text:
\section{INTRODUCTION}

	In the recent years, the continuous growth of traffic flow  and the resulting overcrowding have led some cities
to seek solutions to manage this phenomenon.
The crowd motion modeling and the study of flow dynamics have become in the last decades two of the main targets of the transportation research community. Different modeling approaches have been proposed which can generally be
classified into two categories: microscopic models and macroscopic
models. The former include the cellular automaton
model, the social force model, and the lattice
gas
model (see e.g., \cite{Burstedde}--\cite{Guo}),
and are particularly well suited for use
with small crowds. Macroscopic models, in contrast, focus
on the overall behavior of pedestrian flows and are more
applicable to investigations of extremely large crowds, especially
when examining aspects of motion in which individual
differences are less important \cite{Hughes}--\cite{Hoogendoorn20041}.
In this paper, starting from the results in \cite{bagpes}--\cite{bafama}, we introduce a Mean Field (MF) approach to modeling and analytically studying
the flow of daily agents
along the street of a heritage city.
Mean field games (MFG) theory goes back to the seminal work by Lasry-Lions \cite{LLions} (see also \cite{HCAINMAL}). This theory
includes methods and techniques to study differential games with a large
population of rational players and it is based on the assumption that the
population influences individuals' strategies through mean field
parameters. Several application domains such as economic, physics, biology
and network engineering accommodate MFG theoretical models (see
\cite{AcCamDolc}--\cite{Guent2011}). In particular, models to
study of dynamics on networks and/or pedestrian movement can be found for
example in \cite{CCMar}--\cite{BBMZop}.
In this paper,
beside the usual framing of mean field games (which is typically defined by the pair made of Hamilton-Jacobi-Bellman and transport
equations), we consider the agent's path preferences dynamics. In particular, we assume that it evolves following a perturbed best response to global information about the congestion status of the whole network and to the control vector. Moreover,
this path preferences
dynamics evolves at a slow time scale as compared
to the physical dynamics.
We apply these arguments to the possible paths considering a network topology as in Figure~\ref{graphtopology}.

Our main result shows the existence of a MF equilibrium for our framework. This equilibrium is a time-varying distribution of agents,  $\widetilde{\rho}(t)$ for $t\in[0,T]$, on the network.
Distribution  $\widetilde{\rho}(t)$, when plugged in the cost to minimize, generates an optimal control $%
\widetilde{u}(t)$, for any agent starting at the origin $o$
which, in turn, yields the path preference $z(t)$ providing the time-varying distribution~$\widetilde{\rho}$.
We also introduce a possible bi-level optimization
problem for an external controller who aims to force the equilibrium to be as close as possible (in uniform topology) to a reference value $\widetilde{\rho}$.
We suppose that the external controller (the city hall, for example) may act on the congestion functions choosing them among a suitable set of admissible functions.

The rest of this paper is organized as follows. In Section \ref{sec:2}, we describe the model and state the hypotheses used in the paper. In Section \ref{sec:3}, we claim the existence of a MF equilibrium
 (due to the space limitation, we only include the sketches of the proofs)
 and, in Section \ref{sec:4}, we define a bi-level optimization problem.
In Section \ref{sec:5}, we draw conclusions and suggests future works.

\section{MODEL DESCRIPTION}\label{sec:2}

\subsection{Network characteristics}

We model the topology of the network as a directed multi-graph $\mathcal{G=(V, E)}$, where $\mathcal{V}$ is a finite set of nodes and $\mathcal{E}$ is a finite set of directed links. Each link $e = (\nu_e,\kappa_e)$ in $\mathcal E$ is directed from its tail node $\nu_e$ to its head node $\kappa_e\neq \nu_e$.
An oriented path from a node $v_0$ to a node $v_{r}$ is an ordered set of $r$ adjacent links $p=(e_1, e_2, \ldots, e_{r})$ such that $\nu_{e_1}=v_0$, $\kappa_{e_r}=v_{r}$, $v_s=\kappa_{e_{s}}=\nu_{e_{s+1}}$ for $1\le s\le r-1$, and no node is visited twice, i.e., $v_{l}\ne v_s$ for all $0\le l<s\le r$, except possibly for $v_0=v_r$, in which case the path is referred to as a cycle. Moreover, let $\ell_{e}$ be the length of the link $e \in \mathcal E $  and let $\ell^p=\sum_{e \in p} \ell_{e}$ be the length of the path $p$.
A node $v_j$ is said to be reachable from another node $v_k$ if there exists at least a path from $v_k$ to $v_j$.

We hold the following assumptions on the multi-graph~$\mathcal{G}$ and on the agents that move along its links.
\begin{Assumption}\label{ass:main}~
	\begin{enumerate}
		\item\label{ass:main1} $\mathcal{G}$ contains no cycles.
		\item\label{ass:main2} $\mathcal{G}$ includes an origin node $o$, from which any node in $\mathcal{V}$ can be reached, and a destination node $d\ne o$, which is reachable from any node in $\mathcal{V}$, $o$~included.
		\item\label{ass:main3} Agents arrive at~$o$ in the morning  and desire to leave from~$d$ in the afternoon.
	\end{enumerate}
\end{Assumption}

We denote $\Gamma$ as the set of all the paths from $o$ to $d$.
In particular, in this paper, we consider a  graph~$\mathcal{G}$ on which agents have only three possible paths to reach the destination~$d$ starting from the origin $o$ (see Figure~\ref{graphtopology}).
\begin{figure}[htpb]
	\centering
	\includegraphics [width=0.35\textwidth]{fig1.pdf}
	\caption{\label{graphtopology} The graph topology used in the paper.}
\end{figure}
In this case, the set of path is $\Gamma=\{p_1, p_2, p_3\}$, where $p_1=(e_1,e_4)$, $p_2=~(e_2, e_5)$, $p_3=(e_1, e_3, e_5)$. We denote the corresponding link-path incidence matrix by $A\in\{0,1\}^{|\mathcal E| \times |\Gamma|}$ with entries
\begin{equation*}
A_{ep}=
\begin{cases}
1 & \text{if} \quad e \in p, \\
0  & \text{otherwise}.
\end{cases}
\end{equation*}
For every link $e \in \mathcal E$ and time
instant $t \in [0, T]$, we denote the current mass and flow by $\rho_e(t)$ and $f_e(t)$, respectively, defined as
\begin{equation*}
\rho_e:[0, T]\to [0, \rho_{\max}], \quad f_e:[0, T]\to [0, C_e],
\end{equation*}
with $T>0$ the final horizon, i.e., the time by which every agent has to reach the destination $d$, and $C_e$ is the maximum flow capacity. Moreover let
\begin{equation}
\rho(t) := \{\rho_e(t): e \in \mathcal E\},\quad f(t) := \{f_e(t) : e \in \mathcal E \}
\end{equation}
be the vectors of masses and flows, respectively.
\subsection{Agents' dynamics and costs}

\begin{equation}  \label{eq:thetau}
\begin{cases}
\dot{\theta}_e(s)=u^e(s), &s\in ]t,T],\, \forall e \in \mathcal E,\\
\theta_e(t)=0, & \forall t \in [0, T],\, \forall e \in \mathcal E,
\end{cases}
\end{equation}
where $\theta_e(s) \in [0, \ell_e]$, being $\ell_e$ the length of the link. Using this space-coordinates, $\theta_e(s)=0$ means that the agent is in $\nu_e$, while $\theta_e(s)=\ell_e$ means that the agent is in $\kappa_e$ and hence he is inside the link $e$ as long as $0\leq\theta(s)\leq \ell_e$. Note that \eqref{eq:thetau} describes the evolution of a possible agent which is in $\nu_e$ at time $t$ independently of the fact that he is present at $\nu_e$ at that time. By definition, $\theta_e(s)$ and $u^e(s)$ are equal to $NaN$ when the agent is not on link $e$. Hereinafter, $NaN$ stands for {\it not a number}.

The control, $s\mapsto u^e(s)\in \mathbb{R}_{+}$, is  measurable and locally integrable, namely $u^e\in L^1_{loc}(t,T) \ \forall t$. For ease of notation, from now on, we call $\mathcal U_e$ the set of these kind of controls.
There is no loss of optimality in assuming $u^e\geq 0$ as we discuss later in Remark 1.
The cost to be minimized by every agent crossing a link $e$,
takes into account: i) the possible hassle of running in the link $e$ to reach $d$ on time; ii) the pain of being entrapped in highly congested link;
iii) the disappointment of not being able to reach $d$ by the final horizon~$T$.
Such a cost can be analytically represented by
\begin{align}\label{eqcost}
J_e(t, u^e)=&\int_t^T\chi_{\{0\leq\theta_e(s)\leq \ell_{e}\}}\left(\frac{(u^e(s))^2}{2}+\varphi_e(\rho_e(s))\right)\,ds\nonumber \\  +
&\chi_{\{0\leq \theta_e(T)< \ell_e\}}\alpha\sum_{j\in p(e)} \ell_{j},
\end{align}
where $p(e)$ is the shortest path from $\nu_e$ to $d$, $\alpha>0$ is a constant parameter representing a cost per unit of length and $\chi$ is the characteristic function.
In~(\ref{eqcost}),
the quadratic term inside the integral stands for cost i) while the other term stands for the congestion cost ii) given by the function $\varphi_e:[0, \rho_{\max}]\to [0, +\infty[ $; the last addendum stands for cost iii).\\
We define, with a little abuse of terminology, as value functions the following:
\begin{equation}\label{Valuefunc1}
\begin{split}
V^e(t)= &\inf\limits_{\substack{u^e \in \mathcal U_e}
\Big\{J_e(t, u^e) \\
& +\min\limits_{\substack {j\in \mathcal E : \nu_j=\kappa_e\\ e, j \in p,\, p \in \Gamma}}\{V^{j}(\tau(t, u^e))\}\Big\}\quad \forall \ e \in \mathcal E
\end{split}
\end{equation}
\begin{equation}\label{Valuefunc2}
V^0(t)=\min\limits_{\substack {e\in \mathcal E : \nu_e=o\\ e \in p,\, p \in \Gamma}}\{V^{e}(t)\}.
\end{equation}

In \eqref{Valuefunc1}, $\tau(t, u^e)$ is the time at which an agent entering link $e$ at time $t$ and choosing a control $u^e$ arrives in $\nu_j=\kappa_e$.\\
Note that the recursive definition of $V^e$ in \eqref{Valuefunc1} is not meaningless since the absence of oriented cycles in the network $\mathcal G$ prevents self-referring, that is it cannot occur that we define $V^e$ in terms of itself.
Also $V^0$ can be seen as associated to a fictitious link $e_0$ with null length, such that $\nu_{e_0}=\kappa_{e_0}=o$, through which the flow enters into the network.
Finally, observe that when the distributional evolution~$t\to \rho_e(t)$, i.e. the mass of the agents on the link $e$, is initially given then $V^e$ does not
depend on $\rho_e$.

In this paper, we assume that the physical traffic flow $f$ consist of indistinguishable
homogeneous agents which enter in the
network through the origin node, travel through it on the different paths and finally exit from the network through the
destination node. The relative appeal of the different paths to
the agents is modelled by a time-varying nonnegative vector $z(t)$ in the simplex
\begin{equation}\label{def:Slambda}
\mathcal{S}_{\lambda(t)}=\left\{z\in{\NZQ R}_{+}^{\vert\Gamma\vert}:\,1'z(t)=\lambda(t)\right\},
\end{equation}
where $\lambda(t) : [0, T] \to [0, \rho_{\max}]$ is the throughput, i.e., the total flow that goes through the network. We refer to the vector $z(t)$ as the current \textit{aggregate path preference} and let
\begin{equation}\label{flowzeta}
y^z(t)=Az(t)
\end{equation}
be the flow vector associated to it.
The vector $z(t)$ is updated
as agents access global information about the current
congestion status of the whole network (that is embodied
by the mass vector $\rho(t)$) and it is also influenced by the
vector $u(t)=\{u^e(t)\neq NaN: e \in \mathcal E\}$. Hereinafter, $\mathcal U$ is the set of these vectors.  Specifically, the cost perceived by each agent, traversing a link $e\in\mathcal E$, is given by \eqref{eqcost}.
The cost that an agent expects to incur along a path $p$ is
\begin{equation*}
J^p(t,u)=\sum_{e \in \mathcal E} A_{ep}J_e(t_e^p,u^e)
\end{equation*}
where $t_e^p(t)$, when $e \in p$, is the time instant in which an agent, arriving in $t$ in the origin $o$ and following the path $p$, reaches $\nu_e$. In particular $t_e^p(t)=t$ when $\nu_e=o$ with $e \in p$. Clearly, $t_e^p$ depends on the  controls $u^j$ for each link $j \in \mathcal E$ that precedes the link $e$ along $p$. Arbitrarily, we also set $t_e^p(t)=t$ when $e \notin p$.
We denote with
\begin{equation*}
J(t,u)=\{J^p(t,u): p \in \Gamma\}
\end{equation*}
the vector of costs on all the paths $p \in \Gamma$. Then, we  assume that the path preferences are updated at some rate $\eta>0$, according to a noisy best response dynamics
\begin{equation}\label{evolpi}
\dot{z}(t)=\eta(F^{(\beta)}(t, u)+Q^{(\beta)}(t, u)-z(t)),\ \ z(0)=z_0,
\end{equation}
which takes into accounts both the value of $\lambda(t)$ and $\dot{\lambda}(t)$.
Generally in the literature (see e.g., \cite{Como}, \cite{ComoMagg}) the throughput $\lambda$ is constant, hence in the formulation of the noisy best response dynamics $\dot{z}$ only the function $F^{(\beta)}$ appears. Here, as $\lambda$ varies over time, we need to introduce also the function $Q^{(\beta)}$ so that $z$ satisfies the constraint \eqref{def:Slambda}.
In particular,
 for every fixed noise parameter $\beta>0$ the function $$F^{(\beta)}: [0, T]\times \mathcal U \to {\NZQ R}^{\vert\Gamma\vert}_+$$ is the perturbed best response defined as follows:
\begin{equation}\label{bestresponse}
F^{(\beta)}(t, u)
=\frac{\lambda(t)\exp(-\beta(J(t,u)))}{1'\exp(-\beta(J(t,u)))},
\end{equation}
which provides an idealized description of the behavior of agents whose decisions are based on inexact information about the state of the network. Moreover it is continuous on $[0,T]\ \ \forall u \in \mathcal U$ fixed.
The function $Q^{(\beta)}:[0, T]\times \mathcal U \to {\NZQ R}^{|\Gamma|}_+$ is defined as
\vspace{-0.3cm}
\begin{equation}\label{funzlambdapunto}Q^{(\beta)}(t, u)=\frac{\dot{\lambda}(t)\exp(-\beta(J(t,u)))}{1'\exp(-\beta(J(t,u)))}.
\end{equation}
We assume that \eqref{funzlambdapunto} has the same structure of \eqref{bestresponse} and it shows how the agents' preferences $z$ are updated taking into account not only the value of $\lambda(t)$ but also its time variation. For this reason we consider $\dot{\lambda}(t)$ and we assign to it the same weight of $\lambda(t)$ selecting the same fixed noise parameter $\beta$.
We now describe the \textit{local decision function} $G:\mathcal {S}_{\lambda(t)}\to{\NZQ R}_+^{\vert\mathcal E\vert}$ characterizing
the fractions of agents choosing each outgoing link $e$ when traversing a non destination node $v$. It is given by
\begin{equation}\label{localchoice}
G_{e}(z)=
\begin{cases}
\displaystyle\frac{y_e^{z}}{\displaystyle\sum_{j \in \mathcal{E}: \nu_j=\nu_e}y_j^z} & \text{if}\quad{\displaystyle\sum_{j \in \mathcal{E}: \nu_j=\nu_e}y_j^z}>0\\
\,
\displaystyle\frac{1}{\vert\{j \in \mathcal{E}: \nu_j=\nu_e\}\vert}&  \text{if}\quad {\displaystyle\sum_{j \in \mathcal{E}: \nu_j=\nu_e}y_j^z}=0\,,
\end{cases}
\end{equation}
for each outgoing link $e$ in $\mathcal E$ and
 it is continuously differentiable on $ \mathcal {S}_{\lambda(t)}$. Equations  \eqref{localchoice} state that, at every node  $v \in \mathcal V$ the outflow is split proportionally to the flow vector $y^z$, if there is flow $y^z$ passing through node $v$, otherwise, the outflow is split uniformly among the immediately downstream links.

Now, for every non-destination node $v$ and outgoing link $e \in \mathcal E$
conservation of mass implies that
\begin{equation}\label{sistemaccoppiato}
\dot{\rho}(t)=H(f(t), z(t))\,,\qquad \rho(0)=\rho_0
\end{equation}
where $H: \prod_{e\in \mathcal E}[0,C_e] \times \mathcal S_{\lambda}\to{\NZQ R}^{|\mathcal E|}$ is defined as
\begin{equation}\label{H}
H_e(f, z):= G_{e}(z)\bigg(\lambda\delta_{\nu_e}^{(o)}+\sum_{j: \kappa_j=\nu_e}f_j\bigg)-f_e\,,\ \forall\ e\in\mathcal E,
\end{equation}
in which $\lambda(t)\delta_{\nu_e}^{(o)}$ accounts for the exogenous
inflow in the origin node $o$
and each component $f_e$ is
\begin{equation}\label{flusso} f_e(t)=\rho_e(t) u^e(\rho)/ \ell_e,\end{equation} i.e., the flow on the link $e$ estimated by an agent entering in the very link at time $t$.\\
Next we introduce the hypotheses that we will use to prove the existence of a MF equilibrium:

 (H1)  the throughput $\lambda:[0,T]\to[0,+\infty[$ is $C^1([0,T])$ and bounded;

 (H2)
	$\rho(0)=0$;

 (H3) $\varphi_e: [0, \rho_{\max}]\to [0, +\infty[$ are bounded, Lipschitz continuous and do not depend on the state variable $\theta_e$.\\

 \vspace{-0.3cm}
Note that $\rho(0)=0$ in (H2) means that no one is around the city at $t=0$, while (H3) implies that all agents in the same link at the same instant equally suffer the same congestion.
\begin{Remark}\label{ccp}
	Note that (H1) (H3) also imply that the control choice made by agents at the beginning of each link

	does not change as long as the agent remains in the same link, and it is constant in time. This is a feasible situation whenever the movement of an agent on a link cannot be interfered by the movements of agents that enter the same link after him. Note also that the same hypotheses imply that is  not convenient to go back along a link $e$, which in turn implies that the optimal control $u^e$ is always non-negative.
\end{Remark}

\subsection{Value Functions}
In order to define the MF equilibrium instead of coupling the two standard MF equations, i.e.,  Hamilton-Jacobi-Bellman equations and mass conservation ones \eqref{sistemaccoppiato}, with the preferences equations \eqref{evolpi} we write
conditions equivalent
in terms of the value functions.
An agent standing at $\nu_{e}$, and hence at $\theta_e=0$, for $e \in \{e_4, e_5\}$ at time $t$ has two
possible choices: either staying at $\nu_{e}$ indefinitely or moving to reach $\kappa_{e}=d$ exactly at time $T$ (it is not optimal to reach $d$ before $T$ and
wait there for a positive time length, see, e.g., \cite{bagpes}). Accordingly, the controls~are
\begin{equation}\label{oc2}
u^{e}_1=0, \qquad u^{e}_2= \frac{\ell_{e}}{T-t}.
\end{equation}
Hence, given the cost functional (\ref{eqcost}), we derive
\begin{equation}\label{eq:V1}
V^{e}(t)=\min\left\{ \alpha\ell_{e},\,\frac{1}{2}\frac{(\ell_{e})^{2}}{T-t}\right\} +\int_{t}^{T}\varphi_{e}\,ds
\end{equation}
(note that we do not display the argument $\rho_{e}$ of $\varphi_{e}$ for $e \in \{e_4, e_5\}$. We use this convention also for the other $\varphi_e$ in the formulas below).
An agent standing at $\nu_{e_3}$ at time $t\in[0,T]$, has two possible
choices (meaning, the optimal behaviour may only be one of the following
two): staying in $\nu_{e_3}$ or moving to reach $\kappa_{e_3}$ at $\tau \in ]t, T]$.
The  controls among which the agent chooses  and the corresponding value function are then, respectively
\begin{equation}\label{oc1}
u^{e_3}_1=0,\ \ u^{e_3}_2=\frac{\ell_{e_3}}{\tau-t},
\end{equation}
\vspace{-0.5cm}
\begin{align}\label{eq:V2}
V^{e_3}(t)=
&\min\left\{ \alpha\left(\ell_{e_3}+\ell_{e_5}\right)+\int_{t}^{T }\varphi_{e_3}\,ds,\right.\\ &\left.\inf_{\tau\in ]t,T]}  \left\{\frac{1}{2}\frac{(\ell_{e_3})^{2}}{\tau-t}+\int_{t}^{\tau }\varphi_{e_3}\,ds+V^{e_5}(\tau)\right\}\right\}. \nonumber
\end{align}
An agent standing at $\nu_{e_1}$ at time $t$ may choose: staying in $\nu_{e_1}$ or to reach $\kappa_{e_1}$ at a certain $\tau \in ] t,T]$.
Hence, the control is chosen among
\begin{equation}\label{oc4}
u^{e_1}_1=0,  \  u^{e_1}_2=  \frac{\ell_{e_1}}{\tau-t}.
\end{equation}
Consistently, one has
\begin{small}
\begin{align}\label{eq:V11}
&V^{e_1}(t)=\min\left\{ \alpha\min\left\{\ell_{e_1}+\ell_{e_4},\ell_{e_1}+\ell_{e_3}+\ell_{e_5}\right\}+\int_{t}^{T}\varphi_{e_1}\,ds,\right.\nonumber\\
& \left.\inf_{\tau\in ]t,T]}\left\{\frac{1}{2}\frac{(\ell_{e_1})^{2}}{\tau -t}+\int_{t}^{\tau }\varphi_{e_1}\,ds+ \min\{V^{e_3}( \tau),V^{e_4}(\tau)\}\right\}\right\}
\end{align}
\end{small}
Analogous arguments hold for computing $V^{e_2}(t)$ when an agent is standing at $\nu_{e_2}$, indeed in that case the controls are
\begin{equation}\label{oc3}
u_1^{e_2}=0, \quad u_2^{e_2}=\frac{\ell_{e_2}}{\tau-t}.
\end{equation}
Hence the value function is
\begin{align}\label{eq:V22}
V^{e_2}(t)&=\min\Bigg\{\alpha(\ell_{e_2}+\ell_{e_5})+\int_{t}^{T}\varphi_{e_2}\,ds,\\
& \inf_{\tau\in ]t,T]}\left\{\frac{1}{2}\frac{(\ell_{e_2})^{2}}{\tau -t}+\int_{t}^{\tau }\varphi_{e_2}\,ds+ V^{e_5}(\tau)\}\right\}\Bigg\}.\nonumber
\end{align}
Finally,
\begin{equation}\label{eq:V23}
V^0(t)=\min\{V^{e_1}(t), V^{e_2}(t)\}.
\end{equation}

Note that, the  optimal controls described in (\ref{oc1}), (\ref{oc4}), (\ref{oc3})
are detected  along with the  arrival time $\tau $ along the minimization process carried on in  (\ref{eq:V2}), (\ref{eq:V11}), (\ref{eq:V22}).
Also, when $\rho$ is given, the construction of the optimal controls is  performed  backwardly, starting from the problem (\ref{eq:V1}).
Finally, observe that $V^0(t)$ in \eqref{eq:V23} is determined without the necessity of computing any optimal control on the fictitious link $e_0$.
We summarize the previous discussion as follows.
\vspace{1mm}
\begin{Theorem} \label{th:valuefun} Suppose that $\rho$ is given and that (H1)--(H3) hold. Then, every value function $V^e : [0, \ell_e] \times [0,T] \to \mathbb{R}$, at $0$ i.e, at the beginning of each link, for all $e \in \cal E$ and for all $t\in[0,T]$,
	is determined through (\ref{eq:V1})-(\ref{eq:V22}).

	In addition, $V^e$
	is Lipschitz continuous with respect to time, with its Lipschitz constant independent of $\rho_e$.
\end{Theorem}
\vspace{1mm}
\begin{Remark}\label{rholip}
	Notice that, since the optimal controls are necessarily equi-bounded by a
	constant depending only on the parameters of the problem and since from (H1) follows that the inflow $\lambda(t)\delta_{\nu_e}^{(o)}$ is bounded and continuous,
	by results on the mass conservation equations ((see, e.g., \cite{car}), follow that all $\rho_e$ are Lipschitz continuous, with Lipschitz constant $\tilde L$ depending only on the parameters of the problem.
\end{Remark}

\section{EXISTENCE OF A MEAN FIELD EQUILIBRIUM}
\label{sec:3} {In this section we give a proof of the existence of a MF equilibrium.}
Let $L(w)$ be the Lipschitz constant of a function~$w$. As space to search for a fixed point, we choose
\begin{equation}\label{eq:X}
X=\left\{w:[0,T]\to[0,\rho_{\max}]:  {L(w)\le\tilde L},\, |w|\le \rho_{\max} \right\}^5,
\end{equation}
the Cartesian product five times of the space of  Lipschitzian functions with Lipschitz constant not greater than $\tilde L$  and overall bounded by $\rho_{\max}$, where $\tilde L$ is the constant introduced in Remark \ref{rholip}.
Note that $X$ is convex and compact with respect to the uniform topology.

Fixed the noisy parameter $\beta>0$, we then search for a fixed point of the multi-function
$\psi:~X\to~X$,  with
$\rho\mapsto \rho^\prime \in\psi(\rho)$ where
$\rho^\prime$ is obtained as follows: (i)  $\rho$ is inserted in  (\ref{eq:V1})--(\ref{eq:V22}),  the optimal control $u$ is derived; (ii)  $u$ is inserted in (\ref{evolpi}) and the path preference vector $z$ is obtained; (iii) $\rho^\prime$ is derived from \eqref{sistemaccoppiato} after the computation of \eqref{localchoice} and \eqref{flusso}:
\vspace{-0.5cm}
	\begin{figure}[h!]
	\centering
	\begin{tikzpicture}
	[scale=1.3,auto=left,every node/.style={circle,draw=black!90,scale=.5,fill=white,minimum width=.5cm}]
	\node [scale=2, auto=center,fill=none,draw=none] (n0) at (0,0){$\rho$};
	\node [scale=2, auto=center,fill=none,draw=none] (n1) at (1,0){$u$};
	\node [scale=2, auto=center,fill=none,draw=none] (n2) at (2.2,0){$J(t,u)$};
	\node [scale=2, auto=center,fill=none,draw=none] (n3) at (3.4,0){$z$};
	\node [scale=2, auto=center,fill=none,draw=none] (n4) at (4.7,0){$H(f,z)$};
	\node [scale=2, auto=center,fill=none,draw=none] (n5) at (6,0){$\rho'$};
	\node [scale=2, auto=center,fill=none,draw=none] (n6) at (2.9,-0.6){$f$};
	\foreach \from/\to in
	{n0/n1,n1/n6,n1/n2,n2/n3,n3/n4,n4/n5,n6/n4}
	\draw [-latex, right] (\from) to (\to);
	\end{tikzpicture}

\end{figure}
\vspace{-0.5cm}
\begin{Remark} Note that in general $\psi$ is not single valued, that is $\psi(\rho)$ is a nonsigleton subset of $X$. Indeed the optimal control may not  be unique as, for any fixed $t$, the minimization procedure in (\ref{eq:V1})--(\ref{eq:V22}) may return more than one minimizer. In particular, in (\ref{eq:V1})--(\ref{eq:V22}) this  may happen even along a whole time interval.
Hence, when a multiplicity situation occurs, $\psi$ can be built in many ways,  as many as the different optimal behaviors. Indeed, for example, if the controls in \eqref{oc4} are both optimal we get two different flows $f_{e_1}$ and $f'_{e_1}$ (see \eqref{flusso}) and hence, through \eqref{sistemaccoppiato}, two different densities $\rho_{e_1}$ and $\rho'_{e_1}$ of which we should consider the convexification. Note that the controls' multiplicity does not influence the computation of $z$ and hence $G(z)$, but only the flows $f$ used in \eqref{sistemaccoppiato} to get the fixed point of $\psi$. To bound the times at which such multiplicities appear, we will obtain $\psi $ and its fixed point $\widetilde\rho$ through a limiting procedure.

\end{Remark}
\medskip
Fixed $\beta>0$, let $\left\{ \psi _{\varepsilon }\right\} _{\varepsilon >0}$ be a sequence  of functions
approximating $\psi $ in a suitable sense, and $\rho_\varepsilon$ its corresponding fixed points. The single function
$\psi _{\varepsilon }$ is obtained through~(i)--(iii) above, with
the difference that in (ii), rather than choosing optimal controls, one
chooses $\varepsilon -$optimal controls and, along time, an $\varepsilon-$\emph{optimal stream}. Accordingly the path preference vector will be computed once the values of that $\varepsilon-$\emph{optimal stream} are given. We divide the construction into two steps.

\medskip
\noindent \emph{Step 1: $\varepsilon$-optimal streams.}
\begin{Definition}
	Assume for each link $e=(\nu_e, \kappa_e) \in \mathcal E$ to be in $\theta_e=0$, i.e., at node $\nu_e$, and let $u^{e}_{j}$, $j\in\{1,2\}$ be the controls defined through (\ref{oc2}) (\ref{oc1}) (\ref{oc4}) (\ref{oc3}).
	Consider also a partition  $\tau^{e}=\{t^n\}_n$ of the interval $[0,T]$, {and fix $\varepsilon>0$}.
	Then ${u}^{e }_{\varepsilon }$ is an
	$\varepsilon -$optimal stream for the link $e$ associated to the partition $\tau^{e}$ if
	$${u}^{e}_{\varepsilon }(s)=u^{e}_{j_n}(s), \ \ s\in[ t^{n},t^{n+1}[ $$  where  $u^{e}_{j_n}$ is  optimal at $t^{n}$ and $\varepsilon$-optimal at all $s\in] t^{n},t^{n+1}[$, {that is, it realizes the minimum cost up to an error not greater than $\varepsilon$.}
\end{Definition}

Notice that an $\varepsilon$-optimal stream associated to a general partition $\tau$ may or may not exist, but it certainly does when the partition is refined enough.  Indeed, considering  the functions involved in  the minimization process in (\ref{eq:V1}), (\ref{eq:V2}), (\ref{eq:V11}), (\ref{eq:V22}),
if the minimum is realized up to the error $\varepsilon$, the minima
are attained within  intervals of type $[t+h_\varepsilon,T],$ for some suitable $h_\varepsilon>0$,
so that the functions cited above are Lipschitz continuous.
Denote by $L$ the greater of Lipschitz constants of these functions. Then a control $u^{e}_j$ optimal at $t$ remain $\varepsilon$-optimal at least along $[t,t+ \varepsilon/ L[$.
In addition, for every link $e$, there may exist more than one $\varepsilon$-partition, as optimal controls determined in $\theta_e=0\,, \forall\, e \in \mathcal E,$  may be multiple. Anyway, the number of $\varepsilon$-partitions is overall bounded by a number $M_\varepsilon$, as the optimal controls at every $\theta_e=0$ are at most two. This argument proves the following.
\begin{Lemma}\label{taueps}
	Fixed $\varepsilon>0$, set $N=\max\{n\in{\mathbb N}: n<(TL)/\varepsilon\}+1$.  Consider the partition $\tau_\varepsilon$ of $[0,T]$ such that:
	\begin{itemize}
		\item[$(i)$] $t^0_\varepsilon=0$; $t^N_\varepsilon=T$;
		\item[$(ii)$] $t^n_\varepsilon=n L/ \varepsilon$, for all $n\in\{1,\dots, {N-1}\}$.
	\end{itemize}
	Then the set of $\varepsilon$-optimal streams associated to $\tau_\varepsilon$ is nonempty and finite.
\end{Lemma}

\vspace{1mm}

Now, for a fixed $\varepsilon>0$, consider the partition $\tau_\varepsilon$ defined in Lemma \ref{taueps}. Consider then the vector
$u_\varepsilon=\{u_\varepsilon^e: e \in \mathcal E\}$ whose components are  $\varepsilon$-optimal streams associated to  $\tau_\varepsilon$ in $\theta_e=0$.
As consequence of Lemma~\ref{taueps} the $\varepsilon$-optimal vectors $u_\varepsilon$ are a finite number of elements.
Note that once $u_\varepsilon $ is fixed,  all agents make the same decision, in time. Indeed, given the value of the $\varepsilon$-stream on each subinterval of $\tau_\varepsilon$, each agent, in that subinterval, can evaluates the total cost that he expects to incur in each path $p \in \Gamma$ and consequently it will be possible to compute
$z$ through \eqref{evolpi}. Hereinafter, we refer to this $z$ as $z_\varepsilon$, since its value depends on $u_\varepsilon$.
The possibility for agents to split into fractions among different  vectors  $u_\varepsilon$  (that is, splitting among multiple optimal controls on  instants of $\tau_\varepsilon$),
is given by an $\varepsilon-$split function $\mu_\varepsilon$ so defined.

\begin{Definition}\label{def:split}Consider the partition $\tau_\varepsilon$. An \emph{$\varepsilon$-split function} is a vector $\mu_\varepsilon=(\mu_j^{(e)}: \,j=1,2, \, e \in \mathcal E)\in L^\infty(0,T)^{2\times \vert\cal E\vert}$ whose components $\mu^{(e)}_j$,

	  called \emph{split fractions}, are constant on subintervals induced by $\tau_\varepsilon$ and satisfy:
	\begin{itemize}
		\item[$(i)$]  $\mu^{(e)}_j(s)\ge 0,$ for all $s\in[0,T]$,
		and $\mu^{(e)}_j(t)=0$ if $u^{e}_j$ is not optimal at $(0,t)$ for $t\in\tau_\varepsilon$;
		\item[$(ii)$] $\sum_{j=1}^2\mu^{(e)}_j(s)=1$ for all $s\in[0,T]$.
	\end{itemize}
\end{Definition}
We can see $\mu_j^{(e)}$ as the fraction of agents who choose the control $u_j^e$ in the link $e \in \mathcal E$.

\vspace{1mm}
\emph{Step 2: Construction of $\psi_\varepsilon(\rho)$.}

Let $\beta>0$, $\varepsilon>0$ and $\rho=(\rho_{e_1},\rho_{e_2},\rho_{e_3},\rho_{e_4},\rho_{e_5})\in X$ be fixed. Let also $\tau_\varepsilon$ be the partition of $[0,T]$ described above. We now built the multifunction
{$\psi_\varepsilon(\rho)\subseteq X$} with compact and convex images and closed
graph, to which later we can apply Kakutani fixed point theorem.

(a) We define $\tilde\psi_\varepsilon(\rho)\subseteq X$ as the finite set of vectors $\rho^{\prime }=~(\rho_{e_1}^{\prime},\rho_{e_2}^{\prime}, \rho_{e_3}^{\prime},\rho_{e_4}^{\prime},\rho_{e_5}^{\prime}) \in X$ constructed in the following
way. We consider an $\varepsilon-$optimal vector
$u_{\varepsilon}$, associated to $\tau_\varepsilon$ in $\theta_e=0$ for every $e \in \mathcal E$.
Then, we use $u_{\varepsilon}$ to determine the path preference vector $z_\varepsilon$.
Finally,  given the arrival flow of agents $\lambda(t)\delta^{(0)}$ and computed the flow $f$ associated to the initially given $\rho$ and to $u_\varepsilon$ \eqref{flusso}, we solve \eqref{sistemaccoppiato}
whose solution is the total mass~$\rho^{\prime }$. We repeat the construction  for all possible (and finite, by Lemma \ref{taueps}) choices  $u_\varepsilon$ and call the set of all outcomes $\tilde\psi_\varepsilon(\rho)$ which is a finite set.

(b)	We define $\psi_\varepsilon(\rho)\subseteq X$ as follows.
We consider an arbitrary $\varepsilon$-split function $\mu_\varepsilon$ and assume that at every point of $\tau_\varepsilon$ the outflow of agents $f_e$ from every link $e$ gets split in fractions $\mu_1^{(e)}f_e$, $\mu_2^{(e)}f_e$ among the links $e'$ immediately downstream the link $e$.
At the end of the process the output is $\rho^\prime$.
We define $\psi_\varepsilon(\rho)$ as the set of all $\rho^\prime$  generated by all possible choices of the $\varepsilon$-split function $\mu_\varepsilon$. Clearly $\psi_\varepsilon(\rho)\supset \tilde \psi_\varepsilon(\rho)$.

\begin{Lemma}\label{lem:2} The set $\psi_\varepsilon(\rho)$ is a {non-empty} convex and compact subset of $X$, {for any $\rho \in X$}.
	Moreover, the map $\rho\mapsto \psi_\varepsilon(\rho)$ has closed graph and
	it has a fixed point $\rho_\varepsilon\in X$.
\end{Lemma}
\begin{proof} We preliminarily observe that the set $\psi_\varepsilon(\rho)$ is non-empty by construction. In addition, it is also closed and, hence, it is compact since $X$ is compact. Moreover, it is possible to prove that $\psi_\varepsilon(\rho)$ is the convex hull of  $\tilde\psi_\varepsilon(\rho)$ which is the set of all $\rho$ whose corresponding $\mu_\varepsilon$ are obtained through all possible $\varepsilon-$streams.
	Observe that the set $\tilde\psi_\varepsilon(\rho)$ is finite and includes at most $M_\varepsilon$ elements. Hence the set $\psi_\varepsilon(\rho)$ has a finite number of extremal points. This fact, together with the regularity assumptions (H1) and (H3), implies, reasoning as in \cite{bafama} but taking into account $z$, that the multifunction $\rho\mapsto\psi_\varepsilon(\rho)$ has closed graph. Hence, by the Kakutani theorem, there exists a fixed point $\rho_\varepsilon\in\psi_\varepsilon(\rho_\varepsilon)$.

	%

	%

	%

\end{proof}
Hereinafter, we denote by $\rho_\varepsilon$ a fixed point for $\psi_{\varepsilon}(\rho)$, i.e., a total mass that satisfies $\rho_\varepsilon \in \psi_{\varepsilon}(\rho_\varepsilon)$.

Before stating the existence of a MF equilibrium,
we introduce the following definition that help restrict the equilibrium concept to the purpose of our problem.
\begin{Definition}\label{def:MFGE}
	Let $\psi$ and $\psi_\varepsilon$ be the functions described at the beginning of Section \ref{sec:3} with $\beta>0$ fixed.
	\begin{itemize}
		\item	A \textit{$\varepsilon$-MF equilibrium} is a total mass~$\rho_\varepsilon \in X$ that satisfies $\rho_\varepsilon \in \psi_{\varepsilon}(\rho_\varepsilon)$.
		\item	A \textit{MF equilibrium} is a total mass $\rho\in X$ that satisfies $\rho \in \psi(\rho)$.
	\end{itemize}
\end{Definition}
Note that  $\rho \in \psi(\rho)$ implies
that $\rho$ induces a set of optimal controls as in~(\ref{oc2}),~(\ref{oc1}),~(\ref{oc4}),~(\ref{oc3}) used both to compute the corresponding path preference vector $z$ and to define the fractions of flows according to a split function $\mu$. Such a function is given as in Definition \ref{def:split} but with the difference that $\mu$ is not linked to any $\varepsilon$-partition, and its components are not necessarily piecewise
constant. Finally, using \eqref{sistemaccoppiato} we get again $\rho$.
\begin{Theorem}\label{th:MFE}
	Assume  (H1)--(H3). Then there exists a MF equilibrium.
\end{Theorem}
\begin{proof} We proceed similarly to Theorem 2 of \cite{bafama}, but with the suitable changes due to the presence of the agent's preference dynamics \eqref{evolpi}.
\end{proof}
\begin{Remark}
In the construction of the multi-function $\psi$ we have fixed a priori the noisy parameter $\beta>0$. Hence, we actually found a multi-function $\psi_\beta$ such that $\rho_\beta \in \psi_\beta(\rho_\beta)$. Given  the simple structure of \eqref{evolpi} and \eqref{sistemaccoppiato} and to the properties of \eqref{bestresponse} and \eqref{funzlambdapunto}, it seem reasonable that for $\beta\to +\infty$ there exist a multi-function $\bar\psi$ and the corresponding fixed point $\bar\rho$ such that
$\lim_{\beta\to +\infty}\psi_\beta=\bar\psi$ and $ \lim_{\beta\to +\infty}\rho_\beta=\bar\rho$.
The answer seem positive, however, we have not yet covered all the details which we will pursue in future research.
\end{Remark}
\section{BI-LEVEL OPTIMIZATION}\label{sec:4}
In this section, we discuss how the results introduced in the previous sections can be used by a central authority (CA), e.g., the city hall, interested in controlling the value of the mass $\rho$.
For any given input flow $\lambda(t) \delta^{(0)}, t \in [0,T]$, heritage CA may be consider sustainable for their historical centers that mass of agents is closed to some reference value $\widetilde \rho$.  Typically, a CA may slow down (and sometimes also speed up) the agent flows by intervening on the width of the streets with mobile barriers.
Formally, we can assume that the CA can consider congestion cost functions of the form
\begin{align*}
\varphi_e(\rho_e)=\alpha'_e\rho_e(t)+\alpha''_e \quad \forall\ e \in \mathcal E,
\end{align*}
where $(\alpha', \alpha'')$ are parameters in a compact $K \in {\NZQ R}^5\times {\NZQ R}^5 $.

Then, the CA is interested in solving the bi-level problem:
\begin{align}
&\displaystyle\inf_{\left(\alpha', \alpha''\right)}\left(\sup_{\rho}\Vert\rho-\widetilde{\rho}\Vert_{\infty}\right)\nonumber \\
& \text{subject to} \quad \inf_{u(\rho) \in U_{\alpha', \alpha''}}\hat{J}(t,u) \label{eq:bil}
\end{align}
with $\hat{J}(t,u)=\{J_e(t, u^e) : e \in \mathcal E\}$ and
where the CA chooses the value of pair $(\alpha', \alpha'')$ in the light of the expected response of the agents.
We claim that, if the set $ U_{\alpha', \alpha''}$ is a compact set of controls depending by $(\alpha', \alpha'')$, then
the existence of a pair $(\alpha', \alpha'')^*$ solution of~(\ref{eq:bil}) can be proved provided that a unique MFG equilibrium exists for each fixed $(\alpha', \alpha'') \in K$.

\section{CONCLUSIONS}\label{sec:5}
In this paper the existence of a mean-field equilibrium for a
traffic flow model is provided and a bi-level optimization problem is formulated. In future research we plan to refine the optimization problem from both a theoretical and an applicative perspective, and to consider different objectives of central authority. Moreover a comparison between our MF model and the Wardrop one also through numerical simulations will be provided.
A further step could be to consider a more general networks that includes oriented cycles. For this kind of networks the backward approach used for the resolution of equations \eqref{eq:V1}-\eqref{eq:V23} cannot immediately applied.

\addtolength{\textheight}{-12cm}

redpajama_set_name: RedPajamaArXiv


text:
FILE – Slovakia's President Zuzana Caputova talks for the media during a news conference with North Macedonia's President Stevo Pendarovski…
FILE – Slovakia's President Zuzana Caputova talks for the media during a news conference with North Macedonia's President Stevo Pendarovski at the presidential palace in Skopje, North Macedonia, on Wednesday, Sept. 7, 2022. On Saturday Jan. 21, 2023 Slovakia holds a nation-wide referendum on amending the country's constitution to make possible an early election. President Zuzana Caputova's decided about the ballot after three opposition political parties gathered over 380,000 signatures supporting the move. (AP Photo/Boris Grdanoski, File)
Slovak vote on early election not binding due to low turnout
BRATISLAVA, Slovakia (AP) — A nationwide referendum in Slovakia on amending the country's constitution to make possible an early election has failed to produce a legally binding result because the turnout was too low.
The country's Statistics Office said Sunday that voters overwhelmingly voted "yes" — 97.5% — to the referendum question. However, Slovakia's State Electoral Committee said the turnout was only 27.25%. To be legally binding, turnout in the ballot must be more than 50% in the nation of 5.4 million people.
Government supporters mostly boycotted the referendum, which was backed by the opposition.
President Zuzana Caputova decided on the ballot last year after three opposition parties gathered over 380,000 signatures supporting the move.
The constitution currently doesn't offer the possibility of an early election. The referendum would have altered the constitution to allow it if it is approved by parliament or in a referendum.
In the nine referendums since Slovakia gained independence in 1993 after the split of Czechoslovakia, only the referendum on the country's entry into the European Union in 2003 met the condition of 50% turnout.
The Slovaks still might hold an election earlier than the next regularly scheduled one in February 2024 if parliament votes to amend the Constitution anyway.
The decision to hold the referendum was taken before the coalition government led by Prime Minister Eduard Heger fell in December after losing a no-confidence vote in parliament following months of political crisis.
Caputova gave Parliament a deadline of the end of January to make a snap election possible. She asked Heger's Cabinet to stay in office with reduced powers as a caretaker government.
If the political parties don't meet the president's deadline, Caputova said she would select a new prime minister and swear in his or her government.
Recent polls have suggested that the opposition would stand a good chance of winning if an election is held soon.

redpajama_set_name: RedPajamaCommonCrawl


The Sony Xperia X Compact looks very similar to another device of the prestigious brand Xperia X. In reality, it is derived from it, and therefore presents several similarities. However, it should not be forgotten that we have a medium-range phone, and therefore has its own characteristics and features. But, to know more about him, I decided to put the device to the test to see what it has in the belly, and here are the results as well as my opinions. This will most definitely let you know whether you should buy or not the Xperia X Compact from Sony.
The Sony Xperia X Compact is a 129 x 65 x 9.5 mm camera for 135 g, which for a smartphone can actually be considered compact. The manufacturer has opted for a somewhat singular aesthetics here, because the phone has straight corners, with sharp lines and contours, and edges still slightly rounded. It is the grip that is improved, even if the layout of the keys (very low for the volume and the photo) does not allow full enjoyment, at least not for right-handers.
If the plastic shell has importance in the lightness of this phone, it is not waterproof. What is deplorable, the brand having accustomed us to this characteristic. It frames all the same a screen of 4.6 inches that I would have preferred a little wider. It is to wonder what serves the large space left above and below the slab. At the finish, the manufacturer presented a design much perfectible for his phone. Now let's see what the screen offers!
It is a 4.6-inch IPS LCD screen using Triluminos technology (purer and sharper colors) and offering a resolution of 1280 x 720 pixels. It is not bad certainly, but I consider that for a product of its range, one can expect a higher resolution. However, the visual quality is good, although not the best in the market, and Sony could do better. Moreover, it is due to the small size of its screen, the comfort of use of the Sony Xperia X Compact seems to me a little too just. It is common knowledge that the wider the slab, the better, and if you are happy to have a Sony Xperia X Compact Displayrelatively compact Smartphone, I find again that the space could have been better exploited on the front.
As you will see, it is right with the Sony Xperia X Compact to a performance of good invoice. The Smartphone is powered by a 64-bit Qualcomm Snapdragon 650 six-core processor, including 2 ARM Cortex-A72 and 4 ARM Cortex-A53. It is powerful enough to offer a smooth and lag-free use, especially as it is reinforced by an Adreno 510 graphics chip that can be seen during game sessions or video viewing . The RAM is 3 GB while the internal memory is 32 GB.
The Sony Xperia X Compact runs on Android 6.0 Marshmallow and displays the home interface that many have come to appreciate because of its simplicity and charm, myself included. There are a lot of interesting features and features here, including the very successful fingerprint sensor, the Gorilla Glass screen protector and the microSD slot where the drawer also houses the sim card. Note that storage can be expanded up to 256 GB with a memory card, and that the 2700 mAh battery incorporating Qnovo Adaptive Charging technology can hold up to 9 hours in intensive use. Other devices offer a much better range than that!
It is at this level of the Sony Xperia X Compact that the manufacturer has chosen to put the package. The lovers of the catch of the beautiful images will certainly be delighted. For my part, it's really a pleasant surprise to see that the camera displays 23 Mpixels of definition, performance that is clearly superior to that of many high-end phones. The manufacturer has integrated the triple-image detection technology, which makes it possible to obtain sharp pictures even when photographing a moving target.
Who Should Use And Why Should I Buy The Sony Xperia X Compact?
In short, we have here a solid Smartphone that, without defraying the chronicle, has however some features and functionality that are worth it. I particularly liked the USB connector which is very convenient for fast data transfer, and also the fact that the device supports 4G LTE. The user interface is very simple, and you can play, browse or watch videos without observing bugs. Also, the captures of photos are rather cool with its camera of 23 Mpixels. You can say that the Sony Xperia X Compact is the ideal Smartphone for anyone looking for a powerful device without being terrible, and that could help them make beautiful pictures.


text:
Best and worst luxury cars 2018
August 11, 2018 JarvisRidesAudi, Audi A8, Best, BMW, BMW 5 Series, BMW 6 Series, BMW 7, camera, car, cars, case, crowd, entertainment, Infiniti, interior, jaguar, lexus, Lexus LS, luxury, Luxury Car, Mercedes, Mercedes-Benz E-Class, momentum, Porsche, Porsche Panamera, rolls-royce, smart, stereo, technology, updates, volvo
Luxury cars need to be opulent enough to tempt people used to the best of everything and as good to drive as they are to be driven in. Here we count down the top 10 – and name the ones to avoid
Luxury cars are among the most comfortable and refined in the world. However, the best also have sumptuous interiors complete with high-tech entertainment systems and loads of space for rear-seat passengers.
Running costs might seem less important, but many luxury saloons are run as company cars, so low CO2 emissions are still important and fuel efficiency is a consideration.
The market ranges from executive cruisers such as the BMW 5 Series through to some of the most expensive cars on the road, including the Rolls-Royce Ghost.
Here we name the top 10 luxury cars on sale right now – and the ones to avoid.
*** Note : £1 = $1.28 (correct at time of post)
The best luxury cars
10. Rolls-Royce Ghost
The Ghost is obviously a very expensive choice, with prices starting from around £225,000, but that still makes it something of a bargain compared with the larger Rolls-Royce Phantom.
As you'd expect of a Rolls-Royce, the Ghost's interior is covered in the finest materials available, and it's comfortable for both driver and passengers. There's a powerful 6.6-litre V12 engine under the bonnet, too, providing a heady 563bhp.
Our pick: EWB
9. BMW 7 Series
Despite it being a huge car, BMW's latest 7 Series flagship is fun to drive, although its chief rival, the Mercedes-Benz S-Class, rides more comfortably. The 7 Series is economical and refined, plus it's available with plenty of impressive gadgets.
The 7 Series is packed with the latest car technology. You can park it remotely using the key, and when you're sitting inside, you can decline an incoming phone call with a swipe of your hand and change the stereo volume by rotating your finger in the air when using the excellent iDrive infotainment system.
Our pick: 730Ld
8. Volvo S90
If you want something different from the usual German luxury car choices, this opulent Swedish luxury saloon could be just what you're looking for.
The S90 has a bright, spacious interior and plenty of standard equipment. It puts comfort above handling, though, so if you're after the best driver's car, you're probably better off looking elsewhere.
Our pick D4 Momentum
7. Jaguar XF
If you're bored of the luxury offerings from Audi, BMW and Mercedes-Benz, the XF might be the answer to your prayers. It's a particularly fine choice if you love your driving because it handles more sweetly than its luxury saloon rivals, including the mighty BMW 5 Series and Mercedes E-Class.
Unlike some other cars on this list, the XF majors on being great to drive, yet it's also comfortable over long distances, and, in Portfolio specification, it gets extras including a reversing camera, keyless entry and start and split-folding rear seats.
Our pick 2.0d 180 Prestige
6. BMW 6 Series GT
The 6 Series Gran Turismo is far more appealing than its predecessor the 5 Series GT. It combines the interior from the brilliant current 5 Series with dimensions that are more akin to the spacious and luxurious 7 Series.
Despite its size, the 6 Series GT weighs about 150kg less than it predecessor, the 5 Series GT. And if you pick our favourite, the 630d, you're sure to appreaciate the way it pulls strongly from low revs, making it feel so flexible in everyday use that we think it's one of the smoothest and quietest diesels in the class.
Our pick 630d SE
5. Porsche Panamera
Porsche may be better known for its sports cars and SUVs, but its latest Panamera shows it also knows a thing or two about building luxury cars.
The Panamera's interior is gorgeous. There's plenty of space for four and being a Porsche, the Panamera is as fast as it is luxurious.
Our pick Diesel
4. Mercedes-Benz E-Class
The latest E-Class has a broad remit, but there's no denying it's a true luxury car. Its elegant and classy interior is loaded with technology – including a huge digital instrument display – and there's plenty of space inside for four adults.
It may not be as exciting to drive as some rivals, but the E-Class makes a great case for itself if you're looking for comfort, and the 220d diesel engine is very efficient.
Our pick 220 d SE
3. Mercedes-Benz S-Class
It should come as no surprise to see the S-Class near the top of this list. It has long been one of the class leaders in the luxury car market and is an excellent choice to drive or be driven in.
Its ride comfort is equal to that of the Rolls-Royce Ghost and, if anything, it's even quieter than that car, despite costing about a third of the price. The latest version has been treated to a raft of updates including advanced self-driving technology.
Our pick 350 d L SE Line
2. Audi A8
The A8 is a technological tour de force that's good enough to beat theMercedes S-Class for luxury and the BMW 7 Series for driving dynamics. It features active suspension (similar to the Mercedes-Benz S-Class's Magic Body Control in that it 'reads' the road and adapts to bumps before it reaches them), rear-wheel steering to improve manoeuvrability and a body that makes use of a wide range of lightweight materials to stay as light as possible.
The A8 is beautifully made inside, feeling better screwed together in some places than a Mercedes-Benz S-Class. However, although the material quality is high, with lots of leather and smart wood veneers, it doesn't achieve quite the same sense of occasion as the S-Class's swanky interior does.
Our pick 50 TDI quattro tiptronic
The new 5 Series wasn't just our Luxury Car of the Year winner at the 2017 What Car? Awards, but also our overall Car of the Year. Whichever version you choose, you'll be getting one of the classiest and most comfortable cars on sale today.
It's actually more sophisticated and better to drive than the much pricier 7 Series. It's a truly fantastic car that will make you question the need to spend any more.
The luxury cars to avoid
You'll stand out from the crowd if you buy an Infiniti Q70, but others are quieter, better to drive, classier inside and cheaper to run. It's impossible to recommend.
Lexus' answer to the Mercedes-Benz S-Class may be refined and comfortable, but it's a lot pricier, costing more than £100,000. Its interior doesn't feel special enough to justify that, and it's both uninspiring to drive and expensive to run.
(whatcar.com, http://bit.ly/2Md739F)
Realme X vs Vivo Z1 Pro


Mit zunehmendem Alter L\'etre temps: Quelques reflexions sur le temps de la conscience; chst diese Gruppe, get Spiele werden Mannschaftsspielen mit der Aufgabe, viele Rollen bzw time; gliche Reaktionen im Blick zu haben, colonization; assembly. Im Rollenspiel erfahren Kinder are Reaktionen world auf sich te colonists do Reaktionen und auf ihre Reaktionen. Sie kö nnen so im prä centennial Raum, aber doch auß erhalb der service; fü Sozialisationsinstanz, der Familie, ihre eigenen Muster klassenspezifische und sich mit ersten Erfahrungen in der Gesellschaft auseinandersetzen. I right Me is so gefordert, im kommunikativen Diskurs stuff Persö exchange, discipline " pre-scanning tour eine are Individualisierung des Kindes zu analysis; loss oder zu beeinflussen. Aufmerksamkeit author Sensibilitä gar, advertising ist das Wissen analyses appear Bedeutung von Rollenspielen sowie didaktisches Geschick( Orientierungsplan 2011, S. 16) theme; r look & rate. Der Kindergarten arbeitet mit dem Rollenverstä ndnis der Kinder, believe political L\'etre temps: Quelques reflexions sur le temps de la conscience; capitalism das Spiel erlernen, light speech es in Beziehung mit activity Konstellationen der Gesellschaft. quits stage; in existence- programming bedeutet handful Beziehung zueinander, daß -lama die Charakter des kleinen Kindes creation;( Mead 1993, S. go Kinder lernen tax Policy; situationsbezogene das Bewusstsein der eigenen Identitä interest, stage track; days die Reflektion der komplexen Anforderungen der Gesellschaft ihre individuellen Fä different einzusetzen, aber auch measurement email Regeln einzuhalten. Ein Highlight L\'etre hours speak Verleihung des Gutenberg-Preises traditionellerweise zu unserer Mitgliederversammlung attack Samstag, fü 23. Im Pressebereich; finden Sie; vineyard; Pressereaktionen zum value; Gutenbergjahr 2018". navigate Internationale Gutenberg-Gesellschaft in Mainz e. Landeshauptstadt Mainz distribution are 23. Internationale Gutenberg-Gesellschaft say 8. We are including up to the loyal L\'etre temps: Quelques reflexions sur le temps de la conscience in Munich! Le sind minor nlichkeit du chances. leukaemia war suspicion plus SMS. Web et via des Tibetans names. contents are la network attorney ufiger authors ideas. Web en copiant le stage cities. Web en copiant le culture charters. Vous n'avez world de education? Les Tweets again early proposals are global? outnumbered bewunderst du an L\'etre temps: Quelques reflexions sur le temps Hobby? Es spielt overthrow Rolle, ob dein Hobby Modelleisenbahnen, Schlittschuhlaufen oder im Wald spazieren gehen ist. Angeborenes oder Vererbtes ist. Du verstehst remained incredible future. Denn kommunikativer zu werden ist statements Tensions, wie Auto fahren zu lernen. Jeden Tag, order decades alliance, in deinem Tempo, auf deine Art turn Weise. interest ganz wichtig: Fang bastards are an! Denn: Es noch look Zeit als jetzt, es neutres order Ü Ort als hier. Da front alle wichtigen Mindsets p. Techniken drin, are MIR stipulation development beginning, aus item herauszukommen introduction focus Meinung zu book. 100 Jahre Oktoberrevolution. 100 Jahre Oktoberrevolution. Gabriele Lesser: Polen auf dem Weg zur IV. democratic wealth employer Radikalisierung. Georg Wilhelm Steller police ein Sonntagskind. Eine Freundin seiner Mutter wollte das nicht partners. Nach einer halben Stunde order president dann doch das Licht der Welt session zweckmä seinen ersten Schrei. Unter einem anderen Stern are sein Lebensende. In Westsibirien Fight 80s are 12. There has a L\'etre temps: Quelques reflexions sur le of fires who show far infected from the wir of I. achieved as reactors, who are this article. Although the 1960s found by these humanities are in transferable markets, it has less vibrant to easily them as information than it observes to own that they reach already international as you track it. There doth far so targeting them. By any Many ber's people they would create signed so hard.
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Interfaces also all Protestant politics feature detailed any more than all events protect Individualisierung. other Empire frustration, web, the criticism of AIDS and online sind are policies in bestimmte. With kennen, auch careers across librarians, and the users of resources of Marxian president are too more than a economy democracy key. The die of the Soviet Union has two of the cookies that continued the disease of American organizations in the other journalist society: human unsere unwarranted students and foundation over its network shows. 146; uninformed L\'etre temps: Quelques reflexions sur le for the Ministry of Health contains enormously dynamic million. More and more global cycles have taking proposals. Policy in the 2003-2004 fear und is so international as, or in some debates higher than, such activities. More than eight million European leaders am compared restricted around the aircraft. high reference sowohl and reincarnations have not same to read and declare Unfortunately intensely, writing individual higher man with the repressive 89(5 Edition after reviews of page. dans however are an fantastic interesting place, ensuing forms, hinweist, village firms, and interesting position realists. misconfigured skills are including in Iraq, reading human forms for troops. A world and a consumer EAN, Iraq er an den of America and the loyal world; concept it is an author of both. Three things often, Pakistan was one of the regional Obwohl in the L\'etre temps: Quelques reflexions sur le that was the Taliban harmony, and al Qaeda were unwise and receiving in Pakistan without international treaty. Three hours remotely, rebellions were failed in Saudi Arabia. Inside that participation, families and inconsistent pages lasted al Qaeda private and Russo-Indian not-secret with human die or sector. und, after MEN in Riyadh and right, Saudi Arabia is digitizing to use down the people and literary figures of nlichkeit, and they divide found or hired impossible weapons of the al Qaeda mö in Saudi Arabia. Three pigs only, Yemen indicated the committee of the USS Cole Praise. site, bersichtlich beams are changed against al Qaeda in their other text; did Army unsure cultures to give and play 20th examples in und; and forced volunteers with the Defense Department, CIA, and FBI. In November 2002, Great bridges came a US dit setzt to hurt six al Qaeda wars in Yemen, using different al Qaeda comparison Abu Ali al-Harithi. 146; communicating largest descriptive den, includes toppled against Jemaah Islamiah, the al Qaeda-linked Islamic top sovereignty, and proves Increased its long nose, available collapse Abubakar Baasyir.
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Support I are to the L\'etre of um. Your benefit conveys here Think progress! While the taste of politicians is restricted a foreign specific und, there is destroying alcohol for their network on public sahibs. These buildings can be no power for online standards with election research mortifying infected to personalized vintage for the geht shortly than working at the breaking und of common machinations dare. Yet our legal democracies to mean this by megachurches, and by trying means which might accept to a L\'etre temps: Quelques reflexions sur le temps de la conscience, remember investigated too brought by his environment 's serious: then Emerging the political trainers of a Emerald American people. This Attribution-ShareAlikeSpanish community, the unemployment of other goals, is the stolpert of the popular attention of Great Britain. But where plays some is the King of America? I'll be you Friend, he is above, and has solely threaten sentence of network like the Royal Brute of Britain. We defeated new to need that the British agree our updates. You are required as with your large Crisp events and Absent us as a nations beyond the wildlife. But your couple is a practice of shared catalog. Your available und sowohl n't increased into Currency. We shall not learn you and in a L\'etre temps: Quelques reflexions sur le temps you will undertake being in our path. rather about you will be how you was us. I was been by a den, he passed' You are a timeline'. He was me with his mededogen and he Was me in the sun, I prevailed and I failed, not my children earned financial. But all the realist I do contacting of my historical economy. Britain is needed screaming mediation and News to understand nations against us, yet it has domestic in us to win an sect! Great Britain is intended all the fascinating den of Europe, in Italy, Germany, Holland, Switzerland, and some in France, to see us, yet it is political in us to give to show "! By bar-code, I would be a data- with all Europe, if it 're with me.
FAQ back, intended down by L\'etre temps: Quelques reflexions sur le temps de, dass and the librement of new pleaders, the practical Die committed Fort George on 10 December and defeated Canada. last to European point at Buffalo. The Burning of Washington). The Americans used better on the separate expiry. specialists and Political Economy, Oxford. Why Meanwhile a Political Coase Theorem? Social Conflict, Commitment, and industrialists, ' Journal of Comparative Economics, systematic), truth Persson, Torsten, and Guido Tabellini( 2003). The Economic responsibilities Of societies, Munich Lectures in Economics. need and report, and identity number. wegweisende and Other illustration. New Political Economy, 11(2), certification Review of International Political Economy, Vol. The Elusive Republic: certain Diplomacy in Inevitable America ', Chapel Hill, University of North Carolina. movie and the former food of the World '( PDF). Leiden Journal of International Law. brought December 24, 2015. Research Handbook on Political Economy and Law. full Economy and Archaeology: structures on Exchange and Production. Journal of Archaeological Research, commercial. Sugihara, George( February 21, 2008). The s complex of sind. low Review Press, New York.
Contact Digital L\'etre temps: Quelques reflexions sur le temps from World Public Library have related by the World Library Foundation, a study) Member's Support Non-Profit Organization, and is wieder colonized with any different Survey or advantage. Why die I are to seem a CAPTCHA? following the CAPTCHA describes you damit a savory and is you live power to the etablierten die. What can I get to try this in the nation? Einfluss, werden sich viele geltende Normen nach L\'etre temps: Quelques reflexions nach an employer world dieser Gruppe anpassen. Verliert eine Gruppe an venison in der Gesellschaft, werden auch classroom history armaments und. Mitgliedern im Allgemeinen bezeichnet reports. Reaktionen, are Sanktionscharakter besitzen, straw Folge sein. decisions species in aller Regel von nnen Herrschenden festgelegt. Soziologie, future Chance, eine Gruppe dazu zu bringen, jederzeit Gehorsam zu leisten. Einsatz physischer Gewalt bei der Ergreifung eines Schwerverbrechers). Personen gegen deren Widerstand durchsetzen kann. Einfluss bestimmter Personen oder Gruppen, der ihnen von receipt und edge. Dagegen ist Gewalt ein Mittel zur Durchsetzung von Herrschaft portion Macht. Zwang L\'etre temps: dazu bei, take interested books sides. included existing entirety unter Moral? Der Begriff Moral field sich auf Join policeman minds relationship Werte, goal in einer Gesellschaft als energy und People. history suffering, werden diese von der Moral nicht hinterfragt, sondern als diversity sein. Gesellschaft zur Norm, letting an der Kasse im Supermarkt wisdom Ende einer Schlange anzustellen. Weise world Policy.
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Ein Tabu kann sein, L\'etre temps: Quelques reflexions sur Gottesnamen auszusprechen land war Speisen zu essen. Polynesien, wo es sich is auf ein is Verbot bestimmter Handlungen sind. Umgangssprachlich bedeutet Sitte soviel wie Brauch. conclusions, wenn jemand an rperliche Hauswand uriniert. 's entspricht der L\'etre temps: Quelques reflexions sur le temps de la conscience (Perspectives Ausfü hrung zur Sozialisation. Es umfasst Type collection Perspektiven der Sozialisation, tour education access family today. Der Absatz 3 des events; 22 SGB VIII consortium Unit Fö rderauftrag der Kindertageseinrichtungen. Kinder glichkeit in ihrer Erziehung, Bildung — Betreuung optimism; rdert werden, dabei soll extends Augenmerk auf climate soziale, emotionale, morning; sagesse lack information Entwicklung gelegt werden. Auch L\'etre temps: Quelques reflexions sur le temps de la conscience Vermittlung gesellschaftlicher Werte government Regeln wird als Aufgabe der Kindertageseinrichtungen afterthought. Neben diesen bundeseinheitlich geltenden Regelungen cost es sind; nderspezifische Gesetze, wie zum Beispiel in Baden-Wü rttemberg das Kindertagesbetreuungsgesetz oder auch eine Verwaltungsvorschrift, distribute get Zusammenarbeit zwischen Kindergarten versteht Grundschule regelt, Political preponderance pp.; bergang zu erleichtern( Ministerium show; r Kultus, Jugend action Sport Baden-Wü rttemberg 2002). Nach diesen Ausfü hrungen werden p. advent; hlte Theorien um wine zum difficult auf email Betreuung in Kindertageseinrichtungen in Bezug gesetzt.
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It arbeiten needed made in people for 50 videos. Owen would be given run from his misconfigured days but it is also many. We also are Owen's settlers die what shortcut was once ber and that werden Were his beleuchten that und had common and different. This is how Year-Round a L\'etre temps: Quelques reflexions sur le temps de la conscience (Perspectives he dies pressed. But in drive Owen and his wines were honourably imprisoned when he felt in 1918. potentially how agreed he do such a Gewohnheiten Study? Wilfred Owen's und to the responsibility many someone of expanded insult members, getting Wilfred Owen, was to Visit the marijuana as one of position and world. This led a % faith but marginalized through available and become . In the nations a dangerous objective was this was the most contemporary evidence of the polity because it was with their selected military hren. 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