text:
\section{An Algorithm for Socially Optimal Allocation (Case 3)} \label{sec:case3}

In this section we assume that $g_j(n)$ is decreasing in $n$ for
all $j \in {\cal N}$. For simplicity we assume that the socially
optimal allocation is unique up to the permutations of $\sigma^*$.
When this uniqueness assumption does not hold we need a more
complicated algorithm to achieve the socially optimal allocation.
All users use the Random Selection (RS) algorithm defined in
Fig. \ref{fig:alg1}. RS consists of two phases. Phase 1 is the
learning phase where the user randomizes to learn the interference
functions. Let $B_j(t)$ be the set of distinct payoffs observed
from channel $j$ up to time $t$. Then the payoffs in set $B_j(t)$
can be ordered in a decreasing way with the associated indices
$\{1,2,\ldots,|B_j(t)|\}$. Let $O(B_j(t))$ denote this ordering.
Since the IFs are decreasing, at the time $|B_j(t)|=M$, the user
has learned $g_j$. At the time $|\cup_{j=1}^N B_j(t)|=MN$, the
user has learned all IFs. Then, the user computes $\mathcal{A}^*$
and phase 2 of RS starts where the user randomizes to converge to
the socially optimal allocation.

\begin{figure}[htb]
\fbox {
\begin{minipage}{\columnwidth}
{Random Selection (RS)}
\begin{algorithmic}[1]
\STATE{Initialize: $t=1$, $b=0$, $B_j(1)=\emptyset, \forall j \in
{\cal N}$, sample $\sigma_i(1)$ from the uniform distribution on
$\mathcal{N}$} \STATE{Phase 1} \WHILE{$b<MN$}
\IF{$h_{\sigma_i(t)}(t) \notin B_{\sigma_i(t)}(t)$}
\STATE{$B_{\sigma_i(t+1)}(t+1) \leftarrow O(B_{\sigma_i(t)}(t)
\cup h_{\sigma_i(t)}(t))$} \STATE{$b=b+1$} \ENDIF \STATE{Sample
$\sigma_i(t+1)$ from the uniform distribution on $\mathcal{N}$}
\STATE{$t=t+1$} \ENDWHILE  \STATE{find the socially
optimal allocation $\sigma^*$} \STATE{Phase 2} \WHILE{$b\geq MN$}
\IF{$h_{\sigma_i(t)}(t) < v^*_{\sigma_i(t)}$}
\STATE{Sample $\sigma_i(t+1)$ from the uniform distribution on
$\mathcal{N}$} \ELSE \STATE{$\sigma_i(t+1) = \sigma_i(t)$} \ENDIF
\STATE{$t=t+1$} \ENDWHILE
\end{algorithmic}
\end{minipage}
} \caption{pseudocode of RS} \label{fig:alg1}
\end{figure}

\begin{theorem} \label{thm:theorem1}
Under the assumptions of (C3) if all players use RS algorithm to
choose their actions, then the expected time to converge to the
socially optimal allocation is finite.
\end{theorem}
\begin{proof}
Let $T_{OPT}$ denote the time the socially optimal allocation is
achieved, $T_L$ be the time when all users learn all the IFs, $T_F$ be the time it takes to reach the socially
optimal allocation after all users learn all the IFs. Then $T_{OPT}=T_L+T_F$ and $E[T_{OPT}]=E[T_L]+E[T_F]$.
We will bound $E[T_L]$ and $E[T_F]$. Let $T_{i}$ be the first time
that $i$ users have learned the IFs. Let
$\tau_i=T_i-T_{i-1}, i=1,2,\ldots,M$ and $T_0=0$. Then
$T_L=\tau_1+\ldots+\tau_M$. Define a Markov chain over all $N^M$
possible configurations of $M$ users over $N$ channels based on
the randomization of the algorithm. This Markov chain has a time
dependent stochastic matrix which changes at times $T_1, T_2,
\ldots, T_{M}$. Let $P_{T_0}, P_{T_1}, \ldots, P_{T_M}$ denote the
stochastic matrices after the times $T_0, T_1, \ldots, T_{M}$
respectively. This Markov chain is irreducible at all times up to
$T_M$ and is reducible with absorbing states corresponding to the
socially optimal allocations after $T_M$. Let $\hat{T}_1,
\hat{T}_2, \ldots \hat{T}_{M}$ be the times that all
configurations are visited when the Markov chain has stochastic
matrices $P_{T_0}, P_{T_1}, \ldots, P_{T_{M-1}}$ respectively.
Then because of irreducibility and finite
states $E[\hat{T}_i]<z_1, i=1,\ldots,M$ for some constant
$z_1>0$ . Since $\tau_i \leq \hat{T}_i, i=1,\ldots,M$ a.s. we have
$E[T_L]<Mz_1$. For the Markov chain with stochastic matrix
$P_{T_M}$ all the configurations that do not correspond to the
socially optimal allocation are transient states. Since starting
from any transient state the mean time to absorption is finite
$E[T_F]<z_2$, for some constant $z_2>0$. \qed
\end{proof}

\section{An Algorithm for Socially Optimal Allocation with Sub-linear Regret (Case 2)} \label{sec:case2}

In this section we propose an algorithm whose regret with respect to the socially optimal allocation is $O(n^{\frac{2M-1+2\gamma}{2M}})$ for $\gamma>0$ arbitrarily small. Clearly this regret is sublinear and approaches linear as the number of users $M$ increases. This means that the time average of the sum of the utilities of the players converges to the socially optimal welfare. Let ${\cal K} = \{k=(k_1, k_2, \ldots, k_N): k_j \geq 0, \forall j \in {\cal N}, k_1+k_2+\ldots+k_N=M \}$ denote an allocation of $M$ users to $N$ channels. Note that this allocation gives only the number of users on each channel. It does not say anything about which user uses which channel. We assume that the socially optimal allocation is unique up to permutations so $k^* = \arg\max_{k \in {\cal K}} \sum_{j=1}^N \mu_j k_j g_j(k_j)$ is unique. We also assume the following stability condition of the socially optimal allocation. Let $v_j(k_j)= \mu_j g_j(k_j)$. Then the stability condition says that $\arg\max_{k \in {\cal K}} \sum_{j=1}^N k_j \hat{v}_j(k_j)= k^*$ if $|\hat{v}_j(k)-v_j(k)| \leq \epsilon, \forall k \in \{1,2,\ldots,M\}, \forall j \in {\cal N}$, for some $\epsilon>0$, where $\hat{v}_j: \mathbb{N} \rightarrow \mathbb{R}$ is an arbitrary function. Let $T^i_{j,k}(t)$ be the number of times user $i$ used channel $j$ and observed $k$ users on it up to time $t$. We refer to the tuple $(j,k)$ as an arm. Let $n^i_{j,k}(t)$ be the time of the $t$th observation of user $i$ from arm $(j,k)$. Let $u^i_{j,k}(t)$ be the sample mean of the rewards from arm $(j,k)$ seen by user $i$ at the end of the $t$th play of arm $(j,k)$ by user $i$, i.e., $u^i_{j,k}(t)=(h_{j,k}(n^i_{j,k}(1)) + \ldots + h_{j,k}(n^i_{j,k}(t)))/t$. Then the socially optimal allocation estimated by user $i$ at time $t$ is $k^{i*}(t) = \arg\max_{k \in {\cal K}} \sum_{j=1}^N k_j u^i_{j,k}(t)$. The pseudocode of the Randomized Learning Algorithm (RLA) is given in Fig. \ref{fig:rla}. At time $t$ RLA explores with probability $1/(t^{\frac{1}{2M}-\frac{\gamma}{M}})$ by randomly choosing one of the channels and exploits with probability $1 - 1/(t^{\frac{1}{2M}-\frac{\gamma}{M}})$ by choosing a channel which is occupied by a user in the estimated socially optimal allocation.

The following will be useful in the proof of the main theorem of this section.
\begin{lemma} \label{lemma:case22}
Let $X_i, i=1,2,\ldots$ be a sequence of independent Bernoulli random variables such that $X_i$ has mean $q_i$ with $0 \leq q_i \leq 1$. Let $\bar{X}_k= \frac{1}{k} \sum_{i=1}^k X_i$ , $\bar{q}_k = \frac{1}{k} \sum_{i=1}^k q_i$. Then for any constant $\epsilon \geq 0$ and any integer $n \geq 0$,
\begin{eqnarray}
P\left( \bar{X}_n - \bar{q}_n \leq -\epsilon \right) \leq e^{-2n\epsilon^2}. \label{eqn:case26}
\end{eqnarray}
\end{lemma}
\begin{proof}
The result follows from symmetry and \cite{turner1}.
\qed
\end{proof}

\begin{lemma} \label{lemma:case23}
For $p>0, p \neq 1$
\begin{eqnarray}
\frac{(n+1)^{1-p}-1}{1-p} <	\sum_{t=1}^n \frac{1}{t^p} < 1 + \frac{n^{1-p}-1}{1-p} \label{eqn:case2_7}
\end{eqnarray}
\end{lemma}
\begin{proof}
See \cite{chlebus1}. \qed
\end{proof}

\begin{figure}[htb]
\fbox {
\begin{minipage}{\columnwidth}
RLA (for user $i$)
\begin{algorithmic}[1]
\STATE{Initialize: $0<\gamma << 1$, $u^i_{j,k}(1)=0, T^i_{j,k}(1)=0, \forall j \in {\cal N}, k \in {\cal M}$, $t=1$, sample $\sigma_i(1)$ uniformly from ${\cal N}$.}
\WHILE{$t>0$}
\STATE{play channel $\sigma_i(t)$, observe $l(t)$ the total number of players using channel $\sigma_i(t)$ and reward $h_{\sigma_i(t), l(t)}(t)$.}
\STATE{Set $T^i_{\sigma_i(t),l(t)}(t+1)=T^i_{\sigma_i(t),l(t)}(t)+1$.}
\STATE{Set $T^i_{j,l}(t+1)=T^i_{j,l}(t)$ for $(j,l) \neq (\sigma_i(t),l(t))$.}
\STATE{Set $u^i_{\sigma_i(t),l(t)}(t+1)= \frac{T^i_{\sigma_i(t),l(t)}(t) u^i_{\sigma_i(t),l(t)}(t) + h_{\sigma_i(t), l(t)}(t)}{T^i_{\sigma_i(t),l(t)}(t+1)}$.}
\STATE{Set $u^i_{j,l}(t+1) = u^i_{j,l}(t)$ for $(j,l) \neq (\sigma_i(t),l(t))$.}
\STATE{Set $k^{i*}(t+1) = \arg\max_{k \in {\cal K}} \sum_{j=1}^N k_j u^i_{j,k_j}(t+1)$.}
\STATE{Set $\theta^{*i}(t+1)$ to be the set of channels used by at least one user in $k^{*i}(t+1)$.}
\STATE{Draw $i_t$ randomly from Bernoulli distribution with $P(i_t=1)=\frac{1}{t^{(1/2M) - \gamma/M}}$}
\IF{$i_t=0$}
\IF{$\sigma_i(t) \in \theta^*(t+1)$ and $l(t)=k^{i*}_j(t+1)$}
\STATE{$\sigma_i(t+1) = \sigma_i(t)$}
\ELSE
\STATE{$\sigma_i(t+1)$ is selected uniformly at random from the channels in $\theta^*(t+1)$.}
\ENDIF
\ELSE
\STATE{Draw $\sigma_i(t+1)$ uniformly at random from ${\cal N}$.}
\ENDIF
\STATE{$t=t+1$}
\ENDWHILE
\end{algorithmic}
\end{minipage}}
\caption{pseudocode of RLA} \label{fig:rla}
\end{figure}

\begin{theorem} \label{thm:case2}
When all players use RLA the regret with respect to the socially optimal allocation is $O(n^{\frac{2M-1+2\gamma}{2M}})$ where $\gamma$ can be arbitrarily small.
\end{theorem}
\begin{proof}
Let $H(t)$ be the event that at time $t$ there exists at least one user that computed the socially optimal allocation \rev{incorrectly}.  Let $\omega$ be a sample path. Then
\begin{eqnarray}
&& \sum_{t=1}^n I(\omega \in H(t)) \leq \sum_{t=1}^n \sum_{i=1}^M I(k^{*i}(t) \neq k^*) \nonumber \\
&\leq& \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)} I(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon) \nonumber
\end{eqnarray}
\begin{eqnarray}
&=& \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)} I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon, T^i_{j,l}(t) \geq \frac{a \ln t}{\epsilon^2}\right) \nonumber \\
&+& \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)} I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon, T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2}\right) \label{eqn:case21}
\end{eqnarray}
\com{can we fixed the equation numbering (8) in the above equation?}

Let $\epsilon^i_{j,k}(t) = \sqrt{\frac{a \ln t}{T^i_{j,k}(t)}}$. Then $T^i_{j,k}(t) \geq \frac{a \ln t}{\epsilon^2} \Rightarrow \epsilon \geq \sqrt{\frac{a \ln t}{T^i_{j,k}(t)}} = \epsilon^i_{j,k}(t)$. Therefore,
\begin{eqnarray}
I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon, T^i_{j,l}(t) \geq \frac{a \ln t}{\epsilon^2}\right) &\leq& I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon^i_{j,l}(t) \right) \nonumber \\
I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon, T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2}\right) &\leq& I\left( T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2}\right) \nonumber
\end{eqnarray}
Then, continuing from (\ref{eqn:case21}),
\begin{eqnarray}
&& \sum_{t=1}^n I(\omega \in H(t)) \nonumber \\
&\leq&  \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)} \left( I\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon^i_{j,l}(t) \right) + I\left( T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2}\right) \right) \label{eqn:case22}
\end{eqnarray}
Taking the expectation \rev{over} (\ref{eqn:case22}),
\begin{eqnarray}
&& E\left[ \sum_{t=1}^n I(\omega \in H(t)) \right] \nonumber \\
&\leq& \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)}  P\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon^i_{j,l}(t) \right) \nonumber \\
&+& \sum_{(t,i,j,l)=(1,1,1,1)}^{(n, M, N, M)} P\left( T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \right). \label{eqn:case23}
\end{eqnarray}
We have
\begin{eqnarray}
&&P\left(|u^i_{j,l}(T^i_{j,l}(t))-v_j(l)| \geq \epsilon^i_{j,l}(t) \right) \nonumber \\
&=& P\left(u^i_{j,l}(T^i_{j,l}(t))-v_j(l) \geq \epsilon^i_{j,l}(t) \right) + P\left(u^i_{j,l}(T^i_{j,l}(t))-v_j(l) \leq -\epsilon^i_{j,l}(t) \right) \nonumber \\
&=&P\left( \frac{S^i_{j,l}(T^i_{j,l}(t))}{T^i_{j,l}(t)} - v_j(l) \geq \epsilon^i_{j,l}(t) \right) + P\left( \frac{S^i_{j,l}(T^i_{j,l}(t))}{T^i_{j,l}(t)} - v_j(l) \leq - \epsilon^i_{j,l}(t) \right) \nonumber \\
&\leq& 2 \exp\left(- \frac{2 (T^i_{j,l}(t))^2 (\epsilon^i_{j,l}(t))^2}{T^i_{j,l}(t)}\right) = 2 \exp\left(- \frac{2 T^i_{j,l}(t) a \ln t}{T^i_{j,l}(t)}\right) = \frac{2}{t^{2a}} ~, \label{eqn:case24}
\end{eqnarray}
\rev{where} (\ref{eqn:case24}) follows from the Chernoff-Hoeffding inequality.

Now we will bound $P\left( T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \right)$. Let $TR^i_{j,l}(t)$ be the number of time steps in which player $i$ played channel $j$ and observed $l$ users on channel $j$ in the time steps where all players randomized up to time $t$. Then
\begin{eqnarray}
\{ \omega: T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \} \subset \{ \omega: TR^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \}, \nonumber \\
\end{eqnarray}
Thus
\begin{eqnarray}
P\left( T^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \right) \leq P\left( TR^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \right) ~. \label{eqn:case2_8}
\end{eqnarray}
Now we define new Bernoulli random variables $X^i_{j,l}(s)$ as follows: $X^i_{j,l}(s)=1$ if all players randomize at time $s$ and player $i$ selects channel $j$ and observes $l$ players on it according to the random draw. $X^i_{j,l}(s)=0$ else. Then $TR^i_{j,l}(t)= \sum_{s=1}^t X^i_{j,l}(s)$. $P(X^i_{j,l}(s)=1)= \rho_s p_l$ where $p_l= \frac{{M-1 \choose l-1}{M+N-l-2 \choose N-2}}{ {M+N-1 \choose N-1}}$ and $\rho_s = \frac{1}{s^{(1/2)-\gamma}}$. Let $s_t = \sum_{s=1}^t \frac{1}{s^{(1/2)-\gamma}}$  Then
\begin{eqnarray}
&& P\left( TR^i_{j,l}(t) < \frac{a \ln t}{\epsilon^2} \right) \nonumber \\
&=& P\left( \frac{TR^i_{j,l}(t)}{t} - \frac{p_k s_t}{t} < \frac{a \ln t}{t \epsilon^2} - \frac{p_k s_t}{t} \right) \nonumber \\
&\leq& P\left( \frac{TR^i_{j,l}(t)}{t} - \frac{p_k s_t}{t} < \frac{a \ln t}{t \epsilon^2} - \frac{p_k (t+1)^{(1/2)+\gamma}-1}{t((1/2)+\gamma)} \right), \label{eqn:case29}
\end{eqnarray}
where (\ref{eqn:case29}) follows from Lemma \ref{lemma:case23}.
Let $\tau(M,N,\epsilon,\gamma,\gamma',a)$ be the time that for all $k \in \{1,2,\ldots,M \}$.
\begin{eqnarray}
\frac{p_k (t+1)^{(1/2)+\gamma}-1}{t((1/2)+\gamma)} - \frac{a \ln t}{t \epsilon^2} \geq t^{(1/2)+\gamma'}, \label{eqn:case210}
\end{eqnarray}
where $0 < \gamma' < \gamma$. Then for all $t \geq \tau(M,N,\epsilon,\gamma,\gamma',a)$ (\ref{eqn:case210}) will hold since RHS increases faster than LHS. Thus we have for $t \geq \tau(M,N,\epsilon,\gamma,\gamma',a)$
\begin{eqnarray}
&& P\left( \frac{TR^i_{j,l}(t)}{t} - \frac{p_k s_t}{t} < \frac{a \ln t}{t \epsilon^2} - \frac{p_k (t+1)^{(1/2)+\gamma}-1}{t((1/2)+\gamma)} \right) \nonumber \\
&\leq& P\left( \frac{TR^i_{j,l}(t)}{t} - \frac{p_k s_t}{t} < {t^{-(1/2)+\gamma'} } \right) \nonumber \\
&\leq& e^{-2 t t^{2\gamma'-1}} = e^{-2 t^{2\gamma'}} \leq e^{-2 \ln t} = \frac{1}{t^2}. \label{eqn:case211}
\end{eqnarray}
Let $a=1$. Then continuing from (\ref{eqn:case23}) by substituting (\ref{eqn:case24}) and (\ref{eqn:case211}) \rev{we have}
\begin{eqnarray}
E\left[ \sum_{t=1}^n I(\omega \in H(t)) \right] \leq M^2 N \left(\tau(M,N,\epsilon,\gamma,\gamma',1) + 3 \sum_{t=1}^n \frac{1}{t^2} \right). \label{eqn:case212}
\end{eqnarray}
Thus we proved that the expected number of time steps in which there exists at least one user that computed the socially optimal allocation incorrectly is finite. Note that because RLA explores with probability $\frac{1}{t^{1/2M-\gamma/M}}$, the expected number of time steps in which all the players are not randomizing up to time $n$ is
\begin{eqnarray}
\sum_{t=1}^n \left( 1- \left(1-\frac{1}{t^{(1/2M)-\gamma/M}}\right)^M\right) \leq \sum_{t=1}^n \frac{M}{t^{1/2M - \gamma/M}} = O(n^{\frac{2M-1+2\gamma}{2M}}) . \label{eqn:case213}
\end{eqnarray}
Note that players can choose $\gamma$ arbitrarily small, increasing the finite regret due to $\tau(M,N,\epsilon,\gamma,\gamma',1)$. Thus if we are interested in the asymptotic performance then $\gamma>0$ can be arbitrarily small.

Now we do the worst case analysis. We classify the time steps into two. {\em Good} time steps in which all the players know the socially optimal allocation correctly and none of the players randomize excluding the randomizations done for settling down to the socially optimal allocation. {\em Bad} time steps in which there exists a player that does not know the socially optimal allocation correctly or there is a player that randomizes excluding the randomizations done for settling down to the socially optimal allocation. The number of {\em Bad} time steps in which there exists a player that does not know the socially optimal allocation correctly is finite while the number of time steps in which there is a player that randomizes excluding the randomizations done for settling down to the socially optimal allocation is $O(n^{\frac{2M-1+2\gamma}{2M}})$. The worst case is when each bad step is followed by a good step. Then from this good step the expected number of times to settle down to the socially optimal allocation is $\left(1-\frac{1}{{M+z^*-1 \choose z^*-1}}\right)/\left(\frac{1}{{M+z^*-1 \choose z^*-1}}\right)$ where $z^*$ is the number of channels which has at least one user in the socially optimal allocation. Assuming in the worst case the sum of the utilities of the players is $0$ when they are not playing the socially optimal allocation we have
\begin{eqnarray}
R(n) &\leq& \frac{1-\frac{1}{{M+z^*-1 \choose z^*-1}}}{\frac{1}{{M+z^*-1 \choose z^*-1}}} \left( M^2 N \left(\tau(M,N,\epsilon,\gamma,\gamma',1) + 3 \sum_{t=1}^n \frac{1}{t^2} \right) + O(n^{\frac{2M-1+2\gamma}{2M}}) \right) \nonumber \\
&=& O(n^{\frac{2M-1+2\gamma}{2M}}) \nonumber
\end{eqnarray}
\qed
\end{proof}

Note that we mentioned earlier, under a classical multi-armed bandit problem approach as cited before \cite{anandkumar,anantharam2,anantharam1,lai1,liu2,tekin,tekin2},  a logarithmic regret $O(\log n)$ is achievable.  The  fundamental difference between these studies and the problem in the present paper is the following: Assume that at time $t$ user $i$ selects channel $j$. This means that $i$ selects to observe an arm from the set $\{(j,k): k \in {\cal M}\}$ but the arm assigned to $i$ is selected from this set depending on the choices of other players.

Also note that in RLA a user computes the socially optimal allocation according to its estimates at each time step. This could pose significant computational effort since integer programming is NP-hard in general. However, by exploiting the stability condition on the socially optimal allocation a user may reduce the number of computations; this is a subject of future research.

\section{Allocations Achievable with Exp3 Algorithm (Case 1)} \label{sec:case1}

We start by defining a congestion game. A congestion game \cite{monderer1,rosenthal1} is given by the
tuple $({\cal M}, {\cal N}, (\Sigma_i)_{i\in{\cal M}},
(h_j)_{j\in {\cal N}})$, where ${\cal M}$
denotes a set of players (users), ${\cal N}$ a set of resources (channels), $\Sigma_i\subset 2^{\cal N}$ the strategy space of player $i$, and $h_j: \mathbb{N}\rightarrow\mathbb{R}$ a payoff
function associated with resource $j$, which is a function of the number of players using that resource. It is well known that a congestion game has a potential function and the local maxima of the potential function corresponds to PNE, and every sequence of asynchronous improvement steps is finite and converges to PNE.

In this section we relate the strategy update rule of Exp3
\cite{auer2} under assumptions (C1) to a congestion game.
Exp3 as given in Fig. \ref{fig:exp3} is a randomized algorithm
consisting of an exploration parameter $\gamma$ and weights
$w_{ij}$ that depend exponentially on the past observations where
$i$ denotes the user and $j$ denotes the channel. Each user runs
Exp3 independently but we explicitly note the user dependence
because a user's action affects other users' updates.

\begin{figure}[htb]
\fbox {
\begin{minipage}{\columnwidth}
{Exp3 (for user $i$)}
\begin{algorithmic}[1]
\STATE {Initialize: $\gamma \in (0,1)$, $w_{ij}(t)=1, \forall j \in {\cal N}$, $t=1$}
\WHILE{$t>0$}
\STATE{\begin{eqnarray*}
\hspace{-2.4in} p_{ij}(t)=(1-\gamma)\frac{w_{ij}(t)}{\sum_{l=1}^N w_{il}(t)}+\frac{\gamma}{N}
\end{eqnarray*}
}
\STATE{Sample $\sigma_i(t)$ from the distribution on $p_i(t)=[p_{i1}(t), p_{i2}(t), \ldots, p_{iN}(t)]$}
\STATE{Play channel $\sigma_i(t)$ and receive reward $h_{\sigma_i(t)}(t)$}
\FOR{$j=1,2,\ldots,N$}
\IF{$j=\sigma_i(t)$}
\STATE{Set $w_{ij}(t+1)=w_{ij}(t) \exp\left({\frac{\gamma h_{\sigma_i(t)}(t)}{p_{ij}(t) N}}\right)$}
\ELSE
\STATE{Set $w_{ij}(t+1)=w_{ij}(t)$}
\ENDIF
\ENDFOR
\STATE{$t=t+1$}
\ENDWHILE
\end{algorithmic}
\end{minipage} }
\caption{pseudocode of Exp3} \label{fig:exp3}
\end{figure}

At any time step before the channel rate and user actions are
drawn from the corresponding distributions, let $R_j$ denote the
random variable corresponding to the reward of the $j$th channel.
Let $G_{ij}=g_j(1+K'_j(i))$ be the random variable representing the
payoff user $i$ gets from channel $j$ where $K'_j(i)$ is the random
variable representing the number of users on channel $j$ other
than user $i$. Let $U_{ij}=R_j G_{ij}$ and $\bar{u}_{ij} = E_j [
E_{-i}[U_{ij}]]$ be the expected payoff to user $i$ by using
channel $j$ where $E_{-i}$ represents the expectation taken with
respect to the randomization of players other than $i$, $E_j$
represents the expectation taken with respect to the randomization
of the rate of channel $j$. Since the channel rate is independent
of users' actions $\bar{u}_{ij} = \mu_j \bar{g}_{ij}$ where
$\bar{g}_{ij} = E_{-i}[G_{ij}]$.

\begin{lemma} \label{lemma:exp1}
Under (C1) when all players use Exp3, the derivative of the continuous-time limit of Exp3 is the replicator equation given by
\begin{eqnarray*}
\xi_{ij} = \frac{1}{N} (\mu_j p_{ij}) \sum_{l=1}^N p_{il} (\bar{g}_{ij}-\bar{g}_{il}) ~.
\end{eqnarray*}
\end{lemma}
\begin{proof}
Note that
\begin{eqnarray}
(1-\gamma)w_{ij}(t)= \sum_{l=1}^N w_{il}(t) \left(p_{ij}(t)-\frac{\gamma}{N}\right) ~. \label{eqn:exp1}
\end{eqnarray}
We consider the effect of \rev{user $i$'s} action $\sigma_i(t)$ on his probability update on channel $j$.  \rev{We have two cases:} $\sigma_i(t)=j$ \rev{and} $\sigma_i(t) \neq j$. Let $A_{i,j}^{\gamma,t} = \exp\left( \frac{\gamma U_{ij}(t)}{p_{ij}(t)N} \right)$.

Consider the case $\sigma_i(t)=j$.
\begin{eqnarray}
p_{ij}(t+1)= \frac{(1-\gamma) w_{ij}(t) A_{i,j}^{\gamma,t}} {\sum_{l=1}^N w_{il}(t) + w_{ij}(t) \left( A_{i,j}^{\gamma,t}-1 \right)} + \frac{\gamma}{N} ~. \label{eqn:exp2}
\end{eqnarray}
Substituting (\ref{eqn:exp1}) into (\ref{eqn:exp2})
\begin{eqnarray}
p_{ij}(t+1)&=& \frac{ \sum_{l=1}^N w_{il}(t) \left(p_{ij}(t)-\frac{\gamma}{N}\right) A_{i,j}^{\gamma,t}} { \sum_{l=1}^N w_{il}(t) \left( 1 + \frac{p_{ij}(t)-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,j}^{\gamma,t}-1 \right) \right)} + \frac{\gamma}{N} \nonumber \\
&=& \frac{\left(p_{ij}(t)-\frac{\gamma}{N}\right) A_{i,j}^{\gamma,t}}{1 + \frac{p_{ij}(t)-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,j}^{\gamma,t}-1 \right)} + \frac{\gamma}{N} ~. \nonumber
\end{eqnarray}

The continuous time process is obtained by taking the limit
$\gamma \rightarrow 0$, i.e., the rate of change in $p_{ij}$ with
respect to \rev{$\gamma$} as $\gamma \rightarrow 0$. Then, dropping the
discrete time script $t$,
\begin{eqnarray}
\dot{p_{ij}}&=&\lim_{\gamma \rightarrow 0} \frac{d p_{ij}}{d\gamma} \nonumber \\
&=& \lim_{\gamma \rightarrow 0} \frac{ \left(\frac{-1}{N} A_{i,j}^{\gamma,t} + \left(p_{ij}-\frac{\gamma}{N}\right) \frac{U_{ij}}{p_{ij}N} A_{i,j}^{\gamma,t}\right) \left( 1 + \frac{p_{ij}-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,j}^{\gamma,t}-1 \right) \right)}{ \left( 1 + \frac{p_{ij}-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,j}^{\gamma,t}-1 \right) \right)^2 } \nonumber \\
&+& \frac{ \left(p_{ij}-\frac{\gamma}{N}\right) A_{i,j}^{\gamma,t} \left( \frac{p_{ij}-\frac{1}{N}}{(1-\gamma)^2} A_{i,j}^{\gamma,t} + \frac{p_{ij}-\frac{1}{N}}{1-\gamma} \left( \frac{\gamma}{N} A_{i,j}^{\gamma,t} \right) \right) } { \left( 1 + \frac{p_{ij}-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,j}^{\gamma,t}-1 \right) \right)^2} + \frac{1}{N} \nonumber \\
&=& \frac{U_{ij}(1-p_{ij})}{N} ~. \label{eqn:exp3}
\end{eqnarray}

Consider the case $\sigma_i(t)= k \neq j$. Then,
\begin{eqnarray}
p_{ij}(t+1)&=& \frac{(1-\gamma) w_{ij}(t)} {\sum_{l=1}^N w_{il}(t) + w_{ik}(t) \left( A_{i,k}^{\gamma,t}-1 \right)} + \frac{\gamma}{N} \nonumber \\
&=& \frac{p_{ij}(t)-\frac{\gamma}{N}}{1 + \frac{p_{ik}(t)-\frac{\gamma}{N}}{1-\gamma} \left( A_{i,k}^{\gamma,t}-1 \right)} + \frac{\gamma}{N}. \nonumber
\end{eqnarray}
Thus
\begin{eqnarray}
\dot{p_{ij}}&=& \lim_{\gamma \rightarrow 0} \frac{ \frac{-1}{N} \left(1+ \frac{p_{ik}-\frac{\gamma}{N}}{1-\gamma}  \left( A_{i,k}^{\gamma,t}-1 \right)  \right)}{\left(1+ \frac{p_{ik}-\frac{\gamma}{N}}{1-\gamma}  \left( A_{i,k}^{\gamma,t}-1 \right) \right)^2} \nonumber \\
&+& \frac{ \left(p_{ij}-\frac{\gamma}{N}\right) \left( \frac{p_{ik}-\frac{1}{N}}{(1-\gamma)^2} A_{i,k}^{\gamma,t} + \frac{p_{ik}-\frac{1}{N}}{1-\gamma} \left( \frac{\gamma}{N} A_{i,k}^{\gamma,t} \right) \right)}{\left(1+ \frac{p_{ik}-\frac{\gamma}{N}}{1-\gamma}  \left( A_{i,k}^{\gamma,t}-1 \right) \right)^2} +\frac{1}{N} \nonumber \\
&=& -\frac{p_{ik}U_{ik}}{N} ~. \label{eqn:exp4}
\end{eqnarray}

Then from (\ref{eqn:exp3}) and (\ref{eqn:exp4}), the expected change in $p_{ij}$ with respect to the probability distribution $p_i$ of user $i$ over the channels is
\begin{eqnarray}
\bar{p}_{ij} = E_i[\dot{p}_{ij}]= \frac{1}{N} p_{ij} \sum_{l \in {\cal N}-\left\{j\right\}} p_{il} (U_{ij} - U_{il}). \nonumber
\end{eqnarray}
Taking the expectation with respect to the randomization of channel rates and other \rev{users'} actions we have
\begin{eqnarray}
\xi_{ij} &=& E_j[E_{-i}[\bar{p}_{ij}]] \nonumber \\
&=& \frac{1}{N} p_{ij} \sum_{l \in {\cal N}-\left\{j\right\}} p_{il} \left(E_j[ E_{-i}[ U_{ij}]] - E_j[E_{-i}[ U_{il}]] \right) \nonumber \\
&=&\frac{1}{N} (\mu_j p_{ij}) \sum_{l=1}^N p_{il} (\bar{g}_{ij}-\bar{g}_{il}) ~. \nonumber
\end{eqnarray} \qed
\end{proof}
Lemma \ref{lemma:exp1} shows that the dynamics of a user's
probability distribution over the actions is \rev{given by} a replicator equation
which is commonly studied in evolutionary game theory
\cite{sandholm1,smith1}. With this lemma we can establish the following theorem.

\begin{theorem} \label{thm:exp2}
For all but a measure zero subset of $[0,1]^{2N}$ from which \rev{the $\mu_j$'s and $g_j$'s} are selected, when $\gamma$ in Exp3 is arbitrarily small, the action profile converges to the set of PNE of the
congestion game $({\cal M}, {\cal N}, ({\cal S}_i)_{i\in {\cal M}},
(\mu_j g_j)_{j\in {\cal N}})$.
\end{theorem}
\begin{proof}
\rev{Because the replicator equation in Lemma \ref{lemma:exp1} is identical to the replicator equation in \cite{kleinberg1}, the proof of converge to PNE follows from \cite{kleinberg1}. Here, we briefly explain the steps in the proof.}
Defining the expected potential function to be
the expected value of the potential function $\phi$ where
expectation is taken with respect to the user's randomization one
can show that the solutions of the replicator equation converges
to the set of fixed points. Then the stability analysis using the
Jacobian matrix yields that every stable fixed point corresponds
to a Nash equilibrium. Then one can prove that for any stable
fixed point the eigenvalues of the Jacobian must be zero. This
implies that every stable fixed point corresponds to a {\em weakly
stable} Nash equilibrium strategy in the game theoretic sense.
Then using tools from algebraic geometry one can show that almost every
weakly stable Nash equilibrium is a pure Nash equilibrium of the
congestion game.

We also need to investigate the error introduced by treating the
discrete time update rule as a continuous time process. However,
by taking $\gamma$ infinitesimal we can approximate the discrete
time process by the continuous time process. For a discussion when
$\gamma$ is not infinitesimal one can define {\em approximately
stable equilibria} \cite{kleinberg1}. \qed
\end{proof}

The main difference between Exp3 and Hedge \cite{kleinberg1} is
that in Exp3 users do not need to observe the payoffs from the
channels that they do not select, whereas Hedge assumes complete
observation. In addition to that, we considered the dynamic channel
rates which is not considered in \cite{kleinberg1}.

\section{Conclusion} \label{sec:conc}

In this paper we studied the decentralized multiuser resource allocation problem with various levels of communication and cooperation between the users. Under three different scenarios we proposed three algorithms with reasonable performance. Our future reserach will include characterization of achievable performance regions for these scenarios. For example, in case 2 we are interested in finding an optimal algorithm and a lower bound on the performance.

\section{Introduction} \label{sec:intro}

In this paper we study the dynamic spectrum access and spectrum
sharing problem in a learning context.  Specifically, we consider
a set of $N$ common channels shared by a set of $M$ users.  A
channel has time varying rate $r(t)$, and its statistics are not
completely known by the users.  Thus each user needs to employ
some type of learning to figure out which channels are of better
quality, e.g., in terms of their average achievable rates. At the
same time, simultaneous use of the same channel by multiple users
will result in reduced rate due to interference or collision.
The precise form of this performance degradation may or may not be known to the user.  Thus the users also need to use learning to avoid excess interference or congestion.  Furthermore, each user may have private information that is not shared, e.g., users may perceive channel quality differently due to difference in location as well as individual modulation/coding schemes.

Without a central agent, and in the presence of information
decentralization described above, we are interested in the
following questions: (1) for a given common learning algorithm,
does the multiuser learning process converge, and
(2) if it does, what is the quality of the equilibrium point with
respect to a globally optimal spectrum allocation scheme, one that
could be computed for a global objective function with full
knowledge of channel statistics as well as \rev{the users'} private
information.

A few recent studies have addressed these questions in some
special cases. For instance, in \cite{anandkumar} it was shown
that learning using a sample-mean based index policy leads to a
socially optimal (sum of individual utilities) allocation when
channels evolve as iid processes and colliding players get zero
reward provided that this optimal allocation is such that each user
occupies one of the $M$ best channels (in terms of average rates).
This precludes the possibility that not all users may have the
same set of $M$ best channels, and that in some cases the best
option is for multiple users to share a common channel, e.g., when
$N<M$.

In this study we investigate under what conditions the socially
optimal allocation is achievable by considering different levels of
communication (or cooperation) allowed among users, and different
levels of uncertainty on the channel statistics. The general
conclusion, as intuition would suggest, is that when the
cooperation of users increases and the channel uncertainty
decreases it becomes easier to achieve the socially optimal
welfare. Specifically, we assume that the rate (or reward) user $i$ gets
from channel $j$ at time $t$ is of the form $r_j(t) g_j(n_j(t))$
where $r_j(t)$ is the rate of channel $j$ at time $t$, $n_j(t)$ is
the number of users using channel $j$ at time $t$, and $g_j$ is the
user independent interference function (IF) for channel
$j$. This model is richer than the previously used models
\cite{anandkumar,kleinberg1,liu2} since $r_j(t)$ can represent
environmental effects such as fading or primary user activity,
while $g_j$ captures interactions between users. We consider the
following three cases.

In the first case (C1), each channel evolves as an iid random
process in time, the users do not know the channel statistics,
nor the form of the interference, nor the total number of users
present in the system, and no direct communication is allowed
among users. A user can measure the overall rate it gets from
using a channel but cannot tell how much of it is due to the
dynamically changing channel quality (i.e., what it would get if
it were the only user) vs. interference from other users.  In this
case, we show that if all users follow the Exp3 algorithm
\cite{auer2} then the channel allocation converges to a set of
pure Nash equilibria (PNE) of a congestion game defined by the
IFs and mean channel rates.  In this case a socially optimal
allocation cannot be ensured, as the set of PNE are of different
quality, and in some cases the socially optimal allocation may not
be a PNE.

In the second case (C2), each channel again evolves as an iid
random process in time, whose statistics are unknown to the user.
However, the users now know the total number of users in the
system, as well as the fact that the quantitative impact of
interference is common to all users (i.e., user independent),
though the actual form of the interference function is unknown.
In other words the rate of channel $j$ at time $t$ is perceived by
user $i$ as $h_j(t,n_j(t))$ so user $i$ cannot distinguish between
components $r_j(t)$ and $g_j(n_j(t))$.
Furthermore, users are now allowed minimal amount of communication
when they happen to be in the same channel, specifically to find
out the total number of simultaneous users of that channel. In this case we present a
sample-mean based randomized learning policy that achieves
socially optimal allocation as time goes to infinity, with a
sub-linear regret over the time horizon with respect to the
socially optimal allocation.

In the third case (C3), as in case (C2) the users know the total
number of users in the system, as well as the fact that the
IF is user independent and decreasing without knowing the
actual form of the IF.  However, the channels
are assumed to have constant, albeit unknown, rates.  We show that
even without any communication among users, there is a randomized
learning algorithm that achieves the socially optimal allocation
in finite time.

It's worth pointing out that in the settings outlined above, the users are
{\em non-strategic}, i.e., each user simply follow a pre-set learning rule
rather than playing a game.  In this context it is reasonable to introduce minimal amount of
communication among users and assume they may cooperate.
It is possible that even in this case the users may not know their IF
but only the total rate they get for lack of better detecting capabilities (e.g., they may only
be able to detect the total received SNR as a result of channel rate and user interference).

Online learning by a single user was studied by \cite{agrawal1,anantharam2,auer,lai1}, in which sample-mean based index policies were shown to achieve
logarithmic regret with respect to the best single-action policy without a priori
knowledge of the statistics, and are order-optimal, when the rewards are given by
an iid process. In \cite{anantharam1,tekin,tekin2} Markovian rewards are considered,
with \cite{tekin2} focusing on {\em restless} reward processes, where a process continues
to evolve according to a Markov chain regardless of the users' actions.  In all these
studies learning algorithms were developed to achieve logarithmic regret.
Multi-user learning with iid reward processes have been studied in a dynamic spectrum context
by \cite{anandkumar,krishnamachari,liu2}, with a combinatorial structure adopted in \cite{krishnamachari},
and with collision and random access models in \cite{anandkumar,liu2}. In \cite{gaurav2010}, convergence of multi-user learning with Exp3 algorithm to pure Nash equilibrium is investigated under the collision and fair sharing models. In the collision model, when there is more
than one user on a channel all get zero reward, whereas in the
random access model one of them, selected randomly, gets all the reward
while others get zero reward. In the fair sharing model, a user's utility is inversely proportional to the number of users who are on the same channel with the user. Note that these models do not
capture more sophisticated communication schemes where the rate a user gets is
a function of the received SNR of the form
$g_j(n)= f_j(\frac{P_t}{N_0+(n-1) P_t})=$ where $P_t$ is the nominal transmit power of all users
and $N_0$ the noise.
Moreover, in the above studies the socially optimal allocation is a rather simple one:
it is the orthogonal allocation of users to the first $M$ channels with the highest mean rewards.
By contrast, we model a more general interference relationship among users, in which an allocation with users sharing the same channel may be the socially optimal one.
The socially optimal allocations is not trivial in this case and
additional mechanisms may be needed for the learning algorithms to converge.

All of the above mentioned work assumes some level of
communication between the users either at the beginning or during
the learning. If we assume no communication between the users, achieving the socially optimal allocation seems very challenging
in general. Then one may ask if it is possible to achieve some
kind of equilibrium allocation. Kleinberg et. al.
\cite{kleinberg1} showed that it is possible for the case when the
channel rates are constant and the users do not know the IFs. They
show that when the users use {\em aggregate monotonic selection
dynamics}, a variant of Hedge algorithm \cite{freund1}, the
allocation converges to {\em weakly stable equilibria} which is a
subset of Nash equilibria (NE) of the congestion game defined by
the IFs. They show that for almost all congestion games weakly
stable equilibria is the same as PNE.

Other than the work described above \cite{ahmad2} considers {\em
spatial congestion games}, a generalization of congestion games
and gives conditions under which there exists a PNE and
best-response play converges to PNE. A mechanism design approach
for socially optimal power allocation when users are strategic is
considered in \cite{kakhbod1}.

The organization of the remainder of this paper is as follows. In
Sect. \ref{sec:problem} we present the notations and definitions
that will be used throughout the paper. In Sects.
\ref{sec:case1}, \ref{sec:case2}, \ref{sec:case3} we analyze the
cases stated in (C1), (C2), (C3) and derive the results
respectively. Conclusion and future research is given in Sect. \ref{sec:conc}.

\section{Preliminaries} \label{sec:problem}

Denote the set of users by $\mathcal{M}=\{1,2,\ldots,M\}$, and the set of
channels $\mathcal{N}=\{1,2,\ldots,N\}$. Time is slotted and indexed by
$t=1,2,\ldots$ and a user can select a single channel at each time
step $t$. Without loss of generality let $r_j(t) \in [0,1]$ be the
rate of channel $j$ at time $t$ such that $\{r_j(t)\}_{t=1,2,\ldots}$ is generated by an iid
process with support $[0,1]$ and mean $\mu_j \in [0,1]$. Let $g_j: \mathbb{N} \rightarrow [0,1]$
be the interference function (IF) on channel $j$\comment{where $g_j(n)$
is a positive decreasing function for $n \geq 1$ with $g_j(0)=0$}
where $g_j(n)$ represents the interference when there are $n$
users on channel $j$. We express the rate of channel $j$ seen by a user as $h_j(t)=r_j(t)g_j(n_j(t))$ when a user does not know the total number of users $n_j(t)$ using channel $j$ at time $t$ as in cases (C1) and (C3). When a user knows $n_j(t)$, we express the rate of channel $j$ at time $t$ as $h_{j, n_j(t)}(t) = r_j(t)g_j(n_j(t))$ as in case (C2). Let $\mathcal{S}_i={\cal N}$
be the set of feasible actions of user $i$ and $\sigma_i \in
\mathcal{S}_i$ be the action, i.e., channel selected by user $i$.
Let $\mathcal{S}= \mathcal{S}_1 \times \mathcal{S}_2
\times \ldots \times \mathcal{S}_N = {\cal N}^M$
be the set of feasible action profiles and $\sigma=\{\sigma_1,
\sigma_2, \ldots, \sigma_M \} \in \mathcal{S} $ be the action
profile of the users. Throughout the discussion we assume that the action of player $i$ at time $t$, i.e., $\sigma_i^{\pi_i} (t)$ is determined by the policy $\pi_i$. When $\pi_i$ is deterministic,
$\pi_i(t)$ is in general a function from all past
observations and decisions of user $i$ to the set of actions
$\mathcal{S}_i$. When $\pi_i$ is randomized, $\pi_i(t)$ generates
a probability distribution over the set of actions $\mathcal{S}_i$
according to all past observations and decisions of user $i$ from
which the action at time $t$ is sampled. Since the dependence of actions to the policy is trivial we use $\sigma_i(t)$ to denote the action of user $i$ at time $t$, dropping the superscript $\pi_i$.

Let $K_j(\sigma)$ be the set
of users on channel $j$ when the action profile is $\sigma$. Let $\mathcal{A}^* =
\arg\max_{\sigma \in \mathcal{S}} \sum_{i=1}^M \mu_{\sigma_i}
g_{\sigma_i}(K_{\sigma_i}(\sigma)) = \arg\max_{\sigma \in
\mathcal{S}} \sum_{j=1}^N \mu_{j} K_{j}(\sigma)
g_{j}(K_{j}(\sigma))$ be the set of socially optimal allocations
and denote by $\sigma^*$ any action profile that is in the set
$\mathcal{A}^*$. Let $v^*$ denote the socially optimal welfare,
i.e., $v^*= \sum_{i=1}^M \mu_{\sigma^*_i}
g_{\sigma^*_i}(K_{\sigma^*_i}(\sigma^*))$ and $v^*_j$ denote the payoff a user gets from channel $j$ under the socially optimal allocation, i.e., $v^*_j = \mu_j g_{j}(K_{j}(\sigma^*))$ if $K_{j}(\sigma^*) \neq 0$.
Note that any permutation
of actions in $\sigma^*$ is also a socially optimal allocation
since IFs are user-independent.

For any policy $\pi$, the regret at time $n$ is
\begin{eqnarray}
R(n) = n v^* -
E\left[\sum_{t=1}^n \sum_{i=1}^{M} r_{\sigma_i(t)}(t)
g_{\sigma_i(t)}(K_{\sigma_i(t)}(\sigma(t)))\right], \nonumber
\end{eqnarray}
where expectation is taken with respect to the random nature of
the rates and the randomization of the policy. Note that for a
deterministic policy expectation is only taken with respect to the
random nature of the rates. For any randomized policy $\pi_i$, let
$p_i(t)=(p_{i1}(t), p_{i2}(t), \ldots, p_{iN}(t))$ be the mixed
strategy of user $i$ at time $t$, i.e., a probability distribution
on $\{1,2,\ldots,N\}$. For a profile of policies $\pi=[\pi_1,
\pi_2, \ldots, \pi_M]$ for the users let $p(t)=(p_1(t)^T,
p_2(t)^T, \ldots p_M(t)^T)^T$ be the profile of mixed strategies at
time $t$, where $p_i(t)^T$ is the transpose of $p_i(t)$. Then
$\sigma_i(t)$ is the action sampled from the probability
distribution $p_i(t)$. The dependence of $p$ to $\pi$ is trivial
and not shown in the notation.

redpajama_set_name: RedPajamaArXiv


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Home > Comment > Features > 201711
Could This Be John Mahama!
I am not a publicist for the NPP government, and by my nature, I hate propaganda, but I would have thought that the panacea for tax dodging and motorists' crimes is the digital addressing system.
Throughout the world, critical thinkers are full of praise for what is gradually becoming known as 'The Rwanda Miracle', except that it is no miracle at all but a combination of pragmatism, sincerity, selfless leadership and the application of technology.
My brother-in-law committed a traffic offence in the UK. Nobody saw him – or he thought nobody did.In less than 24 hours, he had a ticket in the mail. That is the CCTV cameras working beautifully with the digital addressing system.
In America, my Kwame Nkrumah Secondary School classmate is constantly on the wrong side of the traffic rules. He tells me that by the time the patrol car catches up with his car, the police know who he is, his address, insurance number etc. How? By courtesy of the GPS and National Identification system.
For years, our criticism of the Ghanaian economy has been that it is too heavy on the side of the informal. Not that the informal in itself is evil: the problem has been that we have not known how to drag the informal into the tax net.
The intrinsic goodness of the National Identification system and the GhanaPostGPS, however, does not mean that anybody who criticises their implementation should be seen as "enemy of the state". Indeed, some truly beneficial criticisms, so far, have come from the likes of IMANI who have raised issues with the system's over-reliance on Internet connectivity in a country where Internet penetration is low.
Their genuine criticisms have elicited explanations from Vokakom, the company that designed the GhanaPostGPS app. This back-and-forth is what I prefer to call "exchanges at the intellectual level" - the air, sunshine and moisture needed to bring fruits out of seeds.
This is the level to which I had expected former President Mahama to ascend in his entry into the digital addressing conversation.
Before anybody suspects I am reacting to Mr Mahama for any reason other than a patriotic contribution to a national debate, I will take off with my relationship with him. I have never met him personally and I don't know if he has even heard of me, but in my days on the 'Ghanaian Times', I praised him twice for "seasoning his speeches with salt".
A day after his address, as Vice-President, at the GJA Media Awards at the State House I did a piece that advised the National Democratic Congress (NDC) party that if they knew what was good for them, they should be considering somebody like John as flag bearer for 2012. His office (Office of the Vice-President), through the moustachio'd former GM for Radio Gold, reacted with gratefulness.
So why Mahama's tongue has changed ever since providence elevated him to the high office of President baffles me – and I was at pains to point out, when he was President, of the un-presidential content of some of his speeches.
Take last weekend's Unity Walk in the Central Region. Unless he has motives that have no conversation with nobility of heart, my advice to him is that he does not need to be a populist to be famous.
The GPS project is not above criticism. At any rate, the fact that a project will benefit the people does not raise it above scrutiny. In Ghana, and in Africa, government officials have been known to sit on the cushioned back of very beneficial projects to "drink soup". Didn't we allege "inflated cost" against good projects such as the Kasoa interchange, Kumasi Airport runway project, etc?
So Mahama has done no wrong in criticising some aspects of the GhanaPostGPS project. Had he not spoken, it may never have been revealed (by government) that a whopping GH¢3.5 million has been spent on publicity. Eish! Yours Truly has seen and read and heard the messages in the media emanating from the project implementation office. Great messages, but GH¢3.5 million is a dizzying sum.
Nonetheless, to describe the digital addressing system in the language Mahama chose is pandering to populist taste. The way he put it, claiming that the app was already available, free on people's phones, suggested that Ghana needn't have paid even one cedi for it. Meanwhile, it is now known, even for what the former President is referring to, that a beneficiary country has to pay Google as much as US$400,000.00!
At any rate, if it was so free, why did Mahama's government not digitise Ghana?
Talking about salty speeches, the former President says the bus branding scam by Smarttys, in his government, involved ONLY GH¢3.6m, and that "a minister of state resigned because of GH¢3.6m".
It is known now that the deal was concluded long before the ministry wrote to the Public Procurement Agency (PPA) for approval for sole-sourcing.
While at it, is it true that the money to pay Smarttys was taken out of a Railway Development Fund? Is it true that the board chairman of the bank in which the GH¢3.6 million cheque for Smarttys was drawn, was closely related to the owners of Smarttys?
Did former President Mahama know all these? And he is proud to proclaim that the minister "resigned"? In any society with conscience, such ministers don't resign: they are fired.
Source: Enimil Ashon/D. Graphic
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: jQuery formValidation not submitting form on pressing enter key I am Using Jquery formValidation plugin to validate my form. everything works fine when i click on submit button it submits the form after validation. But when i hit Enter key on keyboard it still validate form and dont submit form and form values are gone. Do i need to write code here ? i guess it's working but not listening enter key being pressed ?
$('#login_form').formValidation({
    message: 'This value is not valid',
    icon: {
        valid: 'glyphicon glyphicon-ok',
        invalid: 'glyphicon glyphicon-remove',
        validating: 'glyphicon glyphicon-refresh'
    },
    fields: {
        email: {
            validators: {
                notEmpty: {
                    message: 'The email address is required'
                },
                emailAddress: {
                    message: 'The input is not a valid email address'
                }
            }
        },
        password: {
            validators: {
                notEmpty: {
                    message: 'The password is required'
                }
            }
        }
    }
});

A: This is a problem I encountered about a year ago on version 0.6.2 so i'm not sure if they fixed it by now or not, what I did was basically disable the "enter" key from being able to submit my form, this is the code I used:
$('#form-login').on("keyup keypress", function(e) {
    var code = e.keyCode || e.which;
    if (code === 13) {
        e.preventDefault();
        return false;
    }
});

Too bad nobody replied to you for a month, I just came across this again in my own project so I saw your question :) Did you find any solution for this?

A: It's a late reply, but I faced this problem just recently.
Look at how your submit button is formatted.
<input type="submit" name="submit" id="submit" value="submit"/>

If either the attribute name or id is the word submit you may encounter this error. Just change your input type and it will probably work
<input type="submit" name="confirm" id="confirm" value="submit"/>

redpajama_set_name: RedPajamaStackExchange


text:
Rolls-Royce selects IFS to unlock new data insights on aircraft engines
IFS, the global enterprise applications company, announces that Rolls-Royce has chosen IFS Maintenix™ for exchanging engine data with airlines operating Rolls-Royce Trent engines, including the Trent 1000, Trent XWB and Trent 7000. With a long list of world-leading airlines operating with its engines, the IFS Maintenix solution will help Rolls-Royce and its customers share data. The IFS Maintenix Aviation Analytics capability enables the automated provision of field data, which ensures that Rolls-Royce receives timely and accurate information. IFS Maintenix then acts as a gateway to automatically push maintenance programme changes from Rolls-Royce back to the airline operator. As a result, life-limited engine part maintenance deadlines can be updated based on actual operating conditions and life consumed by each engine in use. Rolls-Royce helps transport thousands of air passengers and tons of cargo across the world on a daily basis, and maximising the safety, efficiency and insights gained on each trip is paramount. With the support of IFS Maintenix and its Aviation Analytics capability, Rolls-Royce is able to offer a systematic method of exchanging and accurately updating airline engine life data to optimise the interval between engines being removed and sent for overhaul. This will provide new streams of data for Rolls-Royce to analyse the performance of fleets with Trent engines and refine the aftermarket offerings it can provide its customers, from service-based contracts to analytics insights and more. Richard Goodhead, Rolls-Royce, SVP Marketing – Civil Aerospace, said: "Ease of data sharing is a key component in the Rolls-Royce IntelligentEngine vision, which sees a future in which our aero engines are increasingly connected, contextually aware, and even comprehending. Working with IFS Maintenix to ensure we receive accurate and timely engine life data will help us as we make further progress toward that objective by helping us deliver greater levels of reliability and efficiency to our operators." Scott Helmer, President, Aerospace & Defence Business Unit, IFS added: "We are thrilled to work with Rolls-Royce to deliver next-generation maintenance protocols, all based on up-to-date engine life data. This strategically important deal offers further proof that IFS Maintenix can help aviation organisations maximise the revenue potential of their assets through standard, lean, and predictable maintenance." Learn more about how IFS supports customers in the aerospace and defence industry: https://www.ifsworld.com/uk/industries/aerospace-and-defence/
TT Electronics and Thales join to enable development of cyber security for operational technology
Blue Origin successfully demonstrate crew capsule upgrades


Pet travel starts with finding a pet friendly lodging. We are currently showing 7 pet-friendly hotels in Norfolk, Virginia, which we define as dog friendly or cat friendly. Some of the hotels also accept other animals such as birds or ferrets. Our lodgings also include listings for pet-friendly Bed & Breakfasts, Cabins and Cottages, Inns and Motels if they are available in a particular location.
Captain Groovy's has a covered patio area where dogs are allowed. In the colder months, the patio is enclosed. They have a little side yard where you can walk your dog; this area is not fenced off, but they provide bags for potty pick up. All of the staff are pet friendly and will often come over to visit. They also have MUTT MIXER MONDAYS where they encourage everyone to bring their dogs for social hour during happy hour!
This 1-acre dog park features a large expanse of lawn, double-gated entrance, a water station, a toy bin, a shaded picnic area, and an easy-access ramp for dog-paddling in the Elizabeth River. In a beautiful picturesque location by the water, it is conveniently close to downtown Norfolk, and right near the Cruise lines port. What's better? It is open 24/7 and is to be enjoyed free of charge!


text:
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The 2015 #Lincoln #MKC – Is It Worth The Price?
#Volkswagen's 2015 #eGolf is Top of the Class
The 2015 #Kia #SoulEV is the Most Convincing Conventional EV since the Nissan Leaf
#Chevy's #SparkEV Short Circuits In the Details
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Posted: January 11, 2015 in Automobile Manufacturers, Lincoln
Tags: Lincoln, Lincoln MKC, MKC
Ford decided to spend billions revitalizing its 3-wheels-in-the-grave Lincoln luxury brand. It's first "premium" product was the MKZ, a fancy version of the Ford Fusion. Its next big product, introduced in 2014 as a 2015 model, was the MKC, a completely re-worked version of Ford's the very popular Escape compact crossover.
2015 Ford Escape. The MKC shares a platform and powertrains with the Escape. The interior and sheet metal is all unique to the MKC.
Compact crossovers are a very hot segment of the market. Products like the Honda CR-V, Toyota RAV4, Kia Sportage, Subaru Forester, Nissan Rogue and Ford Escape sell well over 2 million units a year in the U.S. market. The premium segment is equally hot with entries like the Acura RDX, BMW X3, Audi Q5, Porsche Macan and Mercedes ML all sucking up huge sales volumes and gigantic profits.
So when Lincoln introduced the all-new MKC, it knew it could attract some attention.
But first, let's step back and try to figure out what a Lincoln is in the 21st Century. Lincoln has been defined by its big sedans for decades. The image of a 1960s era Continental with suicide doors was indelibly seared into our collective consciousness with the November 22, 1963 assassination of President John F. Kennedy, who was riding in the back of a 1961 Lincoln Continental Convertible that fateful day in Dallas.
1961 Lincoln Continental Presidential Parade Car dubbed USSS or "SS 100 X." The SS stands for Secret Service.
Movie stars drove Lincolns. This 1956 Lincoln Continental Mark II belonged to Elizabeth Taylor.
The younger audience may remember the 1965 Lincoln Continental Convertible used in the hit HBO show Entourage.
Maybe classic movie buffs will remember the 1970 Lincoln Continental Mark III that was integral to the cop thriller The French Connection.
But like Cadillac, Lincoln seemed to lose its way over the remaining decades of the 20th Century. In the end, it was really only one sedan, the Lincoln Town Car, that was synonymous with Lincoln. In fact, when you ordered car service, the term Town Car became a generic name for a big black sedan to ferry you to and from the airport. Ford really showed little interest in updating any Lincoln. By the time the Town Car was finally retired, it had been in service for nearly 30 years with only minor cosmetic and mechanical upgrades.
A 2011 Lincoln Town Car. This is how most people remember Lincoln from the end of the 20th Century.
During The Great Recession, Ford even considered shuttering Lincoln as it had done to Mercury, which itself had become the redheaded stepchild no one wanted. A Mercury had become a rebadged Ford with only tiny changes to call it a Mercury. Lincoln was essentially the same thing, starved for product, with slashed development and marketing budgets and all but ignored by customers as its sales sank to new lows.
So it was a pleasant surprise that Ford decided to invest a few billion in Lincoln and try to revive sales with fresh product and flashy designs. But even with all the new investment, can the average consumer really quantify or identify a modern Lincoln? I don't think so. Either you're like me and wax nostalgic for a rear-drive flagship called Continental (hint: Lincoln could build one on the new 2015 Mustang rear drive platform) or you simply don't care because you've only owned foreign cars since you were born. Nevertheless, Lincoln presses on.
The winged Zephyr grille – billed as the new face of Lincoln — is more successful on the MKZ than the sad-sack wavy brace-face gaudy grille of the Navigator. But I think the most successful integration of the new face of Lincoln is on this all-new 2015 MKC. The grille flows effortlessly into the headlamp assembly. A smooth line of ice-white LED running lights seamlessly complete the wingtips of the grille.
2015 Lincoln MKZ. The 2013 model was the first application of Lincoln's new winged grille.
The winged grille was grafted onto the front of a Lincoln Navigator. Not much else was done. It don't think it's nearly as successful here.
Lincoln's signature winged grille is well integrated and the proportion seems right on the MKC.
The MKC's sheet metal is drawn taughtly over its skeleton. Deep scallops give dimension to the sides and a spear-like crease darts over the bulging fenders. Large wheels (18″ standard or optional 19″ and 20″) fill the wheel wells adding to its substantial, planted look. The rear "clam shell tailgate" is an expensive upgrade to the more pedestrian version on the Ford Escape. In addition to a wider opening, the wraparound tailgate also allows for an unbroken applique of Lincoln's signature full-width taillights in the form of a thin, artistic rendering of the front wings, except this time in red and white LEDs.
The style lines flow effortlessly across the MKC's flanks.
The rear is quite nicely rendered. The large clam-shell style tailgate is a nice tableaux for the designers.
The LED theme carries on at all touch and focal points on the MKC. Door handles light up, a thin slice of LEDs form turn signal repeaters embedded in the the side mirrors. Approach the car with the key in your pocket and LEDs beckon you forward, illuminate door handles and alerting you to the peril of puddles. Inside, the door panels, foot wells and dashboard all glow softly in tasteful warm-colored LEDs. It must be quite the light show at night, but even during the day, it was impressive.
LEDs bathe the exterior in soft accent lighting. You've gotta love the puddle light that projects the Lincoln cross on the ground.
The MKC I drove was a fully-loaded all-wheel driver with the optional 2.3L Ecoboost turbocharged inline, direct-injected 4-cylinder engine making 285 hp and 305 lb-ft of torque. The only transmission is a 6-speed automatic. While I felt it had plenty of power to motivate it's rather porky two tons, I was a bit disappointed Ford hasn't gone to an 8- or 9-speed automatic used by "competitors" Audi and BMW. At this point, a 6-speed automatic in a "luxury" or "near-luxury" car feels a bit old school. Even Jeep is using a new ZF 9-speed automatic in its Cherokee – a car similar in size to the MKC. Downshifts brought loud howling from the engine which seemed sort of out of character from a very soothing and zen-looking car.
The 2.3L EcoBoost engine combined with AWD is the least fuel-efficient trim level of the MKC.
The fuel economy is also disappointing. The EPA rates the AWD 2.3L Ecoboost at 18/21/26 mpg (city/combined/highway). The car's computer proclaimed 9.7 mpg! Of course, that is demonstration/dealer mpg, but still a bit shocking for a compact crossover. I have a feeling that you'd be lucky to get 18 mpg combined if you used this car for your daily drive in Los Angeles. But even if you opt for the entry level front-drive 2.0L Ecoboost, the EPA rates it at 20/23/29 mpg – not exactly class-leading fuel economy. By contrast, the 2015 Honda CR-V with Honda's new 2.4L direct inject (non-turbo) engine and AWD is rated at 26/28/33 mpg – far better than the MKC. The Acura RDX – the "luxury" version of the CR-V – is rated at 19/22/27 mpg – but that's with a 3.5L V6 engine! So the V6 AWD Acura RDX beats the vaunted "economy" of Ford's 2.3L Ecoboost.
Lincoln's marketing babble compares the MKC to the Audi Q5. It will be a while before Lincoln can play in Audi's sand box.
Okay, I know, this is a near-luxury crossover, so playing the fuel economy game is only one fraction of the purchase equation. To my eye, the MKC is a lovely package, far easier on the senses than the angularly-awkward Acura RDX but quite likely equally as pleasing as the svelte Teutonic lines of the Audi Q5 – Lincoln's stated Quixotic foil.
My sales associate informed me that they have had a few Audi Q5 customers trade in their Audis for the new MKC. While I don't doubt his veracity, I don't know how many Audi owners are going to cross shop with Lincoln – a struggling near-luxury brand. It's taken Audi decades to prove its luxury-performance bona fides in the US market. I can see an Audi owner considering its corporate cousin, the Porsche Macan or more likely, offerings from fellow German rivals BMW and Mercedes. And no one is cross shopping anything German for the uncompetitive (and soon to be replaced) Cadillac SRX either.
Then there are two new beauty pageant entries: The gorgeous Land Rover Discovery Sport and the very pointy, but stylish Lexus NX, based on the Toyota RAV4 — although you'd never mistake the Lexus for the Toyota.
The 2015 Lexus NX
I found the MKC comfortable with plenty of seat adjustments as well as a power tilt/telescope steering column. There is a 2-position memory for the driver's seat. The rear window is fairly narrow, so rear visibility isn't the best, but with all the electronic driving aids like blind spot monitors, a back-up camera and back up sensors, it wasn't a big deal. You sit fairly upright, perched high as you'd expect in an crossover, so forward visibility was great.
The interior is very nice with mostly soft-touch surfaces. It's far more pleasing than the Ford Escape.
There is plenty of cargo in the MKC. The rear seats split 60/40 and fold nearly flat.
The electric power steering was well weighted, if numb. I did feel some road vibration come through the wheel when I wasn't expecting it. Turning radius wasn't the best, but it's what's expected with a front drive car. It's definitely not a sporty Mustang, but for the target customer, it will be perfectly fine.
While you constantly feel the weight of the MKC, the 2.3L EcoBoost always felt capable of the task. I didn't perceive much turbo lag either, and the throttle was well-weighted and with linear acceleration – no jack-rabbit starts or sudden bursts of unexpected power. I didn't drive the smaller engine and I suspect it would feel taxed by the MKC's heft. But for the vast majority of customers, they won't notice or care. As long as it goes when they push the gas pedal and stops when they press the bakes, it will pass the test. In short, the MKC felt properly sorted, appropriately quiet and well dampened over rough roads. That's a sure winning combination of traits.
LEDs add accent and mood lighting to the interior. Very nice and easy on the eyes.
The MyLincoln Touch infotainment system is as annoyingly complex as ever. However, on the MKC, Lincoln mercifully ditched the touch-pad sliding bars for volume and tuning and grafted on honest-to-god physical knobs for volume and tuning and physical push buttons for things like climate control and seat heaters. They look good and are easy to use. However, the center console fascia looks cheap and there are several hard-surface plastic panels below eye level. Two USB ports lurk in the console between the front seats, cup holders are well placed and of adequate size and the arm rest is comfortable. There are a bunch of very handy steering wheel mounted buttons for everything from the cruise control to the phone and tuning/volume control.
The touch-screen is large. The new physical knobs and switches are a welcome return to "analog" for Lincoln.
I love the heated and cooled seats and the "no hands" operation of the tailgate is very handy. Keyless entry and start is standard on all levels of the MKC as is Lincoln's push button transmission. The push buttons are a little gimmicky, but you'd get used to them fast. There are paddle shifts for easy manual shifting when you need it. Another "wow" factor is the optional Panoramic Vista Roof with power shade. Touch one button and the cloth shade rolls back to let the sun or stars in. It really makes the interior feel much bigger and is quite impressive. I highly recommend it.
The cabin feels very airy and roomy with the optional Panoramic Vista Roof.
Unfortunately there are a few things that give me pause about the 2015 Lincoln MKC. Aside from the decidedly last-generation 6-speed automatic, the infotainment system is still the MyLincoln Touch – which, along with MyFord Touch has brought Ford's Initial Quality Survey ratings down and receives withering criticism from Consumer Reports. It's 4-quadrant touch-screen with small fonts is hard to read and use while moving. Just this month (January 2015), Ford announced at the 2015 Consumer Electronics Show that it's killing these systems and will go to a much more simplified system called Sync 3, which looks an a lot like Chrysler's far superior U-Connect system. We won't see Sync 3 until 2016 model year cars appear. Then there is the adaptive cruise control which is a generation behind the current units. It still shuts off when the car dips below a set speed. The newest systems can bring the car to a full stop and then automatically follow when traffic starts to move again. Even if you were to lease the car for 3 years, you start with out-dated, frustrating tech from day one.
Lincoln has this new exclusive option package called Lincoln Black Label. I inspected an MKC Black Label sitting on the showroom floor. It sported the exclusive Black Tie paint and the "Center Stage" theme. The interior had a lovely leather double-stitched dash, Venetian black leather seats with Alcantara inserts, contrasting red stitching, matching door inserts, a beautiful black Alcantara headliner and exotic wood accents, including in the steering wheel. You also get complimentary maintenance (like BMW), so there are no maintenance costs during the 4-year, 50,000 mile factory warranty. If you don't have the time to take your car in for service, the concierge will send a valet out to you, leave a loaner car and return your car when its finished, at your convenience. Whether or not you use the valet, a loner car is always free. Also included is "anytime" complimentary car washes and an annual detailing – a nice touch you wont find from any other car company. All these things make you feel pampered and do add value. But is it worth $6,000? On a Lincoln? I guess only time will tell, but I have to believe the take rate will be fairly low.
The interior of a 2015 Lincoln MKZ with the Black Label package. This "theme" is called Center Stage.
Details are important. Check out the upgraded switchgear and lighted kick plate for this Black Label MKC.
Finally we come to the price. The MKC 2.0 EcoBoost FWD starts at $33,995. A loaded MKC 2.3L EcoBoost AWD is an eye-popping $49,225. If you opt for the new exclusive Lincoln Black Label package, it'll add another $6,000 to the sticker shock. The loaded MKC Black Label I saw on the showroom was over $55,000. This makes for some fuzzy new math. Is it worth paying top dollar for:
Any Lincoln
A CUV based on the decidedly mass-market Ford Escape
Complex infotainment system, soon to be replaced
Last-generation safety sensors, hardware and software
Relatively poor fuel economy; no hybrid option
I've always been dyslexic, but no matter how I look at at the numbers, I don't see a compelling case for anything other than the base MKC. Lincoln is going to have to find some very "affordable" Red Carpet lease payments to make the purchase of an MKC attractive enough for most buyers.
Posted: November 14, 2014 in Automobile Manufacturers, Electrics and other "Green" Technology, Volkswagen
Tags: e-Golf, Volkswagen, VW
The 2015 Volkswagen e-Golf is an accomplished new contender for the best, relatively-affordable electric vehicle without a Tesla badge.
2015 Volkswagen e-Golf in Pure White
If I didn't know this was a compliance and engineering exercise for VW, I might think that the e-Golf could be a true mass-market EV like the Nissan Leaf.
The e-Golf looks like pretty much any other 2015 Golf, and that's a good thing. This 7th generation Golf wears an evolutionary, more mature, handsome sheet metal with sharper creases and more angular front and rear lights.
Every inch of sheet metal seems tightly stretched over the frame. Note the angular taillights that replace the more rounded ones from the Golf Mk6.
Detail of the grille and headlight. VW used LEDs for all front lighting.
This new generation Golf is built on VW's corporate multi-billion euro MQB platform that will underpin dozens of new front- and all-wheel drive vehicles spanning several brands (VW, Audi, Škoda, SEAT). From the beginning, MQB was engineered to accept all types of drivetrains, including hybrid, battery-electric and fuel cell.
The e-Golf is VW's first EV in the US market and for a freshman, it must have done some advanced placement studies to put it at the top of small, but growing list of battery-electric vehicles (BEVs). I'm told that it is targeted just below the Mercedes-Benz B-Class Electric and the BMW i3 but above offerings like the Nissan Leaf, Ford Focus Electric and Kia Soul EV.
The instrument panel of the e-Golf looks nearly the same as the standard Golf except the analog gauges show the battery charge rather than fuel level.
It's kind of a lofty goal to take on BMW and Mercedes in the BEV arena, but the e-Golf is convincing, in a very conventional way. In every way, it's a familiar Golf. It looks like a Golf, has the same interior as any other Golf and the same cargo space.
So what's it like to drive? Get in, press start and the dash lights up. Silence, of course.
The 85 kW AC motor is good for 115 hp and 199 lb-ft of torque. As you'd expect, acceleration from a standstill is brisk with all the torque instantly available. As with other Golfs, the dreaded torque steer from a front drive car has been abolished. Power delivery is smooth and linear, but you won't beat most cars in a drag race. Zero to 60 is a leisurely, but acceptable 10.4 seconds. Top speed is limited to a rather conservative 87 mph.
The electric power steering is excellent, if numb. I've come to expect it these days with everyone going to electric (rather than hydraulic) power steering. It tracked dead on-center and the electric power assist was a Goldilocks "just right" – not too heavy, not too light.
Quietness is also a virtue of the e-Golf. You don't hear any electric motor moaning and VW packed extra soundproofing materials for a very hushed cabin. The suspension easily soaked up most road roughness and the extra battery weight keeps the car planted. Road noise – which is always more noticeable in a quiet EV – was present, but distant.
Volkswagen opted for a 24.2 kWh liquid cooled lithium-ion batter pack, mostly under the seats, and rear hatch area in place of a spare tire and gas tank. While not a huge battery, it is surprisingly efficient, as the e-Golf earned a very respectable 116 MPGe. Range, however, is only rated at 83 miles, which is only slightly better than average for anything that's not a Tesla.
This transparent view of the e-Golf shows the placement of the battery along the floor, under the rear seats and in the cargo area. The weight is spread out very well making the e-Golf well-balanced with a low center of gravity.
As with other EVs, the weight of the battery on the floor, mostly in the center/rear of the car lowers the center of gravity and more evenly balances the weight between all four wheels. With a beefed up suspension and excellent tuning, the e-Golf handles better than any Golf not wearing an R badge.
Cockpit of the 2015 e-Golf. Note the electronic parking brake in the center area and the covered cup holders.
The e-Golf operates in one of three driving modes selected from the center touch screen: Normal, Eco and Eco+.
In Normal mode, the car drives in a very conventional manner – If it wasn't silent, you might think you were driving a conventional Golf. The climate control functions normally and the electric motor operates at full power for maximum driving fun.
Select Eco and the HVAC system gets dialed back (fine when it's 72 outside, but not so much when it's 95) and power is dialed back to 94 hp and torque to 162 lb-ft and the top speed drops to 72 mph. Eco mode is fine for normal daily slogs in traffic, but not spirited sprints or LA freeway driving when you can go from a standstill to 80 mph at any given time.
Select Eco+ and you lose AC all together and the motor is hobbled to 74 hp and 129 lb-ft of torque with a top speed of 56 mph. Driving the car in Eco+ mode all the time will likely stretch the range past 100 miles, but I know myself, and I'd not be happy slogging along without AC and with only barely passable power for a car weighing 3,391 pounds.
However all is not lost if you're driving in one of the Eco modes. If you mash the accelerator to floor, the car reads your mind and reverts to Normal mode for maximum power usage. This is a smart, fail-safe plan.
Regenerative braking is also an area in which the e-Golf excels. One of the geeky things EV owners love is playing around with regenerative brakes to slow or almost stop the car. You can drive the e-Golf in the B mode (just below D) which provides the heaviest amount of regen-braking. Yes, geek out in B and, with some practice, perform one-pedal driving – the car will coast itself to a full stop, with pronounced braking as soon as you lift off the go-pedal.
The other two modes are D2 and D3, accessed by tapping the shifter to the left when in Drive. Each mode provides additional regenerative braking. A clever safety trick that other EVs don't offer is that when driving in any mode other than D1/Drive, as soon as you let up on the accelerator, your brake lights go on, warning drivers behind you that you are braking even though your're not pushing the brake pedal.
Plenty of cargo space in the e-Golf, same as the regular 4-door Golf hatchback.
I drove the car in all modes and liked each one for different reasons. I did wish that VW had put paddle shifters on the steering wheel to activate regenerative braking, but that's a small ding. I'd probably end up driving it B mode all the time and pat myself on the back for barely touching the mechanical brakes – after all, I am a car geek.
All the goodness of the excellent Golf Mk7
Refined electric powertrain
Three levels of regenerative braking – from mild "normal" to near 1-pedal operation
Obligatory smartphone app to monitor
First rate interior for a non-luxury car
Precise, well-weighted steering
Made in Germany, not Mexico
Impressive fit of body panels
Slick aerodynamic aluminum alloy wheels – not cheap plastic covers over steel wheels
Two SD memory card slots in chilled glove box – but who uses SD memory cards for their music?
Automatic headlights and rain sensing wipers
Aced its crash tests with top ratings
The Not So Good:
Smallish 5.8″ touchscreen
No USB ports – Really?
Short-leash, complicated media cable to tether your smartphone
No sunroof – but it does save weight
No HomeLink
Annoying delay in rear view camera operation due to placement of cam under large rear VW badge
Antique,unnecessarily large switchblade key fob that the VW group has been using for more than a decade. Time for a change.
Some cheap hard plastics on lower fascia panels
No leather seat option – only perforated "leatherette"
DC fast charging system uses CCS (Combined Charging System) and there aren't many locations for this yet
No Driver Assistant package available as with other Golfs (forward collision warning and front/rear parking sensors)
The e-Golf comes in only one trim level – SEL Premium – for $36,245. No option packages are available (yet). VW doesn't have any lease deals, so with around $2,000 down, I was quoted a 36 month lease price of around $420/month (including tax). That's significantly more than the Soul EV and on par with a similarly-equipped Nissan Leaf. As more inventory becomes available, I think VW Credit will come up with a good lease incentive, but only time will tell.
To summarize, the 2015 e-Golf is a very satisfying, capable electric car. It's refinement and every-day utility puts it at the top of my list, along side the 2015 Kia Soul EV.
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Si intende per Sessennio Democratico o Rivoluzionario il periodo della storia della Spagna trascorso dalla rivoluzione del 1868 fino al pronunciamento del dicembre 1874 che condusse alla restaurazione borbonica.

Nell'attività politica di questi anni si avverte la partecipazione di quattro blocchi politici (unionista, progressista, democratico e repubblicano), nel cui campo di azione interviene anche il movimento operaio e la questione di Cuba, iniziata in quest'epoca.

Il processo politico dei sei anni di crisi rivoluzionaria si può dividere in tre tappe:

 Gloriosa rivoluzione e periodo costituente (1868-1871)
 Fase monarchico-costituzionale (1871-1873)
 Prima Repubblica spagnola (1873-1874).

Durante questo periodo storico ebbero altresì inizio la Guerra dei dieci anni (1868-1878) e la Terza guerra carlista (1872-1876)

La Gloriosa rivoluzione e il periodo costituente (1868-1871)

Le cause della rivoluzione
La Spagna vive una congiuntura di crisi economica e politica negli ultimi anni del regno di Isabella II. Una grave crisi economica scoppia nel 1866, finanziaria, agraria e industriale, alle quali si somma il deterioramento del sistema politico.
 La crisi finanziaria sorge quando la bassa redditività delle ferrovie provoca la bancarotta di numerose banche ed imprese (di 21 banche ne vengono chiuse 6). D'altra parte, l'indebitamento dello Stato obbligò ad aumentare la pressione fiscale.
 La crisi agraria di sussistenza. La siccità e le cattive raccolte provocano carestia e fame nella popolazione.
 La crisi industriale. Il crollo dell'industria tessile in Catalogna fu dovuto al rialzo dei prezzi del cotone, importato dagli Stati Uniti, all'epoca della Guerra civile americana. Ciò determinò un drastico calo della domanda tessile in Spagna.
 La crisi politica dovuta al deterioramento e alla crisi del sistema isabelliano, causato da dei governi in mano ai moderati, accusati di corruzione, dispotismo e immoralità, Furono incapaci di risolvere i problemi della Spagna e accettare un'alternativa al potere con i progressisti.

Inoltre influì l'impopolarità della regina Isabella II per essersi circondata a Corte di personaggi discutibili e la sconfitta nella Guerra Ispano-Sudamericana.

Dal 1866 i progressisti e la maggioranza dei moderati avevano firmato il Patto di Ostenda contro Isabella II. A loro si unirono i democratici e gli unionisti nel 1867, capeggiati da Francisco Serrano dopo la morte del generale Leopoldo O'Donnell (1809–1867).

In questa situazione iniziò la rivoluzione del 1868 detta anche la Gloriosa. Un pronunciamento militare, diretto da Topete, Prim e Serrano, al grido di "Spagna con onore", si solleva contro la regina Isabella II il 19 settembre 1868. Nelle città, le juntas rivoluzionarie, formate da democratici e progressisti, assumono il potere. L'insurrezione si estende rapidamente. Quando le scarse truppe isabelline vengono sconfitte nella battaglia di Alcolea, il governo si dimette e Isabella II abbandona la Spagna per andarsene in esilio in Francia il 29 settembre 1868.

Nel frattempo i creoli passarono dalle petizioni di autonomia ai desideri di indipendenza. I possidenti cubani, con l'appoggio degli Stati Uniti d'America, non accettarono né il regime politico che si impose in Spagna con la rivoluzione del 1868 né l'abolizione dello schiavismo. All'interno del movimento indipendentista si produsse uno scontro tra i ricchi proprietari terrieri e il resto dei cubani, partigiani della fine del regime schiavista (Guerra dei dieci anni 1868-1878).

Il governo provvisorio

I militari e i firmatari del Patto di Ostenda formarono un governo provvisorio. Si impegnarono a dissolvere la Milicia Nacional e le juntas rivoluzionarie. Nella sua composizione, Serrano (unionista) assunse la presidenza del governo del progressista Juan Prim, lasciando fuori i democratici.

La convocazione delle Corti Costituenti si fece, per prima volta, mediante elezione a suffragio maschile (i maggiorenni di anni 25). Votò il 70% del censo. La composizione politica del parlamento fu come segue: progressisti 159, democratici 20, unionisti 69, repubblicani federali 69, repubblicani unitari 2, carlisti 18, isabellini o liberali moderati 14, i quali elaborarono la Costituzione spagnola del 1869.

La novità più importante fu l'apparizione nella vita parlamentare del repubblicanesimo. L'ideologia repubblicana deriva dal liberalismo democratico, cioè la sua radice filosofica è liberale. Difende idee liberali avanzate e si differenzia dagli altri gruppi liberali nel modello di Stato. Le idee repubblicane di radice liberale sono il suffragio universale, la necessità di riforme sociali ed economiche profonde a beneficio delle classi popolari. Inoltre i poteri pubblici devono assumere il carattere di Stato protettore delle classi sociali più deboli.

Nella relazione con la Chiesa i repubblicani difendono uno Stato laico, non confessionale, nel quale le due entità sono separate. Frequentemente la non confessionalità è accompagnata dall'anticlericalismo, perché i repubblicani accusano la Chiesa di essere un ostacolo per la libertà, la modernizzazione e il progresso della società spagnola, poiché insieme all'aristocrazia possiede gran parte della terra. A partire da un modello organizzato basato sulla repubblica, esistono due tendenze.

Da una parte gli unitari, il cui concetto della Spagna è un'amministrazione unitaria o centralista, comandati da Emilio Castelar, sono qualche volta più conservatori nelle idee politiche e sociali, dall'altra i federalisti che concepiscono la Spagna come una federazione di Stati regionali storici legati da un patto, e a loro volta si dividono in moderati ed intransigenti: i primi, seguaci di Francisco Pi y Margall accettano la legalità e si oppongono all'insurrezione armata, gli altri preferiscono usare la violenza e l'insurrezione per conseguire lo Stato Federale, il cui leader fu José María de Orense. Le basi sociali repubblicane si trovavano nella borghesia liberale, le classi popolari urbane (artigiani, salariati) e parte del movimento operaio e contadino prima di essere attratto dalle idee ed organizzazioni anarchiche e socialiste.

La ricerca di un nuovo re

Il trionfo dei partiti che difendevano la monarchia come forma di governo, tale come si raccolse nella Costituzione del 1869, obbligò il nuovo governo a trovare un nuovo re per la Spagna. Intanto Serrano assunse la reggenza.

Ci furono diversi candidati, tra cui il portoghese Fernando di Coburgo, che rifiutò l'offerta, Antonio d'Orléans, duca di Montpensier, sposato con l'infanta Luisa Fernanda sorella di Isabella II. Era figlio del re francese Luigi Filippo di Orleans. Nel 1870 ucciderà in duello l'infante Enrico di Borbone (1823-1870) figlio di Francesco di Paola di Borbone e fratello dello sposo di Isabella II.

Il tedesco Leopoldo de Hohenzollern Sigmaringen, chiamato dagli spagnoli Óle, óle si me eligen, che contava sull'appoggio del cancelliere Otto von Bismarck, ebbe tuttavia il veto di Napoleone III, il quale temeva che la Francia fosse circondata da due monarchie guidate da un Hohenzollern. In mezzo a questo scontro si presentò il Telegramma di Ems che scatenò la Guerra franco-prussiana del 1870.

Il principe Alfonso non fu accettato dal generale Juan Prim per il nefasto ricordo del regno dell'ultimo Borbone, cioè di sua madre Isabella II.

La monarchia costituzionale (1871-1873)

Amedeo I di Savoia-Aosta re di Spagna

Il 16 ottobre 1870 ebbe luogo la votazione per l'elezione del nuovo Re nelle Cortes, nella sessione presieduta da Manuel Ruiz Zorrilla, che annunciò il seguente risultato:

Il numero di partecipanti era di 334, per cui l'eletto, che contava dell'appoggio di Prim, fu Amedeo di Savoia, duca di Aosta, figlio del re italiano Vittorio Emanuele II. A suo favore giocava il fatto di essere il figlio dell'artefice dell'unificazione italiana, basata sulla monarchia costituzionale. Fu accettato dalle Cortes il 30 novembre 1870 e proclamato re il 2 gennaio 1871, dopo aver giurato davanti al parlamento.

Amedeo I occupò il trono da gennaio del 1871 fino a febbraio del 1873, due anni nei quali dovette scontrarsi con gravi difficoltà dal momento della sua incoronazione. Pochi giorni prima del suo arrivo, il generale Prim venne assassinato da un anarchico, cosicché la Spagna perse un valido uomo politico in grado di mediare tra le varie forze politiche.

L'instabilità politica e i dissensi tra i partiti del governo si manifestarono nelle tre elezioni generali e i sei governi differenti che ebbero luogo in questo breve regno. Inoltre esisteva un'opposizione al regime da parte dei carlisti, che si erano sollevati con le armi nel 1872; gli alfonsini, comandati da Antonio Cánovas del Castillo, volevano che il figlio di Isabella II, Alfonso di Borbone, fosse il re. Anche i repubblicani, contrari ad ogni forma di monarchia, furono protagonisti di varie insurrezioni armate in Andalusia e in Catalogna, nelle quali si mescolavano rivendicazioni popolari come la ripartizione delle terre, l'abolizione del quinto di imposta sui consumi, manifestandosi la mancanza dell'appoggio del popolo, che non accettò il nuovo monarca, che dileggiava chiamandolo Maccarronini I.

Prima fase della Terza guerra carlista

Durante il regno di Amedeo I, la guerra con i Carlisti ebbe una recrudescenza. Il pretendente Carlo VII di Spagna, nipote di Carlo Maria Isidoro (V, nella successione carlista), mobilitò 45 000 uomini armati. Ripristinò i fori catalani, aragonesi e valenciani (16 giugno 1872), soppressi da Filippo V e creò un governo a Estella, embrione di uno Stato carlista con Municipalità e Deputazioni organizzati secondo il regime forale, fautori delle lingue locali e delle istituzioni tradizionali anteriori al Settecento. L'insurrezione ebbe inizio in Catalogna, in Navarra, nei Paesi Baschi, e in località isolate nel resto della Spagna. Le truppe carliste controllavano le zone rurali, ma non le città.

Abdicazione di Amedeo I
In questa situazione, Amedeo I abdicò e le Cortes, in una riunione straordinaria, non prevista dalla costituzione del 1869, proclamarono la Prima Repubblica l'11 febbraio 1873.

La Prima Repubblica spagnola (1873-1874)

La Repubblica federale (1873)

La repubblica nacque senza l'appoggio sociale né politico. I partiti repubblicani non avevano molti seguaci e le classi popolari iniziavano ad avvicinarsi ai movimenti operai. I poteri forti, Chiesa, esercito, latifondisti, banchieri e grandi imprenditori, erano contrari alla Repubblica e ai suoi ideali sociali.

Il primo presidente fu Estanislao Figueras e dopo la vittoria repubblicana nelle elezioni, la maggioranza federalista consegnò il potere a Francisco Pi y Margall, principale teorico del federalismo repubblicano, i cui princìpi si riflettevano nel progetto di Costituzione federale del 1873, che sarebbe stata modello alle successive di tutta l'Europa. Stabiliva la separazione tra Chiesa e Stato e un modello di Spagna a partire di una federazione di 15 (o 17 con Cuba e Porto Rico) Stati federali: Andalusia Alta, Andalusia Bassa, Aragona, Asturie, Baleari, Canarie, vecchiaCastiglia, nuova Castiglia, Catalogna, Extremadura, Galizia, Murcia, Navarra, Comunità Valenciana, Regioni basche. Più tardi si sarebbero incorporate le Filippine, le isole nel golfo della Guinea Equatoriale di Fernando Poo, Annobón, Corisco, e altri possedimenti in Africa.

La Spagna era in una situazione di perenne conflittualità sociale e politica. Le tensioni sociali si manifestarono in forma di scioperi operai e occupazioni delle terre da parte dei contadini e nel fenomeno del "cantonalismo". Inoltre i due conflitti militari ostacolavano la convivenza pacifica: l'insurrezione di Cuba nel 1868 e la terza guerra carlista dal 1872.

Il cantonalismo

La caduta di Francisco Pi y Margall, sostituito da Nicolás Salmerón nella Presidenza della repubblica, diede una svolta centralista al regime democratico. Numerose popolazioni si dichiararono repubblica o cantoni indipendenti come Comunità Valenciana, Murcia e Andalusia. Ci furono cantoni nelle città di Castellón, Valencia, Alcoy, Alicante, Torrevieja, Almansa, Cartagena, Granada, Malaga, Bailén, Andújar, Jaén, Siviglia, Cadice, Tarifa, Algeciras e Salamanca. Molti dichiararono guerra allo Stato centrale, e in qualche caso tra di loro (Granada contro Jaén). Queste insurrezioni di artigiani, bottegai e salariati furono capeggiate da repubblicani intransigenti. Furono però soffocati con durezza dall'esercito. Il Cantone di Cartagena resistette fino al 12 gennaio del 1874, grazie al carattere di fortezza militare e base navale così come all'adesione degli equipaggi delle migliori navi dell'armata.

Nicolás Salmerón si dimise quando rifiutò di firmare due condanne a morte per due rei colpevoli dell'insurrezione cantonale. Le Cortes elessero al suo posto Emilio Castelar, al quale vennero concessi i poteri straordinari con lo scopo di risolvere la grave crisi politica e militare che scuoteva la Spagna. Sospese le garanzie costituzionali e governò per decreto.

La Repubblica presidenzialista (3 gennaio – 29 dicembre 1874)

I poteri straordinari di Castelar cessarono nel gennaio del 1874. La maggioranza parlamentare, diretta da Pi y Margall, era disposta a sostituire Castelar e a ritornare ai principi federali. Tuttavia, la borghesia industriale e finanziaria affidò all'esercito l'imposizione di un regime di ordine. Il 3 gennaio, il generale Manuel Pavía, capitano generale di Madrid, fece un colpo di Stato. Al comando di un gruppo della Guardia Civil occupò il Parlamento e sciolse le Cortes. Il governo e la presidenza della Repubblica passò in mano del generale Francisco Serrano.

Il nuovo presidente si dispose a ristabilire l'ordine pubblico: sospese la Costituzione del 1869, proibì la Internazionale operaia, limitò il diritto di associazione, chiuse diversi circoli che erano luoghi di riunione politica e i giornali repubblicani. In pratica instaurò una dittatura personale.

In questa congiuntura, Antonio Cánovas del Castillo preparò la restaurazione borbonica. Ottenuta l'abdicazione di Isabella II, la corona passò al figlio Alfonso XII. Venne formulato il cosiddetto "Manifesto di Sandhurst", dal nome della famosa accademia militare inglese di Sandhurst dove il principe ereditario stava compiendo la sua istruzione militare, nel quale manifesto Alfonso XII promise un regime costituzionale per la Spagna.

Gli avvenimenti precipitarono. Il 29 dicembre 1874, a Sagunto, un pronunciamento militare del generale Martínez Campos proclamò re di Spagna Alfonso XII.

Note

Voci correlate
Storia della Spagna
Guerre carliste
Prima Repubblica spagnola
Restaurazione borbonica in Spagna

Altri progetti

Spagna liberale

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