# Stability analysis of transportation networks with multiscale driver decisions
^{†}^{†}thanks: This work was supported in part by NSF EFRI-ARES
grant number 0735956. Any opinions, findings, and conclusions
or recommendations expressed in this publication are
those of the authors and do not necessarily reflect the views
of the supporting organizations.

###### Abstract

Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the drivers’ route choices are influenced by information at multiple temporal and spatial scales. The considered model involves a continuum of indistinguishable drivers commuting between a common origin/destination pair in an acyclic transportation network. The drivers’ route choices are affected by their, relatively infrequent, perturbed best responses to global information about the current network congestion levels, as well as their instantaneous local observation of the immediate surroundings as they transit through the network. A novel model is proposed for the drivers’ route choice behavior, exhibiting local consistency with their preference toward globally less congested paths as well as myopic decisions in favor of locally less congested paths. The simultaneous evolution 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 evidence 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, evolutionary game theory, and cooperative dynamical systems.

ransportation networks, Wardrop equilibrium, traffic flows, evolutionary game dynamics, route choice behavior, multiscale decisions.

## 1 Introduction

As transportation demand is dramatically approaching its infrastructure capacity, a rigorous understanding of the relationship between the macroscopic properties of transportation networks and realistic driver route choice behavior is attracting renewed research interest. Such an analysis is essential, among other things, for appropriate design of incentives influencing drivers’ behavior in order to induce a desired socially optimal usage of the transportation infrastructure. A particularly relevant issue is the impact of drivers’ en route responses to unexpected events on the overall transportation network dynamics. This issue is particularly significant in a modern real-life transportation network scenario, where recent technological advancements in intelligent traveller information devices have enabled drivers to be much more flexible in selecting their routes to destination even while being en route. While there has been a significant research effort to investigate the effect of such technologies on the route choice behavior of drivers, e.g., see [19, 16], the analytical study of the dynamical properties of the whole network under such behavior has attracted very little attention.

This paper is focused on the stability analysis of transportation networks in a setup where the drivers have access to traffic information at multiple temporal and spatial scales and they have the flexibility to switch their route to destination at every intermediate traffic intersection. Specifically, we consider a model in which the drivers choose their routes while having access to relatively infrequent global information about the network congestion state, and real-time local information as they transit through the network. The drivers’ route choice behavior is then influenced by relatively slowly evolving path preferences as well as myopic responses to the instantaneous observation of the local congestion levels at the intersections. This setup captures many real-life scenarios where unexpected events observed en route might cause drivers 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 drivers’ personal experience, their opinion exchanges with their peers, as well as from information media. However, since the traffic dynamics is significantly influenced by the drivers’ response to real-time local information, such responses can influence the drivers’ path preference thereby modifying their global route choice behavior in the long run. We propose and analyze a novel model for the drivers’ route choice behavior that combines relatively infrequent information about the global congestion status of the network with real-time local observations 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 indistinguishable drivers enters from the origin, and flows through the network until reaching the destination node. Traffic parameters, such as average speed, traffic density, and flow, are modeled as homogeneous quantities on every link, related one to each other by functional dependencies representative of the links’ congestion properties. The dynamics of such traffic parameters is governed by the law of conservation of mass, as well as the drivers’ route choice behavior. In turn, the drivers’ route choice behavior is assumed to be influenced by two factors: the aggregate path preference, measuring the relative appeal of the different routes to the drivers, and local observations of the current congestion levels. The path preference dynamics evolve at a slow time scale (as compared to the traffic dynamics), following a perturbed best response to global information, embodied by the current congestion levels on the whole network. When traversing an intermediate node in the network, drivers behave according to their path preference, if this is consistent with the current, locally observed, aggregate behavior of the other drivers. On the other hand, when there is a discrepancy between the aggregate path preference and the locally observed aggregate behavior, then drivers tend to compensate this by myopically preferring routes which appear to be locally less congested.

The above-described model gives rise to a double feedback dynamics, governed by a finite-dimensional system of coupled ordinary differential equations. Such a dynamical system has two natural time scales, characterizing the dynamics of the drivers’ aggregate path preference and of the traffic parameters on the different links, respectively. We study the long-time behavior of this dynamical system: our main result shows that, in the limit of small update rate of the aggregate path preferences, a state of approximate Wardrop equilibrium [20] is approached. The latter is a configuration in which the delay associated to any source-destination path chosen by a nonzero fraction of the drivers does not exceed the delay associated to any other path. Our results contribute to providing a stronger evidence 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 [13], 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 cooperative^{1}^{1}1Here, the adjective ‘cooperative’ is intended in the sense of Hirsch [9, 10]. 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 evolutionary population games [12, 18] 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 have been widely analyzed with fluid-dynamical and kinetic models: see, e.g., [8], and references therein. As compared to these models (typically described by partial, or integro-differential equations), ours significantly simplifies the evolution of the traffic parameters (treating them as homogeneous quantities on the links, representative of spatial averages), whereas it highlights the role of the drivers’ route choice behavior with its double feedback dynamics, which is typically neglected in that literature.

On the other hand, transportation networks have been studied from a decision-theoretic perspective within the framework of congestion games [3, 17]. In these models, drivers make sequential myopic route choice decisions in pursuit of minimizing their personal travel times, in response to complete information about the whole network. Congestion games are known to belong to a class of games known as potential games, a consequence of which is that, best responses of the drivers are aligned with the gradient of a common potential function and hence the system eventually converges 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 [14]. 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 available to the drivers 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 drivers’ route choice decisions are affected by, relatively infrequent global information, as well as their real-time local information as they transit through the network. As a consequence, classic results of evolutionary game theory and population dynamics [12, 18] are not directly applicable to our framework, and novel analytical tools have to be developed, particularly for the analysis of the fast scale dynamics of the traffic parameters.

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 illustrative numerical experiments. Finally, we conclude in Section 5 and also mention potential future research directions.

Before proceeding, we establish here some notation to be used throughout the paper. Let be the set of reals, be the set of nonnegative reals. Let and be finite sets. Then, will denote the cardinality of , (respectively, ) the space of real-valued (nonnegative-real-valued) vectors whose components are labeled by elements of , and the space of matrices whose real entries labeled by pairs of elements in . The transpose of a matrix , will be denoted by , while to be an identity matrix, and the al-one vector, whose size will be clear from the context. The simplex of probability vectors over will be denoted by . If , will stand for the indicator function of , with if , if . For , is the -norm. By default, let denote the Euclidean norm. Let be the interior of a set , and denote its boundary. Let be the sign function, defined by is 1 if , if , and if . Conventionally, we shall assume the identity to be valid for every , including .

## 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 behavior on a transportation network as a system of coupled ordinary differential equations with two time scales representative of route choice behavior influenced by the two levels of information. 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 these components in detail.

### 2.1 Network characteristics

Let the topology of the transportation network be described by a directed graph (shortly, di-graph) , where is a finite set of nodes and is the set of (directed) links. For every node , we shall denote by , and , the sets of its incoming, and, respectively, outgoing links. A length- (directed) path from to is an -tuple of consecutive links with , and . A cycle is path of length from a node to itself. Throughout this paper, we shall assume that:

###### Assumption 1

The di-graph contains no cycles, has a unique origin (i.e., some such that ), and a unique destination (i.e., such that ). Moreover, there exists a path to the destination node from every other node in .

Assumption 1 implies that one can find a (not necessarily unique) topological ordering of the node set (see, e.g., [6]). We shall assume to have fixed one such ordering, identifying with the integer set , where , in such a way that

We shall model the traffic parameters as time-varying quantities which are homogeneous over each link of the network. Specifically, for every link , and time instant , we shall denote the current traffic density, and flow, by , and , respectively, while

will stand for the vectors of all traffic densities, and flows, respectively. Current traffic flow and density on each link are related by a functional dependence

(1) |

Such functional dependence models the drivers’ speed and lane adjustment behavior in response to traffic density on a particular segment of a road. It will be assumed to satisfy the following:

###### Assumption 2

For every link , the flow-density function is continuously differentiable, strictly increasing, strictly concave and is such that and .

###### Remark 1

Flow-density functions commonly used in transportation theory typically are not globally increasing, but rather have a -shaped graph [8]: increases from until achieving a maximum , and then decreases for . Assumption 2 remains a good approximation of this setting, provided that stays in the interval .

For every link , let

be its maximum flow capacity. Moreover, let

be the sets of local, and, respectively, global admissible flow vectors. Observe that our formulation allows for both the cases of bounded and unbounded maximum flow capacities. As the flow is the product of speed and density, it is natural to introduce the delay function

(2) |

whose components measure the flow-dependent time taken to traverse the different links.^{2}^{2}2Here it has implicitly been assumed, without any loss of generality, that all the links are of unit length.

###### Example 1

A flow-density function that satisfies Assumption 2 is given by

(3) |

where , and . The corresponding delay function is

We shall denote by the set of distinct paths in from the origin node to the destination node , and let

be the link-path incidence matrix of . The relative appeal of the different paths to the drivers will be modeled by a time-varying probability vector over , which will be referred to as the current aggregate path preference, and denoted by . If one assumes, as we shall do throughout this paper, a constant unit incoming flow in the origin node, it is natural to consider the vector

of the flows associated to the current aggregate path preference. Indeed, represents the total traffic flow that a link would sustain in a hypothetic equilibrium condition in which the fraction of drivers choosing any path is given by . Now, let

be the set of feasible path preferences. Here, the term ‘feasible’ refers to the fact that the flow vector associated to any satisfies the capacity constraint for every . Observe that, whenever for every (or, in particular, when link capacities are infinite), the set of admissible path preferences coincides with the whole simplex . In contrast, when for some , is a strict inclusion. On the other hand, whether is empty or not depends solely on the value of the min-cut capacity of the network [1, Ch. 4]

as shown in the following, simply established, result. {proposition} The set is nonempty if and only if . \proofFix a cut-set such that , and . Then, every path contains exactly one link such that , and . Hence, for every , one has that

Minimizing over all cut-sets shows that is necessary for to be nonempty.

For the inverse implication, consider a network with the same topology and link capacities . The min-cut capacity of this network satisfies . Note also that, from our construction, . Therefore, the max-flow min-cut theorem (see, e.g., [1, Thm. 4.1]) implies that there exists some , thus proving that is nonempty.

In the case when it is not hard to show that, since the incoming flow exceeds the outgoing flow of the network, the system will grow unstable, i.e., is unbounded as grows large, for some link . Therefore, throughout this paper we shall confine ourselves to transportation networks satisfying:

###### Assumption 3

The min-cut capacity satisfies .

### 2.2 Route choice behavior and traffic dynamics

We now describe the drivers’ route choice behavior and traffic dynamics on the network. We envision a continuum of indistinguishable drivers traveling through the network. Drivers enter the network from the origin node at a constant unit rate, travel through it, and leave the network from the destination node . While inside the network, drivers occupy some link . The time required by the drivers to traverse link , and the current flow on such link are governed by its congestion properties, as given by (2), and (1), respectively. When entering the network from the origin node , as well as when reaching the tail node of some link , the drivers instantaneously join some link . In this paper, we shall model the choice of such new link to depend on infrequently updated perturbed best responses of the drivers to global information about the congestion status of the whole network as well as on their instantaneous observation of the local congestion levels. We next describe these two aspects of the model in detail.

#### Aggregate path preference dynamics

The drivers’ aggregate path preference , already introduced in Sect. 2.1, models the relative appeal of the different paths to the drivers’ population. The aggregate path preference is updated as drivers access global information about the current congestion status of the whole network. This occurs at some rate , which will be assumed 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 vector . From , drivers can evaluate the vector , whose -th component coincides with the total delay one expects to incur on path assuming that the congestion levels on such path won’t change. Drivers’ are assumed to react to such global information by a perturbed best response

(4) |

where is an admissible perturbation, satisfying the following:

###### Assumption 4

An admissible perturbation is a function where is a closed convex set,
is strictly convex, twice differentiable in , and is such that ,
where is the projected gradient on ^{3}^{3}3We shall use the notation to denote the corresponding projection matrix..

As a result, the aggregate path preference evolves as

(5) |

The perturbed best response function provides an idealized description of the behavior of drivers whose decisions are based on inexact information about the state of the network. In particular, it can be shown that the form of given in (4) is equivalent to the minimization over paths of the expected delay corrupted by some (admissible) stochastic perturbation (see e.g. [11]).

It is easy to establish that the perturbed best response is continuously differentiable on . Moreover, it is well known [18] that, as , and , the perturbed best response converges to the set of best responses.^{4}^{4}4Here, the convergences , and are intended to hold in the Hausdorff metric. (see, e.g., [2, Def. 4.4.11])

###### Example 2

Assume that for all . Then, an example of perturbed best response satisfying Assumption 4 is the logit function with noise level , which is defined as

(6) |

This corresponds to the admissible perturbation function . For any fixed , one has that , with as defined in (6), is a uniform distribution over the set . We refer the reader to [7] for more on the connection between characterized by Assumption 4 and smooth best response functions.

###### Remark 2

In the evolutionary game theory literature, e.g., see [12, 18], the domain of an admissible perturbation function , as well as the one of the minimization in the right-hand side of (4), is typically assumed to be the whole simplex , instead of a closed polytope . Notice that, as already observed in Sect. 2.1, when for every , is a closed polytope, so that one can choose . Therefore, in this case, Assumption 4 does not introduce any additional restriction with respect to such theory.

On the other hand, when for some , then the inclusions of are both strict, so that Assumption 4 does introduce additional restrictions on the admissible perturbations. However, it is worth observing that, in a classic evolutionary 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

(7) |

rather than as in (5). For such dynamics, the fact that whenever , can be shown to imply that would reach a compact in some finite time and never leave it. In contrast, in the two time-scale model of coupled dynamics considered in this paper (see (11)), such more restrictive assumption is needed in order to ensure the same property for the trajectories of (see Lemma 3.2).

#### Local route decisions

We now describe the local route decisions, characterizing the fraction of drivers choosing each link when traversing a non-destination node . Such a fraction will be assumed to be a continuously differentiable function of the local traffic flow , as well as of the current aggregate path preference . We shall refer to

(8) |

as the local decision function at node , and assume that it satisfies the following:

###### Assumption 5

For all , and ,

###### Assumption 6

For all , , and ,

Assumption 5 is a consistency assumption. It postulates that, when the locally observed flow coincides with the one associated to the aggregate path preference , drivers choose to join link with frequency equal to the ratio between the flow and the total outgoing flow .

Assumption 6 instead models the drivers’ myopic behavior in response to variations of the local congestion levels. 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 observing that Assumption 6 is reminiscent of Hirsch’s notion of cooperative dynamical system [9, 10].

###### Example 3

For every non-destination node , and outgoing link , conservation of mass implies that

(10) |

### 2.3 Objective of the paper and main result

The objective of this paper is to study the evolution of the coupled dynamics

(11) |

where is the perturbed best response function defined in (4), is the rate at which global information becomes available, , with defined in (10), and and are related by the functional dependence (1). In particular, our analysis will focus on the double limiting case of small and small . We shall prove that, in such limiting regime, the long-time behavior of the system is approximately at Wardrop equilibrium [20, 15]. The latter is a configuration in which the delay is the same on all the paths chosen by a nonzero fraction of the drivers. More formally, one has the following: {definition}[Wardrop Equilibrium] An admissible flow vector is a Wardrop equilibrium if for some such that, for all ,

(12) |

Existence and uniqueness of a Wardrop equilibrium are guaranteed by the following standard result: {proposition}[Existence and uniqueness of Wardrop equilibrium] Let Assumptions 1-3 be satisfied. Then, there exists a unique Wardrop equilibrium . \proofIt follows from Assumption 2 that, for every , the delay function is continuous, strictly increasing, and such that . The proposition then follows by applying Theorems 2.4 and 2.5 from [15].

The following is the main result of this paper. It will be proved in Sect. 3 using a singular perturbation approach. {theorem} Let Assumptions 1–6 be satisfied. Then, for every initial condition , , there exists a unique solution of (11). Moreover, there exists a perturbed equilibrium flow such that, for all ,

(13) |

where is a nonnegative-real-valued, nondecreasing function of , such that . Moreover, for every sequence of admissible perturbations such that , and , one has

(14) |

Theorem 2.3 states that, in large time limit, the flow vector approaches a neighborhood of the Wardrop equilibrium, whose size vanishes as both the time-scale ratio and the perturbation norm vanish. While a qualitatively similar result is known to hold [18] in a classic evolutionary game theoretic framework (i.e., neglecting the traffic dynamics, and assuming it is instantaneously equilibrated, as in the ODE system (7)), the significance of the above is to show that an approximate Wardrop equilibrium configuration is expected to emerge also in our more realistic model of two-time scale dynamics. Therefore, our results provide a stronger evidence 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 2.3 is proved. First, observe that, thanks to the continuous differentiability of , , and , standard analytical arguments imply the existence and uniqueness of a solution of the the initial value problem associated to the system (11), with initial condition , .

In order to prove the rest of the statement, we shall adopt a singular perturbation approach (e.g., see [13]), viewing the traffic density (or, equivalently, the traffic flow ) as a fast transient, and the aggregate path preference as a slow component. Hence, we shall first think of as quasi-static (i.e., ‘almost a constant’) while analyzing the fast-scale dynamics (10), and then assume that is ‘almost equilibrated’, i.e. close to , and study the slow-scale dynamics (5) as a perturbation of (7). We shall proceed by proving a series of intermediate technical results, gathered in the following subsections.

Before proceeding, we introduce some notation to be used throughout the section. Let

denote, respectively, the density corresponding to the flow associated to the path preference , and the sign of the difference between it and the actual density . Finally, fix some , and define

### 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 and are Lyapunov functions for the fast-scale dynamics (10) with stationary path preference .

The following result is a consequence of Assumptions 5 and 6 on the drivers’ local decision function.

For all , , and ,

where . \proofThroughout this proof, the explicit dependence of on will be dropped. Define , , and let , , and . First, observe that, since , one has that . Now, we are going to show that

(15) |

by writing the difference above as a path integral of first along the segment from to the point with , for and for , and then along the segment from to . In this way, one gets:

(16) |

Assumption 6 implies that for all , and for all . In turn, this implies that along , and along . This and (16) prove (15). In a very similar fashion, one proves that

(17) |

Now, observe that Assumption 5 implies that . From this, (15), and (17), it follows that

which proves the claim.

We now proceed to analyzing, for a fixed global decision , the fast scale dynamics (10). Let

be the -distance between the current flows on the outgoing links of , and the flow associated to the aggregate path preference , and

with and , be the absolute difference between the current flow incoming in node , and the one associated to the aggregate path preference . Also, let .

By combining Lemma 3.1, and Assumption 1, one gets the result below. Recall that we are using the convention , for all . {lemma} For every , and ,

Observe that, thanks to the acyclicity of the graph as per Assumption 1, if for some nodes and , then necessarily . Since , it follows that

for every , and . Hence,

where the last inequality follows from the fact that . Thus, Lemma 3.1 implies that

which proves the claim.

### 3.2 Boundedness of the traffic densities

We shall now prove a couple of results guaranteeing that the traffic density on every link remains bounded in time. We start with the following result, guaranteeing that, on every link , the flow associated to the current path preference, , stays eventually bounded away from the maximum flow capacity . 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.

For every admissible perturbation , there exists , and, for every link , a positive finite constant , dependent on but not on , such that, for every initial condition , ,

The fact that for all follows from the fact that the arrival rate at the origin is unitary. Therefore, for all with or in particular, the lemma follows trivially with and . We now prove the lemma for all with . Recall that, by Assumption 4, the domain of the admissible perturbation is a closed set . This, in particular implies that, for all with ,

For every link with , it follows from (4) that

(18) |

Hence, one has for every link with ,

Then, Gronwall’s inequality implies that

for all and for every link with . The lemma for with now follows from the above, by choosing, e.g., with , and .

The following result shows that the actual flow also stays bounded away from the maximum flow capacity . {lemma} For every admissible perturbation , there exists positive finite constants , and , for every , such that, for every , and every initial condition , ,

Let , and . Observe that, thanks to Lemma 3.2, there exists a positive real constant for every , and , such that, for every ,

(19) |

Since , the above implies that, if for some and , then necessarily for that , we have that , where . Now, let , and . Note that