text:
\section{Introduction}\vspace{-0cm}
The use of demand-side management~(DSM) mechanisms has recently attracted significant attention in the smart grid literature. DSM schemes offer smart grid customers an opportunity to change their demands over time so as to reduce their overall electricity costs. From the utility company's perspective, such a shifting of demand can reduce the peak hour demand on the grid~\cite{hossain2012smart}. A successful implementation of DSM schemes requires customers to actively subscribe to the offers made by the utility company. However, recent empirical studies have shown that customers remain reluctant to subscribe to DSM mechanisms, despite the efforts of utility companies~\cite{FAHEY}. Therefore, it is important to develop a new generation of DSM mechanisms that can improve the penetration of the technology and thus accelerate the deployment of the smart grid.

There has been an abundant body of work dealing with DSM~\cite{mohsenian2010autonomous, atzeni2013noncooperative, QZ00, fadlullah2014gtes, chen2014autonomous, chai2014demand}. The authors in~\cite{mohsenian2010autonomous} proposed a distributed, game-theoretic DSM system, in which each user chooses a daily schedule of household appliances and loads to optimize the energy consumption. The work in~\cite{atzeni2013noncooperative} proposed different game-theoretic approaches to optimize a day-ahead DSM mechanism by using storage units. The overall value of implementing DSM and demand response schemes was studied in~\cite{QZ00} via a Stackelberg game formulation. The work in~\cite{fadlullah2014gtes} investigated how energy consumption may be optimized through a two-step centralized model, in which a power supplier provided consumers with an energy price parameter and consumption summary vector. In~\cite{chen2014autonomous}, the authors adopted an instantaneous load billing scheme so as to shift consumers' peak hour demand and to charge them fairly for their energy consumption. In~\cite{chai2014demand}, utility companies and residential users were modeled in two levels, which reduce demand variation and peak load.

Thus, the works in~\cite{mohsenian2010autonomous, atzeni2013noncooperative, QZ00, fadlullah2014gtes, chen2014autonomous, chai2014demand} study the various economic and optimization aspects of DSM and are representative of the majority of existing works in this area. However, most of those existing works assume that customers will act rationally and subscribe to DSM as long as their objective function can be optimized~\cite{mohsenian2010autonomous, QZ00, fadlullah2014gtes, chai2014demand, atzeni2013noncooperative, chen2014autonomous}. In practice, as observed by numerous real-world experimental studies~\cite{fiegenbaum1988attitudes, barberis2001prospect, levy1997prospect}, the process of individuals can deviate significantly from the rational premise of conventional game theory. This has been corroborated by the relatively low adoption rate of DSM in deployed smart grids~\cite{FAHEY}. In this respect, prospect theory (PT), a Nobel-prize winning theory developed by Kahneman and Tversky, provides the necessary tools to better understand real-world decision making and its deviation from rational behavior~\cite{kahneman1979prospect}. In particular, PT studies have shown that, in real life, users often have subjective behavior when faced with uncertainty of outcome, such as lottery outcomes. Also, it has been shown that customers may have subjective perceptions on how others behave and how this behavior impacts their gains or losses in economic-oriented scenarios. Since the price in DSM strongly depends on aggregate demands and thus on the participation of all customers, the overall perceptions of the customers on each other's DSM decisions will impact their behavior. PT has been widely used in the social sciences~\cite{harrison2009expected}. In addition some recent works~\cite{6310922, Mandayam, okuda2009design, poortoappear} have applied PT to wireless networks. However, to the best of our knowledge, beyond our early work in~\cite{PTICASSP} that studies a two-player storage management PT game, no existing work seems to have investigated how PT considerations impact participation in DSM.

The main contribution of this paper is to propose a new framework for DSM that accounts for realistic customer participation behavior using prospect theory. In particular, the DSM problem is cast as a noncooperative game between customers. In this game, each customer can decide whether or not to participate in a DSM program, while the utility company seeks to reduce the total peak hour demand so as to maintain a desirable target load on the grid. The proposed game uses PT to explicitly incorporate the customers' \emph{subjective perceptions} on DSM decisions and their impact on potential cost savings. Here, subjective perceptions pertain to the way in which each customer evaluates its electricity payment and how that depends on other customers' actions. In this respect, each customer is seeking to minimize a cost function that reflects its one-day electricity bill, under other customers' participation and its impact on the price.

Compared to related works on DSM~\cite{mohsenian2010autonomous, atzeni2013noncooperative, QZ00, fadlullah2014gtes, chen2014autonomous, chai2014demand, saad2012game, 6787063, weaver2009game, coogan2013energy, cui2013game}, this paper brings forward novel ideas from PT in order to explicitly account for realistic customer behavior, which can differ significantly from the classical, rational path predicted by traditional game theory. To solve the game, under both PT and classical game theory, we propose a new learning algorithm that allows the customers to interact and reach an $\epsilon$-mixed Nash equilibrium. We prove the convergence of the algorithm and discuss the properties of the reached operating point. Extensive simulation results based on realistic data show that deviations from rational behavior can strongly impact the overall level of participation in DSM.

The remainder of the paper is organized as follows: Section~\ref{sec:sysmodel} presents the studied system model. In Section~\ref{sec:game}, we formulate the problem as a PT-based game. In Section~\ref{sec:algo}, we introduce the concept of fictitious play and describe our proposed algorithm. Simulation results are presented in Section~\ref{sec:sim}, while conclusions are drawn in Section~\ref{sec:conc}.\vspace{-0cm}

\section{System Model}\label{sec:sysmodel}
\subsection{Demand cost}\label{sec:dp}
Consider a smart grid consisting of a set $\mathcal{N}$ of customers in which each customer $i \in \mathcal{N}$ consumes a certain amount of energy per hour. All customers are offered the opportunity to participate in a demand side management scheme provided by the utility company. For customer $i$, we define an hourly \emph{energy demand scheduling vector} in line with existing works such as,
\begin{equation}\label{eq:x}
\boldsymbol{x}_i=[x_i^1, x_i^2,\ldots, x_i^H],
\end{equation}
where $H=24$ hours. For a certain hour $h \in \mathcal{H}=\{1,2,\ldots, H\}$, $x_i^h$ represents the energy demand of customer $i$. The total energy demand from all customers at time $h$ is thus
\begin{equation}\vspace{-0.1cm}\label{eq:actuald}
d^h=\sum_{i \in \mathcal{N}} x_i^h.
\end{equation}

At a given time $h$, we assume that the price per energy unit charged to a customer $i$ is dependent on its fraction of the total current load as follows~\cite{mohsenian2010optimal}:
\begin{equation}\label{eq:price}
c_i^h(\boldsymbol{x})=B \frac{x_i^h}{\sum_{i \in \mathcal{N}} x_i^h},
\end{equation}
where $B$ could be designed based on notions from a locational marginal pricing (LMP) scheme~\cite{powerbookLMP} in which the price function is not necessarily time-dependent. The electricity price for each customer as given by (\ref{eq:price}) allows one to allocate the price for the amount of energy that a customer consumes. For example, as the number of customers increases, the total demand will increase. Then, the electricity pricing must reflect this change in the demand. For each customer, the individual electricity pricing is dependent on its demand proportion as captured by (\ref{eq:price}). Hence, the total cost of user $i$ over $H$ hours is given by
\begin{equation}
\sum_{h=1}^H c_i^h(\boldsymbol{x}) x_i^h.
\end{equation}

Given the price and local demand, an energy market is set up in which all customers seek to minimize their costs while maintaining their energy demands at a desired level. Here, all customers will interact so as to determine their demands under DSM. These demands include the quantities of energy required and impact the price at a certain time. Instead of fixed reservation prices announced by a utility company, in our model, each user can strategically change its demand, in which its demand fraction of the total demand impacts the underlying electricity price. In this respect, the price in (\ref{eq:price}) can lead to an increased cost for all customers, including those that decide not to participate in load management. Moreover, the demand delayed by DSM will cause a varying electricity price in subsequent hours. Thus, we will develop a DSM mechanism based on load shifting.
\vspace{-0.1cm}
\subsection{Load Management}\label{sec:ls}

In this subsection, we propose a load shifting mechanism to analyze the variations of the demand over time. Inherently, load shifting allows part of the peak hour load to be moved to an off-peak hour, in which such controlled demand is delayed to the following time slot in our formulation. For the proposed model, we assume that, for each participating customer, the demand can be adjusted so as to meet a \emph{predefined/target energy demand} profile (vector) set by the utility company, given by
\begin{equation}\label{eq:gd}\vspace{-0.1cm}
\boldsymbol{G_d}=[g_d^1, g_d^2,\ldots, g_d^H].
\end{equation}
On the one hand, a utility company wants to reduce peak hour demand in order to decrease load on the grid. On the other hand, the company
wants to maintain the amount of load shifted within a reasonable range while avoiding the creation of a new peak hour. For example, if the power company wants to reduce $10\%$ of the load at a given hour $\hat{H}$, the target energy demand could be defined by

\begin{equation}\label{eq:gd2}\vspace{-0.1cm}
g_d^h=\begin{cases}
g^h, &\text{if } h \neq \hat{H},\\
\beta g^h, &\text{if } h = \hat{H},
\end{cases}
\end{equation}
where $\beta=0.9$ and $g_h \in [g^1, g^2,\ldots, g^H]$ can be the \emph{historical demand} referenced by the utility company. Here, we assume that a customer can determine when it starts to participate in DSM. In this respect, customer $i$ will choose its starting time (defined as the \emph{participating time} $a_i$) so as to minimize its cost by observing the difference between the predefined demand per user and its daily demand. In particular, we assume that a customer $i$ would not leave DSM after choosing to participate over $H=24$ hours and thus, the number of participating customers at a given time $h$ is $I^h \triangleq |\mathcal{I}^h|=|\{a_i \ge h\}|$. Here, $a_i$ is the starting hour and we discuss its impact in more detail in Section~\ref{sec:gameconcept}. In this case, the reduced demand of participating customer $i$ is given by
\begin{equation}\label{eq:redde}\vspace{-0.1cm}
r_i^h=\begin{cases}
\gamma_i(x_i^h-\frac{g_d^h-\sum_{i \in \mathcal{N}\setminus \mathcal{I}} x_i^h}{I^h})^+, &\text{if } g_d^h<l^h,\\
0, &\text{if } g_d^h\ge l^h,
\end{cases}
\end{equation}
where $(q)^+:= \max (0, q)$ and $l^h$ is the total demand that includes both $d^h$ and the amount shifted from the previous hour $h-1$, such that $l^h=d^h+\sum_{i \in \mathcal{N}} r_i^{h-1}$. $0<\gamma_i \le 1$ is a factor by which customer $i$ wants to reduce from exceeding its demand, as $\frac{g_d^h-\sum_{i \in \mathcal{N}\setminus \mathcal{I}} x_i^h}{I^h}$ is the averaged demand suggested by the utility company for all participating customers. In particular, if the demand of participating customer $i$ is less than the average demand for participating customers, i.e., $x_i^h<\frac{g_d^h-\sum_{i \in \mathcal{N}\setminus \mathcal{I}} x_i^h}{I^h}$, its reduced demand is $r_i^h=\gamma_i \cdot 0=0$.

Using (\ref{eq:redde}), if customer $i$ participates in DSM at time $h$, its demand will be
\begin{equation}\label{eq:now}\vspace{-0.1cm}
y_i^h=x_i^h-r_i^h.
\end{equation}
Then, the participating customer $i$ moves its shifted load to the following hour, and thus, its DSM demand at time $h<t<H$ is
\begin{equation}\label{eq:later1}\vspace{-0.1cm}
y_i^{t}=(x_i^{t}+r_i^{t-1})-r_i^{t},
\end{equation}
while the demand at the final hour $H$ is
\begin{equation}\label{eq:later2}
y_i^H=x_i^{H}+r_i^{H-1}.
\end{equation}

To analyze such a load management scheme, we next propose a new framework that builds on the analytical tools of classical game theory and prospect theory.

\vspace{-0cm}

\section{Game-theoretic Formulation for Demand-Side management}\label{sec:game}\vspace{-0cm}

In this section, we first formulate a noncooperative game between the customers, and then study the proposed load shifting model using expected utility theory~\cite{GT00} and prospect theory~\cite{kahneman1979prospect}, also discussing their various properties.\vspace{-0.1cm}

\subsection{Noncooperative Game Model}\label{sec:gameconcept}\vspace{-0cm}
In order to analyze the interactions between customers, we use noncooperative game theory~\cite{GT00}, as the strategy choices of the customers are interdependent. We formulate a strategic noncooperative DSM game $\Xi=(\mathcal{N},\{\mathcal{A}_i\}_{i\in\mathcal{N}},\{u_i\}_{i\in \mathcal{N}})$, where $\mathcal{N}$ is the set of players, the action $a_i \in \mathcal{A}_i := \{1, 2, \ldots, H\}$ of customer $i$ is the time (hour) at which customer $i$ would like to begin participation in DSM, and the cost function $u_i$ of customer $i$ captures its electricity payment to the company, using the price in (\ref{eq:price}). Here, we note that, although a customer can participate in load management, its total daily demand remains the same (i.e., $\sum_{h=1}^{H} x_i^t=\sum_{h=1}^{H} y_i^t$). Thus, the utility function (cost) for a player $i \in \mathcal{N}$ that chooses an action $a_i$ is given by
\begin{equation}\label{eq:utility}
\begin{split}
u_i(a_i, \boldsymbol{a}_{-i})=\sum_{h=1}^H c_i^h\biggl(\sum_{i \in \mathcal{N}} y_i^h(a_i,\boldsymbol{a}_{-i}) \biggr)\times y_i^h(a_i,\boldsymbol{a}_{-i}),
\end{split}
\end{equation}
where $\boldsymbol{a}_{-i}=[a_1, a_2, \dots, a_{i-1}, a_{i+1}, \dots, a_N]^T$ is the vector of action choices of all players other than $i$, and $y_i^h(a_i,\boldsymbol{a}_{-i})$ is the DSM demand of user $i$, compared to the initial demand $x_i^h$ in (\ref{eq:x}). For example, each customer can determine a starting hour $a_i$ and shift its load to after that hour.

The goal of each customer $i$ is to choose a strategy $a_i \in \mathcal{A}_i$ so as to minimize its cost as given in (\ref{eq:utility}). For characterizing a desirable outcome for the studied game $\Xi$, one must derive a common solution that enables one to capture the coupling between the individual optimization of each customer. We define $\boldsymbol{a}=(a_i, \boldsymbol{a}_{-i})$ as the vector of all players' strategies. For each such vector $\boldsymbol{a}$, we will have a different electricity price $c_i^h(\cdot)$ in (\ref{eq:price}). Prior to finding a solution for the proposed DSM game, we will introduce expected utility theory and prospect theory.

\vspace{-0.1cm}
\subsection{Expected Utility Theory (EUT)}\vspace{-0cm}
In a smart grid, as the customers may, over a long time period, change their DSM preferences, we are interested in studying the frequency with which they choose a certain time to begin DSM participation. Therefore, we mainly study the proposed game under \emph{mixed strategies}~\cite{GT00} so as to capture the customers long-term, probabilistic choices of a DSM start time. Let $\boldsymbol{p}= [\boldsymbol{p}_1,\ldots,\boldsymbol{p}_N]$ be the vector of all mixed strategies, where, for every customer $i\in \mathcal{N}$, we have $\boldsymbol{p}_i=[p_i(1),\ldots, p_i(H)]^T$ and $p_i(a_i)$ is the probability corresponding to the choice of a pure strategy $a_i \in \mathcal{A}_i$.

Under the conventional EUT model, the expected cost of customer $i$ is captured via the expected value over its mixed strategy. Computing each user's utility requires the vector of all players' strategies. In particular, we assume that the smart grid communication infrastructure will make this information available to users who participate in DSM. Thus, the EUT cost of a player $i$ will be given by \vspace{-0cm}
\begin{equation} \label{eq:multiplayerET}\vspace{-0cm}
\begin{split}
&U_i^{\text{EUT}}( \boldsymbol{p})=\sum_{\boldsymbol{a} \in \mathcal{A}}\bigg(\prod_{l=1}^N p_l(a_l)\bigg) u_i(a_i, \boldsymbol{a}_{-i}),
\end{split}
\end{equation}
where $\boldsymbol{a}$ is the vector of all players' strategies and $\mathcal{A}=\mathcal{A}_1 \times \mathcal{A}_2 \times \dots \times \mathcal{A}_N$. In particular, the mixed-strategy $\boldsymbol{p}(h)=[p_i(h), \boldsymbol{p}_{-i}(h)]^T$ represents the empirical frequencies with which the customers choose a certain participation time $h$. These frequencies capture how often customer $i$ participates in DSM. $\boldsymbol{p}_{-i}(h)$ is the vector of mixed strategies of all players other than $i$, corresponding to $\boldsymbol{a}_{-i}$ in (\ref{eq:utility}).

\vspace{-0.1cm}
\subsection{Prospect Theory}\label{sec:pt}\vspace{-0cm}

As previously mentioned, a player can evaluate its expected utility by using (\ref{eq:multiplayerET}), in which case the customers are assumed to act rationally and objectively under EUT. However, in real life, individuals do not truly behave rationally nor do they trust the rationality of others' behavior. Thus, in order to develop a realistic model of DSM, one must account for the fact that, in practice, customers may not assess their utilities objectively. Indeed, it has been shown that, despite the benefits of DSM, its adoption in practice has remained slow, due to unexpected customer behavior~\cite{FAHEY}.

To study such realistic/practical participation models, we will develop a DSM game model, in which a customer may have subjective views on how the opposing players will choose their strategies. This, in turn, impacts the way in which this customer evaluates its utility in (\ref{eq:utility}) which depends on other customers' strategies. Indeed, it has been observed that in real-life decision-making, people tend to subjectively weight uncertain outcomes~\cite{prelec1998probability}. Uncertainty, here, is defined as the fact that the amount of utility that a customer will obtain, depends on the decision making behavior of other customers, which is probabilistic. For example, a customer cannot be sure of the time at which other customers will begin their participation in DSM. Thus, the customer will not be sure how much economic gain its participation in DSM will yield; since such a participation depends on all the players' strategies. Thus, this customer's EUT evaluation as per (\ref{eq:multiplayerET}) might be \emph{overweighted} or \emph{underweighted} due to the subjective observation of others. Moreover, this customer may need to properly assess whether to shift its demand or not, as it is unsure of other customers' strategies. In this respect, a customer might have its own subjective perception on the participation of other customers in the DSM game (and on the actual price), thus deviating from the rational assumption of conventional game theory and EUT.

In order to analyze such subjective perceptions, we will use the behavioral framework of prospect theory~\cite{kahneman1979prospect}. In this studied model, we mainly focus on how a customer evaluates the strategies of its opponents and, thus, acts accordingly. Our focus is on the prospect-theoretic perspective, which primarily deals with how human decision makers deal with economic outcomes that have some form of uncertainty.

One important PT concept is the so-called \emph{weighting effect}. Weighting effect naturally implies that customers will have subjective views on how their opponents will act (i.e., a weighted view on the probability vector of those players), which, in turn, maps to subjective views on their expected utilities. In the proposed game, we use this weighting effect to incorporate a subjective evaluation for each user's observation of the mixed strategy of its opponents. Thus, under PT, instead of objectively observing the mixed strategy vector $\boldsymbol{p_{-i}}$ chosen by the other players, each user perceives a weighted version of it, $w_i(\boldsymbol{p_{-i}})$. $w_i(\cdot)$ is a nonlinear transformation that maps an objective probability to a subjective one. PT studies have shown that most people will often overweight low probability outcomes and underweight high probability outcomes \cite{kahneman1979prospect}. Without loss of generality, we assume that all players use a similar weighting approach, such that $w_i(\cdot)=w(\cdot),\ \forall i \in \mathcal{N}$. Although many weighting functions exist, we choose the widely used Prelec function (for a given probability $\sigma$)~\cite{prelec1998probability},
\vspace{-0cm}
\begin{equation} \label{eq:weight}\vspace{-0.1cm}
w(\sigma)=\exp(-(-\ln \sigma)^\alpha),\ 0<\alpha \le 1,
\end{equation}
where $\alpha$ is a parameter used to characterize the distortion between subjective and objective probability. Note that when $\alpha=1$, the weighting effect will coincide with the conventional EUT probability. Fig.~\ref{fig:probaweight} illustrates the probability weighting effect. In this figure, we can see that the objective probability and subjective estimation insect at $p=1/e$ and the curve approximates EUT as $\alpha$ increases. Also, weighted summations are not necessarily equal to $1$ (i.e., $w(0.4)+w(0.6)<1$).

\begin{figure}[!t]
\begin{center}
 \vspace{-0.2cm}
 \includegraphics[width=8cm]{probaweight.eps}
  \vspace{-0.5cm}
   \caption{\label{fig:probaweight} The subjective probability under the weighting effect, as the objective probability varies.}
\end{center}\vspace{-0.9cm}
\end{figure}

Under PT, the expected utility achieved by a player $i$, given the weighting effect, is \vspace{-0cm}
\begin{equation} \label{eq:multiplayerPT}\vspace{-0cm}
\begin{split}
&U_i^{\text{PT}}( \boldsymbol{p}) = \sum_{\boldsymbol{a} \in \mathcal{A}}\!\! \bigg(p_i(a_i)\!\!\!\!\!\! \prod_{l \in \mathcal{N} \setminus \{i\}}^N\!\!\!\!\!\! w(p_l(a_l))\bigg) u_i(a_i, \boldsymbol{a}_{-i}).
\end{split}
\end{equation}

Here, we assume that a player has a subjective evaluation only of the other players' strategy probabilities. In this respect, customer $i$'s subjective evaluation of its own probability is equal to its objective probability.

Having defined the cost functions under both EUT and PT, our next goal is to find a solution for the game. Given the set of probability distributions $\mathcal{P}_i$ over its set of strategies $\mathcal{A}_i$, a suitable solution of the game would be a mixed-strategy Nash equilibrium defined as follows:
\begin{definition}
A mixed strategy profile $\boldsymbol{p}^* \in \mathcal{P}=\prod_{i=1}^N \mathcal{P}_i $ is a \emph{mixed strategy Nash equilibrium} (NE) if, for each customer $i \in \{1,2,\ldots, N\}$, we have ($U_i$ represents $U_i^\text{EUT}$ and $U_i^\text{PT}$ under EUT and PT, respectively)\vspace{-0cm}
\begin{equation}
U_i(p_i^*,\boldsymbol{p}_{-i}^*) \le U_i(p_i,\boldsymbol{p}_{-i}^*), \  \forall p_i \in  \mathcal{P}_i.
\end{equation}
\end{definition}
In practice, to avoid slow convergence times and unnecessary overhead, we can consider approximate equilibrium solutions that allow the players to reach the neighborhood of an equilibrium. Hence, we mainly focus on $\epsilon$-Nash equilibria which are defined as follows:
\begin{definition}\label{def:eNE}
A mixed strategy profile $\boldsymbol{p}^* \in \mathcal{P}=\prod_{i=1}^N \mathcal{P}_i $ is an \emph{$\epsilon$-Nash equilibrium ($\epsilon$-NE)} if, for every player $i$, we have\vspace{-0cm}
\begin{equation}\label{eq:epsilonNE}
U_i(p_i^*,\boldsymbol{p}_{-i}^*) \le U_i(p_i,\boldsymbol{p}_{-i}^*)+\epsilon, \  \forall p_i \in  \mathcal{P}_i,
\end{equation}
where $\epsilon$ is a small positive number.
\end{definition}

In order to find the solution for the game $\Xi$ under both EUT and PT, we find a mixed $\epsilon$-NE in strategic form which represents a point within a close neighborhood of the exact equilibrium~\cite{GT00}.

\vspace{-0cm}

\section{Game Solution and Algorithm}\label{sec:algo}\vspace{-0cm}

In this section, we propose a novel algorithm, under both EUT and PT, to solve the studied DSM game and find an equilibrium point. The proposed algorithm builds on classical fictitious play (FP)~\cite{brown1951iterative}. Thus, we propose the following iterative algorithm to find an $\epsilon$-Nash equilibrium of the proposed game:
\begin{equation} \label{eq:algo}
\boldsymbol{p}_i^{(k+1)}=\boldsymbol{p}_i^{(k)}+\frac{\lambda}{k+1}(\boldsymbol{v}_i^{(k)}-\boldsymbol{p}_i^{(k)}),
\end{equation}
where $0<\lambda<1$ is an inertia weight, $k$ is the iteration index, and $\boldsymbol{v}_i^{(k)}=[v_i^{(k)}(a_1), v_i^{(k)}(a_2), \ldots, v_i^{(k)}(a_H)]^T$ is a vector of player $i$'s strategies such that, $v_i^{(k)}(a_l)=1$ if player $i$ chooses the $l$th strategy at iteration $k$ and $v_i^{(k)}(a_l)=0$ for updating the strategies excluding the $l$th strategy. The pure strategy i.e., the $l$th strategy, is the one that minimizes the expected utility with respect to the updated empirical frequencies. Thus, player $i$ can repeatedly choose its strategy and update $\boldsymbol{v}_i^{(k)}$ as follows:
\begin{equation}\label{eq:FPaction}\vspace{-0.1cm}
v_i^{(k)}(a_l^{(k)})\!=\!
\begin{cases}1, \text{if } a_l^{(k)}\!=\!\arg\min\limits_ {a_i \in \mathcal{A}_i}{u}_{i}(a_i, \boldsymbol{p}_{-i}^{(k-1)}),\!\!\!\!\\
0, \text{otherwise,}
\end{cases}
\end{equation}
where the utility here is the expected value obtained by player $i$ with respect to the mixed strategy of the opponent, when player $i$ chooses pure strategy $a_l$ for EUT and its weighted mixed strategy $w(p_l(a_l))$ for PT. It is well known that our algorithm (as a simplified iterative approach of smooth fictitious play (SFP)) is guaranteed to converge to a mixed $\epsilon$-Nash equilibrium~\cite{fudenberg1995consistency}, as the players' beliefs (probabilistic choices) converge to a fixed point. In particular, a player's belief is its mixed strategy set, and $\epsilon$ represents the utility difference based on beliefs. Here, we define $\epsilon_p$ as the mixed strategy difference corresponding to the utility. SFP can reach an $\epsilon$-NE under EUT, in which the belief difference $\epsilon_p$ between two iterations is small when the number of iterations $k$ goes to infinity. However, to our knowledge, such a result has not been extended to PT. Here, we first prove that the proposed algorithm in (\ref{eq:algo}) will converge to a fixed point in beliefs, as follows:

\begin{theorem}\label{th:cov1}
There exists an inertia weight $\lambda$, $0<\lambda<1$, such that the iterative algorithm in (\ref{eq:algo}) converges to a fixed point in belief, as a mixed $\epsilon_p$-equilibrium ($\epsilon_p \le \frac{\sqrt{2}}{k+1}$).
\end{theorem}
\begin{proof}
For the proposed DSM game, the proposed SFP process is guaranteed to converge to a fixed point in beliefs~\cite{fudenberg1995consistency} under EUT. Here, we mainly prove that, under PT, a player $i$'s iterative sequence $\{\boldsymbol{p}_i(k)\}$ converges to a mixed $\epsilon_p$-equilibrium in beliefs. From (\ref{eq:algo}), we have
\begin{equation}\label{eq:pik1}
\begin{split}
\boldsymbol{p}_i^{(k+1)}=&\biggl(1-\frac{\lambda}{k+1}\biggr)\boldsymbol{p}_i^{(k)}+\frac{\lambda}{k+1}\boldsymbol{v}_i^{(k)}\\
=&\biggl(1-\frac{1}{k+1}\biggr)\boldsymbol{p}_i^{(k)}+\frac{1}{k+1}\boldsymbol{v}_i^{(k)}+\frac{1-\lambda}{k+1}\boldsymbol{p}_i^{(k)}\\
&-\frac{1-\lambda}{k+1}\boldsymbol{v}_i^{(k)},
\end{split}
\end{equation}
where the first two terms represent the iteration using FP~\cite{brown1951iterative} and, we define
\begin{equation}\label{eq:epk}\vspace{-0.1cm}
\epsilon_p=\biggl|\frac{1-\lambda}{k+1}\boldsymbol{p}_i^{(k)}-\frac{1-\lambda}{k+1}\boldsymbol{v}_i^{(k)}\biggr|.
\end{equation}
In (\ref{eq:pik1}) and (\ref{eq:epk}), we present the belief difference between the proposed algorithm and FP, where there exists an $\epsilon_p$ difference at the $k$th iteration. In particular, the distance boundary between FP belief and the approached belief can be bounded as follows:
\begin{equation}\label{eq:epdis}
\begin{split}
\epsilon_p = \frac{1-\lambda}{k+1}| (\boldsymbol{p}_i^{(k)}-\boldsymbol{v}_i^{(k)})| \le  \frac{1-\lambda}{k+1}\cdot \sqrt{2},
\end{split}
\end{equation}
where $\epsilon_p$ becomes small as $k$ increases. Indeed, the Euclidean distance in (\ref{eq:epdis}) involves $\boldsymbol{v}_i^{(k)}$ in (\ref{eq:FPaction}), one of whose components is $1$, and the mixed strategy set $\boldsymbol{p}_i^{(k)}$, whose components will sum to $1$. Thus, the maximum Euclidean distance between $\boldsymbol{p}_i^{(k)}$ and $\boldsymbol{v}_i^{(k)}$ is less than $\frac{\sqrt{2}}{k+1}$ at the $k$th iteration.

Within the given boundary, (\ref{eq:pik1}) can be rewritten as
\begin{equation}\label{eq:pik2}\vspace{-0cm}
\begin{split}
\boldsymbol{p}_i^{(k+1)}\!\!=&\biggl(1-\frac{\lambda}{k+1}\biggr)\boldsymbol{p}_i^{(k)}+\frac{\lambda}{k+1}\boldsymbol{v}_i^{(k)}\\
=&\biggl(1-\frac{\lambda}{k+1}\biggr)\biggl(1-\frac{\lambda}{k}\biggr)\boldsymbol{p}_i^{(k-1)}+\frac{\lambda}{k+1}\boldsymbol{v}_i^{(k)}\\
&+\biggl(1-\frac{\lambda}{k+1}\biggr)\frac{\lambda}{k}\boldsymbol{v}_i^{(k-1)}\\
=&\cdots\\
=&\biggl(1-\frac{\lambda}{k+1}\biggr)\biggl(1-\frac{\lambda}{k}\biggr)\cdots\biggl(1-\frac{\lambda}{2}\biggr)\boldsymbol{p}_i^{(1)}\\
&+\frac{\lambda}{k+1}\boldsymbol{v}_i^{(k)}+\biggl(1-\frac{\lambda}{k+1}\biggr)\frac{\lambda}{k}\boldsymbol{v}_i^{(k-1)}+\cdots\\
&+\biggl(\prod_{j=2}^{k}\bigl(1-\frac{\lambda}{j+1}\bigr)\biggr)\frac{\lambda}{2}\boldsymbol{v}_i^{(1)}.
\end{split}
\end{equation}

The first term in (\ref{eq:pik2}) is $\biggl(\prod_{k=1}^{k=+\infty}(1-\frac{\lambda}{k+1})\biggr)\boldsymbol{p}_i^{(1)}$. Since $\prod_{k=1}^{k=+\infty}(1-\frac{\lambda}{k+1})$ is decreasing as $k$ increases and bounded by $|\boldsymbol{p}_i^{(1)}|\le 1$, the first term is convergent as $k$ goes to infinity. In fact, this term converges to $0$.

For the remaining terms in (\ref{eq:pik2}), we can define
\begin{equation}
\begin{split}
h_j=&\biggl(1-\frac{\lambda}{k+1}\biggr)\biggl(1-\frac{\lambda}{k}\biggr)\cdots\biggl(1-\frac{\lambda}{j+2}\biggr)\frac{\lambda}{j+1}\boldsymbol{v}_i^{(j)}\\
=&\biggl(\prod_{j\le k}\bigl(1-\frac{\lambda}{j+2}\bigr)\biggr)\frac{\lambda}{j+1}\boldsymbol{v}_i^{(j)}.
\end{split}
\end{equation}
Similarly, since $|h_j|$ is decreasing as $k$ increases and bounded by $|\boldsymbol{v}_i^{(j)}|=1$, $\sum_{j=1}^{k} h_j$ is convergent as $k$ goes to infinity. Thus, we conclude that, $\boldsymbol{p}_i^{(k+1)}$ will converge to a fixed point, as a mixed $\epsilon_p$-equilibrium in beliefs using (\ref{eq:algo}).
\end{proof}
\begin{remark} \label{rmk:fp}
When $\lambda=1$, the proposed algorithm in (\ref{eq:pik1}) is reduced to FP. Hence (\ref{eq:pik1}) and (\ref{eq:pik2}) can be derived as $\boldsymbol{p}_i^{(k+1)}
=\frac{1}{k+1}\boldsymbol{p}_i^{(1)}+\frac{1}{k+1}(\boldsymbol{v}_i^{(k)}+\cdots+\boldsymbol{v}_i^{(1)})$. Thus, FP ($\lambda=1$) might cycle if $\boldsymbol{v}_i^{(k)}$ repeats after some iteration, i.e., $\boldsymbol{v}_i^{(2)}=\boldsymbol{v}_i^{(4)}=\boldsymbol{v}_i^{(6)}=\cdots$, and $\boldsymbol{v}_i^{(1)}=\boldsymbol{v}_i^{(3)}=\boldsymbol{v}_i^{(5)}=\cdots$, while the proposed algorithm ($0<\lambda<1$) in (\ref{eq:algo}) converges to a fixed point.
\end{remark}

\begin{theorem}\label{th:cov2}
For the proposed DSM game, the proposed algorithm in (\ref{eq:algo}) is guaranteed to converge to a mixed $\epsilon$-NE under both EUT and PT, as its beliefs converge to a mixed $\epsilon_p$-equilibrium.
\end{theorem}

The proof of Theorem~\ref{th:cov2} is given in the appendix.

\begin{table}[!t]\vspace{-0.2cm}
  \centering
  \caption{
    \vspace*{-0.1em}Proposed Load Shifting Solution}\vspace*{-0.5em}
    \begin{tabular}{p{8cm}}
      \hline\vspace*{+0.05em}

\textbf{Phase 1 - Proposed Dynamics:}   \vspace*{.1em}\\
\hspace*{1em}Each customer $i \in \mathcal{N}$ chooses a starting strategy $\boldsymbol{p}_i^{\textrm{init}}$ as its mixed strategy of participation.\vspace*{.1em}\\
\hspace*{2em}\textbf{repeat,}\vspace*{.2em}\\
\hspace*{1em}1) Each customer $i \in \mathcal{N}$ observes others' participation $\boldsymbol{p}_{-i}$\vspace*{.1em}\\
\hspace*{1em}2) Each customer $i\! \in\! \mathcal{N}$ updates its better response strategy using\\
\hspace*{1em}the proposed algorithm in (\ref{eq:algo}):\\
\hspace*{3em}i) The utility operator communicates with the customer\vspace*{.1em}\\
\hspace*{3em}using the grid's two-way architecture communication\vspace*{.1em}\\
\hspace*{3em}(see \cite{EW03} or \cite{hossain2012smart} and references therein).\vspace*{.1em}\\
\hspace*{3em}ii) Customers' loads can be shifted via DSM as in Section~\ref{sec:ls}.\vspace*{.1em}\\
\hspace*{4em}\textbf{Load Shedding}\vspace*{.1em}\\
\hspace*{4em}a) The utility operator advertises all customers' participation\vspace*{.1em}\\
\hspace*{4em}times $a_i \in \mathcal{A}_i$ using their mixed strategies.\vspace*{.1em}\\
\hspace*{4em}b) Each customer publishes its participation $p_i^\text{PT}$ based on a \\
\hspace*{4em}mixed strategy $\boldsymbol{p}^\text{EUT}$, representing the subjective observation\\
\hspace*{4em}with underlying weight effect.\vspace*{.1em}\\
\hspace*{4em}c) After combining probabilities as in  (\ref{eq:multiplayerET}), customer $i$ will\\
\hspace*{4em}receive an expected cost under EUT, sent by the operator,\\
\hspace*{4em}and observe the current vector of strategies $\boldsymbol{p}_{-i}$ so as to\\
\hspace*{4em}assess its subjective utility in (\ref{eq:multiplayerPT}) under PT.\vspace*{.1em}\\
\hspace*{4em}d) Customer $i$ chooses its strategic response in (\ref{eq:algo}). \vspace*{.1em}\\
\hspace*{2em}\textbf{until} convergence to a mixed NE strategy vector $\boldsymbol{p}^*$.\vspace*{.2em}\\
\textbf{Phase 2 - Power Company Strategy}   \vspace*{.1em}\\
\hspace*{1em}1) The operator receives the participation information\vspace*{.1em}\\
\hspace*{1em}given the mixed strategy set as per $\boldsymbol{a}^*$.\vspace*{.1em}\\
\hspace*{1em}2) Actual load shifting is implemented under realistic participation.\vspace*{.1em}\\
   \hline
    \end{tabular}\label{tab:algo}\vspace{-0.7cm}
\end{table}

We summarize the proposed DSM solution in Table~\ref{tab:algo}. To find the solution of this game under both EUT and PT, we use the algorithm in (\ref{eq:algo}) to solve for the mixed $\epsilon$-NE. In the first phase of the algorithm, each customer will set an initial probability vector, while any non-participating customers will set its probability to $0$ throughout the whole DSM process. At the beginning, each participating customer observes others' strategies and evaluates its expected costs of every strategy. In the evaluation process, a participating customer can receive an estimate of the expected costs from the grid operator under EUT, since the grid operator objectively collects information and provides required services/information. Then, the customers will overweight or underweight the information about others' strategies based on the individual weighting effect in (\ref{eq:weight}). Moreover, each customer will subjectively estimate its expected costs of every strategy and then frequently report its participation based on its mixed strategy. The communication between customers and grid operator will continue until the expected costs satisfy the setting $\epsilon$ in utility (\ref{eq:multiplayerET}), (\ref{eq:multiplayerPT}) and (\ref{eq:epsilonNE}). Once a mixed $\epsilon$-NE is reached, the customers will frequently signal their participation decisions based on probabilities and participate in demand-side management in Phase $2$. This phase of the proposed load shifting solution is the practical market operation. Given the submitted information, the grid operator will shift/reduce local loads. The actual process of Phase $2$ is beyond the scope of this paper and will follow real-world contract negotiations that could be interesting to study in future work.

\vspace{-0.1cm}
\section{Simulation Results and Analysis}\label{sec:sim}\vspace{-0cm}
For our simulations, we use the real-world load profile in~\cite{web} which represents customers' initial demands, i.e., the data between April $29$th, 2013, and May $4$th, 2013, from Miami International Airport, since the data of local customer houses, or groups, is large, private and confidential. In all simulations, each customer can choose a starting time to participate in DSM from the time period between $18$:$00$ and $20$:$00$. Alternatively, the customer can decide not to participate; that is, $\mathcal{A}_i = \{18, 19, 20, 24\}, \forall i \in \mathcal{N}$. Also, we set $\beta=0.86$ and $\gamma=0.6$ for all customers.

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{probahistP1.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:histP1} The probability performance of all mixed strategic participations for Customer $1$ under both EUT and PT.}
\end{center}\vspace{-0.2cm}
\end{figure}

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{probahist1900.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:hist1900} The mixed strategic participations for the six customers using both EUT and PT at $19$:$00$.}
\end{center}\vspace{-0.9cm}
\end{figure}

Fig.~\ref{fig:histP1} shows, for a smart grid with six customers, all mixed strategies of a selected Customer $1$ under both EUT and PT. In this figure, we choose $\alpha=0.7$ for all customers to represent their distortion under the weighting effect in (\ref{eq:weight}). Clearly, Customer 1's PT strategy is different than its EUT strategy. Compared to the EUT results (solid lines), the PT participating strategy (dash line) at $20$:$00$ is larger, while the other PT mixed strategies are smaller. In essence, we observe that PT generally enhances (reduces) high (low) frequency EUT strategies. For instance, under EUT, Customer $1$ will have the highest participation strategy at $20$:$00$ due to load shifting as per (\ref{eq:redde}) and the varying price in (\ref{eq:price}). In particular, its largest mixed strategy is greater than the average mixed strategy, i.e., $0.25$ in the proposed four-strategy game. Second, because of the weighting effect in (13), in Fig.~\ref{fig:histP1}, we can see how PT behavior can be different from EUT. In particular, we observe that a PT customer wants to participate more than in the EUT case at times when it has a low hourly energy prices in (\ref{eq:price}), which also corresponds to a situation with high participation under EUT. In other words, for minimizing its cost, under PT, each customer tends to increase its participation at the hours with lower hourly prices in (\ref{eq:price}) and to decrease its participation at the hours with higher hourly prices. This is mainly due to the relationship between shifting the demand and the resulting price as captured by (\ref{eq:price}) and (\ref{eq:utility}). Thus, under realistic behavioral considerations, customers will accentuate the traditional behavior perceived under EUT. This will consequently impact the overall DSM performance as shown in the next figures. Accordingly, under PT, we can observe that higher EUT probabilities will become more pronounced. Thus, the largest mixed strategy using EUT, i.e., at $20$:$00$, would be overweighted via PT observation, and vice versa. This is due to the fact that each PT customer takes a risk by participating when the hourly costs are low.

Fig.~\ref{fig:hist1900} shows, using the same parameters as Fig.~\ref{fig:histP1}, the probability that each customer participates in DSM at $19$:$00$ under both EUT and PT. In this figure, we can first see that the mixed strategy of each customer using PT is different from that resulting from EUT. Under EUT, the rational mixed strategies made by some customers, such as Customers $3$-$6$, are higher than the average mixed strategy, because they have low costs at $19$:$00$. In particular, given the price in (\ref{eq:price}), a customer's lowest demand between $18$:$00$ and $20$:$00$ would cause the lowest costs and the highest participation. This implies that a rational customer wants to participate in DSM when it does not require a lot of energy (or its demand is low) in practice, since the payment is low. Under PT, the customers' realistic decisions will impact the price in (\ref{eq:price}) and change their participation levels. In Fig.~\ref{fig:hist1900}, some customers' mixed strategic components using PT are greater than those of EUT, i.e., Customers $3$-$6$, while Customers $1$ and $2$ have a lower participation probability using PT. If four PT customers, based on their loads between $18$:$00$ and $24$:$00$, decide to shift more load at $19$:$00$, their PT demands will be less than EUT. Then, compared to EUT, the PT prices of Customers $1$ and $2$ will increase because of the changing load fraction in (\ref{eq:price}). Because of the increasing prices, Customers $1$ and $2$'s costs increase and they will be less interested in participating at $19$:$00$. Indeed, there will exist some customers, such as Customers $1$ and $2$, who want to decrease their participation at $19$:$00$ under PT as shown in Fig.~\ref{fig:histP1}. Then, the overall demand increases and Customers $3$-$6$ obtain a decreasing pricing in (\ref{eq:price}) which will make these four customers more likely to participate at 19:00.

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{nonload.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:nonload} The expected nonparticipating load for the six customer game under mixed strategies using both EUT and PT over $24$ hours, when all customers have the same $\alpha=0.7$.}
\end{center}\vspace{-0.9cm}
\end{figure}

Fig.~\ref{fig:nonload} shows the expected nonparticipating load profile in the proposed DSM game as time varies. We choose the same customers as in Fig.~\ref{fig:histP1} with the distortion parameters are set to $\alpha = 0.7$ for all customers. The actual demand is the summation of initial demands in (\ref{eq:actuald}). In Fig.~\ref{fig:nonload}, the nonparticipating load is the minimum expected load that the power company must supply, while the participating load represents the load that can be partly shifted based on each individual customer's action in (\ref{eq:redde}). Here, all customers can start the game from $18$:$00$ and the expected nonparticipating loads between EUT and PT are different. On the one hand, the difference between EUT and PT during $18$:$00$ and $20$:$00$ is due to the change in the customers' decisions, as previously shown in Fig.~\ref{fig:hist1900}. On the other hand, between $21$:$00$ and $23$:$00$, the customers nonparticipating load using PT is always less than that using EUT. Indeed, if we translate the proposed game into a ``participate or not participate'' game, $21$:$00$ can be used to distinguish two such strategies, and customer behaviors before $21$:$00$ can impact their participation at a later time. However, the fact that the nonparticipation level for PT is less than that for EUT after $21$:$00$ directly relates to the choice of a distortion parameter $\alpha$. In particular, a small deviation from the rational strategy for $\alpha=0.7$ leads to an increased competition between the customers due to the fact that a weighted observation increases the costs in (\ref{eq:utility}). Such small deviation represents a case in which a customer participates in DSM but does not trust its view of the opponents' strategy or the information received from the power company. Thus, a slight deviation from the rational path causes increasing costs and customers are more apt to shift their loads under PT to decrease the impact of being non-rational, compared to EUT. As a result, after $21$:$00$, as seen in Fig.~\ref{fig:nonload}, the PT nonparticipating load will be less than that for EUT. At $24$:$00$, the power company needs to deal with the remaining load as defined in (\ref{eq:later2}).

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{nonloaddifal.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:nonloaddifal} The expected nonparticipating load for the six customer game under mixed strategies using both EUT and PT over $24$ hours, when customers have different values of $\alpha$.}
\end{center}\vspace{-0.9cm}
\end{figure}

Compared to Fig.~\ref{fig:nonload}, Fig.~\ref{fig:nonloaddifal} shows the expected nonparticipating load profile using different distortion parameters, as time varies. The distortion parameter $\alpha$ in (\ref{eq:weight}) allow us to measure how a customer perceives the actions of its opponents. A large $\alpha$ implies a small distortion, while a small $\alpha$ represents an excessively subjective perception. In particular, we choose $\alpha=[0.5\ 0.5\ 0.2\ 0.1\ 0.1\ 0.1]^T$. In this figure, we can see that, when some customers have a very irrational observation of their opponents, the PT nonparticipating load between $21$:$00$ and $23$:$00$ will be higher than EUT. This implies that, in reality, if some customers deviate significantly from their rational strategies (for example, a customer forgets to assist the power company in load shifting), the power company will not be able to shift the total load predicted by the rational, objective model. Thus, the power company can use the distortion parameter to decide on how to design its DSM scheme.

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{alpha1900.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:alpha1900} The expected nonparticipating load of all customers at $19$:$00$ as $\alpha$ varies.}
\end{center}\vspace{-0.9cm}
\end{figure}

Fig.~\ref{fig:alpha1900} shows the expected nonparticipating load at $19$:$00$, as the weighting effect parameter $\alpha$ varies. In this figure, we can see that the expected nonparticipating load under EUT is $65.7\%$ of the total load and, the nonparticipating load under PT is less than EUT when $\alpha>0.56$. This is because the majority of PT customers have more interest in participating at $19$:$00$, as shown in Fig.~\ref{fig:hist1900}. Thus, the power company can shift more load in practice, compared to EUT. Also, this figure shows that there exists a distortion threshold, such that, if $\alpha$ is greater (smaller) than the threshold, PT customers will have lower (higher) nonparticipating loads than EUT cases. A large distortion parameter, or a small deviation from EUT, yields an increased competition thus raising the costs to the customers. Consequently, the customers will become risk seeking and more apt to shift their loads and decrease their costs. Thus, the increasing PT costs will force the majority to shift more loads, compared to EUT. However, a small distortion parameter, or a large deviation from EUT, will lead to highly distorting behavior from the customers which will lead to increasingly high competition and decreasing participation, as customers become extremely risk averse and unwilling to participate in the DSM process. In a nutshell, Fig.~\ref{fig:alpha1900} shows that a small deviation from EUT may be beneficial for the power company as it increases customers' participation. In contrast, a significant deviation from EUT will inevitably lead to highly risk averse behavior which will prevent most customers from participating; thus yielding detrimental results for the grid and preventing the operator from reaping the benefits of DSM.

\begin{figure}[!t]
 \begin{center}
 \vspace{-0.2cm}
  \includegraphics[width=8cm]{iter.eps}
 \vspace{-0.5cm}
   \caption{\label{fig:iter} The probability performance of mixed strategy for six customers based on EUT and PT.}
\end{center}\vspace{-0.2cm}
\end{figure}

In Fig.~\ref{fig:iter}, we show all PT mixed strategies of Customer $1$ (corresponding to Fig.~\ref{fig:histP1}), as the number of iterations increases. The proposed algorithm can be shown to converge \emph{sublinearly} by studying its rate of convergence which is defined as $R=\lim\limits_{k \rightarrow \infty}\frac{|\boldsymbol{p}^{(k+1)}-\boldsymbol{p}^*|}{|\boldsymbol{p}^{(k)}-\boldsymbol{p}^*|}$. Here, we mainly test the convergence of the proposed algorithm in (\ref{eq:algo}). In particular, Customer $1$'s initial probability is $\boldsymbol{p}_1^{(0)}=[0.2500\ 0.3333\ 0.2083\ 0.2083]^T$ and it converges to $\boldsymbol{p}_1^{*}=[0.1110\ 0.1480\  0.6484\ 0.0925]^T$. From Fig.~\ref{fig:histP1}, due to the low hourly costs, Customer $1$ will increase its participation at $20$:$00$, corresponding to the highest EUT frequency. In Fig.~\ref{fig:iter}, since the initial probability at $20$:$00$ is small, i.e., $p_1^{(0)}(a_3)=0.2083$, Customer $1$ will choose its best strategy and update $v_1(a_3)=1$ in (\ref{eq:FPaction}). Thus, we can see that the participating probability at $20$:$00$ increases, as the iteration $k$ increases. Similarly, because of the large initial probabilities at other strategies, Customer $1$ will decrease underlying participation via updating $v_1(a_1)=v_1(a_2)=v_1(a_4)=0$ in (\ref{eq:FPaction}). Also, Table~\ref{tab:multi} shows Customer $1$'s all mixed $\epsilon$-Nash equilibria under both EUT and PT. Using different initial probability vectors $\boldsymbol{p}^{(0)}$, the proposed algorithm in (\ref{eq:algo}) will reach different Nash equilibria, and the differences between EUT and PT at two different Nash equilibria are not the same.

\begin{table}[!t]\vspace{-0cm}
\scriptsize
  \centering
  \caption{
    \vspace*{-0em} Customer $1$'s All Mixed $\epsilon$-NE under both EUT and PT}\vspace*{-0.3cm}
\begin{tabular}{|c|c|c|}
\hline
 & EUT & PT  \\ [0.5ex]\hline
$1$ & $[0.1667\ 0.2222\ 0.4722\ 0.1389]^T$ & $[0.1110\ 0.1480\ 0.6484\ 0.0925]^T$ \\\hline
$2$ & $[0.2667\ 0.1333\ 0.5333\ 0.0667]^T$ & $[0.1776\ 0.0888\ 0.6891\ 0.0444]^T$ \\\hline
$3$ & $[0.0392\ 0.1569\ 0.4509\ 0.3530]^T$ & $[0.0261\ 0.1045\ 0.6343\ 0.2351]^T$ \\\hline
$4$ & $[0.1159\ 0.2609\ 0.5072\ 0.1159]^T$ & $[0.0772\ 0.1738\ 0.6718\ 0.0772]^T$\\\hline
$5$ & $[0.2222\ 0.0889\ 0.4666\ 0.2222]^T$ & $[0.1480\ 0.0592\ 0.6447\ 0.1480]^T$\\\hline
\end{tabular}
\label{tab:multi}\vspace{-0.7cm}
\end{table}

\vspace{-0.1cm}

\section{Conclusions}\label{sec:conc}\vspace{-0cm}
In this paper, we have introduced a novel approach for
studying the problem of demand-side management. It is of interest to factor in explicitly the behavior of customers in DSM. In particular, we have developed a game-theoretic approach, based on prospect theory, using which each player subjectively observes other players' actions and determines its own actions so as to minimize a cost function that captures the electricity cost over $24$ hours. Then, we have proposed an algorithm and have proved that it reaches a mixed $\epsilon$-NE. Simulation results have shown that deviations from classical, objective game-theoretic DSM mechanisms can lead to unexpected results and loads on the grid, depending on the level of the customers' subjective perceptions of each others' actions. Therefore, whether a customer participates in DSM or not, depends on this level of rationality of the customer. In a nutshell, the results of this paper have provided important insights into the factors underlying the modest participation in DSM schemes observed in real-world smart grid systems.

\vspace{-0.1cm}

redpajama_set_name: RedPajamaArXiv


text:
'Marriage For Us Can't Be A Prison': Jada Pinkett-Smith Wasn't The Only One In An 'Entanglement'
Posted on September 28, 2021	 - By Tanay Hudson
Source: Albert L. Ortega / Getty
Jada Pinkett-Smith wasn't the only one that had an entanglement. Her hubby Will Smith said he has also had fun outside of their marriage, a habit that has been incorporated into their marriage.
In an interview with GQ, Smith opened up about their non-traditional marriage, saying Pinkett-Smith was the one who didn't agree with following the "rules."
Jada never believed in conventional marriage…Jada had family members that had an unconventional relationship. So she grew up in a way that was very different than how I grew up. There were significant endless discussions about, what is relational perfection? What is the perfect way to interact as a couple? And for the large part of our relationship, monogamy was what we chose, not thinking of monogamy as the only relational perfection. We have given each other trust and freedom, with the belief that everybody has to find their own way. And marriage for us can't be a prison. And I don't suggest our road for anybody. I don't suggest this road for anybody. But the experiences that the freedoms that we've given one another and the unconditional support, to me, is the highest definition of love."
Jada Pinkett Smith Embraces Baldness And Opens Up About Alopecia: 'This Is Going To Be A Little More Difficult To Hide'
Saweetie Is The Latest Baddie To Pop Out With A Baldie
Smith made it clear that Pinkett-Smith wasn't the only one enjoying their open marriage and that he has had his fun, too. He even had women he dreamed of pursuing, including Halle Berry and dancer Misty Copeland. He said he worked through these fantasies of having multiple women with his intimacy coach Michaela Boehm and found that it wasn't what he wanted after all.
I don't know where I saw it or some shit as a teenager, but the idea of traveling with 20 women that I loved and took care of and all of that, it seemed like a really great idea. And then, after we played it out a little bit, I was like, 'That would be horrific. That would be horrific.' I was like, 'Can you imagine how miserable?' What she was doing was essentially cleaning out my mind, letting it know it was okay to be me and be who I was. It was okay to think Halle is fine. It doesn't make me a bad person that I'm married and I think Halle is beautiful. Whereas in my mind, in my Christian upbringing, even my thoughts were sins. That was really the process that Michaela worked me through to let me realize that my thoughts were not sins and even acting on an impure thought didn't make me a piece of s***."
The I Am Legend star also addressed when he and Pinkett Smith went on Red Table Talk to discuss August Alsina's comments about he and Pinkett-Smith's love affair. He said they went public because it was crucial for them to tell their side of the story.
"The pursuit of truth is the only way to be happy in this lifetime," Smith said. "And we sort of came to the agreement that authenticity was the release from the shackles of fame and public scrutiny."
A viral meme of Smith was a result of that interview where Smith looked sad and defeated. He said in reality that wasn't the case at all.
"It was midnight and we were going on vacation the next day," he said. "It was like, no, no, no, guys, I'm not sad. I'm fucking exhausted."
Smith also has a memoir coming out where gets even more transparent about his marriage on November 9 –titled Will.

redpajama_set_name: RedPajamaCommonCrawl


rubrics for judging electronic portfolios?
A colleague of mine asked this question of me in regards to using Mahara as part of the presentation of an adjudicated (pass/fail) e-portfolio (mahara) at the Masters level in a school of public policy.
"Are there rubrics for competency-based portfolio self-assessment and faculty validation-assessment that might be adapted for our purposes (rigor and relevance)?"
I think he wants to know if there is a rubric for judging the structure perhaps completeness of a portfolio in terms of meeting competencies outlined, not perhaps the content. i.e. what makes a good ePort. Of course I would think that is entirely dependent on the outcomes decided upon.
Or perhaps he is asking about the process of assessment using the ePort.
Re: rubrics for judging electronic portfolios?
VALUE = "Valid Assessment of Learning in Undergraduate Education"
I will take a look and also pass this on.
There is no also a rubric plugin available (on Github, screenshot) that you might want to take a look at in order to set up a rubric. I asked the plugin developer if he has some instructions on how to install it and will share them once I have them.
If you want to keep any assessment data within your LMS, it might be best to use the LMS tools for that. One trick that people use who cannot use the Mahara assignment submission plugin because they either don't have Moodle or because they can't / want to install a plugin, is to create a secret URL of the portfolio and then paste that into a regular LMS assignment. And then you can apply any of your grading methods and the lecturer accesses the portfolio via the link.
Another possibility, esp. if you don't want to deal with secret URLs is that that students share their page with the faculty and then put the link into the assignment like above. That way, they can be ensured that only their faculty have access to their portfolio.
The Value rubrics are excellent but intended more as a feedback tool for students than for grading. However, they adapt easily to a graded mark. For each row, I gave column 1 a potential 4 marks, column 2 a potential 3 marks, column 3 a potential 2 marks and column 4 a potential one. If you give this to markers as an Excel sheet, you can have a pre-set cumulative tally for each row as well as a final tally, with a conversion to a % beside it. Easy.
and for my teacher candidates as well.


This training manual provides information about the component function, component location, system installation, and general operation of the FG3000B Medium Intensity Dual Lighting System, part number FG3000B manufactured by Hughey & Phillips, 240 W. Twain Ave., Urbana, OH 43078. The lighting systems described in this manual are Federal Aviation Administration (FAA) type L-864/865.
The following general safety precautions must be observed during all phases of operation, service, and repair of this equipment. Failure to comply with these precautions or with specific warnings elsewhere in this manual violates safety standards of design, manufacture, and intended use of this equipment. Hughey & Phillips assumes no liability for the customer's failure to comply with these requirements, as listed below.
Ensure that only fuses with the required rated current and of the specified type (normal blow, time delay, etc.) are used for replacement. The use of repaired fuses and the short-circuiting of fuse holders must be avoided.
WARNING – The WARNING sign in this manual denotes a hazard. The WARNING calls attention to a procedure or practice which, if not correctly performed or adhered to, could result in injury or loss of life. Do not proceed beyond a WARNING sign until the indicated conditions are fully understood and met.
Safety interlock switches are provided in the Flashhead enclosure to interrupt main AC power to the power supply. These interlock switches are activated when the Flashhead door is opened in a conventional manner. No interlock is provided when other means of access are used. Never tamper with (remove, short circuit) the interlocks in any way.
AC LINE VOLTAGE IS STILL PRESENT WHEN INTERLOCKS ARE ACTIVATED. DISCONNECT POWER AT THE MAIN AC CIRCUIT BREAKERS BEFORE INSPECTING OR SERVICING, UNLESS ABSOLUTELY NECESSARY TO PERFORM MAINTENANCE WITH POWER ON.
DO NOT LOOK DIRECTLY AT THE FLASHHEAD WHILE IT IS IN OPERATION.
The Hughey & Phillips Technical Support department can be reached at (800) 448-8099 or (877) 285-4466. Normal hours are 8:00 am to 5:00 pm Eastern. If you have any questions or are unsure about installation or troubleshooting methods, please call our Technical Support service department. Incorrect wiring can cause permanent damage to the system.
Diagnostic Test: Remove all three circuit boards and check for damage. Remove the photocell wiring from TB1-1 and TB1-2. Perform Flashhead Isolation Test (Section 5) and check for improper resistances. Leave the Flashhead cable disconnected, replace the fuse and apply power. Replace the strobe cable, the hotocell wiring, and the circuit boards one by one to determine which one will blow the fuse.
Diagnostic Test: Remove the flashhead wires TB2-5 and TB2-6 (gray and white), and measure resistance between them – it should be less than 5W.
Diagnostic Test: When the interlock switches are engaged, the K1 relay should energize. If not, measure for 120 VAC across the relay coil. Alternatively, remove the connectors and check resistance across the K1 coil – it should be 300W.
Corrective Action Replace the High Voltage Board.
Diagnostic Test: On TB2 swap the black and blue wires from the flashhead cable so that black is on TB2-4 and Blue is on TB2-3.
Diagnostic Test: Remove the two mounting screws on the control panel. Disconnect DB1, and measure resistance across each diode.
Set meter to Resistance/Diode feature. Place Black lead to line side, red lead to black side. Meter should read ~0.46V. Replace DB1 if any diodes have shorted. Note: The diode board DB1 is located under the control panel where the test switches are.
Possible Cause: Day capacitor/s or 20W resistors faulty, or connections loose/open.
Diagnostic Test: With all conductors attached, check resistance across the terminals of any C2 capacitor. Resistance should be @ 40W.
If resistance is near 0W, one or more C2 capacitors is shorted. Remove conductors to each capacitor, and individually check / replace each capacitor.
Test Procedure: Set meter to Resistance/Diode feature. Place Black lead to line side, red lead to black side. Meter should read ~0.46V. Replace DB1 if any diodes have shorted. Note: The diode board DB1 is located under the Test Switch panel.
Diagnostic Test: Set the day mode switch to Remote, night mode to Test. The K2 relay should energize. If not, measure for 120 VAC across the relay coil. Alternatively, remove the connectors and check resistance across the K2 coil – it should be @ 300W.
Possible Cause: K3 relay near the Motherboard is defective.
Diagnostic Test: Measure K3 coil resistance on motherboard between E15 – E16. Should read approximately 500-600W.
Diagnostic Test: Ensure the DIP switches on the sync monitor board are set correctly per the Operation and Installation Guide. Visually verify that the red LED on the sync monitor board ispulsing at 40 fpm.
Diagnostic Test: Put the system in night mode. Visually check to see that K2 energizes. Check for 120 VAC across the coil of the K2 relay. Alternatively, remove the connectors and check resistance across the K2 coil – it should be @ 300W.
Diagnostic Test: Inspect wiring between power supplies. TB1 position 3 should be daisy-chained between each power supply, per the installation wiring diagram, Figure 4.
Diagnostic Test: Inspect wiring between power supplies. TB1 position 1 should be daisy-chained between each power supply, per the installation wiring diagram for multiple units, Figure 4.
Diagnostic Test: Verify that the relay changes state during transition from Day to Night. Check for 120VAC at K4-A to ground in night mode.
Corrective Action: Replace K4 relay or correct loss of 120VAC to relay.
If the resistance between #5 (gray) and #6 (white) is greater than 5W, or is 'open', suspect that the flashhead interlock switch is not depressed.
For other inconsistencies with the above chart, the probable causes are miswiring, or conductors shorted and/or opened. If possible, disconnect the flashhead cable at both ends and perform a 'Megger' test with a Megohm Meter.
*Note 1: DO NOT MEASURE WITH METER. MAY CAUSE DAMAGE TO EQUIPMENT.
Install a neon lamp (Hughey & Phillips Part # 77-2342) across the trigger output. For the FG3000B, install the neon lamp in TB2-2 and TB2-4 if checking for Day triggers. Install neon lamp in TB2-2 and TB3 if checking for Night triggers.
The neon lamp should be flashing per mode of power supply. With the power supply in day mode, you should see one quick flash. When the system is put into night mode, you should see a long 'burst' of flashes for a few seconds. Then the lamp should begin flashing short 'bursts' of flashes. This happens because the power supply can tell that the red flashtubes are not operating, so it tries to go into backup white night mode.
Located on the control panel (top center) near the test switches. This indicator is illuminated whenever input power is present and the power supply interlock switch is engaged.
The following chart depicts the alarm contact points available for the FG3000B.
Photocell should be cleaned to ensure proper ambient light is seen by the detector.


As the government considers modifying the Law on Personal Income Tax, experts say that it should institute an emergency income tax exemption in the interim.
Taxpayers complain that the law is outdated and does not take recent rises in inflation into account. Meanwhile, the taxable income threshold and dependent deductions remain far too low.
Inflation has continued to soar during the last several years. At the beginning of March, petrol prices jumped more than 17 percent and power prices rose more than 15 percent on February 24.
Despite significant rises in the cost of living, the taxable income threshold has been left unchanged at VND4 million (US$192) per month.
The VnExpress news website on Thursday quoted an "authorized" source from the Finance Ministry as saying the General Department of Taxation would study revisions in the existing law.
Lawmakers approved Vietnam's Personal Income Tax law in late 2007. It took effect in January 2009 and allowed taxpayers to deduct VND1.6 million per dependant. Since then the limits on dependent earnings and deductions have remained the same.
Tran Xoa, director of HCMC-based Minh Dang Quang Law Company, said that the taxable income threshold should be raised soon.
He also said that the VND500,000 monthly salary cap for dependents should be raised to VND1.5 million.
Vietnam adjusts its minimum wage every year. Currently, minimum pay ranges from VND830,000 to VND1.55 million, depending on the location of the employee.
Economist Pham Chi Lan told the Tuoi Tre newspaper that the taxable income threshold should be calibrated at ten times the minimum monthly wage.
Dinh Xuan Thao, director of the National Assembly's Institute for Research on Legislation, said he appreciates the suggestion that a new taxable income threshold should be calculated against the minimum wage so that tax levels remain a fair and accurate reflection of national earnings.
While economists have urged the government to amend the Personal Income Tax Law as soon as possible, no action has been taken to date.
Some experts say that the government should implement some sort of emergency action by either implementing a temporary tax exemption for everyone or halving income taxes across the board.
Despite fears that the exemption would deal a blow to the state budget, 2009 proved a banner year for government revenues thanks to an increase in payments of other taxes.
The National Assembly approved the tax exemption policy for the first half of 2009, saying the move was needed in order to stimulate economic growth during the global downturn.
Dr. Nguyen Van Thuan, dean of the Faculty of Accounting, Finance and Banking at Ho Chi Minh City Open University, said that if the government is still worried that another such temporary tax break would affect revenues for the state budget, it could opt to reduce taxes by 50 percent.


14 October 2016. In a series of improvisations for ITINERANT INTERLUDE #1.111.007, accordion virtuoso Silke Lange offered a musical reflection of Malene Landgreen's four dimensional desires. Using pre-conceived intervallic and melodic patterns, at times, short sequences of only a few notes as starting material, Lange elaborates upon these simple elements, quietly developing them into longer and more intricately organized sequences. The contrast of Landgreen's soft brush strokes with more sharply defined geometries, her warm and vibrant colors, find a musical counterpoint in Lange's alternations of the quiet and static, active and dynamic, clear rhythmic pulses and the suspension thereof. Lange's talent in exploiting the accordion's ability to achieve almost transparent variations in sound color through an interplay of registers, reeds, valves and bellows will find a plastic reflection in the surrounding space, delicately coloring each musical phrase. And, as Landgreen's works ultimately assume final form in relationship to the architectonic space, so too, the subtle modulations of timbre and register very specific to this instrument, and in the hands of this musician, play on perceptions of sound in space, as they react and adapt to the atmosphere of the moment.
A much sought-after accordionist, interpreter of contemporary music and improviser, Silke Lange has been a frequent guest at international festivals of new music such as the Darmstadt Ferienkursen für Neue Musik, Ultraschall and MaerzMusik in Berlin, as well as the Mosel Musikfestival, Festival Mitte Europa and the Young Euro Classic. She regularly appears in theater and performance productions at HAU, Deutsches Theater, Staatsoper Berlin, »club real«, Schauplatz International, Staatsoper Hamburg, Staatstheater Mainz, and Duo Randgruppe. In 2010, together with saxophonist Ruth Veltin, Lange founded the contemporary music ensemble LUX:NM. The group has recorded widely for radio, and CDs have been released by Wergo, DUX und GENUIN classics.


LONDON – Supermodel Claudia Schiffer (search) gave birth to a baby girl in London Thursday.
"I would like to thank my beautiful wife Claudia for giving me an equally beautiful daughter, a little sister for Caspar," said film director Matthew Vaughn (search), who married Schiffer two years ago.
The baby weighed 7 pounds 2 ounces (3.24 kilograms) and mother and daughter were reported to be doing well. The couple did not announce her name.
The newborn, delivered at London's Portland Hospital, is the second child for Schiffer and Vaughn, who married two years ago. They have a 22-month-old son, Caspar.


NOW WHEREVER YOU GO, You can have all the clean, safe, great tasting water you need, from any water source. Slim profile design to maximize refrigerator space. Great for water and fruit juices. Feel the freshness of Fountain water at your home.
* Please wash before use.
* Only for reheatable, not for cooking in direct flames.
* Please be careful when using immediately out of freezer.
* Please be careful as food inside can be split out when opening the lid.


If you want to browse in Japanese, please click here .
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The Tarot Hermit is another card that has remained fairly similar through the centuries. In Waite Smith version from 1910, we see the same robed old man, lifting a lantern on high that we see in the medieval Marseille Tarot. The hermit holds a staff in his other hand. The main differences are that in the Waite Smith version, the hermit is hooded and his robes are gray, rather than brightly coloured. An interesting detail is that we can see the foot of the Hermit in Waite Smith Tarot, whereas in the medieval version, the robe covers the feet.
2. The two most prominent symbols in the traditional Tarot Hermit card are the staff and the lantern. The staff is a symbol of authority and handy tool on any pilgrimage. To those of you versed in Scripture, the passage about Moses striking a rock with his staff to yield water for his people may spring to mind. The staff has been associated with powerful holy men since the days of the Old Testament. The lantern is a symbol of wisdom and in the RWS version, the light emanates from a six-pointed star, the Star of David, symbol of the union between man and God.
3. The astrological correspondence according to the Golden Dawn is Virgo, Mutable Earth. The Virgo correspondence expresses the energy of service, purity, work and health. It also gives us the anatomical correspondence in health readings of all things related to the gut, intestines and digestion. If we think about the 'as below, so above' correlation between the health associations for this card, for the psyche, we get rumination and assimilation of ideas so that we may grow in understanding.
4. The highest spiritual potential of the Hermit Tarot card is to purify our thoughts and actions so that we can be of service to humanity for the Highest Good.
6. The Tarot Hermit card is ruled by Archangel Raphael, ruler of Mercury/Virgo. You may also wish to consider Archangel Cassiel, ruler of Saturn, if you are using a Tarot deck that is showing more of a 'Father Time' approach to the archetype.
Like the mysterious Hermit himself might choose to do, I left out the detail about the foot. What might this symbol tell you? Why is only one foot showing? How is The Hermit keeping his balance? Answer in the comments below, please and don't forget to share if you enjoyed this post!


text:
Donkeys in Covent Garden
Look at an old picture of Covent Garden Market and the chances are you'll see a donkey. A hundred years ago the Piazza was seething with donkeys. Before cars and lorries appeared on the scene they were, together with ponies and horses, the main form of transport. Because of their small size and wiry strength donkeys were useful for squeezing down London's narrow alleyways pulling carts brimming with fruit and vegetables. In prints and paintings, back down the centuries, you often see them standing patiently between the shafts of a wooden cart, their long hairy ears swivelled back, not listening, in the way that elderly people switch their hearing aids off. Meanwhile the activities of the Market - men and women unloading wagons, piling baskets, shouting their wares - swirls around them.
In 1861 Henry Mayhew, describing London's costermongers, wrote that "on a Saturday, the coster's business day - it is computed that as many as 2,000 donkey barrows... visit Covent Garden during the forenoon". The costermongers, who had originally specialised in selling apples (specifically the costard apple which grew in the orchards surrounding London) later broadened their trade to include a great variety of fruit, flowers and fish.
Covent Garden Market was a noisy, bustling, earthy place in the mid 19th century. Here is a contemporary snapshot view of the Piazza: "Against the railings of St Paul's Church are hung baskets and slippers for sale and near the public house is a party of countrymen preparing their bunches of pretty coloured grass - brown and glittering as if it had been bronzed... Under the Piazza the costers purchase their flowers (in pots) which they exchange in the streets for old clothes... Men and women, selling different articles, walk about under the cover of the colonnade. One has seed cake, another small-tooth and other combs, others old caps, or pig's feet, and one hawker of knives, razors and short hatchets may occasionally be seen driving a bargain with a countryman... As you walk away from this busy scene, you meet in every street barrows and costers hurrying home".
The costermongers' home life was inextricably linked to their donkeys. Mayhew wrote that the donkeys were often looked after like pets, treated with kindness. They had their share of the costermonger's dinner: "when bread forms a portion of it, or pudding, or anything suited to the palate of the brute. Those well-used, manifest fondness for their masters are easily manageable".
The costermongers told Mayhew that it was difficult to force a donkey to leave its stable for a second trip during a day unless it had been fed and slept in the interval. A donkey was fed a peck of chaff, a quart of oats and a quart of beans and sometimes a bundle of hay. The costermongers claimed that the donkeys lived well when they themselves made a decent profit.
The reason for the extraordinary popularity of donkeys as a beast of burden lies in their adaptability. Their native habitat is open scrubby terrain, the fringe lands of deserts. Their weird and startlingly loud ee-awing calls can be heard for two miles in such places. They can survive on a sparser diet than ponies and horses. They also live for longer - 40 years is not unusual.
W J Gordon writing in The Horse World of London 1893 describes the costermongers as beginning business selling goods from a basket, "from that he advances to a hand-truck; and from that, when he has amassed sufficient capital, he rises to the dignity of the donkey-cart, which first made an appearance amongst us in the days of Elizabeth, when donkeys first became common in these islands".
Gordon's happy description of the costermongers' donkeys has them "pattering along with a briskness and assurance that can only come of contentment with their work and some of the smallest even are as active and 'packed with power' as one could wish, and with a quiet, fearless outlook, speaking volumes for their master".
Many of the donkeys who pulled carts into Covent Garden Market lived a nice, cosy life in Stepney. In the Guardian newspaper, Dorothy Hartley recalled how, at the turn of the century, in the East End "there were rows of country-type cottages with shut-in gardens at their backs and the costers' donkeys lived there behind the cottages and you could watch them in the morning coming out through the front doors; picking their way, neat-footed, over the babies playing in the narrow passage. Every coster owned a donkey, as one of the family, and those I knew were well cared for. The costers' barrows and carts could be piled up in yards but when released, each donkey trotted off to his home, and if the door was ajar, he'd shove it open and walk in. They were very sociable and reliable donkeys".
The huge increase in traffic - which in 1974 led to the fruit and vegetable market leaving the Piazza for Vauxhall - had earlier turned Covent Garden into a donkey-free zone.
Before the donkeys' final departure they showed their disapproval - charmingly stubborn to the end. An onlooker describes what happened when, overnight, the traffic around Piccadilly Circus became regulated. "The police arrived at 5am - for early Covent Garden Market traffic, and the lorries and heavy dray carts jibbed a bit and a few late night revellers' cabs resented it, but the costers' donkeys just rebelled: they lay down and flatly refused to go the wrong way round. They had trotted up to the Market one way for generations. They knew which was the right way - the way they had always gone".
Leana Pooley, Covent Garden Area Trust


text:
Q: terminal freezes after running meteor I am trying to install meteor for the first time on Mac os x 10.7.5. After entering
cd ~/directory/to/project
meteor

and receiving
Running on: http://localhost:3000/

in the terminal, it freezes. The button on the default page is dead, and I can't use the terminal window any more. I can't seem to find anything in the Meteor documentation about reinstalling or troubleshooting this particular problem. Has anyone else had this problem and know how to fix it?

A: This behavior is as expected.  Meteor is running, and serving your page.
Check out the javascript console on your browser. (on Chrome in OSX you get there by typing command-option-j, check out your own browser's docs for details).  A message should appear there when you hit the button on the default sample app.
When you want Meteor to stop running, you can hit control-c, which will make Meteor quit serving your page end exit.
If, instead, you want to force Meteor into the background and get your commandline back without exiting, hit control-z.  Your terminal will tell you that meteor is stopped, and give you a "job number" to the left of that.  You can use that job number with the bg command to set Meteor running in the background.   When you want it in the foreground again (for example, to control-c out of it) you can use the fg command.
Here is an example of how I put meteor into the background:
Running on: http://localhost:3000/
^Z
[1]+  Stopped                 meteor
gobo:blank naomi$ bg 1
[1]+ meteor &
gobo:blank naomi$

Note that these commands and key combinations are part of my shell (in this case, bash), not part of meteor.

redpajama_set_name: RedPajamaStackExchange


For an overview of the recent history of chemical regulation in America and the proposals to restructure it currently under consideration by the Senate, see my post of last Friday.
This entry was posted in Uncategorized and tagged chemistry, environmental science, EPA, regulation on March 31, 2015 by Evan Hepler-Smith.
Perhaps you have heard: over the past couple of weeks, the Senate Committee on the Environment and Public Works has begun to consider a pair of dueling bills to overhaul the regulation of chemicals in America. Perhaps you have seen this debate described under the heading of "TSCA reform." Perhaps you have wondered what TSCA is, why everybody seems to want to reform it, and what substantive differences lie behind the competing proposals for doing so.
Perhaps you haven't. But you should! The bills under consideration have significant stakes for human health, the environment, and businesses that produce, processe, trade in, or use any of the tens of thousands of chemicals in American commerce. And – not to be glib about such weighty matters – chemical regulation is also pretty fun grist for nerding out.
This entry was posted in Uncategorized and tagged chemistry, environmental science, EPA, regulation on March 29, 2015 by Evan Hepler-Smith.


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Сезон 2014/2015 — второй сезон в истории ФК «Тосно» и первый для клуба в первенстве Футбольной национальной лиги. Сезон охватывал период с 6 июля 2014 года по 30 мая 2015 года. Клуб принимал участие в двух турнирах: Первенстве ФНЛ и Кубке России.

Главные события сезона
По итогам сезона 2013/14 ФК «Тосно», заняв первое место в зоне Запад чемпионата ПФЛ, завоевал путёвку в первенство ФНЛ — второй по рангу футбольный турнир в России. По словам Максима Левченко — одного из владельцев компании Fort Group, генерального спонсора клуба — перед командой была поставлена задача выигрыша первенства и выхода в Премьер-лигу. Для решения поставленной задачи в несколько раз был увеличен бюджет клуба, составивший примерно 300 млн рублей, и серьёзно обновлён состав (в частности, пришли имевшие опыт игры в Премьер-лиге Александр Павленко и Андрей Бочков), при этом тренер Фёдор Щербаченко, анализируя комплектование команд ФНЛ, отметил, что «состав у "Тосно" всё же не для Премьер-лиги, пусть он и добротный». Кроме того, был подписан контракт с новым главным тренером болгарином Николаем Костовым, пришедшим из симферопольской «Таврии» (предыдущий сезон команда доигрывала под руководством начальника команды — ныне генерального директора клуба — Вячеслава Матюшенко.

С первых же туров команда стала одним из лидеров первенства ФНЛ, несколько раз возглавляла турнирную таблицу, однако с конца сентября начался спад: после нескольких поражений клуб опустился на 5-е место в чемпионате и вылетел из Кубка России, в результате чего в начале ноября Николай Костов был уволен. Несколько туров до зимней паузы команда провела под руководством тренера вратарей Кирилла Гашичева, сумевшего поднять ФК «Тосно» на 3-е место, а в декабре новым главным тренером команды стал Александр Григорян, перебравшийся в Ленинградскую область из владивостокского клуба «Луч-Энергия». По словам Григоряна, работа в ФК «Тосно» привлекла специалиста задачей вывода клуба в Премьер-лигу. Интересно, что последний матч в роли тренера «Луча» Григорян провёл как раз против «Тосно» и потерпел поражение со счётом 1:2. Вместе с Григоряном из Владивостока в ФК «Тосно» перешли тренер вратарей Геннадий Тумилович и несколько ключевых футболистов «Луча», в том числе лучший бомбардир Станислав Прокофьев. Помощником Григоряна стал Евгений Перевертайло, признанный лучшим тренером ФНЛ сезона 2012/13 за работу с калининградской «Балтикой».

Помимо смены тренерского штаба в зимнюю паузу в команда значительно усилила состав, пополнившись такими известными игроками, как Сергей Нарубин (многолетний вратарь «Амкара»), Максим Астафьев и игравший в сборной Польши Марцин Ковальчик. Однако на проходившем в Турции Кубке ФНЛ обновлённая команда заняла последнее, 16-е место, после чего Александр Григорян покинул пост, не проведя с командой ни одного официального матча. Официальной причиной отставки были объявлены «семейные обстоятельства», однако по мнению некоторых аналитиков на уход повлияли и разногласия специалиста с руководством клуба. Исполнять обязанности главного тренера стал помощник Григоряна Евгений Перевертайло, с которым впоследствии был подписан контракт, сроком действия до июня 2016 года.. По итогам этого сезона «Тосно» заняло 3 место и, проиграв оба стыковых матча «Ростову», осталось в ФНЛ.

Состав

Трансферы

Зима 2014/2015
В заявочный лист «Тосно» вошли футболисты из числа новичков клуба: вратарь Сергей Нарубин (№ 42), защитники Марцин Ковальчик (№ 16), Евгений Зубейко (№ 22), полузащитники Гурам Тетрашвили (№ 2), Максим Астафьев (№ 80) и Азим Фатуллаев (№ 26) и нападающий Станислав Прокофьев (№ 28).
«На этом заявочная кампания нашего клуба в зимнее трансферное окно завершена», — говорится в сообщении официального сайта клуба из Ленинградской области.

Пришли

Ушли

Сетка плей-офф
Матч за 13-16 места

Матч за 15-16 места

Примечания

Ссылки

Тосно
Сезоны ФК «Тосно»

redpajama_set_name: RedPajamaWikipedia


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Magandang Buhay
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Recently, our Chief Tech has had the most interesting experience with a neighborhood robin, whom we've named, "Rowdy," taking an unusual liking to our company A-Appliance Repair van! One morning, Kerry went out to start up the van and begin his day. He climbed inside the van and started the motor, just like any other morning, but all of a sudden his attention was caught by a loud buzzing sound outside his passenger side door window - a strange vibration indeed!
He looked closer, and to his surprise a large local robin was wildly flapping his wings, buzzing up the side of the window and apparently attacking the van! He thought maybe the bird had inadvertently flown into the window and was stunned or something, so he jumped back out of the van to see what was the matter. The bird then flew off, but not without leaving a hefty bi-product of his recent breakfast behind all over the van's passenger side mirror! Upon closer inspection, Kerry also noted "Rowdy" had left multiple scratch marks on the side door from crazily flapping his wings all over it. Hmmmm.
Now, our Chief Tech was somewhat dismayed, thought it was annoying, cleaned off the mirror, and got back in the van. However, Rowdy immediately returned and perched himself boldly again atop the passenger side mirror, and started loudly cherping / bleeping his unintelligible woes to our Chief Tech. Kerry managed to wave the bird away, drove off and didn't think any more of it as he went about the rest of his day. Unbelievably, the very next morning when he went outside to start up the van, there was Rowdy --- at it again, beating his wings against the side of the van! He chased the bird off, again cleaned off the messed-upon passenger side window, and began to wonder what in the heck this bird had going on in his head. Kerry also mentioned it to me, the author, when he arrived home after work that evening.
I thought it was very odd, but then the thought struck me that it is Springtime, and all the birdies and animals are staking their annual territorial claims as part of their attempts to find new mates. Weirder still -- did Rowdy actually see himself in the mirror and think it was another male robin that he needed to fight off? Or was he just making a lot of buzzzz to attract a new mate? What could it be?
Well, we couldn't figure it out. Rowdy continued his strange morning ritual, complete with mirror-messing, for over a week, and I was then able to catch these camera shots. The bird is bold, let me tell you! Even our two pet kitties watching him from the front window didn't deter him. Then we noticed he was not around for a few days and had huge sighs of relief, hoping Rowdy had finally found a mate and had no more time for van fly-bys.
My Gas Dryer Isn't Heating Enough!
We want to alert our customers that there is a very common reason for improper heating of gas dryers that you can simply and quickly troubleshoot yourself. The most common cause of this problem with gas dryers is actually restricted airflow. These instructions are easy to follow and can save you an appliance repair service call.
Check the dryer vent that's usually located on the outside of the house to test the airflow. Simply put your hand over the vent while the dryer is operating. If you feel little or no hot air coming out of the vent, then you've got an airflow problem!
Next, open the dryer door and check the lint trap, which is often located inside the dryer at the bottom of the door opening. Remove the lint filter and simply clean out any lint build up that you can see in the trap itself. Some people like to use a small cleaning brush to do this - see photo at right.
In addition, if there is any restriction inside your gas dryer's external venting system, the clothes will take forever to dry. So it's important to look behind the dryer to ensure the vent duct hose has connection to both the dryer itself and to the outside vent. Carefully move the dryer away from the wall, if you need to go behind it to more closely inspect the vent hose. If the vent hose has a dent in it or is partially smashed, you will have little to no airflow and will need to try to reform the tubing so that it isn't dented or partially squashed, in order to get maximum airflow.
Now you can look to check the vent for any lint build-up or other obstructions inside the dryer duct.
If there is lint build-up inside the duct, simply remove the lint and clean it out as best you can. It's a good idea anyway to do this general maintenance once yearly to avoid airflow restriction in your gas dryer and potentially costly repairs.
The second place where excess lint causes restriction in airflow is in the actual dryer vent duct tubing. If there's too much lint build-up in the vent exhaust tubing, as in the photo at left (top), and it's too much for your vacuum to clean out, it might be best to just replace the vent tubing altogether. If it does not appear to be that severe, you can try to simply remove the excess lint by vacuuming it out.
If you've performed these troubleshooting steps, and the dryer is still not heating properly, please give us a call to schedule a service call! We will get you on the service schedule quickly and perform any repairs needed for your gas dryer to restore it to proper heating function.


I've lived in the south bay for pretty much my whole life, and I only heard about these gardens a month ago! Where have I been?! The 16 acres of gardens here are absolutely gorgeous, especially when the weather is cooperating (and it usually is). Each garden is separated by brick walls or trimmed hedges, and I think there are 8 or 9 different garden "sections" total. They give you a map of the estate when you buy your admission tickets, so don't worry about getting lost!
It can get pretty crowded here on weekend afternoons, but because the estate is so big, the crowds usually disperse pretty quickly. So don't worry, you'll be able to snag that picture-perfect shot without too many moving bodies in the background. Your admission fee covers entry into both the gardens as well as the main house. If you're on a time crunch, go see the gardens first! The self-guided house tour goes by pretty quickly and you walk through most of the main rooms on the bottom floor (dining room, kitchen, sitting area, ballroom, etc.). They made sure to keep the setting of the rooms/furniture very close to how the Roth/Bourn families (previous owners of the estate) left them, so it really does feel like a trip back in time when you walk through the house! I didn't take pictures inside (lighting was a bit too dark), so the pictures you'll see below are all from the gardens. Enjoy!
Stunning photos Ashley. The gardens are maintained so beautifully. It's funny you have never heard of this place before but at least you have now and we're able to share with us readers of your post.
Such a nice garden. Very refreshing!
Such stunning pictures! I love photographing flowers, especially on the daylight, they always look so pretty!
Me too!! That's why I love going to gardens 🙂 Thanks for stopping by, Ana!
Looks like a lovely place! Cute pictures!
The gardens look incredible! Your photos are fantastic 🙂 Would really love to visit someday!
Thanks Vanessa! I hope you do get to visit! The gardens are 10x better in person!


When I was reading up on the subject it still wasn't clear what the issue was till my dad made the conflict clear: If you switch to a popular vote, a candidate could win by spending all their time in major population centers like New York, California, and Chicago, while ignoring more rural locales like Kansas, Idaho, or Utah. The Electoral College ensures that all states matter, but, still accounts for population variations.
What do you guys think? Email me or leave a comment below saying whether you think we should keep the electoral college or get rid of it and why.


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Q: Add Product Images with Shopify API We have a bunch of disassociated product images in our Shopify store that support could not re-associate. I set up a Postman collection runner to update all these missing images, and it seems to be working in our test environment....
Except, the API call is replacing the default product image with a new image rather than adding the image. I'm using a standard PUT request to the API using the example in their API docs:
PUT /admin/api/2019-04/products/#{product_id}.json

 {
  "product": {
    "id": 632910392,
    "images": [
      {
        "src": "http://example.com/rails_logo.gif"
      }
    ]
  }
}

I get that it's an array I'm sending in, so I'm thinking it's overwriting the array each time rather than adding a new element. I tried using a single element variable of "image" in the JSON but that didn't work.
Any ideas?

A: If you want to add the image use ProductImage resource instead of Product resource.
Product Image Documentation
POST /admin/api/2019-04/products/#{product_id}/images.json
{
  "image": {
    "src": "http://example.com/rails_logo.gif"
  }
}