B ) , ∀ B ∈ B ( X ) (2) where B ( X ) and λ denote the σ -algebra of Borel sets and the Lebesgue measure on X , respectively. Broadly speaking, Assumption 1 implies that the demand is not concentrated within any sub-interval of X .The congestion in the system at time t is defined by the fraction of the total demand that isactive at time t , which is captured by, c t = 1 n n (cid:88) i =1 [ t id , ¯ t ia ) ( t ) , (3)where ¯ t ia denotes the actual arrival time of the i -th player and [ t id , ¯ t ia ) is an indicator function whichreturns 1 if t ∈ [ t id , ¯ t ia ) and 0 otherwise. We assume that the velocity of the system at time t dependson the fraction of travelling users in the system c t which is defined by a strictly decreasing speedfunction V : R + (cid:55)→ R + . Therefore, V represents the mean network speed and is the key collectivebehavioral characteristic of the generalized bathtub model (Jin 2020a). We simply denote thevelocity at time t by v t := V ( c t ) and assume that the velocity is the same for all players who aretravelling at the same time.To determine the travel time of a player, we first define a virtual user who starts his trip at time0. Then, the characteristic travel distance z ( t ), travelled by this virtual user up to time t , is z ( t ) := (cid:90) t v s ds. (4)Since v t > ∀ t ∈ T , z is an invertible function. Let z − denote the inverse function of z . Then, wehave z − (cid:0) z ( t ) (cid:1) = t and z − ( x ) represents the time at which the virtual user has reached x .Now, let T ( t id , x i ) denote the travel time of a player departing at time t id with trip length x i .Considering (4), T ( t id , x i ) can be determined by, T ( t id , x i ) = z − (cid:0) x i + z ( t id ) (cid:1) − t id . (5) Any function with values in [0 , F is countable additive if for all countable family { B i } of pairwise disjoint sets, itholds true that F ( (cid:83) B i ) = (cid:80) F ( B i ), see Billingsley (2012). meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
To determine the optimal departure time, we assume that each player aims to minimize histravel cost. In the DTUE problem, the travel cost is usually defined based on α - β - γ schedulingpreferences (Fosgerau 2015). That means, the cost function is defined as the sum of the travel timeand a penalty cost for arriving at t id + T ( t id , x i ) instead of the desired arrival time. Specifically, weassume that each player’s cost function is given by, J i ( t id , t ia ; t − id ) = αT ( t id , x i ) + β (cid:0) t ia − t id − T ( t id , x i ) (cid:1) + + γ (cid:0) t id + T ( t id , x i ) − t ia (cid:1) + , (6)where α denotes the cost of travelling per unit of time, β and γ denote, respectively, the cost ofearliness and lateness for the traveler arrival. Note that ( y ) + = max { y, } and t − id expresses thedependency of J on the departure times of the other users ( (cid:54) = i ) via their travel times.The cost function defined in (6) captures the fact that travelers prefer not to deviate from theirdesired arrival time (i.e., arrive as close as possible to their desired arrival time) while they donot spend too much time on the traffic. Note that the dependency of the cost function on the triplengths is not emphasized in the notation, while it holds implicitly. Below, we provide the definitionof the optimal strategy that each player adopts to determine his departure time. Definition 1.
The departure time vector ˆ t d := (ˆ t d , . . . , ˆ t nd ) ∈ T nd is a Nash equilibrium (NE) forthe cost function given in (6), if for all i ∈ [ n ] we have J i (ˆ t id , t ia ; ˆ t − id ) ≤ J i ( t, t ia ; ˆ t − id ) , ∀ t ∈ T d . (7)The above definition indicates that at a NE point ˆ t d , no player can decrease his travel cost bydeviating from his departure time. Based on Definition 1, we define DTUE as a NE of the followingDeparture Time Choice Problem (DTCP):min t id ∈T d J i ( t id , t ia ; t − id , x − i ) = αT ( t id , x i ) + β (cid:0) t ia − t id − T ( t id , x i ) (cid:1) + + γ (cid:0) t id + T ( t id , x i ) − t ia (cid:1) + ∀ i (DTCP) s.t. c t = n (cid:80) nj =1 (cid:2) t jd ,t jd + T ( t jd ,x j ) (cid:1) ( t ) ,z ( t ) = (cid:82) t V ( c s ) ds,T ( t id , x i ) = z − (cid:0) x i + z ( t id ) (cid:1) − t id . (8)Similar to (6), DTCP provides the cost function of the i -th player. Note that, given the departuretimes and trip lengths of others ( t − id , x − i ), the player i is able to find his travel time. Specifically,according to the set of equations given in (8), one can derive the characteristic travel distance z ( t )based on the fraction of active trips c t . Then, the travel time function T can be obtained.Since analyzing the DTCP for a large n is arduous due to “curse of dimensionality”, in the nextsection, we examine the behaviour of players in a system where the number of players goes toinfinity, i.e., n → ∞ . This means that we adopt the MFGs approach to determine the DTUE. meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted
3. Mean Field Games Framework
In this section, we discuss the DTCP in the framework of the MFGs. First, recall that the ideabehind the MFGs is to consider a proxy that represents the macroscopic behavior of all the playersat once, instead of taking into account their departure times individually. Therefore, to capturethe information of entering trips from the viewpoint of the i -th player when there are n players(including player i ) in the game, we define the following empirical measures, F in := 1 n − (cid:88) j ∈ [ n ] \ i δ t jd ,x j , (9) E in := 1 n − (cid:88) j ∈ [ n ] \ i δ t jd ,x j ,t ja , (10)where δ is the Dirac delta function. Note that the cost function of each player, defined in (6), is asymmetric function . Further, as n → ∞ the impact of each player on the system vanishes. Thatmeans by changing either the departure time, desired arrival time, or trip length of a player, thevelocity of the system would be left unchanged. Thus, we define the in-flow measure F and thedis-aggregated in-flow measure E as the limits of the sequences { F in } n ∈ N and { E in } n ∈ N , respectively.That is, F := lim n →∞ F in , (11) E := lim n →∞ E in . (12)Note that the limits are independent of i , and F is a probability measure on the product space T d × X , i.e., F ∈ P ( T d × X ), the set of all probability measures define on T d × X . In fact, for all n ∈ N and i ∈ [ n ] the function F in is a probability measure and F is the limit in the weak convergencesense. That means, (cid:82) T d φdF in → (cid:82) T d φdF for all φ ∈ C b ( T d × X ), the set of all bounded continuousfunctions on T d × X (see Billingsley (2013), Carmona, Delarue et al. (2018)). Similarly, one canshow that E ∈ P ( T d × X × T a ). According to (9) and (10), the in-flow measure F depends on E inthe following sense, (cid:90) T d ×X φdF = (cid:90) T d ×X ×T a φ ⊗ T a dE, ∀ φ ∈ C b ( T d × X ) , (13)where φ ⊗ T a is the tensor product of φ and 1 T a . In fact, (13) assures that the in-flow measure F isthe marginal probability measure of the departure times and trip lengths wrt to the dis-aggregated For any i ∈ [ n ], the cost function of the player i satisfies, J i ( t id , t ia ; t − id ) = J ζ ( i ) ( t ζ ( i ) d , t ζ ( i ) a ; t − ζ ( i ) d ) , for all permutation ζ on { , . . . , n } . meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted in-flow measure E , almost surely (a.s.). This linear dependency is continuous by (13) and is denotedby F : P ( T d × X × T a ) → P ( T d × X ) such that, F = F ( E ) . (14)Further, E is constrained by the demand profile m , see (1), such that, (cid:90) T d ×X ×T a T d ⊗ φdE = (cid:90) X ×T a φdm, ∀ φ ∈ C b ( X × T a ) . (15)Constraint (15) restricts the dis-aggregated in-flow measure E to a subset of P ( T d × X × T a ) withmarginal probability measures which is a.s. equal to the demand profile m . Constraints (13) and(15) together yield to the following demand constraint, F ( T d , B ) = m ( B, T a ) , (a.s.) (16)for all Borel measurable subsets of X such as B . Roughly speaking, from the demand viewpoint, thefraction of players having a trip length in B , m ( B, T a ), is the same as the fraction of the departureswith a trip length in B , F ( T d , B ).Now, an arbitrary player with the departure time t d , trip length x and desired arrival time t a can reconsider his cost criteria as a function J : T d × X × T a × P ( T d × X ) (cid:55)→ R + given by, J ( t d ; x, t a ; F ) = αT ( t d , x ) + β (cid:0) t a − t d − T ( t d , x ) (cid:1) + + γ (cid:0) t d + T ( t d , x ) − t a (cid:1) + . (17)To analyze the cost function (17) and determine the equilibrium behavior of travellers, we shouldfirst find the relation between the travel time function T and the in-flow measure F , i.e., therelation between the travel time of a player and others’ departures. The goal of this section is to derive the dynamics of the characteristic travel distance z , defined in(4), based on the in-flow measure F . Then, using (5), we are able to clarify the relation betweenthe travel time function T and the characteristic travel distance z which completes the definitionof (17). To derive the dynamics of the system, in this section, we assume that the in-flow measure F , with the probability density function f , is given as the distribution of the departures.Assuming the number of players in the system goes to infinity ( n → ∞ ), we define the dynamicsof the system according to the fraction of active trips instead of the number of them. Therefore,we denote by ϕ ( t, · ) the probability density function of the remaining trip lengths of active tripsat time t . Then, the out-flow of the system at time t , o t , can be stated as follows, o t = c t v t ϕ ( t, . (18) meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted To derive the dynamics of ϕ ( t, x ), we follow a procedure similar to the generalized bathtub modelgiven in Jin (2020a). Note that in a system with n trips, for a small time interval ∆ t , the numberof active trips at t + ∆ t with a remaining trip length in [ x, x + ∆ x ] is nc t +∆ t ϕ ( t + ∆ t, x )∆ x . Onthe other hand, it is equal to the sum of new departures and trips with remaining trip length in[ x + v t ∆ t, x + v t ∆ t + ∆ x ] at time t . Thus, nc t +∆ t ϕ ( t + ∆ t, x )∆ x ≈ nf ( t, x )∆ x ∆ t + nc t ϕ ( t, x + v t ∆ t )∆ x, which is equivalent to, c t +∆ t ϕ ( t + ∆ t, x ) ≈ f ( t, x )∆ t + c t ϕ ( t, x + v t ∆ t ) . To simplify the system dynamics, we approximate c t +∆ t and ϕ ( t, x + v t ∆ t ) with c t + c (cid:48) t ∆ t and ϕ ( t, x ) + v t ∂ x ϕ ( t,x )∆ t , respectively. Then, dividing both sides by ∆ t and letting ∆ t goes to zero, weget, c t ∂ t ϕ ( t, x ) + c (cid:48) t ϕ ( t, x ) − c t v t ∂ x ϕ ( t, x ) = f ( t, x ) . (19)We use ∂ t and ∂ x to denote, respectively, the partial derivative with respect to time t and space x .Integrating both sides of (19) with respect to x from x to X max , we get, c t ∂ t Φ( t, x ) + c (cid:48) t Φ( t, x ) − c t v t ∂ x Φ( t, x ) = (cid:90) X max x f ( t, ξ ) dξ. (20)Note that Φ( t, x ) denotes the fraction of active trips with the remaining trip lengths more than x at time t , Φ( t, x ) = (cid:90) X max x ϕ ( t, ξ ) dξ . Then, using relation z (cid:48) ( t ) = v t as a direct result of (4), we have, ddt (cid:0) c t Φ( t, x − z ( t )) (cid:1) = c t ∂ t Φ( t, x − z ( t )) + c (cid:48) t Φ( t, x − z ( t )) − c t v t ∂ x Φ( t, x − z ( t )) . Therefore, the equality given in (20) can be written as, ddt (cid:0) c t Φ( t, x − z ( t )) (cid:1) = (cid:90) X max x − z ( t ) f ( t, ξ ) dξ. Thus, by integrating both sides with respect to time from 0 to t and setting y = x − z ( t ), thedynamics of the system can be presented as, c t Φ( t, y ) = (cid:90) t (cid:90) X max y + z ( t ) − z ( s ) f ( s, ξ ) dξds. (21) meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
Note that we assume the system is empty at time 0. Moreover, taking partial derivative with respectto x from both sides, considering that c t is independent of x , and applying Leibniz’s integral rule,we obtain, ∂ x (cid:0) c t Φ( t, x ) (cid:1) = c t ∂ x Φ( t, x ) = − (cid:90) t f (cid:0) s, x + z ( t ) − z ( s ) (cid:1) ds. (22)Finally, using (18) and the definition of in-flow measure F and its probability density function f ,the dynamics of the fraction of the active trips at time t , c t satisfies c (cid:48) t = (cid:90) X max f ( t, x ) dx − o t = (cid:90) X max f ( t, x ) dx + c t v t ∂ x Φ( t, x ) | x =0 . Here, ∂ x Φ( t, x ) | x =0 demonstrates the right derivative at 0 as the left derivative is not defined.Substituting (22), gives, c (cid:48) t = (cid:90) X max f ( t, x ) dx − v t (cid:90) t f (cid:0) s, z ( t ) − z ( s ) (cid:1) ds. Integrating both sides with respect to time and using Tonelli’s theorem (see, e.g., Theorem 18.3 ofBillingsley (2012)), to change the order of the integration, we get, c t = (cid:90) t (cid:90) X max f ( r, x ) dxdr − (cid:90) t (cid:90) r v r f (cid:0) s, z ( r ) − z ( s ) (cid:1) dsdr = (cid:90) t (cid:90) X max f ( r, x ) dxdr − (cid:90) t (cid:90) ts v r f (cid:0) s, z ( r ) − z ( s ) (cid:1) drds. In the light of the equality given in (4) and considering that z (cid:48) ( t ) = v t , the result of the abovediscussion about the characteristic travel distance is summarized in Proposition 1. To state theproposition rigorously, we make the following assumption. Assumption 2.
Let
G > be a constant. Then, for all Borel measurable subset of T d × X suchas B , we assume that the in-flow measure F ∈ P ( T d × X ) satisfies F ( B ) ≤ Gλ ( B ) , (23) where λ is Lebesgue measure on R . Given the demand profile m , defined in (1), and the constant G >
0, let P m,G denote the set of allin-flow measures F ∈ P ( T d × X ) that satisfies Assumption 2 and demand constraint given in (16).Then, by Radon–Nikodym theorem (see e.g. Theorem 32.2 of Billingsley (2012)) any F ∈ P m,G admits a probability density function denoted by f . Further, let M m,G denotes the set of all positivedis-aggregated in-flow measures E ∈ P ( T d × X × T a ) such that F = F ( E ) ∈ P m,G .Considering the definition of P m,G , the system avoids having a mass of departures at the sametime. For small values of G , an in-flow measure F ∈ P m,G is very smooth (without any drastic meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted FIGURE 1 S t includes the area inside the red lines. The dashed red curve indicates a sample path of z ( t ) − z ( τ ) , τ ∈ [0 , t ] which is not included in S t . change in a short interval of time). However, when G gets larger, the feasible in-flow measuresmay have larger fluctuations. Note that Assumption 2 is consistent with the regularity assumptionmade on the demand profile m , i.e., Assumption 1. In the light of definition of P m,G , we have thenext proposition about the dynamics of the characteristic travel distance. Proposition 1.
Consider a traffic system with speed function V and in-flow measure F ∈ P m,G .Then, the characteristic travel distance of the system z F is the solution of the following set ofequations, (cid:40) z F ( t ) = (cid:82) t V (cid:16) F (cid:0) S s ( z F ) (cid:1)(cid:17) ds,S t ( z F ) := (cid:8) ( τ, ξ ) (cid:12)(cid:12) τ ∈ [0 , t ] ∩ T d , ξ ∈ (cid:0) z F ( t ) − z F ( τ ) , ∞ (cid:1) ∩ X (cid:9) . (24)In the set of equations defined in (24), we use subscript F to emphasize the dependency of thevariables on the in-flow measure.Proposition 1 provides the relation between the characteristic travel distance z F and the in-flow measure F , i.e. the distribution of the departures. In fact, S t ( z F ) contains the pairs of thedeparture times and trip lengths of the users that are travelling at time t , in a traffic system withthe characteristic travel distance z F . In other words, for all ( t d , x ) ∈ S t , an agent with departuretime t d and trip length x is in the system at time t . On the other hand, if ( t d , x ) / ∈ S t , the agenthas either not departed or finished her travel before t , as illustrated in Figure 1 .Note that the set of equations given in (24) should be solved simultaneously. Therefore, we shouldinvestigate the existence and uniqueness of the characteristic travel distance derived in Proposition The Borel measurability of S t is obvious. meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
1. To address this problem, we need to introduce some notations. For any compact subset of R n such as C , C ( C ) represents the space of all real valued continuous functions defined on C . Weassume that C ( C ) is equipped with the uniform norm, i.e., ∀ u ∈ C ( C ) : (cid:107) u (cid:107) := sup t ∈ C | u ( t ) | . Also, for a constant
M >
0, we define the following norm on C ( C ) which is equivalent to the uniformnorm on the compact space C , ∀ u ∈ C ( C ) : (cid:107) u (cid:107) M := sup t ∈ C | e − tM u ( t ) | . We denote by d ( · , · ) and d M ( · , · ) the distances associated to (cid:107) · (cid:107) and (cid:107) · (cid:107) M , respectively. Also, wedefine the function U : C ( T ) × P m,G (cid:55)→ C ( T ) such that (cid:40) U ( z, F ) = ˜ z, ˜ z ( t ) = (cid:82) t V (cid:16) F (cid:0) S s ( z ) (cid:1)(cid:17) ds. (25)As demonstrated in the next proposition, systems with smooth in-flow measures, in the senseof Assumption 2, and Lipschitz continuous speed functions, admit a unique characteristic traveldistance. Proposition 2.
For all in-flow measures F ∈ P m,G and Lipschitz continuous speed functions V , there exists a unique function z F ∈ C ( T ) which satisfies the set of the equations given in (24). Banach Fixed-Point Theorem (see Theorem 3.48 of Aliprantis and Border (2006)) enables usto obtain z ∗ , the solution of the set of equations given in (24) which is the characteristic traveldistance of a system with in-flow measure F . The next corollary provides a procedure to obtain z ∗ , based on the successive application of the mapping U defined in (25). Corollary 1.
Fix F ∈ P m,G . Then, starting with an arbitrary element z ∈ C ( T ) , the sequence z l defined as z l := U ( z l − , F ) , l ≥ , (26) converges to z ∗ which is the solution of (24). Consider an arbitrary weakly convergent sequence of probability measures { F k } k ∈ N in P m,G suchthat F k ⇒ F , where ⇒ denotes the weak convergence of the measures. Then, Proposition 3 clarifiesthat the limit probability measures F lies in P m,G , too. Proposition 3.
For any G ∈ R + , P m,G is a closed subset of P ( T d × X ) in the weak convergencetopology. meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted In the following proposition we demonstrate that the characteristic travel distance is continuouswith respect to the in-flow measure. Suppose that z ∗ k is the solution of the set of equations given in(24) for F k . That means z ∗ k is the characteristic travel distance of a system having departures withdistribution F k . Similarly, consider z ∗ as the corresponding solution to the probability measure F . Further, suppose that the probability space P m,G and the set of continuous functions C ( T ) areequipped, respectively, with the weak and uniform convergence topology. Proposition 4.
Suppose that the speed function V is Lipschitz continuous. Then, the charac-teristic travel distance is continuous wrt to the in-flow measure. In other words, if F k ⇒ F then z ∗ k → z ∗ . In the proof of Proposition 4, we provide a convergence bound for the limit of the characteristictravel distances. Thus, the solution of the equations given in (24) is also continuous wrt the dis-aggregated in-flow measure E , and the next corollary can be considered as a consequence of thecontinuity of F (recall that F = F ( E )). Corollary 2.
Suppose that the speed function V is Lipschitz continuous. Then, the character-istic travel distance is continuous wrt to the dis-aggregated in-flow measure E . Note that the dynamics of the system do not depend explicitly on E . The reason is that thetravel time depends on the speed and the speed depends only on the number of agents in thesystem independent of their desired arrival times. In this section, using the results derived in the previous sections, we provide a DTCP formulationbased on the MFGs approach by assuming that the number of travelers goes to infinity in DTCP.Recall that the characteristic travel distance is provided in Proposition 1 and its existence as wellas its uniqueness is demonstrated in Proposition 2. Therefore, considering the objective functionof an arbitrary player given in (17) and the relation between the travel time and the characteristictravel distance provided in (5), we can define the MFGs-DTCP as follows,min t d ∈T d J ( t d ; x, t a ; F ) = αT ( t d , x ) + β (cid:0) t a − t d − T ( t d , x ) (cid:1) + + γ (cid:0) t d + T ( t d , x ) − t a (cid:1) + (MFGs-DTCP) s.t. (cid:40) z ( t ) = (cid:82) t V (cid:16) F (cid:0) S s ( z ) (cid:1)(cid:17) ds,S t ( z ) := (cid:8) ( τ, ξ ) (cid:12)(cid:12) τ ∈ [0 , t ] ∩ T d , ξ ∈ (cid:0) z ( t ) − z ( τ ) , ∞ (cid:1) ∩ X (cid:9) , (27) T ( t d , x ) = z − (cid:0) x + z ( t d ) (cid:1) − t d . (28)Note that in the DTCP model defined in (8), all the three relations should be considered simul-taneously, since the choice of an arbitrary player affects the system significantly. However, in the meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
MFGs-DTCP model, the set of equations given in (27) can be investigated independent of traveltime identity provided in (28). This is due to the fact that, as the number of players n → ∞ , theimpact of a player on the system vanishes. Moreover, note that the MFGs-DTCP model considersthe system at a macroscopic level. That is, we do not need to follow the states and decisions offinitely many players .The following definition clarifies the DTUE as the ε -Mean Field Equilibrium ( ε -MFE) of theMFGs-DTCP. Definition 2.
Given a constant ε ≥ F ( E ∗ ) ∈ P m,G (with E ∗ ∈ M m,G ) is an ε -Mean FieldEquilibrium ( ε -MFE) for the MFGs-DTCP, if the following relation holds, E ∗ (cid:0)(cid:8) ( t d , x, t a ) ∈ T d × X × T a (cid:12)(cid:12) J ( t d ; x, t a ; F ( E ∗ )) ≤ J ( t d ; x, t a ; F ( E ∗ )) + ε, ∀ t d ∈ T d (cid:9)(cid:1) = 1 . Note that, Mean Field Equilibrium (MFE) is ε -MFE with ε = 0. Also, E ∗ can be expressed as thefixed-point of a map H : M m,G (cid:55)→ M m,G defined as, H ( ˆ E ) := (cid:110) E ∈ M m,G (cid:12)(cid:12) E (cid:0)(cid:8) ( t d , x, t a ) ∈ T d × X × T a (cid:12)(cid:12) J ( t d ; x, t a ; F ( ˆ E )) ≤ J ( t d ; x, t a ; F ( ˆ E )) + ε, ∀ t d ∈ T d (cid:9)(cid:1) = 1 (cid:111) . (29)The equivalence of the two definitions provided above holds obviously. In the next section, theexistence of an equilibrium for the MFGs-DTCP will be examined. In this section, we show that there exists an equilibrium solution for the MFGs-DTCP. To provethe existence, we first need to examine whether the cost function given in MFGs-DTCP is jointlycontinuous.
Proposition 5.
Suppose that the speed function is bounded from above and below by V max and V min , respectively, such that V max > V min > . Then, the cost function of the MFGs-DTCP, J ,is jointly continuous on ( T d × X × T a × P m,G ) . Further, the continuity of the cost function on T d × X × T a is Lipschitz. Considering that the velocity V is a function of the congestion c t , the condition V ≥ V min > c t = 1, i.e., the total demand is less than thecapacity of the network.Note that Proposition 5 demonstrates the joint continuity of the cost function only on ( T d ×X × T a × P m,G ). If we assume that the jointly continuity condition is extendable to ( T d × X × T a ×P ( T d × X )), the problem admits a MFE, by Theorem 4.9 in Lacker (2018). Otherwise, we have thenext proposition on the existence of the equilibrium. More precisely, any measure zero subset of agents’ indices set is negligible. meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted Proposition 6.
For an arbitrary ε > , there exists a constant G ∈ R + such that MFGs-DTCPadmits an ε -MFE in the probability space P m,G . This means that there exists an dis-aggregatedin-flow measure E ∗ ∈ M m,G which is the fixed-point of mapping H , given in (29).
4. MFGs Model for the MFGs-DTCP
In this section, we aim to characterize the departure time user equilibrium (DTUE) for the MeanField Games model discussed in the previous section. Recall that Proposition 6 guarantees theexistence of the departure time equilibrium.Consider the optimal behavior of an arbitrary player, assuming that the decisions of the otherplayers are known. Specifically, fix a player with the desired arrival time t a and the trip length x as well as an in-flow measure F as the proxy for the departure times and trip lengths of the otherplayers. By Proposition 2, this system has a unique characteristic travel distance z . Then, basedon Proposition 1, we have v t := V (cid:0) F ( S t ) (cid:1) , which is the velocity of the system at time t . Also, notethat (5) can be written as, (cid:90) t d + T ( t d ,x ) t d v t dt = x. (30)Then, taking derivative with respect to t d from both sides of the above equality implies, (cid:0) ∂ t T ( t d , x ) (cid:1) v t d + T ( t d ,x ) − v t d = 0 , which yields to, ∂ t T ( t d , x ) = v t d v t d + T ( t d ,x ) − . (31)Suppose that α > β . We apply the first order condition of optimality to determine the equilibriumdeparture time considering the cost function J defined in (17). In the case that t a > t d + T ( t d , x ),the third term in the cost function is equal to zero, and we get, α∂ t T ( t d , x ) − β ( ∂ t T ( t d , x ) + 1) = 0 . Substituting (31) in the above equation, we get, v t ∗ d v t ∗ d + T ( t ∗ d ,x ) = αα − β . (32)Similarly, if t a < t d + T ( t d , x ), the second term in the cost function is equal to zero and the followingequality can be derived by applying the first order condition, v t ∗ d v t ∗ d + T ( t ∗ d ,x ) = αα + γ . (33) meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
Based on (32) and (33), it is optimal for an agent who arrives before (after) his desired arrival timeto choose the departure time such that the ratio of the system velocity at departure and arrivaltime be equal to αα − β ( αα + γ ). For on-time agents, based on left and right derivatives of the costfunction we can get, αα + γ ≤ v t ∗ d v t ∗ d + T ( t ∗ d ,x ) ≤ αα − β . (34)Summarizing relations provided in (32), (33), and (34), and similar to Proposition 2 in Lamotteand Geroliminis (2018), we can conclude the following proposition about the optimal choice of anarbitrary agent given the distribution of the others’ departures. Proposition 7.
The optimal departure time t ∗ d of a player having desired arrival time t a andtrip length x with cost function J , given in (17), satisfies the following conditions, αα + γ ≤ v t ∗ d v t ∗ d + T ( t ∗ d ,x ) ≤ αα − β . (35) Further, for an early and late player we have, v t ∗ d v t ∗ d + T ( t ∗ d ,x ) = (cid:40) αα − β , t a > t d + T ( t d , x ) , αα + γ , t a < t d + T ( t d , x ) . (36)Note that the cost function J given in (17) is continuous with respect to departure time on acompact set T d , based on Proposition 5. Therefore, there exists a point at which this function isminimized .Suppose D : T a × X (cid:55)→ T d defines a solution to Proposition 7. That is, D maps the desired arrivaltime and trip length to the departure time t ∗ d which satisfies (35) and (36), specifically, t ∗ d = D ( t a , x ) . (37)Our next goal is to clarify the relation between the demand profile m and the in-flow measure F in terms of D . Consider a population of size n , and let t ia and x i denote the i -th player’s desiredarrival time and trip length, respectively. We assume that { t ia , x i } ni =1 are i.i.d random variableswith the distribution m . By (37), D determines the optimal departure times of the players, i.e., D ( t ia , x i ) = t id . Then, based on Glivenko-Cantelli Law of Large Numbers (see e.g. Section 3.2.2 ofCardaliaguet (2018)), almost surely and in L , F n := n (cid:80) i ∈ [ n ] δ t id ,x i converges weakly to F , whichis the distribution of ( t id , x i ) . This result shows that the limit of (11), F , exists and can be derived The minimum could be achieved on the boundary of T d . But, the cost function includes a term aiming to minimizethe difference between desired and effective arrival time. Thus, we assume that the minimum of J satisfies (35) and(36). We assume D is measurable; thus, { t id , x i } ni =1 are i.i.d RVs, too. meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted based on the demand profile m . Using a similar discussion we can show that the limit of (12), E ,exists and represents the the distribution of ( t id , x i , t ia ).To clarify the relation between in-flow measure F and demand profile m , regarding the function D , consider disaggregated in-flow measure E . Note that E (∆ t d , ∆ x, ∆ t a ) indicates the fraction ofthe trips having departure time in ∆ t d , trip length in ∆ x , and desired arrival time ∆ t a . Then, wecan state the following proposition. Proposition 8.
Suppose D , which is defined in (37), is differentiable with respect to t a and ∂ t D ( t a , x ) > , ∀ t a ∈ T a . Also, assume that the demand profile m satisfies Assumption 1 anddis-aggregated in-flow measure E admits a probability density function e . Then, we have, e ( D ( t a , x ) , x, t a ) = m ( dx, dt a ) ∂ t D ( t a , x ) . (38)Note that (38) provides the relation between the dis-aggregated in-flow measure and the demandprofile that is consistent with the constraint given in (15). In this section, we discuss the MFGs model for the MFGs-DTCP which characterizes the equilib-rium of the system. Note that the goal of the MFGs analysis is to examine the equilibrium behaviorof travelers (i.e., DTUE) not the individual’s optimal departure time. On the other hand, basedon Corollary 1, a generic player would be able to obtain the characteristic travel distance anddetermine, using Proposition 7, his strategy given the in-flow measure F . Therefore, F is the meanfield of the MFGs-DTCP, i.e., F captures the required information for a generic agent to describeand analyze the system. Denoting the actual arrival time by ¯ t a = t d + T ( t d , x ), we can summarizethe discussions and results provided in the previous sections to derive the mean field games model: αα + γ + t a > ¯ t a (cid:0) αα − β − αα + γ (cid:1) ≤ v td v ¯ ta ≤ αα − β + t a < ¯ t a (cid:0) αα + γ − αα − β (cid:1) with solution t d = D ( t a , x ) ,e ( D ( t a , x ) , x, t a ) = m ( dx,dt a ) ∂ t D ( t a ,x ) , f (cid:0) D ( t a , x ) , x ) = (cid:82) T a e ( D ( t a , x ) , x, t a ) dt a , F = (cid:82) f ( t d , x ) dt d dx,z ( t ) = (cid:82) t V (cid:16) F (cid:0) S s ( z ) (cid:1)(cid:17) ds,S t ( z ) = (cid:8) ( τ, ξ ) (cid:12)(cid:12) τ ∈ [0 , t ] ∩ T d , ξ ∈ (cid:0) z ( t ) − z ( τ ) , ∞ (cid:1) ∩ X (cid:9) ,T ( t d , x ) = z − (cid:0) x + z ( t d ) (cid:1) − t d ,F ∈ P m,G . (39)(40)(41)(42)(43)(44)The MFGs-DTCP model given in (39-44) can be explained as follows. Suppose that the decisionof the players are captured by the in-flow measure F . Using equations (41) and (42), and in the Consider two agents with desired arrival times in t a < t a and the same trip length x . Let t a = D ( t a , x ) be thedeparture time of the first player. Then, the virtual user travels a distance of x in the time interval [ t d , t a ). Therefore,since the velocity is positive, it is rational to assume D ( t a , x ) < D ( t a , x ). meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted light of Corollary 1, the associated characteristic travel distance z can be obtained. Then, a genericagent can employ the relation given in (43) to determine his travel time. Subsequently, the playeris able to obtain the optimal departure time based on (39) along with the function D . Finally,the demand profile m , which is known, will be transferred according to (40) that specifies therelation between the in-flow measure, dis-aggregated in-flow measure and demand profile. Thus,the optimal distribution of the departure times will be derived as a function ˆ F . Now, based onDefinition 2, the in-flow measure F would be DTUE of the MFGs-DTCP if ˆ F obtained based onthe above procedure is equal to the initial in-flow measure F .Note that (39) and (40) are the main components of the model. While the former gives theoptimal condition for the decision of a generic player, the latter captures the distribution of thedecisions. The rest are required to make a bridge between relations given in (39) and (40). Remark 1.
The MFGs-DTCP model provided in (39-44) can be extended to capture user spe-cific coefficients α , β , and γ in the cost function where their distributions are given through demandprofile m . This paves the way to consider heterogeneous user preferences when solving the DTCPproblem. In this section, we derive the discrete version of the system of equations given in (39-44) to solve theMFGs model numerically. Let ∆ t and ∆ x denote small intervals in the time and space, respectively,such that ∆ x ≥ V max ∆ t , where V max indicates the maximum of the network free-flow speed. Thismeans that a trip cannot travel more than ∆ x in a time interval ∆ t . We denote the time and spacediscretization as follows, • The time discretization: ( τ ) = [ τ ∆ t, ( τ + 1)∆ t ) , (45) • The space discretization ( κ ) = [ κ ∆ x, ( κ + 1)∆ x ) . (46)All time intervals [ τ ∆ t, ( τ + 1)∆ t ) will be denoted hereafter by ( τ ). A similar interpretation holdsfor ( κ ). Note that these definitions are matched with time horizon T and space set X such that theunion of all the defined intervals is equal to the corresponding set, that is ∪ ( τ ) = T and ∪ ( κ ) = X .Similarly, let ( τ d ) and ( τ a ), respectively, denote the departure and arrival time intervals where theunion of ( τ d ) and ( τ a ) is equal to T d and T a , respectively. Indeed, relations in (40) consider the dependency of the in-flow measure on the dis-aggregated in-flow measure inDefinition 2. meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted We define an equivalent discrete version of the demand profile m as follows, π ( τ a , κ ) := m (cid:0) ( τ a ) , ( κ ) (cid:1) = (cid:90) ( τ a ) (cid:90) ( κ ) m ( dt a , dx ) . (47)We assume that the velocity of the system is constant in each time interval τ and it is capturedby v τ . Then, the discrete analogous of the optimal condition, given in (7), can be presented as, αα + γ + τ a > ¯ τ a (cid:0) αα − β − αα + γ (cid:1) ≤ v τd v ¯ τa ≤ αα − β + τ a < ¯ τ a (cid:0) αα + γ − αα − β (cid:1) , (48)where ¯ τ a := τ d + T ( τ d , κ ) is the actual arrival time interval. Here, with an abuse of notation, T ( τ d , κ )is the travel time of an agent having departure time in ( τ d ) and trip length in ( κ ). Suppose that thefunction D is a solution of (48). That means, τ ∗ d = D ( τ a , κ ) is the optimal departure time intervalfor a traveler having desired arrival time in ( τ a ) and trip length in ( κ ). Additionally, let µ ( τ d , κ, τ a )indicate the fraction of departures in time interval ( τ d ) with trip length in ( κ ) having desired arrivaltime in ( τ a ). Then, similar to (38), we can capture the relation between the demand profile π and µ by, µ ( τ d , κ, τ a ) = π ( τ a , κ )∆ tD ( τ a + 1 , κ ) − D ( τ a , κ ) . We also define the discrete characteristic travel distance by, ζ ( θ ) := ∆ t θ − (cid:88) τ =0 v τ . (49)Then, if we denote by Γ θ ( ζ ) the indices corresponding to the agents that are travelling in theinterval θ , we can get, Γ θ ( ζ ) := (cid:8) ( τ d , κ ) (cid:12)(cid:12) κ > ζ ( θ ) − ζ ( τ ) (cid:9) . Moreover, the velocity in a system with the discrete characteristic travel distance ζ in the timeinterval ( τ ), v τ , would satisfy, v θ = V (cid:16) (cid:88) ( τ d ,κ ) ∈ Γ θ ( ζ ) p ( τ d , κ ) (cid:17) , where p ( τ d , κ ) = (cid:80) τ a µ ( τ d , κ, τ a ) is the fraction of departures in ( τ d ) having trip length in ( κ )independent of the desired arrival time. Similarly, we define the the travel time of an agent havingdeparture time in ( τ d ) and trip length in ( κ ), T ( τ d , κ ). That is, T ( τ d , κ ) := ζ − (cid:0) κ + ζ ( τ d ) (cid:1) − τ d . Here, ζ − shows the inverse of the function ζ , defined in (49). meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
Therefore, the discrete analogous of the MFGs system defined in (39-44) can be represented as, αα + γ + τ a > ¯ τ a (cid:0) αα − β − αα + γ (cid:1) ≤ v τd v ¯ τa ≤ αα − β + τ a < ¯ τ a (cid:0) αα + γ − αα − β (cid:1) with solution τ d = D ( τ a , κ ) ,µ (cid:0) D ( τ a , κ, τ a ) , κ (cid:1) = π ( τ a ,κ )∆ tD ( τ a +1 ,κ ) − D ( τ a ,κ ) , p ( τ d , κ ) = (cid:80) τ a µ ( τ d , κ, τ a ) ,ζ ( θ ) := ∆ t (cid:80) θ − τ =0 V (cid:16) (cid:80) ( τ d ,κ ) ∈ Γ τ ( ζ ) p ( τ d , κ ) (cid:17) , Γ θ ( ζ ) := (cid:8) ( τ d , κ ) (cid:12)(cid:12) κ > ζ ( θ ) − ζ ( τ ) (cid:9) ,T ( τ d , κ ) := ζ − (cid:0) κ + ζ ( τ d ) (cid:1) − τ d , (cid:80) τ d p ( τ d , κ ) = (cid:80) τ a π ( τ a , κ ) , ∀ κ. (50)(51)(52)(53)(54)(55)The set of equations given in (50-55) can be explained similar to the ones provided in (39-44).That means the distribution of the players’ decisions, p , can be treated as the mean-field of thediscrete system. Suppose that the decision of the players are given by p for all τ d and κ . Using (52)and (53), an arbitrary player can derive the discrete characteristic travel distance ζ , and determinehis travel time based on (54). Then, the player could find his optimal departure time using (50)and obtain the function D . Finally, using (51), ˆ p can be obtained as the distribution of reviseddeparture times wrt the function D , which is derived based on (50). Finally, the DTUE is thefixed-point of this procedure, i.e., p = ˆ p . The equilibrium solution for the DTCP cannot be derived directly from the user optimal controlconditions but through an iterative solution method (Zhong et al. 2011). In this section, we presentan algorithm that can be utilized to numerically solve the discrete MFGs framework.Recall that the ε -MFE of the DTCP is the fixed-point of (29). Therefore, we can apply fixed-pointalgorithms with a similar optimality conditions to calculate the equilibrium point of the problem.In the discrete MFGs framework, determining the equilibrium requires obtaining an approximationfor v ¯ τ a given in (50). From the travelers point of view, this approximation enables the playersto predict the travel costs required to choose the optimal departure times. The prediction modelhas to take into account the parameters and evolution of the network, which are captured by thegeneralized bathtub model. Here, to calculate the equilibrium approximation based on the usersdecisions, we propose a heuristic algorithm. Our heuristic algorithm is based on the variationalinequality theory discussed in Noor (1988). The core idea is to use the delay value (¯ t a − t a ) toupdate the departure times in each iteration. In this case, we also consider the travelers meanspeed function with the same desired arrival time as the variable to predict the arrival time of thenext simulation with respect to their trip length. Indeed, we reschedule the departure time for eachtraveler based on equations (50-55).The original algorithm is detailed in Friesz and Han (2019). It is proposed for a continuousdynamic assignment model while we use a discrete version of it based on Ameli, Lebacque, and meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted Leclercq (2020a). Note that in each iteration of the algorithm, a proportion of travelers are selectedfor rescheduling. This proportion is equal to the product of the total demand and a step size. Thestep size is a coefficient between zero and one that is decreasing during the optimization process(Ameli, Lebacque, and Leclercq 2020b). In this study, the step size is fixed to one over the iterationindex. We also add a smart selection process (inspired from Sbayti, Lu, and Mahmassani (2007))to the algorithm in order to speed up the convergence. The process sorts all the trips based ontheir travel cost (17) and then selects the trips with the higher travel costs for the reschedulingprocess. Note that in all numerical examples, the length of the time interval in the discrete modelis considered as one second.
5. Numerical experiments
In order to examine the efficiency of our MFGs model, we first compare its performance with oneof the recently proposed models in the literature. We then apply the MFGs framework to a large-scale test case in order to evaluate its performance and examine how the optimization procedureto determine the DTUE affects the congestion level of the network’s real state.
Lamotte and Geroliminis (2018) used a quadratic function for the network mean speed function.They use a trip-based macroscopic fundamental diagram (MFD) model (Leclercq, S´en´ecat, andMariotte 2017) and not the generalized bathtub model. However, both approaches share a commonground and produce similar results in terms of the traffic dynamics. Here, we apply our proposedframework with the exact same demand profile, the same parameters for the cost function, includinga smooth approximation of the α - β - γ preference, modeled by the marginal utility of the time spentat home h ( t ) = α , the marginal utility of time spent at work w ( t ) = γ − β + arctan(4(¯ t ia − t ia )) γ + βπ )and the same parameters for the mean speed function, i.e., the network capacity and free flowspeed. The description of all simulation parameters are presented in Section 5.1 of Lamotte andGeroliminis (2018). Note that trip-lengths are uniformly distributed between 0 and 3. The solutionmethod in Lamotte and Geroliminis (2018) is conducted on a day-to-day basis using a selectionmethod inspired by Method of Successive Average (MSA) and an optimization method based onthe grid search (detailed in Lamotte and Geroliminis (2016)). Table 2 compares the optimizationresults of this model with the one proposed in this paper based on MFGs. The results show that,considering the relative cost, the proposed MFGs method outperforms the previous approach by5 . meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted that the MFGs framework is consistent with the existing method in the literature for the morningcommute problem.
TABLE 2 The quality of the equilibrium approximation.
Solution method Total numberof iteration Convergence indicator[relative cost] Averagecost Total travel time[sec]MFGs method 259 3.37E-03 12.01 26984
Lamotte and Geroliminis (2018) (a) Time series of accumulation (b) Time series of speed(c) Cumulative departure and arrival curves.
FIGURE 2 Simulation results with heterogeneous trip-length: Speed MFD based framework with grid search(Lamotte and Geroliminis 2018) versus MFGs framework
To further compare the properties of the final solution for both algorithms, we consider thecumulative departure and arrival curves that provide the characteristics of all trips, and the time-evolution of accumulation and mean speed in the network, see Figure 2. The MFGs framework meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted provides a solution with a lower maximum accumulation (Figure 2(a)) and a higher speed (Fig-ure 2(b)) than the grid search algorithm. It means that the system is closer to the system optimum,defined as the solution where the total travel time of all vehicles is minimum. While it is not theobjective function we aim to minimize, it is interesting to notice that reducing further the totalindividual costs has a positive impact on the overall system. Figure 2(c) illustrates how trips arestarted considering trip lengths and departure time. This is a crucial feature as the existing solutionmethods require prior assumptions on such a sorting to reduce the exploration of the solution space.For example, in Lamotte and Geroliminis (2018), partial FIFO sorting conditions are mandatoryto derive the optimal solutions. Our MFGs framework relaxes such conditions and can provide afull exploration of the solution space. Figure 2 exhibits five time periods (the dash line boxes ineach figure) where FIFO patterns are observed in the optimal solution of the grid search solutionbut no sorting pattern in the MFGs solution. In Figure 2(c), the inflow rate of the MFGs solutionis higher than the grid search while the slope of the outflow rate is less than the grid search. Thistest case shows how important it is to relax the sorting assumptions based on the trip lengths toget the optimal solution, which can only be achieved by the prposed MFGs framework. The application of the proposed MFGs framework is easily scalable to much larger instances, whichis the main advantage of MFGs over the classic game theory approaches. In this section, we considera test case corresponding to the northern part of a metropolis in France (Lyon) and all trips duringthe morning peak hours, i.e., more than 60,000 trips in total.
We implement and apply theproposed model to the northern part of Lyon Metropolis (Lyon North). Lyon North network covers25 km and includes 1,883 nodes and 3,383 links. The map is shown in Figure 3(a). The originaldemand setting includes all trips during the morning peak hours from 6:30 AM to 10:40 AM(62,450 trips). It has been calibrated to represent realistic traffic conditions (Krug, Burianne, andLeclercq 2019). All trips have an origin and destination on the real network and a departuretimes. At the link level of the network (Figure 3(b)), the origins set contains 94 points and thedestinations set includes 227 points. In this study, we only keep the original trip lengths as thegeneralized bathtub model does not account for the local traffic dynamics. Some trips have originsor destinations outside the covered area (51,215 trips) and will not be considered in the departuretime optimization. Note that 11235 trips are fully interior. For those, the original departure time isdisregarded and a desired arrival time is assigned. We divide them into seven classes with differentdesired arrival times. The desired arrival time of each user is deduced from the real arrival time meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted (a) Mapping data © Google 2020 (b) The traffic network using in micro-simulation.
FIGURE 3 The northern part of Lyon Metropolis (Lyon North).TABLE 3 Demand scenario for Lyon North with multiple desired arrival time
Class ofthe trips Share ofthe trips Number ofthe trips Mean triplength (km) Arrival timeinterval Estimated Desiredarrival timeClass 1 13.73% 1,543 2.53 6:30-7:15 7:00Class 2 13.84% 1,555 2.58 7:15-7:45 7:30Class 3 15.42% 1,732 2.55 7:45-8:15 8:00Class 4 18.30% 2,056 2.65 8:15-8:45 8:30Class 5 15.05% 1,691 2.63 8:45-9:15 9:00Class 6 11.82% 1,328 2.70 9:15-9:45 9:30Class 7 11.84% 1,330 2.63 9:45-10:30 10:30of the user based on real data (Krug, Burianne, and Leclercq 2019, Alisoltani et al. 2019). Thepercentage of the trips per class and their desired arrival time are presented in Table 3.The network speed function has been calculated in Mariotte et al. (2020). The cost functionparameters, i.e., the α - β - γ scheduling preferences are defined based on the study of Lamotte andGeroliminis (2018): α = 1, β = 0 . . k and γ = 1 . k . In order to consider only the heterogeneityof trip length and desired arrival time distributions, k is fixed to 5 for all trips in this experiment. The optimization process is started with an initial solution wherethe targeted travelers with a higher trip length in all classes start their trip sooner than othersbased on the network free-flow speed ( v max = 13 . m/s ). The heuristic algorithm converges after56 iterations to an equilibrium approximation. The results for the convergence pattern is presentedin Figure 4(a). The final average cost per traveler is 326 .
92, and the figure shows that the finalsolution is stable. As in the previous test case, the MFGs algorithm converges very fast; however,the performance of heuristic/search algorithms depends on the initial solution. meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted Figure 4(b) presents the evolution of the network’s total travel time during the optimizationprocess. Similar to the convergence pattern, the total travel time decreases and converges to astable value. Therefore, the final solution can be an equilibrium approximation for the problem.To provide more insights, we also assess the network and equilibrium features overtime during theoptimization process. (a) Convergence pattern. (b) Evolution of the total travel time in the optimizationprocess.
FIGURE 4 Results of the optimization process: The travel cost is calculated using (17) and the travel time isthe value of T ( t d , x ) for each traveler. Figure 5 presents the results for the accumulation of the network at each time step (∆ t = 1 sec )for the convergence process and the final iteration of the optimization process. In Figure 5(a),every blue extrema indicates the evolution of the accumulation in 4.17 hours simulation at oneiteration. The curve for the next iteration is started right after the previous one. The results showthat the accumulation is also decreasing during the optimization process, and as it is expected,the equilibrium approximation has a low value for the maximum accumulation (red line) of thefinal solution. Remind that the accumulation evolution in Figure 5(a) is drawn for interior trips.The equilibrium accumulation for the full demand is shown in Figure 5(b). The accumulation timeseries associated to the original demand patterns with all given departure times is also presented inthis figure. This curve is above the solution with optimized departure times. Therefore, the DTUEsolution improves the total travel time spent by all users in the system, which is defined by thearea between the accumulation time series.The convergence results regarding the different classes of trips are presented in Figure 6(a). Thealgorithm’s convergence pattern improves in the first three iterations continuously. However, after meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted (a) Evolution of accumulation for the interior trips: Thered line denotes the maximum accumulation of each iter-ation. (b) Accumulation of the real state of the network versusequilibrium approximation for all trips.
FIGURE 5 Results of the network’s performance overtime ( ∆ t = 1 s ) in the optimization process. Each iterationcontains 4.17 hours simulation [6:30 AM - 10:40 AM]. the third iteration, there are small variations for different classes. This is because of our algorithm’sheuristic nature that needs to search (explore) the solution space and then exploit it to find a localor the global optimum solution. Note that the exploration rate of heuristic methods depends onthe complexity of the solution space and the step size. Figure 6(b) illustrates another aspect of theequilibrium approximation where each green diamond represents the departure time, and each redcircle represents the arrival time of a trip. The duration of a trip is represented by a horizontalblue line between departure and arrival time. The trips of each class are sorted based on their triplength. In Figure 6(b), the deformations of the distributions for all classes show that non regularsorting pattern matches with the optimal solutions. So, again we show how important it is to notresort to any prerequisite about the sorting when designing the solution method and defining theoptimal conditions. For instance, in Figure 6(b), the departure and arrival time distributions forclass 4 (with desired arrival time 8:30 and the highest demand level) has a deformation on the triplengths interval [600-850], which illustrates that the partial sorting pattern like FIFO and LIFOdoes not stand.
6. Conclusion
This is the first paper that demonstrates the value of the MFGs approach in transportation equi-librium models. Specifically, this work focuses on modeling and characterizing the departure timechoice equilibrium, which is mathematically challenging for large-scale networks, using the MFGsapproach. We propose a new optimization framework based on the recent findings in transportationsystems and game theory. The framework is designed based on mean field game theory coupled withthe generalized bathtub model. The MFGs theory allows us to consider a large number of players meli et al.:
Departure Time Choice Models Based on Mean Field Games
Submitted (a) Absolute value of the average delay for each class of users. (b) Departure and arrival time distributions of theequilibrium approximation. FIGURE 6 Optimization results regarding the different classes of trips. Note that there are 11235 interior usersin the optimization process. with different desired arrival times. The idea is that each player in the system optimizes its strat-egy with respect to the mean-field of the strategies of the other players. Besides, the generalizedbathtub model can represent more complex interactions of supply and demand in a transportationsystem.Departure time choices of a group of rational travelers on a traffic network are intrinsically relatedto how they predict travel time. In this study, we develop a mathematical model through which ageneric player predicts the other players’ macroscopic behavior. Then, based on this prediction, hederives the dynamics of the system by obtaining the velocity. Having velocity, this player optimizeshis departure time strategy. Since the setting is a game with rational players, they look for a Nashequilibrium that can be obtained by a fixed point argument in the procedure of decision making.Moreover, we implemented the proposed model for the known setting of the morning commuteproblem in the literature, and the morning peak hour of the real traffic network of the LyonNorth. The numerical results for the first test case demonstrate the value of the MFGs frameworkcompared to existing models in the literature. The large-scale test case shows that the proposedframework is able to represent the equilibrium problem with multiple desired arrival times for alarge number of trips that was little addressed before. For the equilibrium calculation, we adapta heuristic fixed-point algorithm that converges very fast. The proposed model also provides agood approximation for the equilibrium. The optimization results on both experiments show thatoptimization based on the mean-field of the users’ strategies performs significantly better thanthe solution methods with Myopia assumption. The gain is much higher when the number ofusers and the scale of the problem are increased. The equilibrium approximation obtained by the meli et al.:
Departure Time Choice Models Based on Mean Field Games Submitted simulation-based optimization contains partial sorting patterns and provide interesting insights onthe prevailing sorting assumptions (FIFO or LIFO). These results are supported by Lamotte andGeroliminis (2018) and underline the importance of the empirical measurements compared to theanalytical studies (e.g., Fosgerau (2015), Daganzo and Lehe (2015)).
Acknowledgments
This work has received funding from the European Research Council (ERC) under the European Union’sHorizon 2020 research and innovation program. (Grant agreement No 646592 – MAGnUM project). We wouldlike to thank Dr. Negin Alisoltani for her helpful discussions and suggestions about the model implementation.
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Appendix. Proofs
Proof of Proposition 2.
Consider the mapping U defined in (25). Note that the space C ( T ) is aBanach space wrt the uniform norm. Therefore, by Banach Fixed-Point Theorem, it is sufficient to show meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted U is a contracting mapping, namely there is Λ ∈ [0 ,
1) such that for all z , z ∈ C ( T ), d (cid:0) U ( z , F ) , U ( z , F ) (cid:1) ≤ Λ d ( z , z ). Hence, consider z , z ∈ C ( T ) arbitrarily. Using triangle inequality we get: (cid:12)(cid:12) ˜ z ( t ) − ˜ z ( t ) (cid:12)(cid:12) ≤ (cid:90) t (cid:12)(cid:12)(cid:12) V (cid:16) F (cid:0) S s ( z ) (cid:1)(cid:17) − V (cid:16) F (cid:0) S s ( z ) (cid:1)(cid:17)(cid:12)(cid:12)(cid:12) dr. Then, since V is Lipschitz, we have: (cid:12)(cid:12) ˜ z ( t ) − ˜ z ( t ) (cid:12)(cid:12) ≤ Lip( V ) (cid:90) t (cid:12)(cid:12)(cid:12) F (cid:0) S s ( z ) (cid:1) − F (cid:0) S s ( z ) (cid:1)(cid:12)(cid:12)(cid:12) ds ≤ Lip( V ) (cid:90) t F (cid:0) S s ( z ) ∆ S s ( z ) (cid:1) ds. Using Assumption 2, we get: (cid:12)(cid:12) ˜ z ( t ) − ˜ z ( t ) (cid:12)(cid:12) ≤ G Lip( V ) (cid:90) t (cid:90) s (cid:12)(cid:12) z ( s ) − z ( s ) (cid:12)(cid:12) + (cid:12)(cid:12) z ( r ) − z ( r ) (cid:12)(cid:12) drds. Multiplying both sides of the above inequality by e − Mt , we obtain: (cid:12)(cid:12) ˜ z ( t ) − ˜ z ( t ) (cid:12)(cid:12) e − tM ≤ G Lip( V ) (cid:90) t (cid:90) s (cid:12)(cid:12) z ( s ) − z ( s ) (cid:12)(cid:12) e − Ms e − M ( t − s ) drds + G Lip( V ) (cid:90) t (cid:90) s (cid:12)(cid:12) z ( r ) − z ( r ) (cid:12)(cid:12) e − Mr e − M ( t − r ) drds ≤ G Lip( V ) d M ( z , z ) M t − e − Mt M , where the second inequality is based on the following relations: • ∀ s ∈ T , (cid:12)(cid:12) z ( s ) − z ( s ) (cid:12)(cid:12) e − Ms ≤ d M ( z , z ), • (cid:82) r e − M ( t − s ) ds = − e − Mr M , • (cid:82) t re − M ( t − r ) dr = Mt − e − Mt M , • (cid:82) t − e − Mr M dr = Mt − e − Mt M .Since Mt − e − Mt M is increasing wrt t >
0, taking supremum over t ∈ T yields to: d M (cid:0) ˜ z , ˜ z (cid:1) ≤ G Lip( V ) M T max − e − MT max M d M ( z , z ) . Now, choose M such that 2 G Lip( V ) MT max − e − MTmax M <
1. Considering the equivalency between (cid:107) · (cid:107) and (cid:107) · (cid:107) M , the proof is complete. (cid:3) Proof of Proposition 3.
Let { F k } k ∈ N ⊂ P m,G such that F k ⇒ F . Using the definition of P m,G andPortmanteau Theorem (see e.g., Theorem 2.1. of Billingsley (2013)), for all open sets O ⊂ T d × X , we have: F ( O ) ≤ lim inf k F k ( O ) ≤ Gλ ( O ) . Now, note that the Lebesgue measure is outer regular in the sense that a measurable set can be approximatedby an open set from outside. That means for all ε > B ⊂ T d × X , there exists anopen set O such that B ⊂ O and λ ( O ) < λ ( B ) + ε . Since B ⊂ O implies that F ( B ) ≤ F ( O ), we can write: F ( B ) ≤ F ( O ) ≤ Gλ ( O ) < Gλ ( B ) + Gε. As ε > F ( B ) ≤ Gλ ( B ) . (56) meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
Additionally, let Q be an arbitrary measurable subset of X . For all k ∈ N , since F k ∈ P m,G , by (16), we have, F k ( T d , Q ) = m ( Q, T a ) , (a.s.) (57)where m is the demand profile. On the other hand, by (56), F (cid:0) ∂Q (cid:1) ≤ Gλ (cid:0) ∂Q (cid:1) = 0, and Q is a F -continuityset . Then, by the weak convergence of F k to F and relation (57), Portmanteau Theorem implies that: m ( Q, T a ) = lim k F k ( T d , Q ) = F ( T d , Q ) , (a.s.) . Therefore, F ∈ P m,G and P m,G is closed under weak convergence. (cid:3) Proof of Proposition 4.
Let Λ = 2 G Lip( V ) M T max − e − MT max M . Following the proof of Proposition 2, we know that, d M ( U ( z , F ) , U ( z , F )) ≤ Λ d M ( z , z ) , ∀ F ∈ P m,G , ∀ z , z ∈ C ( T ) , where U ( z, F ) denotes the characteristic travel distance of a system having in-flow measure F and primarycharacteristic travel distance z , see (25). Then, we have, d M ( z ∗ k , z ∗ ) = d M ( U ( z ∗ , F ) , U ( z ∗ k , F k )) ≤ d M ( U ( z ∗ , F k ) , U ( z ∗ k , F k )) + d M ( U ( z ∗ , F ) , U ( z ∗ , F k )) ≤ Λ d M ( z ∗ , z ∗ k ) + sup ≤ t ≤ T max exp − Mt (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t [ V ( F ( S s ( z ∗ ))) − V ( F k ( S s ( z ∗ )))] ds (cid:12)(cid:12)(cid:12)(cid:12) . Considering that M is chosen such that Λ <
1, we obtain the following bound: d M ( z ∗ k , z ∗ ) ≤ Lip( V )1 − Λ sup ≤ t ≤ T max exp − Mt (cid:90) t | F ( S s ( z ∗ )) − F k ( S s ( z ∗ )) | ds which leads to d M ( z ∗ k , z ∗ ) ≤ Lip( V )1 − Λ (cid:90) T max | F ( S s ( z ∗ )) − F k ( S s ( z ∗ )) | ds. (58)The rhs of (58) is bounded by V ) GT max λ ( T ×X )1 − Λ and converges to 0 as F k ⇒ F by Portmanteau Theorem.Then, using Lebesgue dominated convergence, the rhs of (58) converges to 0 as k → ∞ . Finally, consideringequivalency between (cid:107) · (cid:107) and (cid:107) · (cid:107) M , the proof of the proposition is complete. (cid:3) Proof of Corollary 2.
Due to (13), the bound provided in (58) can then be expressed as d M ( z ∗ k , z ∗ ) ≤ Lip( V )1 − Λ (cid:90) T max | E ( S s ( z ∗ ) × T a ) − E k ( S s ( z ∗ ) × T a ) | ds , (59)which shows that if E k ⇒ E as k → ∞ , then z ∗ k → z ∗ in the uniform norm. (cid:3) Proof of Proposition 5.
The cost function J is defined as follows: J ( t d ; x, t a ; F ) = (cid:40) αT ( t d , x ) + β (cid:0) t a − t d + T ( t d , x ) (cid:1) , t a ≥ t d + T ( t d , x ) αT ( t d , x ) + γ (cid:0) t d + T ( t d , x ) − t a (cid:1) , t d + T ( t d , x ) > t a . In order to prove the proposition, it is sufficient to establish the continuity of T as a function of ( t d , x, t a )and F . Considering that T ( t d , x ) = z − F ( x + z F ( t d )), it suffices to establish the continuity for z F and z − F . ∂A , when A is a set, refers to its boundary. meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted V is bounded from above by V max , z F is Lipschitz with Lip( z F ) ≤ V max . From (59), it follows that forany t, t (cid:48) ∈ T and any F, F (cid:48) ∈ P m,G : | z F ( t ) − z F (cid:48) ( t (cid:48) ) | ≤ | z F ( t ) − z F (cid:48) ( t ) | + | z F (cid:48) ( t ) − z F (cid:48) ( t (cid:48) ) |≤ | z F ( t ) − z F (cid:48) ( t ) | + V max | t − t (cid:48) |≤ e MT max Lip( V )1 − Λ (cid:90) T max ds | F ( S s ( z F )) − F (cid:48) ( S s ( z F )) | + V max | t − t (cid:48) | . The continuity of ( t, F ) → z F ( t ) is thus established and is obviously Lipschitz wrt t . Now, as V is boundedfrom below by V min , it holds that for any F ∈ P m,G : | z F ( t ) − z F ( t (cid:48) ) | ≥ V min | t − t (cid:48) | . Therefore, z F is invertibleand its inverse is Lipschitz with Lip( z − F ) ≤ /V min . To conclude, consider F, F (cid:48) ∈ P m,G , x, x (cid:48) ∈ X , and t = z − F ( x ), t (cid:48) = z − F (cid:48) ( x (cid:48) ). We have: z F (cid:48) ( t ) − z F (cid:48) ( t (cid:48) ) = z F (cid:48) ( t ) − z F ( t ) + x − x (cid:48) . Since | z F (cid:48) ( t ) − z F (cid:48) ( t (cid:48) ) | ≥ V min | t − t (cid:48) | = V min | z − F ( x ) − z − F (cid:48) ( x (cid:48) ) | and | z F ( t ) − z F (cid:48) ( t ) | ≤ e MT max Lip( V )1 − Λ (cid:90) T max ds | F ( S s ( z F )) − F (cid:48) ( S s ( z F )) | , we get: | z − F ( x ) − z − F (cid:48) ( x (cid:48) ) | ≤ | x − x (cid:48) | V min + e MT max Lip( V )(1 − Λ) V min (cid:90) T max ds | F ( S s ( z F )) − F (cid:48) ( S s ( z F )) | . The continuity of ( x, F ) → z − F ( x ) is proved and is Lipschitz wrt x . Since J is a piecewise linear function of t d , t a and travel time function T , it follows that J is Lipschitz continuous wrt t d , x and t a . Note that theLipschitz constant of J depends only on Lip( V ), V min , V max , and α , β , γ . The continuity coefficient for thedependency of J on F is also dependent on M , and Λ, where Λ itself depends on the constant G . (cid:3) Proof of Proposition 6.
First, note that P ( T d × X × T a ) is a convex compact subset of M ( T d × X × T a ),the set of signed measures with bounded variation on ( T d × X × T a ). Further, one can show easily that, forall G ∈ R + , M m,G is a closed subset of P ( T d × X × T a ), with an argument similar to the one given in thethe proof of Proposition 3. Since a closed subset of a compact set is compact, M m,G is a compact subset of M ( T d × X × T a ), too. The convexity of M m,G is trivial by its definition.Now, as M ( T d × X × T a ) is a locally convex topological vector space, we can apply the fixed-point theoremof Kakutani (see Theorem 8.6 of Granas and Dugundji (2003)) to prove H admits a fixed point. The convexityand compactness of H is clear. It remains to show that for all E , H ( E ) is non empty and H is usc (uppersemi-continuous).i) Consider ˜ E ∈ M m,G . We aim to show that H ( ˜ E ) is non-empty. Denote ˜ F = F ( ˜ E ) and ˜ J ( t d ; x, t a ; ˜ F ) = J ( t d ; x, t a ; ˜ F ) − min t ∈T J ( t ; x, t a ; ˜ F ). The function ˜ J is continuous wrt F and Lipschitz continuous wrt t d , x, t a .Further, we have Lip( ˜ J ) ≤ J ). Indeed, the function ( x, t a ) → min t ∈T J ( t ; x, t a ; F ) admits the same Lip-schitz constant as J . Notice that: (cid:12)(cid:12)(cid:12) ˜ J ( t ; x, t a ; ˜ F ) − ˜ J ( t (cid:48) ; x, t a ; ˜ F ) (cid:12)(cid:12)(cid:12) ≤ Lip( J ) | t − t (cid:48) | , ∀ t, t (cid:48) ∈ T . meli et al.: Departure Time Choice Models Based on Mean Field Games Submitted
Denote U = { ( t d , x, t a ) | ˜ J ( t d ; x, t a ; ˜ F ) < ε } and U x,t a = { t | ˜ J ( t ; x, t a ; ˜ F ) < ε } . U is an open set. More precisely,if ( t d , x, t a ) ∈ U , then all ( t (cid:48) d , x (cid:48) , t (cid:48) a ) such that | t d − t (cid:48) d | + | x − x (cid:48) | + | t a − t (cid:48) a | < ε − ˜ J ( t d , x, t a ; ˜ F )2Lip( J )also belong to U . If t d is in U x,t a , i.e., ˜ J ( t, x, t a ; ˜ F ) < ε , then for all t (cid:48) d such that | t d − t (cid:48) d | < ε − ˜ J ( t, x, t a ; ˜ F )Lip( J ) . Thus, for all ( x, t a ) ∈ ( X , T a ), U x,t a has Lebesgue measure greater than ε Lip( J ) . It follows that ( x, t a ) → λ ( U x,t a )is lsc (lower semi-continuous) and bounded from below by ε Lip( J ) where λ denotes Lebesgue measure on R .Now define the function ν on ( T d × X × T a ) by: ν ( t d , x, t a ) = 1 U ( t d , x, t a ) λ ( U x,t a ) . This function is lsc and bounded from above by
Lip( J )2 ε .Finally we define E such that E ( dt d , dx, dt a ) = ν ( t d , x, t a ) λ ( dt d ) m ( dx, dt a ) . By construction, E is positive, has total mass 1 and satisfies the constraint (15). Also, the support of E liesin U by construction; hence, E satisfies E ∈ H ( ˜ E ). It remains to be checked that E ∈ M m,G . It suffices tocheck that F ( E ) satisfies Assumption 2. Consider B ∈ B ( T d × X ). Using Fubini Theorem (see, e.g., Theorem18.3 of Billingsley (2012)) and Assumption 1, regularity condition of m , we have, F ( E )( B ) = E ( B × T a )= (cid:90) B ×T a dt d dm ( x, t a ) ν ( t d , x, t a ) ≤ Lip( J )2 ε (cid:90) B ×T a dt d dm ( x, t a ) ≤ M m Lip( J )2 ε λ ( B ) , where λ is Lebesgue measure on R . The above calculation yields to an estimate from below of G . Thatmeans, choosing G ≥ M m Lip( J )2 ε assures that E ∈ M m,G . This completes the proof that H ( ˜ E ) is non-empty.ii) We next show that H is usc (upper semi-continuous). Note that J depends on E via F . In order tosimplify the notations, in this paragraph we will write J ( E ) for J ( F ( E )). We rewrite the definition of H asfollows, H ( E ) = (cid:110) e ∈ M m,G (cid:12)(cid:12)(cid:12) e (cid:16) ˜ J ( E ) ≤ ε (cid:17) = 1 (cid:111) . In order to show that H is usc it is required to prove that for any open set W ∈ M m,G , the set H − W = { E | H ( E ) ⊂ W } is open, see Page 166 of Granas and Dugundji (2003). Conversely, denoting W c and ( H − W ) c the respective complements of W and H − W , it suffices to show that if W c is closed, ( H − W ) c is closed.Consider a convergent sequence { E n } n ∈ N of elements of ( H − W ) c , and let E be the limit of this sequence.We now show that E ∈ ( H − W ) c . For all n ∈ N , H ( E n ) (cid:54)⊂ W and there exists e n ∈ H ( E n ) ∩ W c . By compact-ness, we can assume, after extracting a sub-sequence, that the sequence { e n } n ∈ N converges weakly towardssome e ∈ W c ⊂ M m,G . It remains to show that e ∈ H ( E ). meli et al.: Departure Time Choice Models Based on Mean Field Games
Submitted η > N ( η ) such that for n ≥ N ( η ), | ˜ J ( E n ) − ˜ J ( E ) | < η (uniformly in C ( T d × X ×T d )). It implies that { ˜ J ( E n ) ≤ ε } ⊂ { ˜ J ( E ) ≤ ε + η } , n ≥ N ( η ) , and consequently for all n ≥ N ( η ), e n (cid:16) ˜ J ( e ) < ε + η (cid:17) = 1 . Since e n = ⇒ n →∞ e , it follows by Portmanteau Theorem that e (cid:16) ˜ J ( e ) ≤ ε + η (cid:17) ≥ lim sup n →∞ e n (cid:16) ˜ J ( e ) ≤ ε + η (cid:17) , and thus: e (cid:16) ˜ J ( e ) ≤ ε + η (cid:17) = 1 , ∀ η > . Finally, η (cid:55)→ { ˜ J ( e ) ≤ ε + η } is monotone decreasing; thus, by Monotone Convergence Theorem, we have:lim η → e (cid:16) ˜ J ( e ) ≤ ε + η (cid:17) = e (cid:16) ˜ J ( e ) ≤ ε (cid:17) = 1 . Therefore, we proved that E ∈ ( H − W ) c . (cid:3) Proof of Proposition 8.
Fix t a ∈ T a and x ∈ X . Consider h and l as the small changes in time andspace, respectively. The demand with desired arrival time in ∆ t a := ( t a − h , t a + h ) and trip length in∆ x := ( x − l , x + l ) is equal to m (∆ x, ∆ t a ) which can be approximated by m ( dx, dt a ) hl .On the other hand, since D is increasing wrt the desired arrival time, the departure time of the agentswith desired arrival times in ∆ t a and the trip length x is in the interval (cid:0) D ( t a − h , x ) , D ( t a + h , x ) (cid:1) . Then,we approximate the fraction of agents having departure time in (cid:0) D ( t a − h , x ) , D ( t a + h , x ) (cid:1) , trip length in∆ x , and desired arrival time in ∆ t a by e ( D ( t a , x ) , x, t a ) (cid:0) D ( t a + h , x ) − D ( t a − h , x ) (cid:1) l. But, we have: e ( D ( t a , x ) , x, t a ) (cid:0) D ( t a + h , x ) − D ( t a − h , x ) (cid:1) l ≈ m ( dx, dt a ) hl. Letting h →0 yields to the desired result.