Existence of optima and equilibria for traffic flow on networks

This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.


Introduction
Consider a model of traffic flow where drivers travel on a network of roads. We denote by A 1 , . . . , A m the nodes of the network, and by γ ij the arc connecting A i with A j . Following the classical papers [13,14], along each arc the flow of traffic will be modeled by the conservation law Here t is time and x ∈ [0, L ij ] is the space variable along the arc γ ij . By ρ = ρ(t, x) we denote the traffic density, while the map ρ → v ij (ρ) is the speed of cars as function of the density, along the arc γ ij . We assume that v ij is a continuous, nonincreasing function. If v ij (0) > 0 we say that the arc γ ij is viable. It is quite possible that two nodes i, j are not directly linked by a road. This situation can be easily modeled by taking v ij ≡ 0, so that the arc is not viable. The conservation laws (1.1) are supplemented by suitable boundary conditions at points of junctions, which will be discussed later.
We consider n groups of drivers traveling on the network. Different groups are distinguished by the locations of departure and arrival, and by their cost functions. For k ∈ {1, . . . , n}, let As in [2,3,4], we consider a set of departure costs ϕ k (·), and arrival costs ψ k (·) for the various drivers. Namely, a driver of the k-th group departing at time τ d and arriving at destination at time τ a will incur in the total cost ϕ k (τ d ) + ψ k (τ a ). (1.5) In this framework, the concepts of globally optimal solution and of Nash equilibrium solution considered in [2,3] can be extended to traffic flows on a network of roads.
Definition 2. An admissible family {U k,p } of departure distributions is globally optimal if it minimizes the sum of the total costs of all drivers.
Definition 3. An admissible family {U k,p } of departure distributions is a Nash equilibrium solution if no driver of any group can lower his own total cost by changing departure time or switching to a different path to reach destination.
The main goal of this paper is to prove the existence of a globally optimal solution and of a Nash equilibrium solution, under natural assumptions on the costs and on the flux functions.
In the case of a single group of drivers traveling on a single road, the existence and uniqueness of such solutions were proved in [2]. We highlight the main features of the present analysis.
• Following the direct method of the Calculus of Variations, a globally optimal solution is constructed by taking the limit of a minimizing sequence {U (ν) k,p } ν≥1 of admissible departure distributions. The existence of the limit is guaranteed by the "tightness" of the sequence of approximating measures. Namely, for each ε > 0 there exists T > 0 (independent of n) such that the total amount of drivers departing at times t / ∈ [−T, T ] is less than ε.
• Toward the existence of a Nash equilibrium, the results in [2,3] used the assumption ψ k ≥ 0, meaning that the arrival cost functions are nondecreasing. We now strengthen this assumption to ψ k > 0, so that the arrival costs are strictly increasing. This apparently minor change in the hypotheses has an important consequence. Namely, it allows us to prove a crucial a priori bound on all departure rates, in any Nash equilibrium solution.
• The proofs in [2,3] relied on a monotonicity argument. Indeed, the departure distribution U (·) for a Nash equilibrium was obtained as the (unique) pointwise supremum of a family of admissible distributions, satisfying an additional constraint. On the other hand, the present existence result is proved by a fixed point argument. By its nature, this topological technique cannot yield information about uniqueness or continuous dependence of the Nash equilibrium.
The paper is organized as follows. In Section 2 we describe more carefully the traffic flow model, explaining how to compute the admissible solutions using the Lax-Hopf formula. In Section 3 we prove the existence of a globally optimal solution, while Section 4 is devoted to the existence of a Nash equilibrium solution.
For the modeling of traffic flow we refer to [1,6,13,14]. Traffic flow on networks has been the topic of an extensive literature, see for example [5,8,9,10] and references therein. A different type of optimization problems for traffic flow was considered in [11].

Analysis of the traffic flow model
In our model, x ∈ [0, L ij ] is the space variable, describing a point along the arc γ ij . Here L ij measures the length of this arc. The basic assumptions on the flux functions F ij (ρ) = ρ v ij (ρ) and on the cost functions ϕ k , ψ k are as follows.
Remark 2.1. By (A1), the flux function u = F ij (ρ) is continuous, concave and strictly increasing on the interval [0, ρ * ij ]. Therefore, it has a continuous inverse: ρ = g ij (u). As shown in fig. 1, the function u → g ij (u) is convex and maps the interval Remark 2.2. According to (A2), the cost for early departure is strictly decreasing in time, while cost for late arrival is strictly increasing. The assumption that these costs tend to infinity as t → ±∞ coincides with common sense and guarantees that in an equilibrium solution the departure rates are compactly supported.
Remark 2.3. In the engineering literature (see for example [9]) it is common to define the travel cost as the sum of the travel time plus a penalty if the arrival time does not coincide with the target time T A : Here D(t) denotes the total duration of the trip for a driver departing at time t, while Ψ is a penalty function. Calling τ a (t) = t + D(t) the arrival time of driver departing at t, the cost function (2.3) can be recast in the form (1.5). Indeed, In order that the assumptions (A2) be satisfied, it suffices to require that the function Ψ be continuously differentiable and 2.1 Traffic flow with an absolutely continuous departure distribution.
We now describe more in detail how the traffic flow on the entire network can be uniquely determined, given the departure distributions U k,p . As a first step, we consider the absolutely continuous case, so that (1.4) holds.
Along any arc γ ij , the traffic density ρ ij satisfies a boundary value problem of the form where u − ij (t) describes the incoming flux at x = 0. Following the approach in [2,3], we switch the roles of the variables t, x, replacing the above boundary value problem (2.5) with a Cauchy problem for the conservation law describing the flux u ij = F ij (ρ ij ): is the inverse of the function F ij , as in Remark 2.1. Consider the integrated functions Then U ij (t, x) provides a solution to the Hamilton-Jacobi equation (2.7) The viscosity solution to the above Cauchy problem is given by the Lax-Hopf formula [7] U ij (t, x) = min Here g * ij denotes the Legendre transform of g ij , namely For any given node A i , in general there will be several incoming arcs γ ,i , ∈ I(i) and several outgoing arcs γ i,j , j ∈ O(i) (see fig. 2). To determine the flux at the entrance of the arc γ ij , one needs to know how many drivers, after reaching the node A i , actually want to take the road γ ij . For this purpose, we need to introduce the distribution functions = total number of drivers of the k-th group, traveling along the path Γ p , that have entered the arc γ ij (possibly joining a queue at the entrance) within time t.
= total number of drivers of the k-th group, traveling along the path Γ p , that have exited from the arc γ ij (reaching the node A j ) within time t.
Of course, U ± kp,ij ≡ 0 if the path Γ p does not contain the arc γ ij .
The entire flux along the network is entirely determined by the functions U ± kp,ij . To recursively compute these functions, observe that the total number of drivers who have entered the arc γ ij (possibly joining a queue) within time t is determined by Notice that the first sum accounts for the drivers that transit through the node A i , while the second sum accounts for the drivers that initiate their journey from the node A i . Using the solution formula (2.8) with x = L ij we obtain Notice that the function U + ij is nondecreasing and Lipschitz continuous. Indeed, the rate at which drivers arrive at the end of the arc γ ij is computed by Among the drivers who reach the end of the arc γ ij within time t, we still need to compute how many belong to the various groups.
In the case where all departure rates U k,p with d(k) = i are absolutely continuous, the functions U + kp,ij can be computed as follows. For a.e. t there exists a unique time τ enter (t) such that A driver entering the arc γ ij at time τ enter (t) thus reaches the end of the arc at time t. The first-come first-serve assumption now implies In other words, among all drivers who exit from the arc γ ij at time t, the percentage of (k, p)drivers must be equal to the percentage of (k, p)-drivers that enter γ ij at time τ enter (t). In turn, the arrival distributions are computed by (2.14) Remark 2.4. The above equations can be solved iteratively in time. Namely, let be the minimum time needed to travel along any viable arc of the network. Given the departure ratesū k,p , if the functions U ± kp,ij (t) are known for all t ≤ τ , by the above equations (2.10)-(2.14) one can uniquely determine the values of U ± kp,ij (t) also for t ≤ τ + ∆ min .
Let G k,p be the total number of drivers of the k-th group who travel along the path Γ p . The admissibility condition implies G k,1 + · · · + G k,N = G k . We use the Lagrangian variable β ∈ [0, G k,p ] to label a particular driver in the subgroup G k,p of k-drivers traveling along the path Γ p . The departure and arrival time of this driver will be denoted by τ d k,p (β) and τ a k,p (β), respectively. Let U depart k,p (t) = U k,p (t) denote the amount of drivers of the subgroup G k,p who have departed before time t. Similarly, let U arrive k,p (t) be the amount of (k, p)-drivers who have arrived at destination before time t. For a.e. β ∈ [0, G k,p ] we then have With this notation, the definition of globally optimal and of Nash equilibrium solution can be more precisely formulated.
Definition 2 . An admissible family of departure distributions {U k,p } is a globally optimal solution if it provides a global minimum to the functional Definition 3 . An admissible family of departure distributions {U k,p } is a Nash equilibrium solution if there exist constants c 1 , . . . , c n such that: Here A k,p (τ ) is the arrival time of a driver that starts at time τ from the node A d(k) and reaches the node A a(k) traveling along the path Γ p .
In other words, condition (i) states that all k-drivers bear the same cost c k . Condition (ii) means that, regardless of the starting time x, no k-driver can achieve a cost < c k .

Traffic flow with general departure distribution
If all departure distributions U k,p are absolutely continuous, the previous analysis shows that the first-come first-serve assumption on the queues completely determines the traffic pattern. This is no longer true if a positive amount of drivers of different groups initiate their journey exactly at the same time. For example, assume that i = d(k) = d(k ) is the departure node for both k-drivers and k -drivers. Let γ ij be the first arc in the paths Γ p and Γ p , and assume that, at the instant t 0 , a positive amount of drivers in the subgroups G k,p and G k ,p initiate their journey. Since all these drivers join the queue at the entrance of the arc γ ij at the same time, additional information must be provided to determine their relative position in the queue. For this purpose, we follow the approach introduced in [3].
Given an arc γ ij , call the family of all drivers that initiate their journey from the node A i , traveling along γ ij as first leg of their journey. The total number of these drivers is In other words, U depart ij (t) is the total number of drivers in the family G ij that depart before time t. The relative position of these drivers in the queue at the entrance of the arc γ ij will be determined by a set of prioritizing functions.  We now show that, by choosing one set of prioritizing functions, the traffic flow on the entire network is entirely determined. Indeed, for all k, p, i, j the values U + kp,ij (t), determining how many drivers of the subgroup G k,p reach the end of the arc γ ij within time t, are computed as follows.
CASE 1: The only drivers traveling on the arc γ ij are those who started their journey from the node A i . In this case, as in (2.11) the total number of drivers arriving at the end of the arc γ ij within time t is Given the prioritizing functions B k,p , the values U + kp,ij (t) are immediately obtained by the formula (2.25) CASE 2: The arc γ ij is also traveled by drivers who transit through the node A i , departing from other nodes. In this case, drivers originating from A i have to merge with drivers in transit, coming from other nodes. The cumulative distribution function, accounting for the total number of drivers that have entered the arc γ ij within time t is As before, the total number of drivers who have exited from the arc γ ij before time t is given by (2.11). To determine how many of these drivers belong to each subgroup G k,p , we proceed as follows.
Consider the driver who exits from the arc γ ij at time t. This driver will have entered the arc γ ij at an earlier time τ = τ enter (t), such that Notice that τ enter (t) is uniquely determined, for all but countably many times t. For all subgroups of drivers in transit, the first-come first-serve priority assumption implies is the total number of drivers that reach the end of the arc γ ij before time t, while β counts how many of these drivers start their journey from the node A i . In turn, B k,p (β) determines how many belong to the subgroup G k,p .

Globally optimal solutions
In this section we establish the existence of a globally optimal solution. The proof follows the direct method of the Calculus of Variations, constructing a minimizing sequence of solutions and showing that a subsequence converges to the optimal one. Theorem 1 (existence of a globally optimal solution). Let the flux functions F ij and the cost functions ϕ k , ψ k satisfy the assumptions (A1)-(A2). Then, for any n-tuple (G 1 , . . . , G n ) of nonnegative numbers, there exists an admissible set of departure distributions U k,p and prioritizing functions B k,p which yield a globally optimal solution of the traffic flow problem.
Proof. 1. By (A2), all functions ϕ k + ψ k are bounded below. By possibly adding a constant, it is not restrictive to assume that ϕ k (t)+ψ k (t) ≥ 0 for every time t. Calling m 0 the infimum of all total costs in (2.16), taken among all admissible departure distributions {U k,p }, this implies m 0 ≥ 0. In addition, it is clearly not restrictive to assume G k > 0 for all k ∈ {1, . . . , n}.
By choosing a subsequence, we can assume Moreover, by Helly's compactness theorem we can assume that, as ν → ∞, one has the pointwise convergence while Ascoli's theorem yields the uniform convergence In (3.3), we have extended the prioritizing functions to the entire real line by setting In the remainder of the proof we show that the set of departure distributions U k,p together with the prioritizing functions B k,p yield a globally optimal solution.
2. Let ε > 0 be given. We claim that there exists a large enough constant T , independent of ν such that for all ν sufficiently large. Indeed, by (A2) there exists T such that If any one of the two conditions in (3.4) fails, then the total cost would be > m 0 + 1. Since by assumption as ν → ∞ the total cost approaches the infimum m 0 , this proves our claim.
By (3.4) it follows that the limit functions satisfy In particular, the limit departure distribution is admissible.

4.
It remains to prove that the limit departure distribution is optimal. Recall that, without loss of generality, we are assuming ϕ k (t) + ψ k (t) ≥ 0 for all k, t. The total costJ determined by the set of departure distributions {U k,p } and prioritizing functions {B k,p } can be estimated byJ (3.7) Fix ε > 0. By (3.4) all departures and arrivals of β-drivers with β ∈ [ε, G k,p − ε] take place in a uniformly bounded interval of time, say [−T, T ]. On this interval, all functions ϕ k , ψ k are uniformly continuous. Hence the pointwise convergence (3.6) yields Together with (3.7), this impliesJ ≤ m 0 , completing the proof.

Remark 3.2.
A natural conjecture is that, in a globally optimal solution, all departure rates u k,p = d dt U k,p are uniformly bounded and have compact support. Hence the corresponding solution should be uniquely determined without need of prioritizing functions. In the case of one group of drivers traveling on a single road, this fact was proved in [2]. In the general case, a proof of the above conjecture will likely require a more detailed study of the globally optimal solution, establishing necessary conditions for optimality.

Nash equilibria
In this section we prove the existence of a Nash equilibrium solution for traffic flow on a network. For our model, it turns out that in a Nash equilibrium all departure rates must be uniformly bounded and have compact support. As a consequence, the corresponding solution is uniquely determined without need of prioritizing functions.
Theorem 2 (existence of a Nash equilibrium). Let the flux functions F ij and the cost functions ϕ k , ψ k satisfy the assumptions (A1)-(A2).
(i) For any n-tuple (G 1 , . . . , G n ) of nonnegative numbers, there exists at least one admissible family of departure rates {u * k,p } which yields a Nash equilibrium solution.
(ii) In every Nash equilibrium solution, all departure rates are uniformly bounded and have compact support.
Before proving the theorem, we establish a "modulus of continuity" for the exit time. Namely, for drivers traveling along a given path Γ p , the arrival time τ p (t) is a uniformly continuous function of the departure time t. In the following, we first consider a single arc γ ij with flux function F ij (·) satisfying (A1). As in figure 1, g ij denotes the inverse function while g * ij is the Legendre transform. For a driver who enters the arc γ ij at time t (possibly joining a queue), we denote by τ ij (t) his exit time.
Lemma 4.1. Given constants G, M > 0, there exists a continuous function φ ij : R + → R + depending on the flux function F ij in (2.1) and on M, G, such that φ ij (0) = 0 and moreover the following holds. Let U − ij (·) be a Lipschitz continuous departure distribution, such that 0 ≤ u ij (t) = d dt U − ij (t) ≤ M for a.e. t and u ij (t) dt ≤ G. Then the exit time τ ij (·) satisfies Proof. 1. Let L ij be the length of the arc γ ij . Following [2], consider the function Since the Legendre transform g * ij is convex, the function h ij is concave. Moreover, calling µ ij . = L ij /v ij (0) the minimum time needed to drive across the arc γ ij (= length of the arc divided by the maximum speed), one has Using the solution formula (2.24), the amount of drivers that exit from the arc γ ij before time t is computed by (see Fig. 3) In other words, U + ij (τ ) is the amount by which one can shift upward the graph of h(· − τ ), before hitting the graph of U − ij (·). In turn, the exit time of a driver who enters the arc γ ij at Figure 3: A geometric construction of the exit distribution t → U + ij (t) from the entrance distribution U − ij , using the formula (4.4). time t is given by where we used the notation a ∨ b . = max{a, b}.
2. Let t 1 < t 2 be given. Two cases can be considered.
CASE 1 (Fig. 4, left). The minimum is attained at a points ∈ [t 1 , t 2 ]. In this case we have CASE 2 (Fig. 4, right). The minimum in (4.6) is attained at a points < t 1 . In this case we have In this case, we conclude where the continuous function φ ij is implicitly defined by 3. The two functions φ ij , φ ij defined at (4.8), (4.10) are both continuous and vanish at the origin. Defining the conclusion of the lemma is satisfied. Then, for any viable path Γ p , there exists continuous function φ p : Proof.
The assumption (4.12), together with the fact that the flux through each arc γ ij cannot be greater than the maximum flux F max ij , implies that the maximum incoming flux through each arc is bounded by some constant M . For each arc γ ij , let φ ij be the modulus of continuous dependence constructed in Lemma 4.1. If Γ is the path in (1.2), obtained as the concatenation of the arcs γ i( −1),i( ) , = 1, . . . , ν, it now suffices to define the function φ p as the composition of the corresponding functions φ i( −1),i( ) , namely Our final lemma shows that the arrival times, and hence the cost functions, depend continuously on the departure rates. 14) for every ν ≥ 1. Assume that for all k, p one has the weak convergence For each viable path Γ q , call τ ν q (t), τ * q (t) the corresponding arrival times of a driver who departs at time t and travels along Γ q . Then, as ν → ∞, one has the uniform convergence Proof. 1. We first consider the case of a single arc γ = γ ij . Assume that the departure rates and converge weakly: u ν u. Define the integrated functions Our assumptions imply the convergence U ν (t) → U (t), uniformly for t ∈ R. In turn, this implies that the exit distributions Fix a time t ∈ [−T, T ]. To estimate the difference |τ ν (t) − τ (t)| between the corresponding arrival times, consider the modified departure distribution (4.20) Call τ (t) the arrival time of a driver departing at time t, relative to the distribution U . By Lemma 4.1, the function t → τ (t) satisfies a uniform modulus of continuity say for some continuous function φ with φ(0) = 0. In particular, Observing that for all ν ≥ν , s ≤ t + ε and using the representation formula (4.5) with h ij given by (4.2), we obtain τ (t + ε) Switching the roles of U, U ν and hence defining we obtain the converse inequality for all ν ≥ν . 3. For each arc γ ij , the inequalities (4.22)-(4.23) show that the functions t → τ ν ij (t) converge uniformly to the corresponding functions t → τ ij (t). Given any path Γ q , the uniform convergence τ ν q → τ * q is now obtained by a straightforward argument, using induction on the number of arcs contained in Γ q .
We are now ready to prove the main result of this paper.
Proof of Theorem 2.
1. We claim that there exists a time interval I = [−T 0 , T 0 ] so large that, in any Nash equilibrium, no driver will depart or arrive at a time t / ∈ I. Indeed, given the n-tuple (G 1 , . . . , G n ), the travel time along any viable path Γ p = γ i(0),i(1) , . . . , γ i(ν−1),i(ν) is a priori bounded by

(4.24)
Here and in the sequel, we call G . = G 1 + · · · + G n (4.25) the total number of drivers. Notice that, in each summand on the right hand side of (4.24), the first term is an upper bound for the time spent waiting in the queue (total number of drivers divided by the maximum flux) while the second term is an upper bound on the actual travel time (length divided by the minimum speed). Let Therefore, in a Nash equilibrium no driver will depart or arrive outside [−T 0 , T 0 ]. Otherwise, he would achieve a strictly lower cost by departing at time t = 0. Observe that ψ min > 0, because of the assumption (A2). We claim that, in a Nash equilibrium, all departure rates u k,p must satisfy the priori bound u k,p (t) ≤ κ . = ϕ max · F max ψ min for a.e. t . (4.27)

Let
Indeed, consider drivers of the k-th family traveling along the path Γ p . Let t 1 < t 2 be any two departure times, and call τ 1 < τ 2 the corresponding arrival times. The total costs for these two drivers must be the same, hence ϕ(t 1 ) + ψ(τ 1 ) = ϕ(t 2 ) + ψ(τ 2 ) .
On the other hand, the upper bound on the flux implies We thus conclude Since this bound is valid for every interval [t 1 , t 2 ] ⊆ I, the pointwise bound (4.27) must hold. Moreover, for t / ∈ I we already know that u k,p (t) = 0. The last statement of the Theorem is thus proved. (4.29) It is understood that u k,p ≡ 0 if the path Γ p does not connect A d(k) with A a(k) . Notice that U is a closed convex subset of L 1 (R; R n×N ).
For each fixed ν ≥ 1, we consider a finite dimensional subset U ν ⊂ U consisting of all u = (u k,p ) which are piecewise constant on time intervals of length T /ν. Introducing the points t ν .

4.
Given u = (u k,p ) ∈ U, let τ q (t) be the arrival time of a driver starting at time t and traveling along the path Γ q . Clearly, this arrival time depends on the overall traffic conditions, hence on all functions u k,p . If this driver belongs to the j-th family, his total cost is Therefore, for every ν sufficiently large we can find two intervals