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1 Politecnico di Torino Porto Institutional Repository [Article] Stability analysis of transportation networks with multiscale drier decisions Original Citation: Como, Giacomo; Sala, Ketan; Acemoglu, Daron; Dahleh, Munther A.; Frazzoli, Emilio (2013). Stability analysis of transportation networks with multiscale drier decisions. In: SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ol. 51 n. 1, pp ISSN Aailability: This ersion is aailable at : since: Noember 2015 Publisher: SIAM Published ersion: DOI: / Terms of use: This article is made aailable under terms and conditions applicable to Open Access Policy Article ("Public - All rights resered"), as described at html Porto, the institutional repository of the Politecnico di Torino, is proided by the Uniersity Library and the IT-Serices. The aim is to enable open access to all the world. Please share with us how this access benefits you. Your story matters. (Article begins on next page)

2 STABILITY ANALYSIS OF TRANSPORTATION NETWORKS WITH MULTISCALE DRIVER DECISIONS GIACOMO COMO, KETAN SAVLA, DARON ACEMOGLU, MUNTHER A. DAHLEH, AND EMILIO FRAZZOLI Abstract. Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the driers route choices are influenced by information at multiple temporal and spatial scales. The considered model inoles a continuum of nonatomic indistinguishable driers commuting between a common origin/destination pair in an acyclic transportation network. The driers route choices are affected by their, relatiely infrequent, perturbed best responses to global information about the current network congestion leels, as well as their instantaneous local obseration of the immediate surroundings as they transit through the network. A noel model is proposed for the driers route choice behaior, exhibiting local consistency with their preference toward globally less congested paths as well as myopic decisions in faor of locally less congested paths. The simultaneous eolution of the traffic congestion on the network and of the aggregate path preference is modeled by a system of coupled ordinary differential equations. The main result shows that, if the frequency of updates of path preferences is sufficiently small as compared to the frequency of the traffic flow dynamics, then the state of the transportation network ultimately approaches a neighborhood of the Wardrop equilibrium. The presented results may be read as a further eidence in support of Wardrop s postulate of equilibrium, showing robustness of it with respect to non-persistent perturbations. The proposed analysis combines techniques from singular perturbation theory, eolutionary game theory, and cooperatie dynamical systems. Key words. Transportation networks, Wardrop equilibrium, traffic flows, eolutionary game dynamics, route choice behaior, multiscale decisions. 1. Introduction. As transportation demand is fast approaching its infrastructure capacity, a rigorous understanding of the relationship between the macroscopic properties of transportation networks and realistic drier route choice behaior is attracting renewed research interest. Such an analysis is essential, among other things, for appropriate design of incenties influencing driers behaior in order to induce a desired socially optimal usage of the transportation infrastructure. A particularly releant issue is the impact of driers en route responses to unexpected eents on the oerall transportation network dynamics. This issue is particularly significant in modern transportation network settings, where recent technological adancements in intelligent traeller information deices hae enabled driers to be much more flexible in selecting their routes to destination een while being en route. While there has been a significant research effort to inestigate the effect of such technologies on the This work was supported in part by NSF EFRI-ARES grant number and AFOSR grant number FA The first author was partially supported by the Swedish Research Council, through the the junior research grant Information Dynamics in Large-Scale Networks and the Linnaeus excellence center LCCC. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the iews of the supporting organizations. Department of Automatic Control, Lund Uniersity, BOX 118, SE Lund, Sweden (giacomo.como@control.lth.se). Sonny Astani Department of Ciil and Enironmental Engineering, Uniersity of Southern California, Los Angeles (CA), , US (ksala@usc.edu). Department of Economics, Massachusetts Institute of Technology, 77 Mass Ae, Cambridge (MA), 02139, US (daron@mit.edu). Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Mass Ae, Cambridge (MA), 02139, US (dahleh@mit.edu). Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Mass Ae, Cambridge (MA), 02139, US (frazzoli@mit.edu). 1

3 route choice behaior of driers, e.g., see [26, 22], the analytical study of the dynamical properties of the whole network under such behaior has attracted ery little attention. This paper is focused on the stability analysis of transportation networks in a setup where the driers hae access to traffic information at multiple temporal and spatial scales and they hae the flexibility to switch their route to destination at eery intermediate traffic intersection. Specifically, we consider a model in which the driers choose their routes while haing access to relatiely infrequent global information about the network congestion state, and real-time local information as they transit through the network. The driers route choice behaior is then influenced by relatiely slowly eoling path preferences as well as myopic responses to the instantaneous obseration of the local congestion leels at the intersections. This setup captures many real-life scenarios where unexpected eents obsered en route might cause driers to take a temporary detour, but not necessarily to change their path preferences. Such path preferences may instead be updated, e.g., on a daily, weekly, or longer time basis, in response to information about the global congestion state of the different origin-destination paths collected from the driers personal experience, their opinion exchanges with their peers, as well as from information media. Howeer, since the traffic dynamics is significantly influenced by the driers response to real-time local information, such responses can influence the driers path preference thereby modifying their global route choice behaior in the long run. We propose and analyze a noel model for the driers route choice behaior that combines relatiely infrequent information about the global congestion status of the network with real-time local obserations as explained below. In our model, the network is represented by a directed acyclic graph with one origin and one destination. A continuous constant flow of nonatomic indistinguishable driers enters from the origin, and flows through the network until reaching the destination node. Traffic parameters, such as aerage speed, traffic density, and flow, are modeled as homogeneous quantities on eery link, related to each other by functional dependencies representatie of the links congestion properties. The dynamics of such traffic parameters is goerned by the law of conseration of mass, as well as the driers route choice behaior. In turn, the driers route choice behaior is assumed to be influenced by two factors: the aggregate path preference, measuring the relatie appeal of the different routes to the driers, and local obserations of the current congestion leels. The path preference dynamics eole at a slow time scale (as compared to the traffic dynamics), following a perturbed best response to global information, embodied by the current congestion leels on the whole network. When traersing an intermediate node in the network, driers behae according to their path preference, if this is consistent with the current, locally obsered, aggregate behaior of the other driers. On the other hand, when there is a discrepancy between the aggregate path preference and the locally obsered aggregate behaior, then driers tend to compensate this by myopically preferring routes which appear to be locally less congested. The model described aboe gies rise to a double feedback dynamics, goerned by a finite-dimensional system of coupled ordinary differential equations. Such a dynamical system has two natural time scales, characterizing the dynamics of the driers aggregate path preference and of the traffic parameters on the different links, respectiely. We study the long-time behaior of this dynamical system: our main result shows that, in the limit of small update rate of the aggregate path preferences, a 2

4 state of approximate Wardrop equilibrium [27] is approached. The latter is a configuration in which the delay associated to any source-destination path chosen by a nonzero fraction of driers does not exceed the delay associated to any other path. Our results contribute to proiding a stronger eidence in support of the significance of Wardrop s postulate of equilibrium for a transportation network. They may also be read as a sort of robustness of such equilibrium notion with respect to non-persistent perturbations of the network. The analytical arguments we propose mainly rely on three ideas: adopting a singular perturbation approach [16], by considering the aggregate path preference as quasi-static when studying the fast scale dynamics of the traffic parameters, and the traffic parameters as almost equilibrated when analyzing the slow scale dynamics of the aggregate path preference; exploiting the inherent cooperatie 1 dependence of the route choice function on the local traffic parameters in order to establish exponential stability of the fast scale dynamics of the traffic parameters; adapting results from eolutionary population games [15, 24] in order to establish stability properties of the slow scale perturbed best response dynamics of the aggregate path preference. Our work is naturally related to two streams of literature on transportation networks. On the one hand, traffic flows on networks hae been widely analyzed with fluid-dynamical and kinetic models: see, e.g., [11], and references therein. As compared to these models (typically described by integro- or partial differential equations), ours significantly simplifies the eolution of the traffic parameters (treating them as homogeneous quantities on the links, representatie of spatial aerages), whereas it highlights the role of the driers route choice behaior with its double feedback dynamics, which is typically neglected in that literature. On the other hand, transportation networks hae been studied from a decisiontheoretic perspectie within the framework of congestion games [3, 23]. In these models, driers make sequential myopic route choice decisions in pursuit of minimizing their personal trael times, in response to complete information about the whole network. Congestion games are known to belong to the class of potential games [19], a consequence of which is that, best responses of the driers are aligned with the gradient of a common potential function and hence the system eentually conerges to a critical point of this potential function, which, under appropriate monotonicity conditions of the congestion properties of the links of the network, corresponds to a Wardrop equilibrium. Such an approach has been used, for example in [18]. Dynamical systems framework for stability analysis of transportation equilibria hae also been deeloped in [25, 5, 20]. The stability of Wardrop equilibrium in the context of communication networks has been studied in [4]. It is important to note that the two salient features of a typical congestion game setup are that information is aailable to the driers at a single temporal and spatial scale, and that the dynamics of traffic parameters are completely neglected by assuming that they are instantaneously equilibrated. In contrast, we study the stability of Wardrop equilibrium in a setting where the dynamics of the traffic parameters are not neglected, and the driers route choice decisions are affected by relatiely infrequent global information, as well as their realtime local information as they transit through the network. As a consequence, classic results of eolutionary game theory and population dynamics [15, 24] are not directly applicable to our framework, and noel analytical tools hae to be deeloped, particularly for the analysis of the fast scale dynamics of the traffic parameters. For such dynamics, the most noel technical feature of our approach consists in proing local 1 Here, the adjectie cooperatie is intended in the sense of Hirsch [12, 13]. 3

5 contraction properties which follow from the cooperatie nature and other structural properties of the system. The rest of the paper is organized as follows. In Section 2, we formulate the model and state the main result. Section 3 is a technical section that contains the proofs for the main result including intermediate results. In Section 4, we report results from illustratie numerical experiments. Finally, we conclude in Section 5 and also mention potential future research directions. Before proceeding, we establish here some notations to be used throughout the paper. Let R be the set of reals, R := {x R : x 0} be the set of nonnegatie reals. Let A and B be finite sets. Then, A will denote the cardinality of A, R A (respectiely, R A ) the space of real-alued (nonnegatie-real-alued) ectors whose components are labeled by elements of A, and R A B the space of matrices whose real entries are labeled by pairs of elements in A B. The transpose of a matrix M R A B, will be denoted by M R B A, while I will be an identity matrix, and 1 the all-one ector, whose size will be clear from the context. We shall use the notation Φ := I A 1 11 R A A to denote the projection matrix on the space orthogonal to 1. The simplex of probability ectors oer a finite set A will be denoted by S(A) := {x R A : 1 x = 1}. If B A, 1 B : A {0, 1} will stand for the indicator function of B, with 1 B (a) = 1 if a B, 1 B (a) = 0 if a A \ B. For p [1, ], p is the p-norm. By default, let := 2 denote the Euclidean norm. Let int(x ) be the interior of a set X R d, and X denote its boundary. Let sgn : R { 1, 0, 1} be the sign function, defined by sgn(x) is 1 if x > 0, sgn(x) = 1 if x < 0, and sgn(x) = 0 if x = 0. By conention, we shall assume the identity d x /dx = sgn(x) to be alid for eery x R, including x = 0. Finally, we shall adopt the conention that the gradient f of a function f : D R, where D R A, be a column ector in R A, while f := Φ f will stand for the projected gradient on S(A). 2. Model formulation and main result. In this section, we formulate the problem and state the main result. In our formulation, we represent the dynamics of the traffic and the route choice behaior on a transportation network as a system of coupled ordinary differential equations with two time scales representatie of route choice behaior influenced by the two leels of information. The dynamics of the physical ariables, i.e., density and flow on each link, eole at the fast time scale and are drien by local information on the current physical state of the network, whereas the aggregate path preferences eole at the slow time scale in response to global information on the current physical state of the network. The key components of our model are: network topology, congestion properties of the links, path preference dynamics, and node-wise route choice decision. We next describe each of these components in detail Network characteristics. Let the topology of the transportation network be described by a directed graph (in short, di-graph) G = (V, E), where V is a finite set of nodes and E V V is the set of (directed) links. For eery node V, we shall denote by E, and E, the sets of its incoming, and, respectiely, outgoing links. A length-l (directed) path from u V to V is an l-tuple of consecutie links {( j 1, j ) E : 1 j l} with 0 = u, and l =. A cycle is path of length l 1 from a node to itself. Throughout this paper, we shall assume that: Assumption 1. The di-graph G contains no cycles, has a unique origin (i.e., some V such that E = ), and a unique destination (i.e., some V such that E = ). Moreoer, there exists a path to the destination node from eery other node in V, as well as from the origin node to any other node in V. 4

6 Assumption 1 implies that one can find a (not necessarily unique) topological ordering of the node set V (see, e.g., [8]). We shall assume to hae fixed one such ordering, identifying V with the integer set {0, 1,..., n}, where n := V 1, in such a way that E 0 u< E u, = 0,..., n. We shall model the traffic parameters as time-arying quantities which are homogeneous oer each link of the network. Specifically, for eery link e E, and time instant t 0, we shall denote the current traffic density, and flow, by ρ e (t), and f e (t), respectiely, while ρ(t) := {ρ e (t) : e E}, f(t) := {f e (t) : e E} will stand for the ectors of all traffic densities, and flows, respectiely. Current traffic flow and density on each link are related by a functional dependence f e = µ e (ρ e ), e E. (2.1) Such functional dependence models the driers speed and lane adjustment behaior in response to traffic density on a particular segment of a road. It will be assumed to satisfy the following: Assumption 2. For eery link e E, the flow-density function µ e : R R is continuously differentiable, strictly increasing, strictly concae and is such that d µ e (0) = 0, lim µ e (ρ e ) <. ρe 0 dρ e Remark 1. Flow-density functions commonly used in transportation theory typically are not globally increasing, but rather hae a -shaped graph [11]: µ e (ρ e ) increases from µ e (0) = 0 until achieing a maximum C e = µ e ( ρ e ), and then decreases for ρ e ρ e. Assumption 2 remains a good approximation of this setting, proided that ρ e stays in the interal [0, ρ e ). It should be noted that the fact that the support of the flow function is unbounded, i.e., that the density can grow as large as possible, preents the (fast time scale) dynamics (2.11) of the physical ariables to take into account backward propagation of perturbations. For eery link e E, let C e := sup{µ e (ρ e ) : ρ e 0} = lim µ e(ρ e ) ρ e be its maximum flow capacity. Moreoer, let F := [0, C e ), F := e E[0, C e ) e E be the sets of local, and, respectiely, global feasible flow ectors. Obsere that our formulation allows for both the cases of bounded and unbounded maximum flow capacities. As the flow f e is the product of speed and density, it is natural to introduce the delay function if f e C e T : R E [0, ] E, T e (f e ) := µ 1 e (f e )/f e if f e (0, C e ), (2.2) 1/ dµe dρ e (0) if f e = 0, 5

7 whose components measure the flow-dependent time taken to traerse the different links. 2 Example 1. A flow-density function that satisfies Assumption 2 is gien by µ e (ρ e ) = C e ( 1 e θ eρ e ) e E, (2.3) where C e > 0, and θ e > 0. The corresponding delay function is T e (f e ) = 1 C e log. θ e f e C e f e We shall denote by P the set of distinct paths in G from the origin node 0 to the destination node n. Let { A R E P 1 if e p, A ep = 0 if e / p, be the link-path incidence matrix of G. The relatie appeal of the different paths to the driers will be modeled by a time-arying probability ector oer P, which will be referred to as the current aggregate path preference, and denoted by π(t). If one assumes, as we shall do throughout this paper, a constant unit in-flow in the origin node, it is natural to consider the ector f π := Aπ of the flows associated to the current aggregate path preference. Indeed, the e-th entry, fe π = p A epπ p, represents the total traffic flow that a link e E would sustain in a static condition in which the fraction of driers choosing any path p P is gien by π p. Now, let Π := {π S(P) : f π e < C e, e E} be the set of feasible path preferences. Here, the term feasible refers to the fact that the flow ector f π associated to any π Π satisfies the capacity constraint fe π < C e for eery e E. Obsere that, wheneer C e > 1 for eery e E (or, when link capacities are infinite), the set of feasible path preferences Π coincides with the whole simplex S(P). In contrast, when C e 1 for some e E, Π S(P) is a strict inclusion. On the other hand, the following result shows that whether Π is empty or not depends solely on the alue of the min-cut capacity of the network [1, Ch. 4]. Let C := min C U, C U := C e, U V: 0 U,n/ U where E U := {e = (u, ) E : u U, V \ U}. Proposition 2.1. The set Π is nonempty if and only if C > 1. Proof. Fix a cut-set U V such that 0 U, and n / U. Then, eery path p P contains exactly one link (u, ) p such that u U, and V \ U. Hence, for eery π Π, one has that C U = π p = 1. length. e E U C e > p e E U e E U A ep π p = p 2 Here, it has implicitly been assumed, without any loss of generality, that all the links are of unit 6

8 Minimizing oer all cut-sets U shows that C > 1 is necessary for Π to be nonempty. For the inerse implication, consider a network with the same topology G and link capacities c e = max{c e E 1 (C 1), 0}. The min-cut capacity of this network satisfies c C (C 1) = 1. Note also that, from our construction, C e > c e 0. Therefore, the max-flow min-cut theorem (see, e.g., [1, Thm. 4.1]) implies that there exists some π Π, thus proing that Π is nonempty. In the case when C 1 it is not hard to show that the system will grow unstable, i.e., ρ e (t) is unbounded as t grows large, for some link e E. Therefore, throughout this paper, we shall confine ourseles to transportation networks satisfying: Assumption 3. The min-cut capacity satisfies C > Route choice behaior and traffic dynamics. We now describe the driers route choice behaior and traffic dynamics on the network. We enision a continuum of indistinguishable driers traeling through the network. Driers enter the network from the origin node 0 at a constant unit rate, trael through it, and leae the network from the destination node n. While inside the network, driers occupy some link e E. The time required by the driers to traerse link e, and the current flow on such link are goerned by its congestion properties, as gien by (2.2), and (2.1), respectiely. When entering the network from the origin node = 0, as well as when reaching the head node {1, 2,..., n 1} of some link e / En, the driers instantaneously join some link e E. In this paper, we shall model the choice of such new link to depend on infrequently updated perturbed best responses of the driers to global information about the congestion status of the whole network as well as on their instantaneous obseration of the local congestion leels. We next describe these two aspects of the model in detail. Aggregate path preference dynamics. The driers aggregate path preference π(t), already introduced in Sect. 2.1, models the relatie appeal of the different paths to the driers population. It is updated as driers access global information about the current congestion status of the whole network. This occurs at some rate η > 0, which could be thought of as being small with respect to the time-scale of the network flow dynamics. Information about the current status of the network is embodied by the current traffic flow ector f(t). From f(t), driers can ealuate the ector A T (f(t)), whose p-th entry, e A ept e (f e (t)), coincides with the total delay a drier expects to incur on path p assuming that the congestion leels on that path won t change during her journey. 3 Driers are modeled as reacting to such global information by updating their path preferences independently at rate η according to some feasible path preference F h (f(t)) Π, so that the aggregate path preference π(t) eoles as d dt π(t) = η ( F h (f(t)) π(t) ). (2.4) Here F h : F Π is a perturbed (or smoothed) best response function, as per Assumption 4 formulated below. First, let us introduce the notion of admissible perturbation. 3 The delay that a drier would actually incur taking path p at time t would in fact possibly differ from e AepTe(fe(t)), since by the time t t the drier reaches a certain link e p, the delay on that link, T e(f e(t )), might well hae changed from its alue T e(f e(t)) at time t, as a result of the fast-scale dynamics of the physical ariables ρ e(t) and f e(t). 7

9 Definition 2.2. An admissible perturbation is a function h : Π h R where Π h Π is a closed conex set, h( ) is strictly conex, twice differentiable in int(π h ), and is such that lim π Πh h(π) =. Example 2. Let D := {e E : C e 1} be the, possibly empty, set of links with capacity not exceeding 1. For a ε (0, min e C e ), and β > 0, let and define h : Π h R by Π h := {π Π : f π d C d ε, d D}, h(π) := β 1 p π p log π p β 1 d (C d ε f π d ) log(c d ε f π d ), where the summation indices p and d run oer the sets P and D, respectiely, and the standard conention 0 log 0 := 0 is adopted. It can be readily erified that Π h Π is a nonempty conex polytope, and lim π Πh h(π) =. Hence, h is an admissible perturbation. Obsere that, if C e > 1 for all e E, then D is empty, Π h = Π, and h(π) = β 1 p π p log π p reduces to the standard negatie entropy function. Obsere that compactness and conexity of Π h, together with strict conexity of h(ω) imply existence and uniqueness of a minimizer of ω A T (f) h(ω) in Π h. This supports the following. Assumption 4. The function F h : F Π is a perturbed best response, i.e., F h (f) := argmin { ω A T (f) h(ω) }, f F, (2.5) ω Π h where h : Π h R is an admissible perturbation (as per Definition 2.2). In fact, Assumption 4 and Definition 2.2 imply that F h (f) int(π h ) and that F h (f) is continuously differentiable on F. The perturbed best response function F h (f) proides an idealized description of the behaior of driers whose decisions are based on inexact information about the state of the network. In particular, it can be shown that the form of F h (f) gien in (2.5) is equialent to the minimization, oer paths p P, of the expected delay e A ept e (f e ) corrupted by some (admissible) stochastic perturbation (see e.g. [14]). Moreoer, it is well known [24] that, as h 0, and Π h Π, the perturbed best response F h (f) conerges to the set argmin{ω A T (f) : ω Π} of best responses. 4 Example 3. Assume that C e > 1 for all e E, and fix a noise parameter β > 0. Define a perturbed best response by putting Π h = Π, and h(π) = β 1 p π p log π p for all π Π, as in the special case of Example 2. Then, the corresponding perturbed best response is the logit function F h p (f) = exp( β(a T (f)) p ) q P exp( β(a T (f)) q ), p P. (2.6) For any fixed f F, one has that lim β F h (f), with F h (f) as defined in (2.6), is a uniform distribution oer the set argmin{(a T (f)) p : p P}. We refer the reader to [10, 15] for more on the connection between F h characterized by Assumption 4 and smoothed best response functions. Remark 2. The micro-foundations of the aggregate path dynamics in (2.4) can be heuristically justified by looking at it as the mean-field limit of a stochastic finite 4 Here, the conergences Π h Π, and {F h (f)} argmin{ω A T (f) : ω Π} are intended to hold in the Hausdorff metric. (see, e.g., [2, Def ]) 8

10 population model along the following lines. Consider a model with a large but finite drier population, with each drier updating her path preference at the clicking of an independent Poisson clock of rate η by choosing a new preferred path p with probability Fp h (f(t)). Then one could show that the conditional aerage ariation of the aggregate path preference from time t to time t ε, for small ε > 0, is gien by η ( F h (f(t)) π(t) ) εo(ε). As the stochastic elements of the driers updating mechanisms are idiosynchratic, one may expect such stochastic influences to be aeraged away as the population size grows large, by appealing to some law of large numbers in the spirit of Kurtz s theorem [17], [9, Chapter 11]. We will not attempt to formally justify the micro-foundations of the model discussed in this paper, but rather leae it as a topic for future work. Remark 3. In the eolutionary game theory literature, e.g., see [15, 24], the domain of an admissible perturbation function h, as well as the one of the minimization in the right-hand side of (2.5), is typically assumed to be the whole simplex S(P), instead of a closed polytope Π h Π S(P). Notice that, as already obsered in Sect. 2.1, when C e > 1 for eery e E, Π = S(P) is a closed polytope, so that one can choose Π h = Π. Therefore, in this case, Definition 2.2 does not introduce any additional restriction with respect to such theory. On the other hand, when C e 1 for some e E, then the inclusions of Π h Π S(P) are both strict, so that Definition 2.2 does introduce additional restrictions on the admissible perturbations. Howeer, it is worth obsering that, in a classic eolutionary game theoretic framework, the dynamics of the aggregate path preference would be autonomous rather than coupled to the one of the actual flow. In particular, perturbed best response dynamics in that framework would read as d dt π(t) = F h (f π (t)) π(t), (2.7) rather than as in (2.4). For such dynamics, the fact that T e (fe π ) = wheneer fe π C e, can be shown to imply that π(t) reaches a compact Π h Π in some finite time and neer leae it. In contrast, in the two time-scale model of coupled dynamics considered in this paper (see (2.13)), such more restrictie assumption is needed in order to ensure the same property for the trajectories of π(t) (see Lemma 3.4). Local route decisions. We now describe the local route decisions, characterizing the fraction of driers choosing each link e E when traersing a non-destination node. Such a fraction will be assumed to be a continuously differentiable function G e(f E, π) of the local traffic flow f E := {f e : e E }, as well as of the current aggregate path preference π. We shall refer to G : F Π S(E ) (2.8) as the local decision function at node {0, 1,..., n 1}, and assume that it satisfies the following: Assumption 5. For all 0 < n, and π Π, ( j E f π j ) G e ( f π, π E ) = fe π, e E. Assumption 6. For all 0 < n, π Π, and f E F, f e G j (f E, π) 0, j e E. 9

11 Assumption 5 is a consistency assumption. It postulates that, when the locally obsered flow coincides with the one associated to the aggregate path preference π, driers choose to join link e E with frequency equal to the ratio between the flow fe π and the total outgoing flow j E f j π. Assumption 6 instead models the driers myopic behaior in response to ariations of the local congestion leels. It postulates that, if the congestion on one link increases while the congestion on the other links outgoing from the same node is kept constant, the frequency with which each of the other outgoing links is chosen does not decrease. It is worth obsering that Assumption 6 is reminiscent of Hirsch s notion of cooperatie dynamical system [12, 13]. Example 4. An example of local decision function G satisfying Assumptions 5 and 6 is the i-logit function. The i-logit route choice function with sensitiity γ 0 is gien by G e(f E, π) = fe π exp( γ(f e fe π )) j E f j π exp( γ(f j fj π, (2.9) )) for eery e E, 0 < n. Obsere that in the extreme case γ = 0, (2.9) reduces to G e(f E, π) = fe π j E f j π, (2.10) which models a situation where the driers do not take into account the local obseration on the current flow, and always act in a way that is consistent with their aggregate path preference. For eery non-destination node {0, 1,..., n 1}, and outgoing link e E, conseration of mass implies that where d dt ρ e(t) = H e (f(t), π(t)), (2.11) H e (f, π) := { G e (f E, π) f e if = 0 ( j E f j)g e(f E, π) f e if 1 < n, (2.12) for all π Π and f F Objectie of the paper and main result. The objectie of this paper is to study the eolution of the coupled dynamics d dt π(t) = η ( F h (f(t)) π(t) ) d ρ(t) = H(f(t), π(t)), dt (2.13) where: F h is the perturbed best response function defined in (2.5); η > 0 is the rate at which global information becomes aailable; H(f, π) = {H e (f, π) : e E}, with H e defined in (2.11); f and ρ are related by the functional dependence (2.1). In particular, our analysis will focus on the double limiting case of small η and small h. We shall proe that, in such limiting regime, the long-time behaior of the system is 10

12 approximately at Wardrop equilibrium [27, 21]. The latter is a configuration in which the delay is the same on all the paths chosen by a nonzero fraction of driers. More formally, one has the following: Definition 2.3 (Wardrop Equilibrium). A feasible flow ector f W F is a Wardrop equilibrium if f W = Aπ for some π Π such that, for all p P, π p > 0 = (A T (Aπ)) p (A T (Aπ)) q, q P. (2.14) Existence and uniqueness of a Wardrop equilibrium are guaranteed by the following standard result: Proposition 2.4 (Existence and uniqueness of Wardrop equilibrium). Let Assumptions 1-3 be satisfied. Then, there exists a unique Wardrop equilibrium f W F. Proof. It follows from Assumption 2 that, for eery e E, the delay function T e (f e ) is continuous, strictly increasing, and such that T e (0) > 0. The proposition then follows by applying Theorems 2.4 and 2.5 from [21]. The following is the main result of this paper. It will be proed in Sect. 3 using a singular perturbation approach. Theorem 2.5. Let Assumptions 1 6 be satisfied. Then, for eery initial condition π(0) int(s(p)), ρ(0) (0, ) E, there exists a unique solution of (2.13). Moreoer, there exists a perturbed equilibrium flow f (h) F such that, for all η > 0, lim sup f(t) f (h) δ(η), (2.15) t where δ(η) is a nonnegatie-real-alued, nondecreasing function of η > 0, such that lim η 0 δ(η) = 0. Moreoer, for eery sequence of admissible perturbations {h k } such that lim k h k = 0, and lim k Π hk = Π, one has lim k f (h k) = f W. (2.16) Theorem 2.5 states that, in the large time limit, the flow ector f(t) approaches a neighborhood of the Wardrop equilibrium, whose size anishes as both the time-scale ratio η and the perturbation norm h anish. While a qualitatiely similar result is known to hold [24] in a classic eolutionary game theoretic framework (i.e., neglecting the traffic dynamics, and assuming it is instantaneously equilibrated, as in the ODE system (2.7)), the significance of the aboe is to show that an approximate Wardrop equilibrium configuration is expected to emerge also in our more realistic model of twotime scale dynamics. Therefore, our results proide a stronger eidence in support of the significance of Wardrop s postulate of equilibrium for a transportation network. In fact, they may be read as a sort of robustness of such equilibrium notion with respect to non-persistent perturbations. 3. Proofs. In this section, Theorem 2.5 is proed. First, obsere that, thanks to the continuous differentiability of F h, G, and µ, standard analytical arguments imply the existence and uniqueness of a solution of the initial alue problem associated to the system (2.13), with initial condition ρ(0) (0, ) E, π(0) int(s(p)). In order to proe the rest of the statement, we shall adopt a singular perturbation approach (e.g., see [16]), iewing the traffic density ρ (or, equialently, the traffic flow f) as a fast transient, and the aggregate path preference π as a slow component. Hence, we shall first think of π as quasi-static (i.e., almost a constant ) while analyzing 11

13 the fast-scale dynamics (2.11), and then assume that f is almost equilibrated, i.e., close to f π, and study the slow-scale dynamics (2.4) as a perturbation of (2.7). We shall proceed by proing a series of intermediate technical results, gathered in the following subsections. Before proceeding, we introduce some notation to be used throughout the section. Let ρ π e := µ 1 e (f π e ), σ e := sgn (ρ e ρ π e ) = sgn (f e f π e ) denote, respectiely, the density corresponding to the flow associated to the path preference π, and the sign of the difference between it and the actual density ρ e. Finally, fix some α (0, 1), and define n 1 V (f, π) := α =0 e E n 1 f e fe π, W (ρ, π) := α =0 e E ρ e ρ π e. (3.1) 3.1. Stability of the fast-scale dynamics. We gather here a few properties of the fast-scale dynamics. Our results will essentially amount to showing that V (f, π) and W (ρ, π) are Lyapuno functions for the fast-scale dynamics (2.11) with stationary path preference π. The following result is a consequence of Assumptions 5 and 6 on the driers local decision function. Lemma 3.1. For all π Π, {0,..., n 1}, and f E F, ) σ e (λ π G e(f E, π) fe π 0, e E where λ π := e E f π e. Proof. Throughout this proof, the explicit dependence of G e on π will be dropped. Define J := {e E : f e > f π e }, K := {e E : f e < f π e }, and let G J := j J G j, G K := k K G k, and G J c := e E \J G e. First, obsere that, since e E G e = 1, one has that G J = G J c. Now, we are going to show that G J (f π ) G E J (f E ) 0, (3.2) by writing the difference aboe as a path integral of G J ( ) first along the segment S J from f E to the point f R E with fj := fj π, for j J and f e := f e for e E \ J, and then along the segment S K from f to f π. In this way, one gets: G J (f π ) G E J (f E ) = G J ( f E ) d f E G J ( f E ) d f E S J S K = G J c( f E ) d f E S J G J ( f E ) d f E. S K (3.3) Assumption 6 implies that G J c/ ρ j 0 for all j J, and G J / ρ k 0 for all k K. In turn, this implies that G J c d f E 0 along S J, and G J d f E 0 along S K. This and (3.3) proe (3.2). In a ery similar fashion, one proes that G K (f E ) G K (f π ) 0. (3.4) E 12

14 Now, obsere that Assumption 5 implies that λ π G e(f π, π) = f π E e. From this, (3.2), and (3.4), it follows that 0 λ π = e E ( ) ( G J (f E ) G J (f π ) λ π E ( σ e λ π G e(f E ) λ π G e(f π ) E ) = σ e (λ π G e(f E ) fe π e E which proes the claim., G K (f E ) G K (f π E ) We now proceed to analyzing, for a fixed global decision π Π, the fast scale dynamics (2.11). Let V (f, π) := e E f π e f e, = 0, 1,..., n 1, be the l 1 -distance between the current flows on the outgoing links of, and the flow associated to the aggregate path preference π, and V (f, π) := λ π λ, = 1, 2,..., n, with λ π := e E f e π and λ := e E f e, be the absolute difference between the current flow incoming in node, and the one associated to the aggregate path preference π. Also, let V0 (f, π) := 0. Lemma 3.2. For all = 0, 1,..., n 1, π Π, and f F, e E σ e H e (f, π) V (f, π) V (f, π). ) ) Proof. Writing G e for G e(f E, π), and using Lemma 3.1, one gets that σ e H e (f, π) = e E which proes the claim. e E σ e (λ G e f e ) = σ e (λ λ π )G e σ e (λ π G e fe π ) σ e (fe π f e ) e E e E λ λ π e E f π e f e = V (f, π) V (f, π), By combining Lemma 3.2, and Assumption 1, one gets the result below. Recall the definition of W (ρ, π) from (3.1), and that we are using the conention d x /dx = sgn(x) for all x R. Lemma 3.3. For eery f = µ(ρ) F, and π Π, ρ W (ρ, π) H(f, π) (1 α)v (f, π). e E 13

15 Proof. Obsere that, thanks to the acyclicity of the graph as per Assumption 1, if e E E w for some nodes and w, then necessarily w 1. Since α < 1, it follows that α 1 E (e)1 E w (e) α w1 1 E (e)1 E w (e), for eery 1 n, and 0 w n 1. Hence, α V (f, π) α f e fe π 0 <n = 0 <n e E 1 <n 0 w<n e E α w1 0 w<n e E α α w f e fe π 0 w<n = αv (f, π), α 1 E (e)1 E w (e) f e fe π e E w 1 E w (e) f e f π e 1 <n 1 E (e) where the last inequality follows from the fact that n 1 =1 1 E (e) n =1 1 E (e) = 1, and we recall (3.1) for the definition of V (f, π). Thus, Lemma 3.2 implies that ρ W (ρ, π) H(f, π) = α σ e H e (f, π) which proes the claim. 0 <n 0 <n e E α V (f, π) αv (f, π) V (f, π), 0 <n α V (f, π) 3.2. Boundedness of the traffic densities. We shall now proe a couple of results guaranteeing that the traffic density on eery link remains bounded in time. We start with the following result, guaranteeing that, on eery link e E, the flow associated to the current path preference, f π e (t), stays eentually bounded away from the maximum flow capacity C e. Its proof relies on Assumption 4. Recall that our formulation allows for both the cases of finite and infinite maximum flow capacity on a link. Lemma 3.4. For eery admissible perturbation h, there exists t 0 R, and, for eery link e E, a positie finite constant C e, dependent on h but not on η, such that, for eery initial condition π(0) int (S(P)), ρ(0) (0, ) E, f π e (t) C e < C e, t t 0, e E. Proof. The fact that f π e (t) 1 for all e E follows from the fact that the arrial rate at the origin is unitary. Therefore, for all e E with C e > 1 (and, hence also for C e = ), the claim follows triially with C e = 1 and t 0 = 0. We now proe the lemma for all e E with C e < 1. Recall that, by Definition 2.2, the domain of the admissible perturbation h is a closed set Π h int(π). This, in particular implies that κ e := C e sup{(aω) e : ω Π h } > 0. 14

16 It follows from (2.5) that C e κ e = sup{(aω) e : ω Π h } sup {(A argmin{ω A T (f) h(ω) : ω Π h }) e : f F} = sup {( AF h (f) ) e : f F}. (3.5) Hence, one has This implies that d dt f π e (t) = η ( A(F h (f(t)) π(t)) ) e η (C e κ e f π e ). f π e (t) C e κ e (f π e (0) C e κ e )e ηt e ηt, t 0, (3.6) where the last inequality follows from the fact that f π e (0) = p A epπ p (0) 1, and C e κ e. The lemma for e E with C e < 1 now follows from (3.6), by choosing, e.g., C e := C e κ/2 with κ := min{κ e : e E s.t. C e < 1}, and t 0 := η 1 log(κ/2). The following result shows that the actual flow f e (t) also stays bounded away from the maximum flow capacity C e. Lemma 3.5. For eery admissible perturbation h, there exists η > 0, and a positie finite constant C e, for eery e E, dependent on h but not on η, such that, for eery η < η, and eery initial condition π(0) int (S(P)), ρ(0) (0, ) E, for all t 0 and e E. Proof. For t 0, let us define f e (t) C e < C e, ζ(t) := W (ρ(t), π(t)), χ(t) := V (f(t), π(t)). Obsere that, thanks to Lemma 3.4, there exists t 0 0, and a positie constant C e for eery e E, such that, for eery t t 0, ρ π e (t) ρ e, ρ e := µ 1 e (C e ), e E. (3.7) Since ρ π e (t) 0, the aboe implies that, if ρ e (t) ρ π e (t) 2ρ e for some t t 0, then necessarily ρ e (t) 2ρ e for t t 0. Hence, f e (t) f π e (t) χ e for all t t 0, where χ e := µ e (2ρ e) C e. Obsere that, since µ e is strictly increasing by Assumption 2, one has χ e = µ e (2ρ e) C e > µ e (ρ e) C e = 0. Now, let Notice that ζ := 2 E max{ρ e : e E}, χ := α n 1 min{χ e : e E}. W (ρ, π) E max{ ρ e ρ π e ) : e E}, V (f, π) α n 1 f e f π e, e E. Therefore, it follows that, for any t t 0, if ζ(t) ζ, then for some e E, we hae that ρ e (t) ρ π e 2ρ e for t t 0. This in turn implies that χ(t) χ e χ. Therefore, in summary, ζ(t) ζ = χ(t) χ > 0, t t 0. (3.8) 15

17 On the other hand, obsere that (3.7) implies that there exists some l > 0 such that 0 <n α e E 1 µ e(ρ π e (t)) l, t t 0. By combining the aboe with Lemma 3.3, one finds that, for any u, t t 0, ζ(t) ζ(u) = t u 0 <n t u t t u α ( d σ e ds ρ e d ) ds ρπ e ds e E ρ W (ρ, π) H(f, π)ds u 0 <n α e E ( (1 α)χ(s) 2ηl) ds. η (AF h µ e(ρ π (f π )) e (Aπ) e ds e ) (3.9) Now, let us define η := (1 α)χ /(2l). By contradiction, let us assume that lim sup t f e (t) C e for some e E. Since f e (t) = µ e (ρ e (t)) < C e for eery t 0, this implies that lim sup t ρ e (t) =. This, together with (3.7) implies that lim sup t ζ(t) =. Then, in particular, the set T := {t > 0 : ζ(t) > ζ(s), s < t} is an unbounded union of open interals, with lim t T,t ζ(t) =. This, and (3.8) imply that there exists a non negatie constant t t 0 such that χ(t) χ, t T [t, ). For eery η < η, Equation (3.9) and the aboe gie ζ(t) ζ(u) t u ( (1 α)χ(s) 2ηl) ds t u ( (1 α)χ 2ηl) ds < 0 for eery t > u t such that t and u belong to the same connected component of T. But this contradicts the definition of the set T. Hence, if η < η, then lim sup t f e (t) < C e for eery e E. Since on eery compact time interal I R, one has sup t I f e (t) = f e (ˆt) < C e for some ˆt I, the foregoing implies the claim. The result below is a consequence of Lemma 3.5, and will proe useful in the sequel. Proposition 3.6. There exists K > 0, and t 1 0 such that, for eery initial condition π(0) int (S(P)), ρ(0) (0, ) E, π h(π(t)) K for all t t 1. Proof. First, obsere that, thanks to Lemma 3.5, there exists T > 0 such that T (f(t)) T for all t 0. Thanks to this, and Assumption 4, one has that F h (f(t)) int(π h ), and π h(f h (f(t))) = ΦA T (f(t)), where recall that Φ = I P 1 11 is the projection matrix corresponding to the projected gradient with respect to π on S(P). Hence, π h(f h (f(t))) Φ A T, which implies that there exists a conex compact K int(π h ) such that F h (f(t)) K for all t 0. Define (t) := η 1 e ηt t 0 16 e η(t s) F h (f(s))ds.

18 As (t) is an aerage of elements of the conex set K, necessarily (t) K for all t 0. Then, π(t) = e ηt π(0) (1 e ηt ) (t) approaches K, which implies that, for large enough t, π(t) K 1 int(π h ), where K 1 is a closed subset of int(π h ) that contains K. Hence, after large enough t, say t 1, π h(π(t)) stays bounded Estimating the distance between the current density and the one associated to the current path preference. We analyze here the behaior in time of W (ρ(t), π(t)). First, we hae the following result, characterizing the ariation of W (ρ, π) as a function of π. Recall that π = Φ π denotes the projected gradient with respect to π on S(P). Lemma 3.7. There exists l > 0, and t 0 0, such that, for eery initial condition π(0) int (S(P)), ρ(0) (0, ) E, π W (ρ(t), π(t)) (F h (f(t)) π(t)) 2l 1 α, t t 0. Proof. First, obsere that, thanks to Lemma 3.4, one has that there exists t 0 0 such that l e := sup{1/µ e(ρ π e (t)) : t t 0 } <. Put l := max{l e : e E}. Then, for eery path p P, and eery t t 0, one has W (ρ, π) π p = = α 0 <n e E α 0 <n e E α 0 <n 0 <n l 1 α, σ e σ e ρ π e πp ) µ 1 e πq (q A epπ q A ep 1 µ e(ρ π e ) e E α A ep l e e E (3.10) where the third inequality follows from the fact that, thanks to Assumption 1 on the acyclicity of the network, each path p P passes through at most one link e E. Therefore, 2l 1 α Fp h (f) W (ρ, π) π p p π p W (ρ, π) π p p Fp h (f) W (ρ, π) π p W (ρ, π) π p p π p p = π W (ρ, π) (F h (f) π), where the first inequality follows upon recalling that both F h (f), and π are probability ectors oer the path set P, and by using (3.10). We can now combine Lemmas 3.3 and 3.7, in order to get the following estimate of the behaior in time of W (ρ(t), π(t)). Lemma 3.8. There exist l > 0, L > 0, η > 0 and t 0 0 such that, for eery initial condition π(0) int (S(P)), ρ(0) (0, ) E, W (ρ(t), π(t)) 2ηlL (1 α) 2 ( W (ρ(t 0 ), π(t 0 )) 17 2ηlL ) (1 α) 2 e (1 α)(t t0)/l

19 for eery t t 0 and η < η. Proof. Define ζ(t) := W (ρ(t), π(t)). Notice that, thanks to Lemmas 3.4 and 3.5, there exist L > 0, η > 0 and t 0 0, such that, for any η < η, This in particular implies that ρ e (t) ρ π e (t) L f e (t) f π e (t), e E, t t 0. V (f(t), π(t)) 1 L W (ρ(t), π(t)) = 1 L ζ(t), η < η, t t 0. Obsere that W (ρ, π) is a Lipschitz function of ρ and π, while both ρ(t) and π(t) are Lipschitz on eery compact time interal. Therefore, ζ(t) is Lipschitz on eery compact time interal, and thus absolutely continuous. Hence, dζ(t)/dt exists for almost eery t 0, and, thanks to Lemmas 3.3 and 3.7, it satisfies d dt ζ(t) = d W (ρ(t), π(t)) dt = ρ W (ρ, π) H(f, π) η π W (ρ, π) (F h (f) π) (1 α)v (f, π) 2ηl 1 α (1 α) 2ηl ζ(t) L 1 α. Then, the claim follows by integrating both sides Proof of Theorem 2.5. We now proceed to proing Theorem 2.5. Let us introduce the function Θ : Π R, Θ(π) := e E f π e 0 T e (s) ds (3.11) and obsere that Θ(π) = ΦA T (f π ), π int(π). (3.12) In game-theoretic terminology, equation (3.12) implies that Θ(π) is the potential function [19] for the continuous-population congestion game with action space P and payoff ector function A T (f π ). 5 5 In fact, (3.12) is equialent to A eqt e(fe π ) A ept e(fe π ) = e e π q Θ(π) π p Θ(π) for eery p, q P, i.e., the difference between the total delays associated to the flow f π on paths q and p equals the limit incremental ratio of Θ(π) with respect to an infinitesimal mass transfer in π from path p to path q. Intuitiely, if a nonatomic drier, whose weight is infinitesimal in the continuum population model, switches path from p to q, the potential Θ increases by an infinitesimal amount equal to the product of the increase in the drier s delay cost times the drier s weight. 18

20 Obsere that, since T e (f e ) is increasing, one has that each term f π e T 0 e (f e ) df e is conex in fe π. Hence, the composition with the linear map π fe π = p A epπ p is conex in π, which in turn implies conexity of Θ oer Π. Then, for any admissible perturbation h : Π h R, Definition 2.2 implies strict conexity of Θ(π) h(π). Therefore, since Π h is compact and conex, there exists a unique minimizer π h := argmin {Θ(π) h(π) : π Π h }. (3.13) Let f (h) := f πh. Then, we hae the following: Lemma 3.9. Let {h k } be any sequence of admissible perturbation functions such that lim k h k = 0, lim k Π hk = Π. Then, lim f (hk) = f W. k Proof. Write π k for π h k, F k for F h k, and Π k for Π hk. Since {Aπ k } AΠ, and AΠ is compact, there exists a conerging sub-sequence {Aπ kj : j N}. Let us denote by f := lim j Aπ kj AΠ its limit, and choose some π Π such that f = Aπ. Notice that, since sup{t e (fe π ) : π Π h } < for all e E, Definition 2.2 implies that the minimizer in (3.13) has to be in the interior of Π h. As a consequence, one finds that necessarily π h(π kj ) = ΦA T (Aπ kj ), which in turn implies that F kj (Aπ kj ) = π kj. Then, using (2.5), one finds that (Aπ kj ) T (Aπ kj ) h kj (π kj ) (Aπ kj ) T (Aπ kj ) h kj (ω), (3.14) k for all ω Π kj. Now, fix any π Π. Since Π k Π, one has that there exists a sequence { π j } such that π j Π kj for all j, and lim j π j = π. Hence, taking ω = π j in (3.14), and passing to the limit as j grows large, one finds that (π ) A T (Aπ ) π A T (Aπ ), π Π. In turn, the aboe can be easily shown to be equialent to the condition (2.14) characterizing Wardrop equilibria. From the uniqueness of the Wardrop equilibrium, it follows that necessarily f = f W. Then the claim follows from the arbitrariness of the accumulation point f. We shall now estimate the time deriatie of Θ h (π) along trajectories of our dynamical system. For this, define Γ(t) := Θ(π(t)) h(π(t)), ψ(t) := ΦA T (f π (t)) π h(π(t))). (3.15) Then, using (3.12), one has d ( dt Γ(t) = π Θ h(π(t)) ) d dt π = ηψ(t) ( F h (f(t)) π(t) ) (3.16) = ηψ(t) ( F h (f π (t)) π(t) ) ηψ(t) ( F h (f(t)) F h (f π (t)) ). Lemma 3.8 implies that there exists t 2 0, η > 0 and M 1 > 0 such that, for any η < η, W (ρ(t), π(t)) ηm 1 for all t t 2. From the definition of W, it also follows that W (ρ, π) α n 1 ρ ρ π 1 for all ρ, π. Moreoer, following Assumption 2, with 19