text:
\section{Introduction}

Since the mid 1960s, computer scientists have referred to Chess as the \emph{Drosophila} of AI \cite{ensmenger2012chess}. Similar to the role of fruit fly in genetics studies, Chess provides an accessible, familiar, and relatively simple test bed that nonetheless can be used to produce wide knowledge about other  complex environments.
As a result, Chess have long been used as benchmarks for the development of AI for decades. The earliest attempt dated back to Shannon's interest in 1950 \cite{shannon1950xxii}.
In the past decades, remarkable progress has been made on developing AIs that can demonstrate super-human performance on playing Chess; this includes IBM DeepBlue \cite{campbell2002deep} and AlphaZero \cite{silver2017mastering}.

Despite the algorithmic achievements on Chess, we still have limited knowledge about the geometric landscape of the strategy space of Chess.
Tied to the strategic geometry are the concepts of \emph{transitivity} and \emph{non-transitivity} \cite{czarnecki2020real}.
Transitivity refers to how one strategy is  better than other strategies. In transitive games, we know that if strategy A beats strategy B and B beats C, then A can surely beat C.
On the other hand,
non-transitive games refers to those where there exists a loop in the preference of strategies. For example, if A can beat B and B beats C, then there is still possibility that  that C can still lose to A in such games.
One of the simplest purely non-transitive game is Rock-Paper-Scissor game. Yet,
games in the real world are far more complex; their strategy space often have variations in terms of both transitive and non-transitive components.
Judging on the strategies that are generated by AIs, researchers have hypothesised that the majority of real-world games possess a spinning top geometry in the strategy space, and in such geometry  (see Figure \ref{fig:teaser}(a)),   the radial axis represents the non-transitivity degree and the upright axis represents the transitivity degree.
The middle region of the transitive dimension is accompanied by a highly non-transitive dimension, and it gradually diminishes as the skill level either grows towards the Nash Equilibrium \cite{nash1951non},  or degrades to worst-performing  strategies.

While the idea of non-transitivity is intuitive, how to quantify and measure the size of its dimension remains an open challenge. Early attempt has been made in quantifying non-transitivity \cite{czarnecki2020real}, based on which they proposed the spinning top hypothesis.  However, their sampled strategies are all artificial, in the sense that they only study strategies that are generated by  different algorithmic models of   AIs. While one can argue that both artificial and human strategies are sampled from the same strategy space, they could demonstrate completely different inherent  play styles. For example, \cite{mcilroy2020aligning} showed that Stockfish AI \cite{romstad2017stockfish} did poorly in mimicking the play style of weak human players, despite being depth-limited to mimic that player's strength. Therefore, there is no guarantee that the conclusions from \cite{czarnecki2020real} would apply to human strategies. Consequently, the geometric profile of the strategy space of Chess is still unclear, this directly motivates us to investigate and quantify the degree of non-transitivity on human strategies.

Apart from its popularity in history,
Chess is also one of the most popular classical games;  it provides the largest amount of well-maintained  human games that are  publicly available in various sources. These databases include Lichess \cite{lichess_about}, which contains games played since 2013, and the Free Internet Chess Server (FICS) which contains games played since 1999 \cite{fics_about}.  By studying over one billion records of human games on Lichess and FICS, in this paper, we measure the non-transitivity of Chess games on real-world data, and investigate its potential implications for training effective AI agents as well as on human skill progression.
Specifically, we performed two ways of non-transitivity measurements by adopting Nash Clustering and counting the number of Rock-Paper-Scissor cycles.
Our findings positively indicate that the strategy space occupied by real-world Chess strategies demonstrates a spinning top geometry. More importantly, there exists a strong connection between the degree of non-transitivity and the progression of a Chess player's rating. In particular, high degrees of non-transitivity tend to  prevent human players from making progress on the Elo rating, whereas progressions are easier to make at the level of ratings where the degree of non-transitivity is lower.
Additionally, we also investigate the implication of the degree of non-transitivity for  population-based training methods. By considering \emph{fixed-memory Fictitious Play} as a proxy, we reach the conclusion that maintaining large-size and diverse populations \cite{yang2021diverse} of strategies is imperative to training effective AI agents in solving Chess type of games, which also matches the empirical findings that are observed based on artificial strategies \cite{czarnecki2020real}.

\section{Related Work} \label{section: related works}

Our work is mostly related to \cite{czarnecki2020real}, who proposed the Game of Skill geometry (i.e., the spinning top hypothesis), in which the strategy space of real-world games resembles a spinning top in 2-dimension, where the vertical axis represents the transitive strength, and the horizontal axis represents the degree of non-transitivity. Such hypothesis has  empirically been verified on several real-world games like Hex, Go, Tic-Tac-Toe, and Starcraft. The approach taken by \cite{czarnecki2020real} is to sample strategies uniformly along the transitive strength, and strategies are generated by first running solution algorithms like Minimax Algorithm with Alpha-Beta pruning \cite{russell2002artificial} and Monte-Carlo Tree Search (MCTS) \cite{chaslot2008monte}. Sampling strategies uniformly in terms of transitive strength is then conducted by varying the depths up to which Minimax is performed, and by varying the number of simulations for MCTS. Finally, for Starcraft, sampling strategies are done by using the agents from the training of AlphaStar AI \cite{vinyals2019grandmaster}. However, such sampling of strategies means that the empirical verification is performed  on artificial strategies, and there have been works that suggest an inherent difference in the play styles of AI and humans. For example, \cite{mcilroy2020aligning} showed that Stockfish AI \cite{romstad2017stockfish} did poorly in mimicking the play style of  human players, despite being depth-limited to mimic that player's strength. This motivates us to perform similar investigations, but on human strategies.

This paper focuses the non-transitivity quantification on real-world Chess strategies. However, Chess players are rated primarily using the Elo rating system, which is only valid for the pool of players being considered. As such, there are various systems of player rating computations implemented in the world. The two most used implementations are the ones by the Fédération Internationale des Échecs (FIDE) \cite{fide_charter} and the United States Chess Federation (USCF) \cite{uscf_about}. In its most basic implementation, the Elo rating system consists of two steps, i.e., assigning initial ratings to players and updating the ratings after matches. USCF and FIDE determine a player's starting rating by taking into account multiple factors, including the age of the player and their ratings in other systems \cite{uscf_rating_system, fide_handbook_B022017}. However, the exact rule used in determining the starting rating differs between the two organisations. Furthermore, the update rules used also differs noticeably. These are defined in the handbook of each organisation \cite{uscf_rating_system,fide_handbook_B022017}. On the other hand, online Chess platforms like Lichess and the FICS, which are the source of data for this paper, typically use a refined implementation of the Elo rating system, namely the Glicko \cite{glickman1995glicko} system for FICS, and Glicko-2 \cite{glickman2012example} for Lichess. These systems typically set a fixed starting rating, and then include the computation of a rating deviation for Glicko while updating the player rating after matches. Glicko-2 includes the rating deviation and the rating volatility, both of which represent the reliability of the player's current rating.

\section{Preliminaries and Notations}

\textbf{Notations. }
For any positive integer $k$, $[k]$ denotes the set $\{1, ..., k\}$. $\Delta_k$ denotes the set of all probability vectors with $k$ elements. For a set $A$ and some function $f$, $\{f(a)\}_{a \in A}$ denotes the set constructed by applying $f$ to each element of $A$. Furthermore, for a set $A= \{A_1, ..., A_{\vert A \vert}\}$, $\{s_i\}_{i \in A}$ denotes $\{s_{A_1}, ..., s_{A_{\vert A \vert}}\}$. Moreover, for an integer $k$, $\{s_i\}_{i = 1}^k$ denotes $\{s_1, ..., s_k\}$. $\mathbb{N}_1$ denotes the set $\{1, 2, ...\}$, i.e., natural numbers starting from $1$. For a set $A = \{A_1, ..., A_{\vert A \vert}\}$, the notation $A_{-j}$ denotes $A$ without the $j^{th}$ element, i.e., $A_{-j} = \{A_i\}_{i \in [\vert A \vert], i \neq j}$. Hence $A = \{A_j, A_{-j}\}$ $\forall j \in [\vert A \vert]$. Furthermore, we use $I[a > b]$ for $a,b \in \mathbb{R}$ to denote the indicator function, where $I[a > b] = 1$ if $a > b$ and $0$ otherwise. Finally, for any integers $m$ and $k$ where $k \leq m$, let $\mathbf{e^m_k}$ denote a vector of length $m$, where the vector is all zeroes, but with a single 1 at the $k^{th}$ element. Likewise, $\mathbf{0}_k$, $\mathbf{1}_k$ denote the vector of zeroes and ones of length $k$ respectively.
\begin{definition}
\label{definition: game theory game}
A game consists of a tuple of $(n, S, M)$ where:
\begin{itemize}
    \item $n \in \mathbb{N}_1$ denotes the number of players.
    \item $S = \prod_{i=1}^n S_i$ denotes the joint strategy space of the game, where $S_i$ is the strategy space of the $i^{th}$ player for $i \in [n]$. A strategy space is a set of strategies a player can adopt.
    \item $M = \{M_i\}_{i = 1}^n$, where $M_i: S \rightarrow \mathbb{R}$ is the payoff function for the $i^{th}$ player. A payoff function maps the outcome of a game resulting from a joint strategy, to a real value, representing the payoff received by that player.
\end{itemize}
\end{definition}

\textbf{Normal Form (NF) Games. }
In NF games, every player takes a single action simultaneously. Such games are fully defined by a payoff matrix $\boldsymbol{\mathcal{M}}$ where $$\boldsymbol{\mathcal{M}}_{d_1, d_2, ..., d_n} = \big\{M_1(a_1, a_2, ..., a_n), ..., M_n(a_1, a_2, ..., a_n)\big\}$$
for $a_k \in A_k$, $d_k \in [\vert A_k \vert]$, $k \in [n]$, $A_k$ denoting the action space of the $k^{th}$ player, and $d_k$ is the index corresponding to $a_k$. $\boldsymbol{\mathcal{M}}$ fully specifies an NF game because it contains all the components of a game by Definition \ref{definition: game theory game}. In an NF game, the strategy adopted by a player refers to the single action taken by that player. Therefore, the action space $A_k$ is equivalent to the strategy space $S_k$ for $k \in [n]$.

An NF game is zero-sum when $\sum_{z \in \boldsymbol{\mathcal{M}}_{d_1, ..., d_n}} z = 0$ for $d_k \in [\vert A_k \vert]$, $k \in [n]$. For two-player zero-sum NF games, the entries of $\boldsymbol{\mathcal{M}}$ can therefore be represented by a single number, since the payoff of one player is the negative of the other. Additionally, if the game is symmetric, i.e., both players have identical strategy spaces, then the matrix $\boldsymbol{\mathcal{M}}$ is \textbf{skew-symmetric}.

A well-known solution concept that describes the equilibrium of NF games is Nash Equilibrium (NE) \cite{nash1951non,deng2021complexity}. Let $\mathbf{p_i}(s)$ where $s \in S_i$, $\mathbf{p_i} \in \Delta_{\vert S_i \vert}$, $i \in [n]$, denote the probability of player $i$ playing pure strategy $s$ according to probability vector $\mathbf{p_i}$. Also, let $p = \{\mathbf{p_1}, ..., \mathbf{p_n}\}$, and $p[j:\mathbf{k}]$ denoting changing the $j^{th}$ element in $p$ to the probability vector $\mathbf{k}$, where $\mathbf{k} \in \Delta_{\vert S_j \vert}$.

\begin{definition}
\label{NE Constraint}
The set of distributions $p^*$, where $p^* = \{\mathbf{p_1^*}, ..., \mathbf{p_n^*}\}$ and $\mathbf{p_i^*} \in \Delta_{\vert S_i \vert}$ $\forall i \in [n]$, is an NE if $\sum_{s \in S} p^*(s) M_i(s) \geq \sum_{s \in S} p^*[i:\mathbf{e_k^{\vert S_i \vert}}](s) M_i(s)$ $\forall i \in [n]$, $k \in [\vert S_i \vert]$, where $p^*(s) = \prod_{j \in [n]} \mathbf{p_j^*}(s_j)$ and $s = \{s_1, ..., s_n\}$, $s_i \in S_i$ $\forall i \in [n]$.
\end{definition}

Definition \ref{NE Constraint} implies that under NE, no player can increase their expected payoff by unilaterally deviating to a pure strategy. While such NE always exists \cite{nash1951non}, it might not be unique. To guarantee the uniqueness of the obtained NE, we apply the maximum entropy condition when  solving for the NE. The maximum-entropy NE is proved to be unique \cite{balduzzi2018re}.

\begin{theorem}
\label{theorem: symmetric maxent ne}
Any two-player zero-sum symmetric game always has a unique maximum entropy NE given by $\{\mathbf{p^*}, \mathbf{p^*}\}$ where $\mathbf{p^*} \in \Delta_{\vert S_g \vert}$, and $S_g$ is the strategy space for each of the two players \cite{balduzzi2018re}.
\end{theorem}

As a consequence of Theorem \ref{theorem: symmetric maxent ne}, finding the maximum entropy NE of a two-player zero-sum symmetric game amounts to solving the Linear Programming (LP) problem shown in Equation \ref{Special Case Maxent NE LP}.
\begin{align}
    & \mathbf{p^*} = \argmax_\mathbf{p} \sum_{j \in [\vert S_g \vert]} -\mathbf{p}_j \text{ log } \mathbf{p}_j \nonumber \\
   & \text{s. t. }  \ \ \ \
     \boldsymbol{\mathcal{M}} \mathbf{p} \leq \mathbf{0}_{\vert S_g \vert} \nonumber \\
    & \ \ \ \ \ \ \ \ \ \ \ \mathbf{p} \geq \mathbf{0}_{\vert S_g \vert} \nonumber \\
    & \ \ \  \ \ \ \ \ \ \ \  \mathbf{1}_{\vert S_g \vert}^\top \mathbf{p} = 1
\label{Special Case Maxent NE LP}
\end{align}
where $\boldsymbol{\mathcal{M}}$ is the skew-symmetric payoff matrix of the first player

\textbf{Extensive Form (EF) Games. }
EF games \cite{hart1992games} model situations where participants take actions sequentially and interactions happen for more than one step. We consider the case of perfect information, where every player has knowledge of all the other player's actions, and finite horizon, where interactions between participants end in a finite number of steps (denoted as $K$).
In EF games, an action refers to a single choice of move of some player at some stage of the game. Therefore, the term action space, is used to define the available actions to a player at a certain stage of the game.
\begin{definition}
\label{definition: history EF games}
A history at stage $k$, i.e., $h_k$ is defined as $h_k = \{a_1, ..., a_{k-1} \}$, where $a_j = \{a_j^1, ..., a_j^n \}$ for $j \in [k-1]$. $a_j$ is the action profile at stage $j$, denoting actions taken by all players at that stage.
\end{definition}
The action space of player $i$ at stage $k$ is therefore a function of the history at that stage, i.e., $A_i(h_k)$ for $k \in [K]$, $i \in [n]$, since the available actions for a player depends on how the game goes.
\begin{definition}
\label{definition: strategy EF games}
Let $H_k$ be the set of all possible histories at stage $k$. Let $A_i(H_k) = \bigcup_{h_k \in H_k} A_i(h_k)$ be the collection of actions available to player $i$ at stage $k$ from all possible history at that stage. Let the mapping $s_i^k : H_k \rightarrow A_i(H_k)$ be a mapping from any history at stage $k$ (i.e., $h_k$) to an action available at that stage due to $h_k$, i.e., $s_i^k(h_k) \in A_i(h_k)$  $\forall h_k \in H_k$. A strategy of a player $i$ is defined as $s_i = \bigcup_{k \in [K]} s_i^k$, and $s_i \in S_i$ where $S_i$ is the strategy space of player $i$.
\end{definition}
In EF games, behavioural strategies are defined as a contingency plan for any possible history at any stage of the game.
According to the Kuhn's theorem \cite{kuhn2009extensive}, each behavioural strategy has a realisation-equivalent mixed strategy in EF games of perfect recall.
Finally, the payoff function of an EF game is defined as the mapping of every terminal histories, i.e., history at stage $K + 1$, to a sequence of $n$ real numbers, denoting the payoffs of each player, i.e., $\boldsymbol{\mathcal{M}}_i : H_{K+1} \rightarrow R$ for $i \in [n]$, where $H_{K+1}$ is the set of all terminal histories. However, any terminal history is produced by at least one joint strategy adopted by all players. Therefore, the payoff of a player is also a mapping from the joint strategy space to a real number, consistent with Definition \ref{definition: game theory game}.

An EF game can be converted into a NF game. When viewing an EF game as an NF game, the "actions" of a player in the corresponding NF game would be the strategies of that player in the original EF game. Therefore, a $n$-player EF game would have a corresponding $\boldsymbol{\mathcal{M}}$ matrix of $n$ dimensions, where the $i^{th}$ dimension is of length $\vert S_i \vert$. The entries of this matrix correspond to the sequence of $n$ real numbers representing the payoff of each player as a consequence of playing the corresponding joint strategies.

\textbf{Elo Rating.} The Elo Rating system is a rating system devised by Arpad Elo \cite{elo1978rating}. It is a measure of a participant's performance relative to others within a pool of players. The main idea behind Elo rating is that given 2 players, A and B, the system is to assign a numerical rating to each of them, i.e., $r_A$ and $r_B$ where $r_A, r_B \in \mathbb{R^+}$, such that the winning  probability of $p(A>B)$ is approximated by
\begin{equation}
\label{Elo formula}
    p(A > B) \approx \frac{1}{1 + \text{exp}(-k(r_A - r_B))} = \sigma \big(k({r_A - r_B}) \big)
\end{equation}
where $k \in \mathbb{R}^+$, $\sigma(x) = {1}/{1 + \text{exp}(-x)}$. Finally, it is worth noting that a game whose payoff function is defined by Equation \ref{Elo formula} is a purely transitive game.

\section{Non-Transitivity Quantification}

We employ two methods to quantify the degree of non-transitivity; namely, by counting strategic cycles of length three, and by counting the size of Nash Clustering \cite{czarnecki2020real}.
Before elaborating  the principle of each one of them, we start by introducing the techniques we applied for payoff matrix construction.

\subsection{Payoff Matrix Construction} \label{subsection: payoff matrix construction}

The raw data for the experiments were obtained from two sources. The primary source is the Lichess \cite{lichess_about} database, which contains games played from 2013 until 2021.
The second dataset is sourced from the FICS \cite{fics_about} database. For FICS, we use games from 2019 which contains the most number of games.
Altogether, we collected over one billion records of match data from human players.
To proceed with such a large amount of data, we introduce several novel procedures to turn match records into payoff matrices.

The first processing step would be to extract the required attributes from every game data. These would be the outcome of the game (to create the payoff matrix) and the Elo ratings of both players (as the measurement of transitive strength).

The second step concerns the number of games taken from the whole Lichess database. As of 2021, Lichess database contains a total of more than 2 billion games hosted from 2013. Therefore, a sampling method is required. Games are sampled uniformly across every month and year, and the number of games sampled every month is 120,000 since in January 2013, the database contains only a little more than 120,000 games. To avoid the influence of having a different number of points from each month, the number of games sampled per month is thus kept constant. In earlier years such as 2013 to 2015, sampling uniformly across a month is trivial since games from an entire month can be load into a single data structure. However, for months in later years, games in any given month could reach as many as 10 million. Therefore, a two-stage sampling method is required. The pseudocode for the algorithm can be found in Algorithm \ref{algo: two-stage uniform sampling} of Appendix \ref{appendix: two-staged sampling}. For FICS, the number of games in every month is relatively small and can be sampled directly.

The third processing step concerns discretisation of the whole strategy space to create a skew-symmetric payoff matrix representing the symmetric NF game. For this purpose, a single game outcome should represent the two participants playing two matches, with both participants playing black and white pieces once. We refer to such match-up as a two-way match-up. A single strategy in the naive sense would thus be a combination of one contingency plan (as defined in Definition \ref{definition: strategy EF games}) if the player plays the white pieces with another one if the same player plays the black pieces. However, such naive interpretation of strategy results in a very large strategy space. We thus discretise the strategy space along the transitive dimension. Since we employ real-world data from human players, we use the Elo rating of the corresponding players to measure transitive strength. Therefore, given a pair of Elo ratings $a$ and $b$, the game where the white player is of rating $a$ and black player of rating $b$, and then another game where the black player is of rating $a$ and the white player of rating $b$, corresponds to a single two-way match-up data. Discretisation is conducted by binning Elo ratings, such that given 2 bins, $b_1$ and $b_2$, any two-way match-up where one player's rating falls into $b_1$ and the other player's rating falls into $b_2$, will be treated as a two-way match-up between $b_1$ and $b_2$. The resulting payoff matrix of the NF game is thus outcomes of all two-way match-ups between every possible pair of bins. Since two-way match-ups are considered, the resulting payoff matrix would be skew-symmetric.

However, when searching for the corresponding two-way match-up between 2 bins of rating in the dataset, there could either be multiple games or no games corresponding to this match-up. Consider a match-up between one bin $a = [a_1, a_2]$ and another bin $b = [b_1, b_2]$. To fill in the corresponding entry of the matrix, a representative score for each case is first calculated. Let $rs_{a,b}$ denote the representative score for when the white player rating is in $a$ and the black player rating is in $b$, and $rs_{b,a}$ to denote the other direction. For argument's sake, consider finding $rs_{a,b}$. Since games are found for bin $a$ against $b$, scores must be expressed from the perspective of $a$. The convention here is $1$, $0$, and $-1$ for a win, draw, and loss respectively. This convention ensures that the entries of the resulting payoff matrix are skew-symmetric. In the case that there are multiple games where the white player's rating falls in $a$ and black player's rating falls in $b$, the representative score is then taken to be the average score from all these games. On the other hand, if there are no such games, the representative score is then the expected score predicted from the Elo rating, i.e.,
\begin{align}
\label{eqn:Expected representative score}
    \mathbb{E}_s[a > b] = 2p(a > b) - 1
\end{align}
where $\mathbb{E}_s[a>b]$ is the expected score of bin $a$ against bin $b$, and $p(a>b)$ is computed using Equation \ref{Elo formula}, by setting $k$ as $\frac{\text{ln}(10)}{400}$, $r_a = \frac{a_1 + a_2}{2}$, and $r_b = \frac{b_1 + b_2}{2}$. Since this score is predicted using Elo rating, it is indifferent towards whether the player is playing black or white pieces. After computing the representative score for both cases, the corresponding entry in the payoff matrix for the two-way match-up between bin $a$ and $b$ is simply the average of the representative score of both cases, i.e., $\frac{rs_{a,b} + rs_{b,a}}{2}$.
The payoff matrix construction procedure is summarised in Algorithm \ref{algo: Payoff Matrix Construction}.
\FloatBarrier
\begin{algorithm}[t!]
\SetAlgoLined
\textbf{Inputs}: Dataset of games $D$, set of $m$ tuples $B = \{b_1, ..., b_m\}$; \\
Initialise $\boldsymbol{\mathcal{M}}$ as $m \times m$ matrix of zeroes;\\
\For{$i \in [m]$}{
    \For{$j \in [m]$}{
        \uIf{$i \geq j$}{
            \textbf{continue};\\
            }
        \Else{
            $D_{W,B} = D[W \in b_i, B \in b_j]$;\\
            \uIf{$D_{W,B} = \emptyset$}{
                $r_i, r_j = \frac{b_i^L + b_i^H}{2}, \frac{b_j^L + b_j^H}{2}$;\\

                $E_{W,B} = 2p(i > j) - 1$;\\
            }
            \Else{
                $E_{W,B} =$ Av$(D_{W,B})$;\\
            }
            $D_{B,W} = D[B \in b_i, W \in b_j]$;\\
            \uIf{$D_{B,W} = \emptyset$}{
                $r_i, r_j = \frac{b_i^L + b_i^H}{2}, \frac{b_j^L + b_j^H}{2}$;\\

                $E_{B,W} = 2p(i > j) - 1$;\\
            }
            \Else{
                $E_{B,W} =$ Av$(D_{B,W})$;\\
            }
            $\boldsymbol{\mathcal{M}}_{i,j}, \boldsymbol{\mathcal{M}}_{j,i} = \frac{E_{W,B} + E_{B,W}}{2}, -\frac{E_{W,B} + E_{B,W}}{2}$;\\

        }
    }
}
Output $\boldsymbol{\mathcal{M}}$ as the payoff matrix;\\
\caption{Chess Payoff Matrix Construction}
\label{algo: Payoff Matrix Construction}
\end{algorithm}
\FloatBarrier
In Algorithm \ref{algo: Payoff Matrix Construction}, $D[W \in b_i, B \in b_j]$ collects all games in $D$ where the white player's rating $W$ is $b_i^L \leq W \leq b_i^H$ and the black player's rating $B$ is $b_j^L \leq B \leq b_j^H$. Av$(D_{W,B})$ averages game scores from the perspective of the white player, i.e., the score is $1$, $0$, $-1$ if the white player wins, draws, or losses respectively. Finally, in computing $p(i > j)$ following Equation \ref{Elo formula}, $r_i$ and $r_j$ used in the equation follows from the $r_i$ and $r_j$ computed in the previous line in the algorithm.

\subsection{Nash Clustering}

\begin{figure*}[!t]
\vspace{-5pt}
    \centering
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2019-Nash-and-RPS-Elo.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2020-Nash-and-RPS-Elo.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-FICS-2019-Nash-and-RPS-Elo.pdf}}
    \vspace{-10pt}
    \caption{Nash Clustering (left of each plot) and RPS Cycles (right of each plot) on Lichess 2019 (a), Lichess 2020 (b) and FICS 2019 (c). See Figure \ref{fig: full results} of Appendix \ref{appendix: full results} for the full version of the results, which include Lichess 2013 - 2020 and FICS 2019. The spinning top geometry is observed in both Nash Clustering and RPS Cycles plots, for all the years. Furthermore, both Nash cluster size and number of RPS cycles peaks at a similar Elo rating range of 1300 to 1700 for all the years. This provides comprehensive evidence that the strategies adopted in real-world games do not vary throughout the years or from one online platform to another, and that the non-transitivity of real-world strategies is highest in the range of 1300 to 1700 Elo rating.}
    \label{fig: nash clustering and rps cycles elo}
    \Description{Nash Clustering and counting RPS Cycles on Lichess data from 2019 and 2020, and FICS data from 2019; empirical evidence to illustrate the existence of the spinning top geometry in real-world Chess strategy space.}
    \vspace{-0pt}
\end{figure*}

\begin{figure}[!t]
    \centering
    \includegraphics[width = \linewidth]{LR-Rating.pdf}
    \vspace{-20pt}
    \caption{Chess Rating Mappings between Lichess, USCF, and FIDE. The relation between Lichess, USCF, and FIDE ratings are largely linear and monotonically increasing, and the linear model fits well to the data.}
    \Description{Chess player rating mapping between Lichess system, FIDE system, and USCF system.}
    \label{fig: lr regression}
    \vspace{-0pt}
\end{figure}

Following the payoff matrix constructed from the previous step, the first method of quantifying non-transitivity is through Nash Clustering \cite{czarnecki2020real}. Nash Clustering is derived from the layered game geometry, which states that the strategy space of a game can be clustered into an ordered list of layers such that any strategy in preceding layers is strictly better than any strategy in the succeeding layers. A game in which the strategy space can be clustered in this way is known as a layered finite game of skill.

\begin{definition}
\label{definition: k-layered}
A two-player zero-sum symmetric game is defined as a $k$-layered finite game of skill if the strategy space of each player $S_g$ can be factored into $k$ layers, i.e., $L_i$ $\forall i \in [k]$ where $\bigcup_{i \in [k]} L_i = S_g$, $L_i \cap L_j = \emptyset \ \forall i \neq j$, $i,j \in [k]$, and the layers are such that $M_1(s_1, s_2) > 0$ $\forall s_1 \in L_i$, $s_2 \in L_j$, $\forall i,j \in [k]$, $i < j$ and $\exists z \in \mathbb{N}_1$ such that $\forall i < z$, $\vert L_i \vert \leq \vert L_{i+1} \vert$, $\forall i \geq z$, $\vert L_i \vert \geq \vert L_{i+1} \vert$.
\end{definition}

A consequence of Definition \ref{definition: k-layered} is that strategies in earlier layers would have higher win-rate than those in succeeding layers. For any strategy $s \in S_g$, the corresponding win-rate $TS(s)$ is
\begin{equation}
\label{strategy win-rate equation}
    TS(s) = \frac{1}{\vert S_g \vert - 1}\sum_{z \in S_g} I[M_1(s,z) > 0].
\end{equation}
Therefore, if one define the transitive strength of a layer to be the average win-rate of the strategies within that layer, the transitive strength of the layers would be ordered in descending order. Hence, intuitively, one can see the $k$-layered finite game of skill geometry as separating the transitivity and non-transitivity components of the strategy space. The non-transitivity is contained within each layer, whereas the transitivity variation takes place across layers. It is thus intuitive to use the layer sizes as a measure of non-transitivity.

Furthermore, from Definition \ref{definition: k-layered}, the layer sizes are such that it increases monotonically up to a certain layer, from which it then decreases monotonically. Thus, when the transitive strength of each layer is plotted against the size, a 2-dimensional spinning top structure would be observed if the change in sizes are strictly monotonic and $k > 2$. It is also because of this reason, \cite{czarnecki2020real} named this geometry the spinning top geometry.

For every finite game, while there always exists $k \geq 1$ for which the game is a $k$-layered finite game of skill, a value of $k > 1$ does not always exist for any game. Furthermore, the layered geometry is not useful if $k = 1$. Therefore, an alternative relaxation to the $k$-layered geometry is proposed where a layer is required to be "overall better" than the succeeding layers. However, it is not necessary that every strategy in a given layer can beat every strategy in any succeeding layers. Such layer structure is obtained by Nash Clustering.

Let $\text{Nash}(\boldsymbol{\mathcal{M}} \vert X)$ where $X \subset S_g$ denote the symmetric NE of a two-player zero-sum symmetric game, when both players are restricted to only the strategies in $X$, and $\boldsymbol{\mathcal{M}}$ be the payoff matrix from the perspective of player 1. Furthermore, let $\text{supp}(\text{Nash}(\boldsymbol{\mathcal{M}} \vert X))$ denote the set of pure strategies that are in support of the symmetric NE. Finally, let the set notation $A \backslash B$ denote the set $A$ but excluding the elements that are in set $B$.

\begin{definition}
\label{Nash Clustering Definition}
Nash Clustering of a finite two-player zero-sum symmetric game produces a set of clusters $C$ defined as
\begin{align}
C &= \{C_j : j \in \mathbb{N}_1, C_j \neq \emptyset\},  \ \  \text{where}  \nonumber\\ C_0 &= \emptyset, \ \  C_i = \text{supp}\Big(\text{Nash}\big(\boldsymbol{\mathcal{M}} \big| S_g \backslash \bigcup_{k = 0}^{i-1} C_k \big)\Big), \ \ \  \forall i \geq 1.
\end{align}
\end{definition}
Furthermore, to ensure the Nash Clustering procedure is unique, this paper uses maximum entropy NE  in Equation \ref{Special Case Maxent NE LP}. Following Nash Clustering, the measure of non-transitivity would thus be the size or number of strategies in each cluster.

One can  apply the Relative Population Performance (RPP) \cite{balduzzi2019open} to determine if a layer or cluster is overall better than another cluster. Given 2 Nash clusters $C_i$ and $C_j$, $C_i$ is overall better or wins against $C_j$ if $\text{RPP}(C_i, C_j) > 0$. When using RPP as a cluster relative performance measure, the requirement that any given cluster is overall better than any succeeding clusters proves to hold \cite{czarnecki2020real}. Specifically, one can use the fraction of clusters beaten, or win-rate, defined in Equation \ref{RPP win-rate equation}, as the measure of transitive strength of each cluster.
\begin{equation}
\label{RPP win-rate equation}
TS(C_a) = \frac{1}{\vert C \vert - 1} \sum_{C_i \in C} I \Big[\text{RPP}\big(C_a, C_i\big) > 0\Big]
\end{equation}
where  $I$ is the indicator function. We refer to the above win-rate as the RPP win-rate. Since any layer  beats all succeeding layers, the win-rates of the Nash clusters would occur in descending order.

Alternatively, we can apply the average Elo rating in Equation \ref{Elo formula} of the pure strategies inside each Nash cluster to measure transitive strength of the cluster.
Specifically, given the payoff matrix from  Section \ref{subsection: payoff matrix construction},  each pure strategy corresponds to an Elo rating bin, and the transitive strength of a Nash cluster is thus computed as the average Elo rating of the bins inside that cluster.
Let $C_k \in C$ be a Nash cluster, and $c_k$ be the integer indexes of the pure strategies in support of $C_k$, $B = \{b_1, ..., b_m\}$ be the bins used when constructing the payoff matrix, where each bin $b_i$ is has a lower and upper bound of $(b_i^L, b_i^H)$, and the bins corresponding to $C_k$ is $\{b_j\}_{j \in c_k}$. The average Elo rating of Nash cluster $C_k$ is defined as
\begin{equation}
\label{eqn: Nash cluster average label}
    TS_{\text{Elo}}(C_k) = \frac{1}{\vert C_k \vert} \sum_{j \in c_k} \frac{b_j^L + b_j^H}{2}.
\end{equation}

Given the average Elo rating against the cluster size for every Nash cluster from Nash Clustering, the final step would be to fit a curve to represent the spinning top geometry. We use the method of fitting an affine-transformed skewed normal distribution to minimise Mean Squared Error (MSE). The results of applying Nash Clustering is shown in Figure \ref{fig: nash clustering and rps cycles elo}.

Furthermore, we also investigate using win-rates as a measure of transitive strength, defined in Equation \ref{RPP win-rate equation}. However, instead of using RPP to define relative performance between clusters, we use an alternative, more efficient measure, that achieves the same result when applied on Nash clusters. This measure is named the Nash Population Performance (NPP).
\begin{definition}
\label{NPP definition}
Let $C = \{C_1, ..., C_{\vert C \vert}\}$ be the Nash Clustering of a two-player zero-sum symmetric game with a payoff matrix of $\boldsymbol{\mathcal{M}}$. $\text{NPP}(C_i, C_j) = \mathbf{p_i}^\top \boldsymbol{\mathcal{M}} \mathbf{p_j}$ where $\mathbf{p_k}= \text{Nash}(\boldsymbol{\mathcal{M}} \vert \bigcup_{r = k}^{\vert C \vert} C_r$ for $k \in [\vert C \vert]$ and $C_i, C_j \in C$.
\end{definition}
Using NPP also satisfies the requirement that any cluster is overall better than the succeeding clusters.
\begin{theorem}
\label{NPP arrangement theorem}
Let $C$ be the Nash Clustering of a two-player zero-sum symmetric game. Then $\text{NPP}(C_i, C_j) \geq 0$ $\forall i \leq j$, $C_i, C_j \in C$ and $\text{NPP}(C_i, C_j) \leq 0$ $\forall i > j$, $C_i, C_j \in C$.
\end{theorem}
\begin{figure*}[!t]
\vspace{-5pt}
    \centering
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2019-Nash-and-Training.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2020-Nash-and-Training.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-FICS-2019-Nash-and-Training.pdf}}
    \vspace{-10pt}
    \caption{(Left of each plot) Nash Clustering using RPP win-rate as the measure of transitive strength on Lichess 2019 (a), 2020 (b) and FICS 2019 (c). The spinning top geometry persists despite using a different measure of transitive strength compared to Figure \ref{fig: nash clustering and rps cycles elo}. (Right of each plot) Fixed-Memory Fictitious Play on Lichess 2019 (a), Lichess 2020 (b) and FICS 2019 (c). The training process does converge for all the above cases, provided that the starting population size is 50 or above. This is because the Nash cluster sizes are never much larger than 50, and therefore a population size of 50 ensures good coverage of the Nash clusters around any transitive strength. Such convergence behaviour also provides additional evidence that the spinning top geometry is present in the strategy space of real-world Chess. See Figure \ref{fig: appendix 2} of Appendix \ref{appendix: alternative measure results} and Figure \ref{fig: full results} of Appendix \ref{appendix: full results} for full versions of the results on Nash Clustering with RPP win-rate and fixed-memory Fictitious Play respectively.}
    \label{fig: nash clustering and training}
    \Description{Training process using Fixed-Memory Fictitious Play on Lichess games from 2019, 2020 and FICS games from 2019, compared with their corresponding non-transitivity plots using Nash Clustering.}
    \vspace{-5pt}
\end{figure*}

The proof of Theorem \ref{NPP arrangement theorem} is given in Appendix \ref{appendix: theorem proofs}. This theorem suggests that applying NPP to compute Equation \ref{RPP win-rate equation} instead of RPP would result in an identical win-rate for any Nash cluster, and the descending win-rate guarantee is preserved. NPP is more efficient since in Definition \ref{NPP definition}, $\mathbf{p_k}$ for any Nash cluster is the corresponding NE solved during Nash Clustering to obtain that cluster. However, this also means that NPP is more suitable as a relative performance measure of Nash clusters, and less so for general population of strategies. For the latter, RPP would be more suitable.

Figure \ref{fig: nash clustering and training} shows the results of applying Nash Clustering with win-rate as the measure of transitive strength. Again, the spinning top geometry persists for all cases, suggesting that Nash Clustering is agnostic to the measure of transitive strength.

\subsection{Rock-Paper-Scissor Cycles}

The second method to quantify non-transitivity is to measure the length of the longest cycle in the strategy space, since this stems directly from the definition of non-transitivity. A cycle is defined as the sequence of strategies that starts and ends with the same strategy, where any strategy beats the strategy next to it in the sequence. However, calculating the length of the longest cycle in a directed graph is well-known to be  NP-hard \cite{bjorklund2004approximating}. Therefore, we use the number of cycles of length three (i.e., Rock-Paper-Scissor or RPS cycles), passing through every pure strategy as a proxy to quantify non-transitivity.

To obtain the number of RPS cycles passing through each pure strategy, an adjacency matrix is first formed from the payoff matrix. Letting the payoff matrix from Section \ref{subsection: payoff matrix construction} be $\boldsymbol{\mathcal{M}}$, the adjacency matrix $\boldsymbol{\mathcal{A}}$ can be written as $\big\{\boldsymbol{\mathcal{A}}_{i,j}\big\}_{i,j = 1}^{\vert S_g \vert}$ where $\boldsymbol{\mathcal{A}}_{i,j} = 1$ if $\boldsymbol{\mathcal{M}}_{i,j} > 0$, and $0$ otherwise.

\begin{theorem}
\label{closed walks theorem}
For an adjacency matrix $\boldsymbol{\mathcal{A}}$ where $\boldsymbol{\mathcal{A}}_{i,j} = 1$ if there exists a directed path from node $n_i$ to node $n_j$, the number of paths of length $k$ that starts from node $n_i$ and ends on node $n_j$ is given by $(\boldsymbol{\mathcal{A}}^k)_{i,j}$, where the length is defined as the number of edges.
\end{theorem}
The proof of Theorem \ref{closed walks theorem} can be found in Appendix \ref{appendix: theorem proofs}. By Theorem \ref{closed walks theorem}, the number of RPS cycles passing through a strategy can therefore be found by taking the diagonal of $\boldsymbol{\mathcal{A}}^3$. Note that we \textbf{cannot} go beyond three because the diagonal element of $\boldsymbol{\mathcal{A}}^n$ includes cycles with repeated nodes for $n > 3$.
This completes the quantification of non-transitivity via counting RPS Cycles. The results are shown in Figure \ref{fig: nash clustering and rps cycles elo}.
Finally, we also investigate the use of RPS cycles but using strategy win-rate defined in Equation \ref{strategy win-rate equation}, as a measure of transitive strength instead.
Figure \ref{fig: appendix 2} of Appendix \ref{appendix: alternative measure results} shows the results of this method. Again, the spinning top geometry persists, showing that the RPS cycles method is agnostic towards the measure of transitive strength used.

\subsection{Connections to Existing Rating Systems}

As explained in Section \ref{section: related works} and Equation \ref{Elo formula}, there are various Chess player rating systems throughout the world, and each system is only valid for the pool of players over which the system is defined. Since the results in Figure \ref{fig: nash clustering and rps cycles elo} use game data from Lichess and FICS, they are not directly applicable to different rating systems such as USCF and FIDE. An investigation into the mapping between these rating systems is thus required.

We conduct a short experiment to obtain the mapping between the rating system of Lichess, USCF and FIDE. The dataset for this experiment is obtained from ChessGoals \cite{chessgoals_database}, which consists of player ratings in four sites, i.e., Chess.com \cite{chesscom_about}, FIDE, USCF, and Lichess.
To investigate the mapping, 3 plots are first created, i.e., Lichess to USCF, Lichess to FIDE, and USCF to FIDE. A linear model is then fitted for each plot by MSE minimisation. The results are  shown in Figure \ref{fig: lr regression}. Since the mapping between these 3 rating systems are monotonic as demonstrated in Figure \ref{fig: lr regression}, the spinning top geometry observed in Figure \ref{fig: nash clustering and rps cycles elo} would still hold when mapped to USCF or FIDE rating systems. This is because the relative ordering of the Nash clusters or pure strategies based on Elo rating will be preserved. Therefore, our results can be  direct translated and cross-validated to different rating systems.

\section{Implications  for Learning} \label{section: implications of non-trans}

\begin{figure*}[!t]
    \centering
    \vspace{-15pt}
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2019-Nash-and-Histogram.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-Lichess-2020-Nash-and-Histogram.pdf}} %
    \subfloat[]{\includegraphics[width = 0.33\textwidth]{Publish-FICS-2019-Nash-and-Histogram.pdf}}
    \vspace{-10pt}
    \caption{(Left of each plot) Histogram of Player Elo Ratings and (Right of each plot) Nash Clustering on Lichess 2019 (a), Lichess 2020 (b) and FICS 2019 (c). See Figure \ref{fig: full results} of Appendix \ref{appendix: full results} for the full version of the results. The peak of the histogram as well as the bulk of the data consistently lies in the $1300$ to $1700$ Elo rating range, which tallies with the range of Elo rating in which players tend to get stuck as seen in \cite{chesscom_rating_stuck, lichess_rating_stuck, lichess_rating_stuck_2}, suggesting that players tend to get stuck in this range. Furthermore, the peaks of the histograms coincides with the peak of the fitted skewed normal curve on the Nash Clustering plots. This implies that high non-transitivity in the $1300$ to $1700$ rating range is one potential cause of players getting stuck in this range.}
    \label{fig: nash clustering and histogram}
    \Description{Histogram of player ratings from Lichess 2019 and 2020 games, as well as FICS 2019 games, compared with their corresponding non-transitivity plots using Nash Clustering.}
    \vspace{-5pt}
\end{figure*}

Based on the discovered geometry, we  investigate the impact of non-transitivity on multi-agent reinforcement learning \cite{yang2020overview}.
Spcficially we design experiments to verify the influence of non-transitivity on a particular type of MARL algorithm: population-based learning \cite{lanctot2017unified,perez2021modelling,liu2021unifying,dinh2021online}, a technique that underpins many existing empirical success on developing game AIs \cite{zhou2021malib,lanctot2019openspiel}.
  In fact, on simulated strategies \cite{czarnecki2020real}, it has been found that if the spinning top geometry exists in the strategy space, then the larger the non-transitivity of the game is in terms of layer size (for $k$-layered geometry) or Nash cluster size (for Nash Clustering), the larger is the population size required for methods such as fixed-memory fictitious play to  converge. We attempt to verify if the above conclusion still holds when real-world data is used instead. Furthermore, observing the above convergence trend with respect to the starting population size provides an even stronger evidence for the existence of the spinning top geometry in the strategy space of real-world Chess.

As a proxy for population-based training method, fixed-memory Fictitious Play starts with a population (or set) of pure strategies $P^0$, of a fixed size $k$. The starting pure strategies in $P^0$ are picked to be the $k$ weakest pure strategies, based on the win-rate against other pure strategies, as shown in Equation \ref{strategy win-rate equation}. At every iteration $t$ of the fictitious play, the oldest pure strategy in the population $P^{t-1}$ is replaced with a pure strategy that is not in $P^{t-1}$. The replacement strategy is required to beat all the strategies in the current population on average, i.e., suppose that $\pi$ is the replacement strategy at iteration $t$. Then $\sum_{s \in P^{t-1}} M(\pi, s) > 0$. If there are multiple strategies satisfying this condition, then the selected strategy is the one with the lowest win-rate, i.e., $\pi = \argmin_{s \in S_a} TS(s)$, where $S_a = \{s \in S_g \backslash P^{t-1}, \sum_{r \in P^{t-1}} M(s, r) > 0\}$.

To measure the  performance at every iteration of the fictitious play, we use the expected average payoff of the population at that iteration. Suppose at any iteration $t$, the probability allocation over population $P^t$, which is of size $k$, is $\mathbf{p^t}$. Given the payoff matrix $\boldsymbol{\mathcal{M}}$ is of size $m \times m$, the performance of $WR(P^t)$ is written as
\begin{equation}
WR(P^t) = \sum_{i \in [k]} \frac{\mathbf{p^t}_i}{m} \sum_{j \in [m]} M(P^t_i, S_j),
\end{equation}
where $S_j$ is the pure strategy corresponding to the $j^{th}$ column of $\boldsymbol{\mathcal{M}}$. Fixed-memory fictitious play is then carried out for a number of starting population sizes. The results are shown in Figure \ref{fig: nash clustering and training}.

However, the minimum population size needed for training to converge does not always correspond to the largest Nash cluster size, and there are  two contributing factors. The first factor is that covering a full Nash cluster, i.e., having a full Nash cluster in the population at some iteration, only guarantees the replacement strategy at that iteration to be from a cluster of greater transitive strength than the covered cluster. There are no guarantees that this causes subsequent iterations to cover Nash clusters of increasing transitive strength. The second factor is that the convergence guarantee with respect to the $k$-layered game of skill geometry defined in Definition \ref{definition: k-layered}. Since Nash Clustering is a relaxation of this geometry, covering a full Nash cluster does not guarantee that the  minimal population size requirement will be satisfied.

\section{Link to Human Skill Progression}
Section \ref{section: implications of non-trans} demonstrates that non-transitivity has implications on the performance improvement of AI agents during training. On the other hand, one of the most common issue faced by Chess players is the phenomenon of being stuck at a certain Elo rating \cite{chesscom_rating_stuck, lichess_rating_stuck, lichess_rating_stuck_2}. Having performed experiments using real-data, we also investigate whether non-transitivity has implications on human skill or rating progression, in particular, whether there exists a link between the non-transitivity of the strategy space of real-world Chess to the phenomenon of players getting stuck in certain Elo ratings. This is conducted by comparing the histogram of player ratings at every year using the data  from Section \ref{subsection: payoff matrix construction}, with the corresponding Nash Clustering plot. The results are shown in Figure \ref{fig: nash clustering and histogram}.

A plausible explanation is that in the $1300$-$1700$ Elo rating range, there exists a lot of RPS cycles from Figure \ref{fig: nash clustering and rps cycles elo}, which means there exists large number of counter strategies. Therefore, to improve from this Elo rating range, players would need to learn how to win against many strategies, for example, by dedicating more efforts to learn  diverse opening tactics. Otherwise, their play-style would be at risk of getting countered by some strategies at that level. This is in contrast to ratings where non-transitivity is low. With a lower number of counter strategies, it is arguably easier to improve, since there are relatively fewer strategies to adapt towards.

\section{Conclusion}
The goal of this paper was to measure the non-transitivity of Chess games, and investigate its potential implications for training effective AI agents, as well as on human skill progression.
Our work is novel in the sense that we, for the first time, profile the geometry of the strategy space based on the real-world data from human players; this
in turn prompts the discovery of the spinning top geometry that was previously only observed  on AI-generated strategies, and more importantly,  the discovery of the relationship between the degree of non-transitivity and  the  progression of human skills in playing Chess.
Additionally, the discovered relationship also indicates that in order to overcome the non-transitivity in the strategy space, it is imperative to maintain large population sizes and enforcing diversity when applying population-based  methods in training AIs.
Throughout our analysis, we managed to tackle several practical challenges. For example, despite the variations of Elo rating systems for Chess, we introduced effective mappings between those rating systems on over one billion match records; this enabled direct translations and cross-validations of  our findings  across different rating systems.
Finally, we argue that although we only study Chess in this paper, our conclusions  should also hold  for other complex zero-sum  games such as Go, DOTA and StarCraft; we leave the verification on real-world data for future work.

\clearpage
\bibliographystyle{ACM-Reference-Format}

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40 Ways to Support Struggling Readers in Conten...
"The book's major strengths are its ease of use and the range of approaches to address many different reading issues. You can read straight through for a host of ideas, or you can pinpoint exactly which kind of strategy to explore."
—Kristie Mary Betts, English Teacher
Peak to Peak High School, Lafayette, CO
"Bottom line: This book is reader friendly! Teachers in the content areas can quickly and easily find specific ideas to help students."
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Help for students who are overwhelmed, feel confused, can't remember, lack language skills, or just don't get it.
In today's era of accountability, teachers are expected to help all secondary students understand complex concepts and ideas and demonstrate proficiency on high-stakes tests. To promote success for struggling ...
Chapter 11: Use and Teach Concept Maps
Elaine K. McEwan
In:40 Ways to Support Struggling Readers in Content Classrooms, Grades 6–12
Subject:Secondary English
[Page 51]
Use and Teach Concept Maps
Highly meaningful learning that includes novel problem solving and creativity is only possible in domains of knowledge where the learner has considerable, well-organized prior knowledge.
—Novak (1998, p. 24)
Graphic organizers are very popular and when constructed by students during independent reading, they are highly effective in helping them to make sense of challenging or poorly organized text. Less understood and appreciated for its power to help struggling readers to more easily construct long-lasting knowledge is the concept map: a graphic representation that identifies key concepts and joins them with connecting words showing the nature of their relationships.
Concept maps differ from traditional graphic organizers in that they require students to specify precisely how concepts are related to one ...
Teach Students How to Graphically Organize Text and Concepts by Chunking
Teach Students How to Mark Text as They Read
McEwan, E. K. (2007). Use and teach concept maps. In 40 ways to support struggling readers in content classrooms, grades 6–12 (pp. 51-54). Corwin Press, https://www.doi.org/10.4135/9781483329703.n11
McEwan, Elaine K. "Use and Teach Concept Maps." In 40 Ways to Support Struggling Readers in Content Classrooms, Grades 6–12, 51-54. Thousand Oaks, CA: Corwin Press, 2007. http://dx.doi.org/10.4135/9781483329703.n11.
McEwan, E. K. 2007. Use and Teach Concept Maps. In: 2007. 40 Ways to Support Struggling Readers in Content Classrooms, Grades 6–12, Thousand Oaks, CA: Corwin Press. pp. 51-54 Available at: <http://www.doi.org/10.4135/9781483329703.n11> [Accessed 15 Jan 2021].
McEwan, Elaine K. "Use and Teach Concept Maps." 40 Ways to Support Struggling Readers in Content Classrooms, Grades 6–12. Thousand Oaks: Corwin Press, 2007, pp. 51-54. SAGE Knowledge. 15 Jan 2021, doi: http://www.doi.org/10.4135/9781483329703.n11.
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Secondary English - Innovative Approaches to Poetry

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There's also the psychological component. Dogs are like us—running outside, ideally on trails, does more than just burn off steam. It keeps us in contact with the kind of primal connection that we in the athletic-running-outdoor-fitness realm use copiously to fend off the brand of misery that can come from too much time indoors, too much time staring into screens, too much in the grip of the 24-hour news cycle, too much time in the high-tech comfort zone of modern life. Take it all a step further, this past August, at the Spartan Honors Series event weekend at West Point, Spartan launched the Spartan Canine Heat Presented by Nulo. The Canine Heat allowed owners of service dogs to participate together on the Spartan Kids course. In future events, those with trained dogs from police departments, government agencies, military personnel and those with service/support dogs will be invited to take part in an event with obstacles designed for the Canine Heat.
It should be mentioned that Nulo founder Michael Landa started his company to help mitigate the alarming numbers of overweight pets and pets requiring insulin treatments for diabetes resulting from sedentary lifestyles. So the partnership between Spartan and Nulo with this new series is a great fit and one we are all determined to see make a difference.
Learn more about how Nulo and Spartan teamed up to launch the first ever Spartan Canine Race held at West Point in September.


When my enemies are driven back, they stumble and perish at your presence.
you sat on your throne giving righteous judgement.
The enemy was utterly laid waste.
You uprooted their cities; their very memory has perished.
For he shall rule the world with righteousness and govern the peoples with equity.
Then will the Lord be a refuge for the oppressed, a refuge in the time of trouble.
And those who know your name will put their trust in you, for you, Lord, have never failed those who seek you.
They shall return to the land of darkness, all the nations that forget God.
Put them in fear, O Lord, that the nations may know themselves to be but mortal.
Then Arioch quickly brought Daniel before the king and said to him: 'I have found among the exiles from Judah a man who can tell the king the interpretation.' The king said to Daniel, whose name was Belteshazzar, 'Are you able to tell me the dream that I have seen and its interpretation?' Daniel answered the king, 'No wise men, enchanters, magicians, or diviners can show to the king the mystery that the king is asking, but there is a God in heaven who reveals mysteries, and he has disclosed to King Nebuchadnezzar what will happen at the end of days. Your dream and the visions of your head as you lay in bed were these: To you, O king, as you lay in bed, came thoughts of what would be hereafter, and the revealer of mysteries disclosed to you what is to be. But as for me, this mystery has not been revealed to me because of any wisdom that I have more than any other living being, but in order that the interpretation may be known to the king and that you may understand the thoughts of your mind.
'You were looking, O king, and lo! there was a great statue. This statue was huge, its brilliance extraordinary; it was standing before you, and its appearance was frightening. The head of that statue was of fine gold, its chest and arms of silver, its middle and thighs of bronze, its legs of iron, its feet partly of iron and partly of clay. As you looked on, a stone was cut out, not by human hands, and it struck the statue on its feet of iron and clay and broke them in pieces. Then the iron, the clay, the bronze, the silver, and the gold, were all broken in pieces and became like the chaff of the summer threshing-floors; and the wind carried them away, so that not a trace of them could be found. But the stone that struck the statue became a great mountain and filled the whole earth.
Then King Nebuchadnezzar fell on his face, worshipped Daniel, and commanded that a grain-offering and incense be offered to him. The king said to Daniel, 'Truly, your God is God of gods and Lord of kings and a revealer of mysteries, for you have been able to reveal this mystery!' Then the king promoted Daniel, gave him many great gifts, and made him ruler over the whole province of Babylon and chief prefect over all the wise men of Babylon. Daniel made a request of the king, and he appointed Shadrach, Meshach, and Abednego over the affairs of the province of Babylon. But Daniel remained at the king's court.
God, the Saviour of all, you sent your bishop Willibrord from this land to proclaim the good news to many peoples and confirm them in their faith: help us also to witness to your steadfast love by word and deed so that your Church may increase and grow strong in holiness; through Jesus Christ your Son our Lord, who is alive and reigns with you, in the unity of the Holy Spirit, one God, now and for ever. Amen.


The Vision of the American Football Coaches Association ("AFCA") is to provide education for American football coaches so that they can better serve the public, not only through the enhancement of their technical skills and coaching capabilities, but also through role modeling and character development for young men and young women. AFCA provides substantial benefits to its member coaches, at an extremely low cost, including the annual convention that all have the opportunity to attend and from which all can benefit. In order to provide these benefits, the AFCA must have available a large amount of capital.
The Vision of the American Football Coaches Foundation (the "Foundation") is to provide funds to support AFCA's education vision and activities. The Foundation was created to provide financial resources for AFCA through various fund development initiatives.
The AFCA is a tax-exempt trade organization; however, direct contributions to the AFCA do not qualify for an income or estate tax charitable deduction to the donor. The Foundation is a publicly supported, tax-exempt charity. Donations to the Foundation do qualify for an income and estate tax charitable deductions to the donor. In order to maintain this favorable tax status, the Foundation must meet certain guidelines in raising funds. A significant portion of these guidelines relate to the source of the Foundation's financial support. A publicly supported charity like the Foundation maintains its favorable tax status by receiving a large portion of its financial support from a lot of different donors. All coaches who benefit from AFCA should be supporters of the Foundation. Permissible sources of this public support can come not only from contributions from coaches but also from people within the sphere of influence of the coaches. Donations from these sources are subject to specific rules and restrictions.
Because the Foundation is a publicly supported charity and must receive a large portion of its support from a broad segment of the general public, the Foundation cannot rely solely on contributions from a few wealthy individuals, family foundations or corporations to maintain its favorable tax status. This rule was intended by the IRS to make sure that a publicly supported charity is responsive to a broad segment of the public, and is not controlled by a small group of wealthy individuals, foundations or corporations.
Small donations are as important as large donations because every dollar given by a small donor (including AFCA coaches), the Foundation can receive two more dollars from a wealthy donor or the donor's foundation, without endangering the Foundation's favorable publicly supported tax-exempt charity status. The Foundation needs your personal contribution and needs you to encourage those within your sphere of influence to give also.
The ability of the Foundation to tax efficiently accept donations from large individual donors is dependent upon whether the Foundation receives support from a broad number of small individual donors. The Foundation must receive one-third of its support from a broad cross-section of the general public in order to maintain its favorable tax status. A single donor's gift, including the gift of a donor's family foundation, in excess of 2% of the total contributions received by the Foundation, will not qualify in satisfying the one-third public support test. That is why small donations from a large number of contributors are vital to the Foundation's success in maintaining its favorable publicly supported tax-exempt charity status.
An efficient and cost effective way for coaches and those within the coach's sphere of influence to make gifts to the Foundation is with life insurance. A coach or someone within his sphere of influence can make a significant gift to the Foundation by naming the Foundation as the beneficiary or owner/beneficiary of a life insurance policy on the person's life. Here is how it works.
A person who believes in the Foundation Vision applies for and receives a life insurance policy on his life. When the application for the insurance is signed, the person names the Foundation as the beneficiary of the death benefit of the life insurance policy. During his life, the person (the insured under the life insurance policy) owns the policy and pays the premiums on the policy. No income tax deduction is allowed the insured for his payment of the life insurance premiums. Upon the insured's death, the death benefit of the life insurance policy is paid to the Foundation. The payment of the death benefit to the Foundation is estate tax free to the insured and income free to the Foundation.
As an alternative, the insured, upon applying for the life insurance policy, immediately transfers ownership of the life insurance policy to the Foundation. The Foundation names itself as beneficiary of the death benefit of the life insurance policy. The insured makes income tax deductible gifts annually to the Foundation to pay the premiums on the life insurance policy owned by the Foundation on his life. Upon the insured's death, the death benefit of the life insurance policy is paid to the Foundation. The payment of the death benefit to the Foundation is estate tax free to the insured and income tax free to the Foundation.
The Foundation's Vision is an ambitious undertaking. It is strategic to the future success of AFCA. The accomplishment of the Foundation's Vision will require contributions from many different sources. Other tax and financial planning techniques to support the Foundation will be discussed in future installments of this column.


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Our G.A.N.G.
Smart Investment
GRIDIRON GROUP
Push Forward
EXPERIENCE GREATNESS
Team Inspire
DREAM.HOPE.BELIEVE
Impressive Return
OUR YOUTH.OUR COMMUNITIES.OUR RESULTS
Gridiron Group	GIadmin2014	2019-05-10T23:03:42+00:00
Gridiron Group is inspired by youth that have dreams greater than their circumstances and the potential to achieve them when given the opportunity.
We uniquely teach, empower, and equip communities and their youth to reach championship levels of success that leave a legacy of winners.
Our mission is to ignite and foster opportunities for youth and families that embrace athletics and academic empowerment as the framework for success.
I just wanted to take a minute to thank you personally for the donation of a signed Rex Ryan Football to our gift auction. I am very excited to add it to the auction I am sure it will garner a great deal of enthusiasm and ultimately funds for our foundation.I was just on your website and then FB page and I am so impressed with what you do for today's youth. As a fellow, albeit at a MUCH lower level, athlete I am all too familiar with the benefits of Athletics on the lives of our youth. Your mission is a proud and necessary one. Thanks for all you do, and for your support of my cause.If I can ever repay the favor don't hesitate to ask.
Brian A. Flaherty, Denise Marie Flaherty Mem. Foundation Inc., President 2013
"I wish to once again thank Damian and Darian for giving our young people a wonderful experience this past Wednesday. The young people truly had a great time and the rain did not damper their spirits. Look forward to discussing some possibilities with you guys in the fall. Once again, thank you!"
Dordy Jourdain, Bedford Stuyvesant - YMCA 2013, Executive Director
"I had a great time at Damian's camp this year. It's such a joy to see players that have done and been through so much still come back and help the youth. To see so many former guys I played with and against on the gridiron, join forces to give back to the community, not because they have to but because they want to, was a great feeling of purpose. The way those kids listened and were attentive to the knowledge was very inspiring to know that the youth can and will have a future. Keep it up My Brotha, The World and the Community need more like you!"
Todd J. Duckett, Marketing & Business Development, Former NFL Running Back
"My experience with GridIron Group, has been a blessing, a learning experience, inspiring, and a joy to be apart of. The Love and Passion that GridIron's founder and President Damian Gregory has for everything he does is palpable, just being around him you can feel his emotion for the growth and education of the kids, the city of Lansing, Sexton H.S., and his firm belief in using Football and other sports as a tool to achieve other goals and become more successful in the game of life. I really enjoy having the opportunity to teach the kids some of the lessons I've learned over the years of being involved in sports, and hope that I gave each of them something to hold on to, remember and learn from.
I am grateful for them taking the time out of their summer and coming to a football camp and fighting through the heat because they didn't have to be there but they wanted to be and they competed through everything I or any of the other coaches threw at them. Though they came to learn from me, I also learned a lot from them. Once again I'd like to thank Damian and GridIron Group."
Jesse R. Miller , Offensive Lineman Michigan State University, 2004-2008
"I am very proud to be a partner and affiliate with GridIron G.A.N.G. Inc. With so many opportunities available for our youth to go astray and get in to destructive behaviors it was awesome to get together a group of professional athletes, coaches, and speakers to work with children from all backgrounds to inspire our youth first, through Hard Work & Discipline, and to dream, and to dream big. I really enjoyed the experience and can't wait until the next."
Clarence Love, Super Bowl Champion '00 Baltimore Ravens - AFC Champion '02 Oakland Raiders
Thank you for a great day! Touchdown for Success is always fun for our students and staff, but I think each year it gets better. Thank you for all of your hard work and enthusiasm. We look forward to field day every year because of the energy and inspiration you and the guys give us.
Thank you for a fun day. These days help remind us why we do what we do.
Megan LoBue, The Calais School Board Trustee, The Calais Foundation
While knowledge and information are extremely important, our quality of life and relationships really define who we are. Gridiron Institute focuses on the core components of excelling at life, by balancing life with leadership, dedication, learning to win and lose, pushed by motivation.
The Institute teaches through experience by combining athletics (Our Gridiron Football Camps) and academic empowerment (Life Skills Training) to assists youth at every level by motivating, developing self- esteem, reducing risky behavior, communicating with diplomacy, overcoming stigma, and discovering support channels that challenge them to achieve at their greatest ability.
Lansing Camp Registration
Gridiron G.A.N.G. is non-profit under the Gridiron Group umbrella. Our goal is to bring together leaders of today and tomorrow to build a flourishing educational network for communities in need. Our organization aims to enrich the lives of the young men and women we come into contact with, using sports as a common ground and education as the foundation.
Gridiron G.A.N.G. values parents as the cornerstone of a successful and productive community. Proper growth and development of children within the family structure from prenatal to adolescence is a proven asset to any community. Our physical engagement program seeks to address these developmental needs of youth by providing athletic and educational mentors that work with each youth during Gridrion Institute Football Camp.
Our G.A.N.G. has a collective of individuals that have experience in the NFL , Medical & Health, and Organizational Program Development. Each person brings unique skills serving in different capacities with Gridiron Institute Football Camp, and Gridiron G.A.N.G., giving added-value to Gridrion Group umbrella as a whole.
Damian Gregory, Founder & Former NFL Player
Damian Gregory former NFL Player known for his athletics on the field and his championships off the field believed in a dream greater than himself. Damian knew from a young child he wanted nothing more than to be an agent of change, so he did just that. He set out to challenge the leaders of tomorrow, today's youth, by providing what is now the well-known Gridiron Group. Throughout the U.S., Gridiron provides training and practical resources that link support and education together with a touch of inspiration Giving Adolescents New Goals, a new life and the faith to fly!
James Lamar, Medical Marketing
James Lamar has a medical business marketing background with seven years in the Home Care Health field, as well marketing to different medical fields, and doctors to coordinate with patients primary doctors and specialty doctors ensuring the quality of care.
Over the years James Lamar has worked for several HMO and CMS insurance to better ensure the outcome for patients and their insurance as well as several other companies as consultant for relationships between organizations, partnering now with Gridiron Group with Patient Provider Solutions.
Clarence Love, Super Bowl Champion
Clarence Love a Super Bowl Champion and GANG member shares his humility with everyone he comes into contact with as he challenges youth to overcome every barrier that stands in the way of their dreams. Love attended college at the University of Toledo and was drafted by the Philadelphia Eagles in the 4th round of the 1998 NFL Draft. He played with the Eagles, the Baltimore Ravens where he won Super Bowl XXXV, and the Oakland Raiders. Clarence's passion for youth keeps him extremely active in his community.
Ben Kelly, Super Bowl Champion
Ben Kelly a Super Bowl Champion and GANG member is a former American football cornerback of the National Football League, Arena Football League and Canadian Football League. He was drafted by the Miami Dolphins in the third round of the 2000 NFL Draft. He played college football at Colorado.
Kelly was also a member of the New England Patriots, Denver Broncos, Los Angeles Avengers, Calgary Stampeders and Grand Rapids Rampage. He earned a Super Bowl ring with the Patriots in Super Bowl XXXVI. Kelly is well represented in the Aloha Bowl record books after he returned a kickoff for a TD against Oregon on Dec. 25, 1998
Ron Rice, President of the NFL Alumni Detroit Chapter
Ron Rice a former NFL player and President of the NFL Alumni Detroit Chapter, and National Board member of Charity Ron's passion is for players and their family as their advocate for success and quality of life.
Under the NFL Alumni Association, the NFL Alumni Foundation is a 501 (c)(3), non-profit organization composed of former professional football players, owners, coaches, administrators and cheerleaders who have joined together to support youth charities. Additionally, Associate Members play an integral role in supporting the charity and educational goals of the NFL Alumni.
Mark Word, Former NFL Player
Mark Word is a former professional American football player who played defensive end for three seasons for the Kansas City Chiefs and the Cleveland Browns. While he was signed to the Cleveland Browns, Mark suffered a shoulder injury during training camp and was forced to spend the rest of the season on the injured reserve list. In 2005, Mark finished out his football career playing for the Montreal Alouettes of the CFL.
Since then Mark has been involve with numerous programs in various communities working with youth specifically on and off the field. Over the past several years Mark has been a part the Gridiron G.A.N.G. helping with Gridiron Institute Football Camps and other initiatives with the Gridrion Group.
Gridiron Institute & Bayshore Education Center (B.E.C.)
7th Annual Youth Football Camp
Lely High School | 1 Lely High School Blvd. | Naples, FL 34113
Calais Foundation "Touchdown For Success" Youth Empowerment Day
June 5th 2019 | Calais School | 5 Highland Avenue | Whippany, NJ 07981
Calais School
5 Highland Avenue | Whippany, NJ 07981
10th Annual Gridiron Institute Football Camp
July 20, 2019 | Southside Community Center | 5815 Wise Road | Lansing, MI 48911
July 20, 2019 | 9am-3pm
Southside Community Center
5815 Wise Road | Lansing, MI 48911
$20,000 – $50,000 | GridIron "Big Game" Presenting Multi City Tour
Present Gridiron's proven Health, Fitness and Nutrition program to markets and cites of your choice in collaboration with your organization's cause marketing program or community relation efforts. Event can also be done in the same market as "The Big Game" for maximum impact and exposure.
The presenting sponsor is our partner and "co host" "The Big Game" Tour.
Media day and kickoff event to announce partnership and tour
Opportunity for up to 20 organization representatives to participate in drills and work alongside GridIron coaches during the camp. Includes tee shirt and lunch for all attendees.
Custom logo and or message can be included on tee shirts, towels and water bottles. • Custom step and repeat featuring your organization's logo
Presenting field signage with logo and/or message
Group photo with participants and coaches
Logo, ad and message of GridIron Website and social media accounts.
Mention in press release and media interviews
Named as the "Presenting Sponsor" on your GridIron Football Camp promotional materials and flyers • Initiatives
Private after party dinner and reception for sponsor, coaches, and VIP invited guests
Big Game Sponsor
$15,000 | GridIron "Overtime" Presenting Two Day Camp Sponsor
The presenting sponsor is our partner and "co host" of a two day GridIron Football Camp. Includes Stadium, Game Day Gear, Energy, & Hydration sponsorship benefits.
Named as the "Presenting Sponsor" on your GridIron Football Camp promotional materials and flyers
Private after party dinner and reception for sponsor, coaches and VIP invited guests
Overtime Sponsor
$10,000 – GridIron "Game Day" Presenting Camp Sponsor
The presenting sponsor is our partner and "co-host" of the GridIron Football Camp. Includes Stadium, Game Day Gear, Energy, & Hydration sponsorship benefits.
Custom logo and or message can be included on tee shirts, towels and water bottles.
Custom step and repeat featuring your organization's logo • Presenting field signage with logo and/or message
Game Day Sponsor
$5000 – Stadium Sponsor
Provide sponsorship of our field, referees and medics.
Custom step and repeat featuring your organization's logo
Prominent field signage with logo and/or message
Opportunity for up to 5 organization representatives to participate in drills and work alongside GridIron coaches during the camp. Includes tee shirt and lunch for all attendees.
Group photo with camp participants and coaches.
Logo, ad and message on GridIron website and social media accounts
Logo included on GridIron Football Camp promotional materials and flyer
Stadium Sponsor
$2500 – Gear Sponsor
Provide the official tee shirts, towels & water bottles for camp participants and coaches.
Custom logo ad and/or message to be included on all sponsored merchandise and apparel
Group photo with camp participants and coaches
Gear Sponsor
$1250 – Energy Station Sponsor
Provide healthy lunch, snacks and beverages for the camp participants and coaches.
Energy Station Naming Rights
Custom logo, ad, and/or message displayed in at Power Station (subject to approval)
Logo, ad and message of GridIron website and social media accounts
Energy Station Sponsor
$500 – Hydration Station Sponsor
Sponsor unlimited ice, water, cups, and energy drinks from our campers.
Hydration Station Naming Rights
Custom logo, ad, and/or message displayed at Hydration Station (subject to approval)
Hydration Station Sponsor
$250 –Youth Experience Sponsor
Sponsor one child from an underserved population to participate in a GridIron Football Camp. Ignite and foster a unique opportunity for youth that embrace athletics and academic empowerment as the framework for success.
Youth Experience Sponsor
Other of $1 or more
If you would like to give a personally donate for whatever amount. You may do so by giving a donating of at least $1 or more.
Other of a $1 or more
Download Sponsorship Packet
473 Mountain View Lane
For more information or questions email info@gridirongroup.org
Feature Articles & Pictures
LANSING, Mich. (WILX) – Using the game of football to give life lessons and create bonds within the community. News 10's Marcus Dash spoke with a Lansing native and former NFL player on how he's using football to teach life values to the area's youth.
A rainy Saturday morning wasn't stopping dozens of Lansing's kids from coming to play football at the Schmidt Southside Community Center in Lansing. A free camp put on by The Gridiron Institute led by former NFL player Damian Gregory. A way to give back to the community he grew up in. Read Full Article Here…
3rd Annual BEC Gridiron Football skills Camp at Lely High school in Naples, Fla. We had an amazing turnout of kids of all ages. The camp was a resounding success, the Collier Citizen wrote a very compelling article about the weekends events and impact on the Naples youth…Read full article here
The Touchdown for Success has become a vital part of that mission. The interactive, day long program includes NFL style drills, a nutritious lunch, motivational message and fun gift bag. In addition the Touchdown for Success initiative provides ongoing health, fitness, and wellness support year round by offering programming and resources for youth with multiple learning disabilities."…Read Full Article Here
LANSING, Mich (WLNS) – 102 inner-city youth participate in the 6th annual Gridiron Institute Football Camp at Southside Community center. The city of Lansing funds the event and speakers like Lamar Wilson encourage the youth to stay educated and work towards success…Read Full Article & Watch Video Here
Damian Gregory, 39, credits his grandparents, his tight-knit community and Magic Johnson in shaping his life. And he promised himself that as soon as he became successful, he would pay it forward.After his football career, he founded Gridiron G.A.N.G. (Giving Adolescents New Goals), a 501(c)(3) nonprofit that hosts free football day camps with an educational component…Read Full Article & Watch Video Here
Copyright 2016 Gridiron Inc. | www.gridirongroup.com | All Rights Reserved | Site by Matthew J. Thorpe


The Chief Executive of Central and North West London NHS FT (CNWL), Claire Murdoch, has been appointed by Simon Stevens as the new NHS National Mental Health Director.
NHS England's independent Mental Health Taskforce chaired by Paul Farmer of MIND recently published its report setting out a vision and costed improvement plan for mental health over the coming five years.
Claire Murdoch, a registered mental health nurse for 34 years, will play a key role in leading the implementation of the report. Ms Murdoch, CEO and nurse in one of the largest integrated community and mental health trusts in the country, has been an HSJ top 15 Chief Executive for the last two years and has a passion for driving improvements in care and support, and better prevention and self-care.
Her trust provides physical, palliative care, mental health, learning disability, sexual health, prison, dentistry and a range of specialist services to many counties and London boroughs and has doubled in size and diversity since she took over in 2007.
She is also a Director of Imperial Academic Health Science Partnership, a Non-Executive Director at Health Education England NW London and Chair of the Cavendish Square Group (the Ten London Mental Health Service providers).
Her appointment – made following a public job advert and an open competition – follows the earlier appointments by Mr Stevens of Cally Palmer, CEO of the Royal Marsden to become NHS England's National Cancer Director, and Pauline Philip, CEO of Luton and Dunstable Hospital, to become National Urgent and Emergency Care Director. Like Cally and Pauline, Claire will undertake her new national role on secondment from CNWL where she will remain chief executive.
Fantastic to read that the implementation will be lead by a mental health professional.
I look forward to reading the updates and information over the next few months, years.
Just came across this article whilst going through Twitter and am absolutely impressed by the women who have been appointed to posts that are so vital to improve both mental and physical health. As a mental health nurse and always looking for role models Claire is certainly one who will become mine. I will look forward to following her on Twitter as well as follow her progress and hopefully the difference she makes. It's so vital that those who shape mental health services has her experience and background. Best wishes to her.
Will she use her influence to remove the chief executive and board of Southern Healthcare who have undermined the confidence of the patients and general population of the south. This trust and board do not have the confidence of the population they serve and their continued clinging to power is bringing mental healthcare and the NHS into disrepute.
What a superb appointment. Those of us involved in workforce planning, education and training know well how Claire brings her skills, insight and commitment across boundaries to tackle complex problems and find sustainable solutions. A terrific (and courageous) strategic thinker.
This is an excellent appointment which will hearten those of us who have need of MH services.
The creation of a clinical director for MH, rather than this so important area of care being an addition to long term conditions, sends an important message and can only be for the good.


BioOne is a natural biodegradable bacteria that will keep drains clean without harming pipes or finishes on fixtures. 2 ounces once a month down each drain will help keep them flowing freely. BioOne is also safe for septic systems and will help prolong pump outs. Stop by Perrysburg Plumbing to purchase for $65 each.


Baer's Home Outfitters was located at 313 S. Michigan St. in South Bend, Indiana. Founded in 1945 by Melvin and Lucille Baer, the store was a popular home furnishings place for quite a few years in downtown South Bend. Baer's left Indiana years ago but is still going strong with 16 different showrooms in Florida.
This 1967 Christmas mailer is from the Winter of 1967 and is a great glimpse into 1960s interior design and furniture.
Find fact, I currently own two of the items listed in this catalog.


When you are involved in an accident events can be confusing and important information can be overlooked. That is why we made it simple for you to record vital details of the accident. You can print the accident form in the tools and resources section. If you have further questions please contact us.
If you have an accident follow these simple steps to speed up the processing of your claim. The more information we have, the faster and more fairly and more fairly we can settle your claim.
Do not discuss the accident with anyone but the the Police at the scene.
Do not discuss detail of your insurance coverage or your policy limits.
Get a Police Report Number if possible.
Write down the names, addresses and phone numbers of others involved in the accident and witnesses.
Write down the license numbers and insurance company names and policy numbers of everyone involved in the accident.
Use the form to record your own recollection of the accident as soon as you can, while details are still fresh in your memory. Write the date and time of the accident and the weather and road conditions. Write down all details you can think of even if they do not seem important to you. IMPORTANT - If you have a camera or a Cell Phone with picture capability take some shots of the scene, vehicle damage etc.
Were you involved in the accident?
Did anyone appear to be hurt?
Who, if anyone, do you feel was responsible?
Your Information that is needed.
Passengers Information that is needed.
What to do in case of an acccident. Print this form and place in glove box.


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Doucoure flattered by PSG interest as he outlines desire to leave Watford
The French midfielder wants to move on from Vicarage Road to experience Champions League football and boost his international prospects
Click Here: mochila fjallraven
Abdoulaye Doucoure is intent on leaving Watford amid reported interest from Paris Saint-Germain.
The French midfielder has been strongly linked with the Ligue 1 champions , who are apparently looking at bolstering their midfield options by signing the 26-year-old.
And Doucoure says he's "flattered" by the reported interest, insisting that Watford are aware of his ambitions to play for a team in the Champions League.
He told Canal+ : "I think I am going to leave Watford. The club know my ambitions and want to help me get to the next level.
"I'm flattered by the interest of PSG but for the moment there's nothing concrete. I will focus on my performance for my club. I've not had contact with PSG and they've not contacted Watford.
"My goal is to play well for Watford and then maybe join a Champions League team.
"This winter we'll study all possibilities on offer. The most important thing for me is to continue to progress and with my performances.
"Staying in England is a possibility and the Italian league is becoming more interesting. There are clubs who have returned to show interest in me again.
"The first criteria for me is that the club plays in the Champions League."
Watford signed Doucoure from Rennes back in 2016, with the midfielder having impressed with his performances in the Premier League.
He's since played 79 games in England's top flight, scoring a total of 11 goals and claiming eight assists.
The Hornets are currently seventh in the table and are unbeaten in their last four games in all competitions, including Saturday's 2-1 win away at Crystal Palace that saw Tom Cleverley score a 74th-minute winner.
Next up for Javi Gracia's side is a home meeting with Burnley, who have won their last four matches in all competitions.
Super Rugby 2016: Five wallabies bolters
What is Neymar's release clause & transfer value? PSG & Brazil star's price tag


I can spot a good cook at a thousand meters. Good cooks give themselves away without knowing it. Of course you have to be able to recognize the behavior and it differs depending on their surroundings. In a grocery store, a good cook spends next to no time in the centre of the store. Canned goods and pre-made food rarely make it into their grocery carts. They are the perimeter shoppers. They hover in the fresh food aisles where vegetables, fruit, fresh herbs, seafood, meats and cheeses live. It's actually quite comical watching a cook fondle food. Everything needs to be touched, gently squeezed and most importantly smelled.
Now if you happen upon a good cook in a restaurant, they are either euphoric or owly because their meal is either perfect or sadly pedestrian. They hate feeling like they should have stayed home and made it themselves.
You have to understand that fifteen years ago you couldn't buy prosciutto anywhere in Saint John except at our shop and I had just gotten my first order in. I wanted to know how she knew that I was carrying prosciutto before I had advertised and more importantly who she was. A girl that was willing to hike over building debris for imported food was a girl that I needed to meet.
And meet we did and have been friends ever since. Her love of food and cooking has had a profound influence on my cooking journey. To give you a little perspective into our friendship, when she had her Viking state of the art stove installed we paid it homage by sitting on the floor in front of the oven watching a prime rib roast while drinking Prosecco. She also introduced me to my favorite bubbly not to mention Jamie Oliver, The River Cafe girls, London and Greece! We've cooked together for years. She would be one of those good cooks that would be fun to follow around the grocery store!
Linda is queen of the apps. Her 'I'm just going to put out a few munchies while I get dinner ready' would rival most full course dinners. She served Ralph and I this dip on a Sunday afternoon while the guys watched football and we played in the kitchen. I've since served it many times and it's always a home run!
In a food processor, pulse the ginger until finely chopped. Add the 3/4 cup cashews; process until smooth, 2 to 3 minutes.
Add the yogurt, basil, brown sugar, and curry powder; season with salt. Process until incorporated, 1 to 2 minutes, scraping down the sides as needed.
Transfer to a serving bowl. Crush the remaining cashews I used my mortar and pestle then sprinkle crushed nuts over the dip.
THE LOVE: The original recipe called for cilantro. I've used basil, Italian parsley and oregano, all with great results!
This entry was posted in appetizer and tagged Cashew, Dips, food, friends, green curry, life, recipes by Michelle. Bookmark the permalink.
Oh I have been waiting for this….as one who has tasted this dip…I can tell you with 100% certainty that this will be the best you will ever taste. Thanks for posting!
As someone who has had the pleasure of tasting this dip…I can assure you 100% that this will be the best you have tried. Delicious.
I was wondering who you were talking about! Great post mom, and I love this dip!
Wonderful! I think this is a must make dip.
I'm one of those 'food handlers' My husband says it takes me forever to shop for groceries because I have to touch it all. There is no such thing as running in to grab something. It's one of life's great pleasures!
I tried this tonight and it was great. Ann and I are going to your favourite restaurant this week for our 43rd wedding anniversary. Always enjoy looking at your recipes.