Nash equilibria for a model of traffic flow with several groups of drivers

DOI:10.1051/cocv/2011198 www.esaim-cocv.org

NASH EQUILIBRIA FOR A MODEL OF TRAFFIC FLOW WITH SEVERAL GROUPS OF DRIVERS

Alberto Bressan

and Ke Han

Abstract.Traffic flow is modeled by a conservation law describing the density of cars. It is assumed that each driver chooses his own departure time in order to minimize the sum of a departure and an arrival cost. There areN groups of drivers, Thei-th group consists ofκidrivers, sharing the same departure and arrival costsϕi(t), ψi(t). For any given population sizesκ1, . . . , κn, we prove the existence of a Nash equilibrium solution, where no driver can lower his own total cost by choosing a different departure time. The possible non-uniqueness, and a characterization of this Nash equilibrium solution, are also discussed.

Mathematics Subject Classification. 35E15, 49K20, 91A12.

Received August 20, 2011.

Published online 16 January 2012.

1. Introduction

Consider a model of traffic flow wheren groups of drivers travel along a freeway, from a locationA(say, a residential neighborhood) to a locationB (the city center). Each driver seeks to optimize his departure time, given a cost for starting early, and a cost for arriving late. Different groups have different departure and arrival costs. For example, there may be three groups: drivers in the first group need to arrive at destination by 8:00 am., those in the second group need to arrive by 9:00 am., while drivers in the third group can arrive at any time between 9:00 and 12:00, and simply wish to minimize the time spent on the road.

Following the classical papers [13,14], traffic flow will be modeled by the conservation law

ρ_t+ [ρ v(ρ)]_x = 0. (1.1)

Hereρ=ρ(t, x) is the density of cars, whilev(ρ) is their speed. In our model, a basic assumption is that drivers of different groups are distinguished only by their costs, not by the vehicles they drive. In other words, the speed v(·) as a function of traffic density is the same for all groups. Callingρ_i=ρ_i(t, x) the density of drivers of the i-th group, the scalar function ρ=ρ₁+· · ·+ρ_n provides an entropy solution to the conservation law (1.1).

Aim of this paper is to study Nash equilibria. These are solutions where no single driver can lower his own cost by changing his own departure time. In the casen= 1, where all drivers share the same cost, the existence, uniqueness, and a characterization of Nash solutions were recently established in [2].

Keywords and phrases.Scalar conservation law, Hamilton-Jacobi equation, Nash equilibrium.

1 Department of Mathematics, Penn State University University Park, 16802 Pa, USA.bressan@math.psu.edu;

kxh323@psu.edu

Article published by EDP Sciences c EDP Sciences, SMAI 2012

The main features of the general case n ≥ 2, and the new results proved in the present paper, can be summarized as follows.

• A solution to the traffic flow problem with n groups of drivers can be determined by assigning a scalar densityρ=ρ(t, x), satisfying the conservation law (1.1), together with a “prioritizing function”. Among the firstβ drivers who have departed within a given timet, this function specifies how many belong to the first group, how many to the second group, etc.;

• for any n-tuple of costs c = (c₁, . . . , c_n), there exists at least one Nash equilibrium solution where all drivers of the i-th group share the same total cost c_i. The corresponding density function ρ in (1.1) is uniquely determined byc. However, for the samen-tuple of costs (c₁, . . . , c_n), there can be several different decompositionsρ=ρ₁+· · ·+ρ_n, yielding different Nash equilibria. In general, the map

c → κ(c) .

=

(κ₁, . . . , κ_n); κ_i .

=

ρ_i(t, x) dx is the the total number ofi-drivers in some Nash solution with costs (c₁, . . . , c_n)

(1.2)

is a multivalued function. On the other hand, the mapc→κ₁+· · ·+κ_n is always single valued;

• the multifunctionc→κ(c)⊂Rⁿin (1.2) has closed graph and nonempty, compact, convex values. Relying on Cellina’s approximate selection theorem [3], a topological argument shows that

c∈Rⁿκ(c) =Rⁿ₊. Otherwise stated, for everyn-tuple (κ₁, . . . , κ_n) with κ_i ≥0 for eachi, there exists a vectorc = (c₁, . . . , c_n) of costs and a Nash equilibrium solution where the total number of drivers of thei-th group isκ_i and eachi-driver pays the same total costc_i. Whenn≥2 this solution may not be unique.

In our model,x∈[0, R] is the space variable, describing a point along the road. A solution of (1.1) is determined by assigning the incoming flow

ρ(0, t)v(ρ(0, t)) = ¯u(t) (1.3)

at the entrance of the freeway. As remarked in [2], it is more convenient to switch the roles of the independent variables t, x, and rewrite the boundary value problem (1.1), (1.3) as a Cauchy problem for the flux of cars u=ρv(ρ), namely

u_x+f(u)_t = 0, u(t,0) = ¯u(t). (1.4)

Here u→f(u) =ρis defined as a partial inverse of the mapρ→ρ v(ρ) =u, as in Figure1.

The remainder of the paper is organized as follows. In Section 2 we clarify all the assumptions and state the main result. Section 3 is devoted to the analysis of Nash equilibria for a given vector of costs (c₁, . . . , c_n), while in Section 4 we prove the existence of an equilibrium solution for given population sizes (κ₁, . . . , κ_n).

For a general introduction to scalar conservation laws and the Lax formula we refer to [11,12,15]. A short introduction to control theory and Hamilton-Jacobi equations can be found in [4]. All the basic results on set- valued functions, including Cellina’s approximate selection theorem, can be found in [1]. Various optimization problems for traffic flow, based on the Lighthill-Whitham conservation law model, have been considered in [7–10].

For an introduction to differential games and for recent applications to traffic flow on networks we refer to [5]

and [6], respectively.

2. Statement of the main results

Consider the conservation law

u_t+f(u)_x = 0 (t, x)∈[0, T]. (2.1)

In our traffic model, after interchanging the roles of the variables,xwill denote time whilet∈[0, T] describes points along a highway of total length T. The dependent variable u=u(t, x) denotes the traffic flow at time

ρ

0 0 f (0)’ p

f (p)^*

0

M ρ v( )ρ

ρ^* ρ

u

M ρ

f(u)

u

*

Figure 1.Left: the functionρ→ρ v(ρ) describing the flux of cars. Middle: the functionf, implicitly defined byf(ρv(ρ)) =ρand extended according to (2.2). Right: the Legendre transformf^∗.

xacross the pointton the highway. In the Lighthill-Whitham model one hasu=ρv(ρ), whereρis the density of cars while v(ρ) is the speed of cars (see Fig. 1). The function u →f(u) = ρ is obtained by inverting the function ρ → ρv(ρ) = u, over the interval [0, ρ^∗] where _∂ρ^∂ [ρv(ρ)] ≥ 0. As in [2], we extend f to a function f :R→R∪ {+∞}, by setting (see Fig.1)

f(u) .

=

f(0+)u if u <0,

+∞ if u > M. (2.2)

Introducing the integral function

U(t, x) .

= _x

−∞

u(t, s) ds, (2.3)

the conservation law (2.1) can be equivalently written as a Hamilton-Jacobi equation

U_t+f(U_x) = 0. (2.4)

Throughout the following,f denotes the flux function extended to the entire real line as in (2.2), while f^∗(p) .

= max

u {pu−f(u)} (2.5)

is the Legendre transform off. In our model,U(t, x) describes the total number of cars that have crossed the pointt along the highway during the time interval ]− ∞, x].

As initial data we shall consider any non-decreasing, functionQ:R→R⁺, with

Q(−∞) = 0, Q(+∞) = κ < +∞. (2.6)

Here Q(x) describes the total number of drivers that have initiated their journey (possibly joining a queue at the entrance of the highway, if there is any) at time≤x.

Notice thatQis continuous except for countably many timesx. To fix the ideas, we shall consider the left- continuous version whereQ(x) =Q(x−) coincides with its left limit at everyx. When needed, we shall denote byQ(x+) = lim_y→x+Q(y) the right limit of Qat x.

For a givenQ(·), consider the Lipschitz continuous function U(x) .

= inf

Q(y) +M(x−y); y≤x

≤ Q(x). (2.7)

The difference Q(x)−U(x) can be interpreted as the length of the queue at the entrance of the highway at time x, whileU(x) denotes the total number of drivers that have actually departed (after clearing the queue) up to timex.

Fort >0, the entropy-admissible solution to the Cauchy problem (2.4), (2.7) is provided by the Lax formula:

U(t, x) = min

t f^∗ x−y

t +U(y); y∈R

= min

t f^∗ x−y

t +Q(y); y∈R

. (2.8)

Observe that the last two expressions in (2.8) are equal because (f^∗)(p)≤M for allp, and (f^∗)(p)→M as p→+∞. Moreover,U(0+, x) = U(x).

As in [2], we consider a costϕ(x) for early departure and a costψ(x) for late arrival. The basic assumptions will be:

(A1) The flux function f : [0, M] →R is continuous. Moreover, it is twice continuously differentiable on the open interval ]0, M[ and satisfiesf(0) = 0, together with

f(u)≥b₁>0, f(u)≥b₂>0 for 0< u < M, lim

u→M−f(u) = +∞. (2.9) (A2) The cost functionals ϕ, ψare locally Lipschitz continuous and satisfy

⎧⎨

⎩

ϕ(x) ≤ −δ₀ < 0, ψ(x) ≥ 0, ψ(x) ≥ 0,

⎧⎪

⎨

⎪⎩

x→−∞lim ϕ(x) = +∞,

x→+∞lim

ϕ(x) +ψ(x)

= +∞.

(2.10)

According to (A2), the cost for early departure is decreasing in time, while cost for late arrival is increasing.

The assumption that these costs tend to infinity ast→ ±∞coincides with common sense and guarantees that in an equilibrium solution the departure rate is compactly supported.

Given an initial dataQ(·) as in (2.6), forβ ∈ [0, κ[ we define the pointsx^q(β),x^d(β), andx^a(β) by setting

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

x^q(β) = max{x∈R; Q(x)≤β}, x^d(β) = max{x∈R; U(0+, x)≤β}, x^a(β) = max{x∈R; U(T, x)≤β}.

(2.11)

In the application to traffic flow, β is a Lagrangian variable labeling a particular driver. In this case, x^q(β) accounts for the time where this driver joins the queue, x^d(β) is the actual departure time and x^a(β) is the arrival time. We observe that, for all except countably many β, the points x^q(β) and x^a(β) are uniquely determined by the relations

Q(x^q(β)) ≤ β ≤ Q(x^q(β)+), U(T, x^a(β)) = β. (2.12) It will be convenient to introduce the function

h(s) .

= −T f^∗

−s T

, (2.13)

wheref^∗ is the Legendre transform off, as in (2.5). For a.e.β the arrival timex^a(β) can also be characterized as

x^a(β) = inf

x; Q(y)≥β+h(y−x) for ally≤x

. (2.14)

Defining the constant

μ = [length of the highway]

[maximum speed] = T f(0), (2.15)

for a driver that departs at an arbitrary timexwe define the arrival time as A(x) .

= max

x+μ, sup

β<Q(x)x^a(β)

. (2.16)

According to (2.16), if there is no traffic at all, then the total time needed for the trip isμ. On the other hand, if the driver starting at timexencounters traffic, his arrival time will be the supremum among the arrival times of all cars departed earlier.

The following definition was introduced in [2].

Definition 2.1. We say that a bounded, nondecreasing initial dataQ(·) satisfying (2.6) yields aNash solution of the Cauchy problem (2.4)–(2.7) with initial and terminal cost functionsϕ, ψif there exists a constantcsuch that:

(i) for almost everyβ∈[0, κ] one has

ϕ(x^q(β)) +ψ(x^a(β)) = c; (2.17)

(ii) for allx∈R, there holds

ϕ(x) +ψ(A(x)) ≥ c. (2.18)

In connection with the traffic model, condition (i) states that all drivers bear the same costc. Condition (ii) says that, regardless of the starting timex, no one can achieve a cost< c.

Next, assume that there are different groups of drivers that need to reach destination at different times. For i = 1, . . . , n we thus consider cost functions ϕ_i(x), ψ_i(x), all satisfying the assumptions (A2). We seek Nash equilibrium solutions in this more general situation.

Let Q₁(·), . . . , Qn(·) describe the departure distributions of the various groups of drivers. In other words, Q_i(x) is the total number of drivers of the i-th group that have started their journey (possibly joining the queue at the entrance of the highway) within time x. We assume that all these functions are left continuous and non-decreasing, with

Q_i(−∞) = 0, Q_i(+∞) = κ_i. (2.19)

Fori= 1, . . . , n, the valueκ_i≥0 accounts for the total number of drivers of thei-th group.

We will define a solution (U₁, . . . , U_n)(t, x) to the traffic flow problem, corresponding to the above departure pattern. Here U_i(t, x) describes the total number of drivers of the i-th group that have crossed the pointt on the highway within timex. Setting

Q(x) .

= n i=1

Q_i(x), U(t, x) .

= n

i=1

U_i(t, x), (2.20)

it is clear that the function U should provide an admissible solution to equation (2.4) with initial dataQ(·).

This implies that the functionU is uniquely determined by the Lax formula (2.8). In general, however, that this is not enough to uniquely determine the componentsU_i. For example, ifQ_iandQ_j both have a jump atx=x₀, this means that at the same instant of timex₀ a positive amount of drivers join the queue, both from thei-th and from thej-th group. Different solutions can then be obtained, depending on the priority that we assign to drivers in each group. To resolve this issue, we consider a family of nondecreasing functionsBi: [0, κ]→[0, κ_i] with the following properties:

n i=1

Bi(β) = β for allβ ∈[0, κ], (2.21)

Bi(Q(x)) = Q_i(x) for allx∈R, i∈ {1, . . . , n}. (2.22)

Remark 2.2. Given the left continuous functions Q₁, . . . , Q_n as in (2.19), an n-tuple of functions Bi satis- fying the above properties can be constricted as follows. Consider the left-continuous, nondecreasing function x→ Q(x) .

=

iQ_i(x)∈[0, κ]. Ifβ =Q(x) for somex, we then define Bi(β) = Q_i(x).

On the other hand, ifQ(x)< β≤Q(x+), we define Bi(β) .

= Q_i(x) + β−Q(x) Q(x+)−Q(x)

Q_i(x+)−Q_i(x) . It is now easy to check that both conditions (2.21)–(2.22) are satisfied.

Given an initial data Q(·) as in (2.6) and the prioritizing functions Bi : [0, κ] → [0, κ_i], the functions U_i in (2.20) are now uniquely determined by the formulas

U_i(t, x) = Bi(U(t, x)). (2.23)

Here, fort >0,

U(t, x) = min

t f^∗ x−y

t +Q(y); y∈R

(2.24) is the solution of (2.4) with initial dataQ(·), computed by the Lax formula.

We wish to introduce a concept of Nash equilibrium solution, extending Definition 2.1 to the case of several groups of drivers with different costs. Toward this goal, we first need to determine the starting time (possibly joining the queue at the entrance of the highway) and the arrival time of each driver in the i-th group. Let ζ∈[0, κ_i] label this particular driver. We then define

⎧⎨

⎩

x^q_i(ζ) = max{x∈R; Q_i(x)≤ζ},

x^a_i(ζ) = max{x∈R; U_i(T, x)≤ζ}. (2.25) Notice that, for all but countably manyζ, these values are uniquely determined by the relations

Q_i(x^q_i(ζ)) ≤ ζ ≤ Q_i(x^q_i(ζ)+), U_i(T, x^a_i(ζ)) = ζ. (2.26) Definition 2.3. Consider a left continuous, nondecreasing initial dataQ(·) as in (2.6), and letBi: [0, κ]→[0, κ_i] be non-decreasing, surjective maps satisfying (2.21).

We say that the starting distribution Q(·) together with the prioritizing functions B₁, . . . ,B_n yield a Nash equilibrium solution to the traffic flow problem with cost functionsϕ_i, ψ_iif there exist constantsc₁, . . . , c_nsuch that, fori= 1, . . . , n, the following holds.

(i) For almost everyζ∈[0, κ_i] one has

ϕ_i(x^q_i(ζ)) +ψ_i(x^a_i(ζ)) = c_i; (2.27) (ii) for allx∈R, one has

ϕ_i(x) +ψ_i(A(x)) ≥ c_i. (2.28)

Herex→A(x) is the function in (2.16), describing the arrival time of a driver starting at timex. According to (i), every driver of thei-th group pays the same costc_i, while by (ii) he cannot decrease this cost by choosing any other starting time. We can now state our main result of existence of Nash equilibria.

Theorem 2.4. Let the flux function f satisfy (A1), and let the cost functions ϕ_i, ψ_i for the various groups of drivers satisfy (A2), for every i∈ {1, . . . , n}. Then, for every n-tuple(κ₁, . . . , κ_n)of non-negative numbers there exists a Nash equilibrium solution, where κ_i is the total number of drivers of thei-th group.

3. Nash equilibria with given costs

As a first step toward the proof of Theorem 2.4, given constants c₁, . . . , c_n, in this section we construct a Nash equilibrium solution where all drivers of thei-th group pay the same costc_i.

Letx→Λ_i(x) be the map implicitly defined by

ϕ_i(Λ_i(x)) +ψ_i(x) = c_i. (3.1)

In other words, if an i-driver arrives at destination at timex, in order to pay a total cost c_i he must depart at time Λ_i(x). The assumption (A2) implies that the maps Λ_i : R → R are well defined, locally Lipschitz continuous, and nondecreasing.

We now consider an auxiliary problem where all drivers have the same starting and arrival costs, namely ϕ(x) .

= −x, ψ(x) .

= min

i=1,...,nΛ_i(x). (3.2)

Lemma 3.1. Let the flux function f satisfy (A1), and assume that each pair of cost functions (ϕ_i, ψ_i), i = 1, . . . , n satisfies (A2). Then the Hamilton-Jacobi equation (2.4) has a unique Nash equilibrium solution corresponding to the cost functionsϕ, ψ in (3.2), where all drivers pay zero total cost.

Proof. In the case all drivers have the same starting and arrival costs, under the assumptions (A1)–(A2) the existence and uniqueness of the Nash equilibrium solution were established by Theorem 3 in [2]. We remark that the functionsϕ, ψ in (3.2) may not satisfy all the assumptions stated in (A2). However, all the arguments used in the original proof in [2] remain valid. Indeed,

• since every mapx→Λ_i(x) is locally Lipschitz continuous and nondecreasing, the same is true for the map ψin (3.2);

• since the Nash equilibrium is not affected if we add a constant to the cost functions, in the proof of the existence theorem, the conditionψ≥0 can be easily replaced by the assumptionψis bounded below.

To establish a lower bound onψ= min_i{Λ_i(x)} we proceed as follows. IfΛ_i(x))≤0, from the inequalities c_i ≥ c_i−ψ_i(x) = ϕ_i(Λ_i(x)) ≥ ϕ_i(0)−δ₀·Λ_i(x)

we deduce

Λ_i(x) ≥ ϕ_i(0)−c_i δ₀

=. a_i. (3.3)

Therefore,

ψ(x) ≥ a .= min{a₁, . . . , a_n}; (3.4)

• in the proof of Theorem 3, the assumption lim_x→+∞(ϕ(x) +ψ(x)) = +∞ is only used to show that all departures must occur within somea-priori bounded interval of time. Therefore, all the arguments in the proof remain valid if we can show that there exists an interval [a, b] such that

x /∈ [a, b] =⇒ ψ(x) > x. (3.5)

If all couples (ϕ_i, ψ_i) satisfy the assumptions (A2), we can chooseaas in (3.4). We then chooseb > alarge enough so thatϕ_i(x) +ψ_i(x)≥c_i for allx > b. This yields (3.5).

We recall that initial data for the Nash equilibrium solution admits the representation

Q^∗(x) = sup{Q(x); Q∈ Q₀}, (3.6)

where Q₀ is the family of all nondecreasing maps Q(·) as in (2.6), such that the corresponding solution U in (2.8), (2.11) satisfies

ϕ(x^q(β)) +ψ(x^a(β)) ≤ 0 for a.e.β ∈[0, Q(+∞)].

Otherwise stated,Qshould be the initial data for a solutionU of (2.4) where a.e. driver has a total cost≤0.

Next, let U =U(t, x) be the unique Nash equilibrium solution of (2.4) with costs (3.2). We will show that every Nash equilibrium solution of the problem for several groups of drivers with costs ϕ_i, ψ_i can be obtained as follows. Consider the unit simplex

Δ .=

(θ₁, . . . , θ_n);

n i=1

θ_i = 1, θ_i≥0 for all i

. (3.7)

LetΘ be the family of all measurable maps

x → θ(x) = (θ₁(x), . . . , θ_n(x)) ∈ Δ such that

θ_i(x) = 0 whenever Λ_i(x) > min

j=1,...,nΛ_j(x). (3.8)

Letκ .=U(T,+∞). For each mapθ∈Θ, define the prioritizing functionsB_i^θ(·) as follows. For all but countably manyβ∈[0, κ], there exists a unique arrival timex^a =x^a(β) such thatU(T, x^a) =β. We then set

B_i^θ(β) .

=

_x^a_(β)

−∞

θ_i(x)U_x(T, x) dx. (3.9)

Since all functions B^θ_i are Lipschitz continuous with Lipschitz constant 1, they can be uniquely extended to the entire interval [0, κ] by continuity. It can be easily checked that the functions B^θ_i are nondecreasing and satisfy (2.21). We then define

κ^θ_i .

= B_i^θ(κ), Q^θ_i(x) .

= B_i^θ Q(x)

, U_i^θ(t, x) .

= B^θ_i

U(t, x) . This clearly yields

n i=1

Q^θ_i(x) = ⁿ

i=1

B^θ_i(Q(x)) = Q(x), ⁿ

i=1

U_i^θ(t, x) = ⁿ

i=1

B_i^θ(U(t, x)) = U(t, x).

Theorem 3.2. Let the flux function f satisfy (A1), and assume that each pair of cost functions (ϕ_i, ψ_i), i= 1, . . . , n satisfies (A2). LetU =U(t, x) be the unique Nash solution constructed in Lemma 3.1, with initial dataQ(·), and letΘ be the set of all measurable maps θ:R→Δsatisfying (3.8). Then the following holds.

(i) For everyθ∈Θ, the initial dataQ(·)together with the prioritizing functionsB^θ_i defined at (3.9)provides a Nash equilibrium solution to the traffic flow problem with costsϕ_i, ψ_i;

(ii) if the starting distributionQ(·) together with the prioritizing functionsBi provide a Nash equilibrium solution with costsc= (c₁, . . . , c_n), then Q=Qand there exists θ∈Θ such that Bi=B_i^θ for each i= 1, . . . , n.

Proof.

1. To prove (i), fix anyθ∈Θ andj∈ {1, . . . n}. Since the functions

x → U(T, x), β → B^θ_j(β), x → U_j^θ(T, x) = B_j^θ(U(T, x))

are all continuous and non-decreasing, for all but countably many ζ ∈ [0, κ^θ_j] the following holds. The point x^a_j(ζ) is uniquely determined by the equation

U_j^θ(T, x^a_j) = ζ, (3.10)

moreover there exists a unique valueβ ∈[0, κ] such thatζ=B^θ_j(β), and the pointx^a(β) is uniquely determined by the equation

U(T, x^a) = β. (3.11)

The above uniqueness properties together yield

x^a_j(ζ) = x^a_j(B^θ_j(β)) = x^a(β). (3.12)

2. We now derive a similar identity for the initial timex^q when drivers join the queue. Since the functions x→Q(x) andx→Q^θ_j(x) =B^θ_j(Q(x)) are non-decreasing, for all but countably manyζ∈[0, κ^θ_j] the following holds. The pointx^q_j(ζ) is uniquely determined by the relations

Q^θ_j(x^q_j) ≤ ζ ≤ Q^θ_j(x^q_j+), (3.13) moreover there exists a unique valueβ∈[0, κ] such thatζ=B^θ_j(β), and the pointx^q(β) is uniquely determined by the relations

Q(x^q) ≤ β ≤ Q(x^q+). (3.14)

The above uniqueness properties together yield

x^q_j(ζ) = x^q_j(B_j^θ(β)) = x^q(β). (3.15)

3. We claim that, ifζ∈[0, κ_j] is a value for which all the uniqueness assumptions in the previous two steps hold, then

Λ_j x^a_j(ζ)

= min

i=1,...,nΛ_i x^a_j(ζ)

. (3.16)

Suppose that, on the contrary,Λ_j x^a_j(ζ)

> min

i=1,...,nΛ_i x^a_j(ζ)

. By continuity there existsε >0 such that Λ_j(y) > min

i=1,...,nΛ_i(y) for ally ∈ I_ε = [x. ^a_j(ζ), x^a_j(ζ) +ε].

By (3.8) and (3.9), no driver of thej-th group arrives during the time intervalI_ε. HenceU_j(T,·) is constant on this interval. For ally∈I_εwe thus have

U_j^θ(T, y) = B_j^θ(U T, y)

= _y

−∞

θ_j(x)U_x(T, x) dx =

_x^a_j(ζ)

−∞

θ_j(x)U_x(T, x) dx

=

_x^a_(β)

−∞

θ_j(x)U_x(T, x) dx = B^θ_j(β) = ζ.

In this case, every pointy∈I_εprovides a solution to the equationU_j^θ(T, y) =ζ. This contradicts the uniqueness assumption on the solution of (3.10). Hence (3.16) must hold. Since the uniqueness assumptions stated in the previous steps hold for all but countably many values ofζ, we conclude that (3.16) is satisfied for a.e.ζ∈[0, κ_i],

4. From the identities (3.12)–(3.15) it now follows ϕ

x^q_j(ζ) +ψ

x^a_j(ζ)

= ϕ x^q(β)

+ψ x^a(β)

= 0. (3.17)

By (3.2) and (3.16) for a.e.ζ∈[0, κ_i] we have x^q_j(ζ) = min

i=1,...,nΛ_i x^a_j(ζ)

=Λ_j x^a_j(ζ)

. Hence ϕ_j

x^q_j(ζ) +ψ_j

x^a_j(ζ)

= ϕ_j Λ_j

x^a_j(ζ) +ψ_j

x^a_j(ζ)

= c_j. This proves (2.27).

5. We now prove (2.28). Assume that, on the contrary, there existsx∈Rand an indexj∈ {1, . . . , n}such that

ϕ_j(x) +ψ_j A(x)

< c_j. Sinceϕ_j is monotone decreasing, this impliesx > Λ_j

A(x) . Then ϕ(x) +ψ

A(x)

< ϕ

Λ_j(A(x)) +ψ

A(x)

= −Λ_j A(x)

+ min

i=1,...,nΛ_i A(x)

≤ 0.

This contradicts to the fact that U(t, x) provides a solution to the Nash equilibrium problem with a single population of drivers and costsϕandψas in (3.2). Hence (2.28) must hold.

6. In the remaining steps we prove part (ii) of the theorem. Assume that the initial distributionQ together with the prioritizing functionB= (B₁, . . . ,Bn) provide a Nash equilibrium solution. We begin by proving that Q(·) = Q(·). For this purpose, it suffices to prove thatQ(·) satisfies (2.17) and (2.18), with c= 0 and costsϕ, ψ given by (3.2). The identityQ=Qwill then follow from the uniqueness of the Nash equilibrium solution for a single group of drivers.

Let U = U(t, x) be the solution of (2.4) with initial data Q, and fix an index j ∈ {1, . . . , n}. Since Bj : [0,κ]˜ →[0,˜κ_j] is non-decreasing, for all except countably many values ofζ ∈[0,˜κ_j] the following holds. There is a uniqueβ ∈[0,˜κ] such thatζ .=Bj(β). Moreover, the departure and arrival times

x˜^q_j(ζ) = ˜x^q(β), x˜^a_j(ζ) = ˜x^a(β) are uniquely determined by the identities

Q(˜ x^q(β)) ≤ β ≤ Q(˜ x^q(β)+), U(T, x˜^a(β)) = β, (3.18) and satisfy

ϕ_j(˜x^q(β)) +ψ_j(˜x^a(β)) = c_j. SinceQ, Byield a Nash equilibrium, by (2.27) and (3.1) it follows that

x^q(β) = ˜x^q_j(ζ) = Λ_j x˜^a_j(ζ)

= Λ_j x˜^a(β)

. (3.19)

Next, we claim that for this particular value ofβ one has Λ_j

x˜^a(β)

= min

i=1,...,nΛ_i x˜^a(β)

.

Indeed, assume that there exists k ∈ {1, . . . , n} with Λ_k x^a(β)

< Λ_j x^a(β)

. Since the cost functionϕ_k is continuous and strictly decreasing, choosing a starting timeΛ_k(˜x^a(β))< y <x˜^q(β) we achieve

ϕ_k(y) +ψ_k(˜x^a(β)) < ϕ_k(Λ_k(˜x^a(β)) +ψ_k(˜x^a(β)) = c_k. Therefore, the total cost to ak-driver starting at time yis

ϕ_k

y) +ψ_k(A(y)) ≤ ϕ_k(y) +ψ_k x˜^a(β)

< c_k. This yields a contradiction with (2.28), proving our claim.

7.We can now prove that the initial distributionQ yields a Nash solution to the problem with costsϕ, ψ as in (3.2). Since the mapsB_iare non-decreasing andB₁(β) +· · ·+B_n(β) =β for allβ∈[0,˜κ], for a.e.β one can find an indexj such thatζ=Bj(β) satisfies the conditions in the previous step. In this case, we have

x˜^q(β) = ˜x^q_j(ζ) = Λ_j(˜x^a_j(ζ)) = Λ_j(˜x^a(β)) = min

i=1,...,nΛ_i(˜x^a(β)) = ψ(˜x^a(β)).

Recalling thatϕ(x) =−x, for a.e.β∈[0,˜κ] we thus have

ϕ(˜x^q(β)) +ψ(˜x^a(β)) = 0.

Hence (2.17) is satisfied.

Finally, to show (2.18), assume by contradiction that there existsx∈Rsuch that ϕ(x) +ψ A(x)

<0. By the definition (3.2), this implies

i=1,...,nmin Λ_i A(x)

< x.

Choose an indexj such thatΛ_j A(x)

= min

i=1,...,nΛ_i A(x)

. Sinceϕ_j is strictly decreasing, one has ϕ_j(x) +ψ_j

A(x)

< ϕ_j

Λ_j(A(x)) +ψ_j

A(x)

= c_j against the assumption (2.28).

This completes the proof that Q yields a Nash solution, in connection with the cost functionsϕ, ψin (3.2).

By the uniqueness of the Nash solution, we conclude thatQ(·) = Q(·).

8.Thanks to the two previous steps, we can now writeQ,U andκin place of Q, U and ˜κ. LetJ ⊆[0, κ] be the set of allβ ∈[0, κ] such that (i) for everyi= 1, . . . , n, the differentialB_i(β) = dBi/dβat the pointβ is well defined, and (ii) the arrival timex^a(β) of theβ-driver is uniquely determined by the equationU(T, x^a) =β.

We observe that meas([0, κ]\J) = 0. Moreover, introducing the set of arrival times X(J) .

= {x^a(β); β∈J},

we have

x /∈X(J)

U_x(T, x) dx = 0. (3.20)

We now define the measurable function θ= (θ₁, . . . , θ_n) : [0, κ]→Δ_n by setting θ_i(x) .

= B_i

U(T, x)

if β=U(T, x)∈J, (3.21)

and choosingθ(x)∈Δ_n arbitrarily, subject only to the condition (3.8), when x /∈X(J). For alli∈ {1, . . . , n} and everyβ∈J using (3.20) we obtain

B_i^θ(β) .

=

_x^a_(β)

−∞

θ_i(x)U_x(T, x) dx =

_x^a_(β)

−∞

B_i

U(T, x)

U_x(T, x) dx

=

_{U(T, x}^a_(β))

0

B_i(s) ds = _β

0

B_i(s) ds = Bi(β).

SinceJ is dense in [0, κ], by continuity the identity Bi(β) =B_i^θ(β) remains valid for allβ∈[0,κ].˜

4. Nash equilibria with given population sizes

Assume that Q(·) together with the prioritizing function B = (B₁, . . . ,Bn) yields a Nash equilibrium. Let κ .=Q(+∞) be the total number of drivers. The size of thei-th group of drivers is thenκ_i =Bi(κ)

For a given vectorc∈Rⁿ, letK(c)⊂Rⁿ denote the set of alln-tuples (κ₁, . . . , κ_n) which correspond to some Nash solution with costs c = (c₁, . . . , c_n). Our eventual goal is to construct a Nash equilibrium solution for a givenn-tuple (κ₁, . . . , κ_n) describing the sizes of the various groups of drivers. This will be achieved by showing that the multifunctionc→K(c) is onto,i.e.

c∈Rⁿ

K(c) = Rⁿ₊ .

=

(κ₁, . . . , κ_n); κ_i≥0 for all i .

The next lemma provides a key step in this direction.

Lemma 4.1. The multifunctionc→K(c)is upper semicontinuous, with compact convex values.

Proof.

1. We first show that the multifunction has compact convex values. Fix c= (c₁, . . . , c_n) and let U(t, x) be the unique Nash solution with costsϕ, ψ as in (3.2), constructed in Lemma 3.1. By Theorem 3.2, we have the following characterization:

K(c) =

(κ₁, . . . , κ_n); κ_i = _+∞

−∞

θ_i(x)U_x(T, x) dx,

θ₁, . . . , θ_n

∈Θ

. (4.1)

By Oleinik’s inequality, we have 0≤U_x(T, x)≤Lfor some constantLand allx∈R. ThereforeU_x(T,·)∈L^∞ and the map K : (θ₁, . . . , θ_n) → (κ₁, . . . , κ_n) defined by the integral in (4.1) is a continuous linear operator from L¹(R;Rⁿ) into Rⁿ. The operatorK is trivially compact, because the range is finite dimensional. Since the domain Θ defined at (3.8) is a closed convex, bounded subset of L¹(R;Rⁿ), we conclude that its image K(c) =K(Θ) is a compact, convex subset ofRⁿ.

2. To prove that the mapc→K(c) is upper semicontinuous,i.e.it has closed graph, consider a sequence of Nash equilibrium solutions corresponding to costsc^ν = (c^ν₁, . . . , c^ν_n), ν ≥1. According to Definition 2.3, these solutions are determined by an initial dataQ^ν(·) together with the prioritizing functions:B_i^ν: [0, κ^ν]→[0, κ^ν_i], withκ^ν=Q^ν(+∞), so that the conditions (2.27)–(2.28) hold.

Assume that, for everyi= 1, . . . , n,

c^ν_i → c¯_i, κ^ν_i →κ¯_i as ν → ∞. Of course, this implies

κ^ν = n i=1

κ^ν_i → n i=1

¯κ_i = ¯κ as ν→ ∞.

We need to show that there exists a Nash equilibrium solution with costs ¯c= (¯c₁, . . . ,¯c_n) and total population sizes (¯κ₁, . . . ,¯κ_n).

By (2.21), all mapsB_i^ν: [0, κ^ν]→Rare Lipschitz continuous with constant 1. Therefore, by possibly taking a subsequence, we can achieve the pointwise convergenceB_i^ν(β)→ Bi(β) for allβ ∈ [0,κ[ . This convergence¯ is uniform on every compact subinterval. By possibly taking a further subsequence, by Helly’s compactness theorem we can assume thatQ^ν(x)→Q(x) pontwise, for everyx∈R. Clearly,Q(+∞) = ¯κ.

Next, for everyi∈ {1, . . . , n}andν ≥1, recallκ^ν_i =Q^ν_i(+∞) and consider the functionsβ_i^ν : [0, κ^ν_i]→[0, κ^ν] defined as

β^ν_i(ζ) = min{β; B_i^ν(β) =ζ}.

Observe that this construction yields

B_i^ν(β_i^ν(ζ)) = ζ for allζ∈[0, κ^ν_i].

With obvious meaning of notation, we also consider the departure and arrival times x^q,ν_i (ζ) = x^q,ν(β_i^ν(ζ)), x^a,ν_i (ζ) = x^a,ν(β_i^ν(ζ)).

Since all these functions are bounded and monotone, by possibly taking a further subsequence we can assume the pointwise convergence

β_i^ν(ζ) → β¯_i(ζ) for allζ∈[0,κ¯_i[,

x^q,ν_i (ζ) → x¯^q_i(ζ) = ¯x^q( ¯β_i(ζ)), x^a,ν_i (ζ) → x¯^a_i(ζ) = ¯x^a( ¯β_i(ζ)), with

Bi( ¯β_i(ζ)) = ζ for a.e.ζ∈[0,κ¯_i].

From the identities

ϕ_i(x^q,ν_i (ζ)) +ψ_i(x^a,ν_i (ζ)) = c^ν_i for a.e. ζ∈[0, κ^ν_i], lettingν → ∞and using the continuity ofϕ_i, ψ_i we conclude

ϕ_i(¯x^q_i(ζ)) +ψ_i(¯x^a_i(ζ)) = ¯c_i for a.e.ζ∈[0,¯κ_i].

Hence (2.27) holds.

To complete the proof, it remains to show that ϕ_i(x) +ψ_i

A(x)

≥ ¯c_i for allx∈R, (4.2)

whereAis the arrival map corresponding to the departure distributionQ:

A(x) .

= max

x+μ, sup

β<Q(x)

x¯^a(β)

. (4.3)

If (4.2) fails, by continuity we can findy∈Randδ >0 such that ϕ_i(y−δ) +ψ_i

A(y)

≤ ¯c_i−δ. (4.4)

We now consider the arrival maps

A^ν(x) .

= max

x+μ, sup

β<Q^ν(x)x^a,ν(β)

(4.5) and observe that

lim sup

ν→∞ A^ν(y−δ) ≤ A(y). (4.6)

From the inequalities

ϕ_i(y−δ) +ψ_i

A^ν(y−δ)

≥ c^ν_i, lettingν → ∞and using (4.6), sinceϕ_i≤0 while ψ_i≥0, we conclude

ϕ_i(y−δ) +ψ_i A(y)

≥ c¯_i.

This yields a contradiction with (4.4), completing the proof.

In order to prove the existence of Nash equilibrium solutions to the traffic problem with given population sizes ¯κ= (¯κ₁, . . . ,κ¯_n), we shall use a topological argument, based on Bruwer’s fix point theorem.

In the following, we denote by{e1, . . . ,en}the standard basis in Rⁿ, and let Δ .=

x∈Rⁿ; , ⁿ

i=1

ei·x≤1, ei·x≥0 for alli= 1, . . . , n

(4.7) be the unit simplex inRⁿ. Moreover, we consider the vector

¯e .

= n i=1

ei = (1,1, . . . ,1) ∈Rⁿ. (4.8)

Lemma 4.2. The multifunctionc→K(c)satisfies the following properties.

(i) If(κ₁, . . . , κ_n)∈K(c₁, . . . , c_n)and one of the costs satisfies c_j ≤ γ₀ .

= min

i=1,...,n min

x∈R

ϕ_i(x) +ψ_i(x)

, (4.9)

thenκ_j = 0;

(ii) For everyM >0, there exists a constantC_M such that:

if n i=1

c_i ≥ C_M and (κ₁, . . . , κ_n) ∈ K(c₁, . . . , c_n), then

n i=1

κ_i ≥ M. (4.10)

Proof.

(i) In view of assumption (A2),γ₀is well defined real number. Since any driver starting at timexmust arrive at some timeA(x)> x, the total cost to a driver of thej-th group satisfies the inequality

ϕ_j(x) +ψ_j(A(x)) > ϕ_j(A(x)) +ψ_j(A(x)) ≥ γ₀ ≥ c_j. Since noj-driver can achieve a cost≤c_j, the total number ofj-drivers must be zero.

(ii) Let ˜ube any solution to the conservation law (2.1) with total mass

˜u(t, x) dx = M and having compact support, say, contained in the domain [0, T]×[a, b]. We claim that (4.10) is satisfied by choosing the constant C_M ≥0 large enough so that

C_M ≥ n· max

i=1,...,n{ϕ_i(a) +ψ_i(b)}. (4.11)

Indeed, assume that the vector of costs c = (c₁, . . . , c_n) satisfies c_i ≥ C_M, and let (κ₁, . . . , κ_n) ∈ K(c).

Consider the functionsΛ_i(·) as in (3.1). Observe that the choice of the constantC_M in (4.11) yields

i=1,...,nmin Λ_i(x) ≤ min

i=1,...,nΛ_i(b) ≤ a for all x∈[a, b]. (4.12)

In view of Theorem 3.2, it suffices to prove that the Nash solution U^∗, for a single group of drivers with cost functionsϕ, ψ as in (3.2) and total cost zero, contains a total number of drivers≥M.

We observe that, in the above solution with density ˜u(t, x), every driver departs and arrives at some times x^q, x^a ∈[a, b], and pays a total cost

ϕ(x^q) +ψ(x^a) = −x^q+ min

i=1,...,nΛ_i(x^a) ≤ −x^q+a ≤ 0.

c

c

C Δ γ (γ,γ)

κ

p

q

M Δ

κ

κ

0

g

Φ

Figure 2.The simplexesΔ^∗=Δ^inner∪Δ^outer, andΔγ.

Therefore, the integral function U(0, x) .

=_x

−∞u(0, y) dy˜ is contained in the family Q₀ of all departure distri- butions where each driver bears a total cost≤0.

CallingQ^∗(·) the initial data for the Nash solution, the representation (3.6) now yields

κ₁+· · ·+κ_n = Q^∗(+∞) ≥ U(0,+∞) = M.

4.1. Proof of Theorem 2.4

With the aid of the previous lemmas, we can now give a proof of Theorem 2.4. Clearly, it suffices to prove that the upper semicontinuous, compact convex valued multifunctionc→K(c)⊂Rⁿ₊ is surjective, namely

c∈Rⁿ

K(c) = Rⁿ₊. (4.13)

We shall argue by contradiction. Assume that, for some vector ¯κ= (¯κ₁, . . . ,¯κ_n)∈Rⁿ₊ is not contained in the left hand side of (4.13). It is not restrictive to assume that ¯κ_i >0 for every i. Indeed, in the general case we can simply takec_j =γ₀ for every indexj such that ¯κ_j = 0, and consider a lower dimensional problem.

1. Using Lemma 4.2, fix anyM >

iκ¯_i and chooseC_M so that (4.10) holds. We then chooseγ < γ₀ and C > C_M −γ₀. Consider the two n-dimensional simplexes (see Fig.2)

Δ_γ .

=

(c₁, . . . , c_n);

n i=1

c_i ≤ C, c_i ≥ γ for alli

, (4.14)

Δ^∗ .

=

(κ₁, . . . , κ_n);

n i=1

κ_i ≤ M, κ_i ≥ 0 for alli

. (4.15)