\section{Introduction}
\subsection{Research question}
We investigates the reality and accuracy of
evolutionary game dynamics theory \cite{2011Sandholm}\cite{dan2016book}
in high dimensional human game behavior experiments \cite{Behavioral2003}.

In game statics theory,
the mixed strategy Nash equilibrium
established around 1950 is the central concept.
Until 1987, a representative game experiment,
the O'Neill game\cite{1987Nonmetric},
provided the first illustration
that laboratory human strategy behavior
can be accurately captured by the central concept.
Table 1 shows the payoff matrix of
the experiments involving a long-run repeated,
discrete time, two-person, zero-sum game.
Since then, this game has been extensively
repeated in various experimental settings
 \cite{Binmore2001Minimax}\cite{Yoshitaka2013Minimax}
 and analysis \cite{Brown90}.
So far, the literature has focused on the long time strategy distributions
and the time dependence of individual behaviors,
but seldom explored social dynamics structure.
Here, the dynamics structure means
the geometrical pattern of
the evolutionary trajectories in the game state space.


Intuitively, two reasons hinder the study of the dynamics pattern.
First, the game has a high-dimension strategy space;
the means to handle the high dimensional dynamic pattern
remains an unresolved problem \cite{dan2020}.
Second, in a laboratory game experiment,
human strategy decision-making is highly stochastic,
and testing the dynamic pattern is a difficult task
\cite{Behavioral2003}\cite{1999What},
especially in a discrete time game \cite{dan2014}.
Past decade has seen the improvement on test out cyclic dynamical pattern
in some presentative games, like rock-paper-scissors \cite{dan2014,wang2014social}
and matching pennies-like $2\times 2$ game \cite{wang2014}.
But actually, all of these are two dimensional, not high dimensional.
In a real game, there always many players
and many strategy involve,
then the evolution trajectory will be in high dimensional state space.

To study high dimension game dynamics structure is not trivial.
As a cutting-edge question, not only in laboratory game experiments,
it appears in seeking the regularity in real game dynamics processes.
It is meaningful for the natural science and social science,
as well as in engineering and artificial intelligent visual world
\cite{dan2016book}\cite{2011Sandholm}\cite{dan2020}.
One can say that, in a real or visual world,
wherever the game theory can be applied,
during game playing, there always exists evolutionary processes;
Then in the processes which appearing as geometrical trajectories,
the regularity always a curious question.





\subsection{Background of eigencycle approach}
We thus develop an approach, the \textbf{eigencycle set}
to identify the high dimensional dynamics.
Our approach bases on
dynamical systems theory, a branch of mathematics.
As established before,
for the local dynamics near the equilibrium,
the linearization and
analytical solutions based on the
eigenvalues and eigenvectors
can be applied.
Suppose that an initial probability distribution can be expressed as
a linear combination of the eigenvectors $\xi$ as \cite{Lu5083},
\begin{equation}\label{eq:eigdeco1}
 {p}(0) =  a_1 {\xi}_1 + a_2 {\xi}_2 + ... + a_k {\xi}_k,
\end{equation}
wherein $\xi_i$ is associated eigenvector
of the eigenvalue  $\lambda_i$;
The probability will evolve in time according to
\begin{equation}\label{eq:eigdeco2}
{p}(t) = e^{\lambda_1t} a_1 {\xi}_1 + e^{\lambda_2 t} a_2 {\xi}_2
         + ... + e^{\lambda_k t} a_k {\xi}_k,
\end{equation}
where, the coefficients
$(a_i, i \in [1,k])$ and
$({\xi}_i, i \in [1,k])$ are independent of time $t$,
and the only time depending term is $e^{\lambda_i t}$.
Instated of the coefficients $a_i$  or the eigenvalue $\lambda_i$, we concern
the components of the eigenvectors.
Suppose a normalized eigenvector having $s$ components
and denoting as
${\xi}_i = (\eta_1, ..., \eta_m, ..., \eta_n, ... \eta_s)$,
we can present the $m$-th dimension evolution as
\begin{equation}\label{eq:eigdeco2}
{p_m}(t) = e^{\lambda_1 t} a_1 {\eta_m}_1 + e^{\lambda_2 t} a_2 {\eta_m}_2
         + ... + e^{\lambda_k t} a_k {\eta_m}_k.
\end{equation}
Similary, the $n$-th dimension evolution as
\begin{equation}\label{eq:eigdeco2}
{p_n}(t) = e^{\lambda_1 t} a_1 {\eta_n}_1 + e^{\lambda_2 t} a_2 {\eta_n}_2
         + ... + e^{\lambda_k t} a_k {\eta_n}_k.
\end{equation}
Obviously, $p_m(t)$ and $p_n(t)$ will form a trajectory in the two dimensional space
(hereafter denoted as $\Omega^{mn}$
which is a  subspace of the $s$ dimensional space).

Our starting point comes from the synchronization
between two components belong to a given eigenvector.
There exists two time invariant:
First, the phase angle difference of any two components
$\left(\arg(\eta_m)-\arg(\eta_n)\right)$ is invariant along time;
Second, the amplitude of each component $\eta_i$, $||\eta_i||$,
is invariant along time. Then we study
the  inference of two components in general conditions.
By which,
we develop a new theoretical expected observation,
called as eigencycle set,
to identify human dynamics behaviors in a high-dimensional game.


The main contributions are as follows:
(1) As illustrated in Figure \ref{fig:fine_structure},
we find the fine and hyperfine structures
in high-dimensional game dynamics pattern in a historic human game experiment data,
with the cycle measurement accuracy increased by an order of magnitude.
(2) We propose a tool, the eigencycle set analysis,
for general dynamics system analysis. We illustrate its validation
 in the O' Neill game data who firstly illustrated the accuracy of Nash equilibrium in 1987.   

Inspired by the finding in experiments as well as the approach
which deeply rooted in standard dynamics system theory,
we suggest that, the central concept of
game dynamics theory needs to be reconsidered,
which will be discussed later.


\subsection{Contents organization}
The rest of this paper is organized as follows:
In section 2,
we introduce the O'Neill game as the example,
and analytical calculate the eigen system
of the replicator dynamics equations; Then
we illustrate the  eigencycle set approach
and the theoretical results, 
from which the fine structure 
and hypefine structure are predicted;
In section 3,
we introduce the six human behavior game experiment data,
and report the experimental results of
the high dimensional cycle motion
projection on the two dimensional subspaces;
In section 4,
we validate the theoretical eigencycle set
 approach for bridging theory and experiments.
 In this section, the discovery of the fine structure and
 the significant evidence of the hyperfine structure are reported.
In section 5,
we summarize our contributions, the implication of our finding and the approach
and present the scope for future research. 
At the last, by numerical results comparisons, we suggest that, 
the 'invariant manifold' can be  
the potential central concept for game dynamics theory.  























\section{Theoretical results on the eigencycle set}
\subsection{The eigenvector in the O'Neill game}
The O'Neill game is a zero-sum, 4$\times$4 game. Table \ref{tab:gamemodel}
shows the payoff matrix:
\begin{table}[h!]
\caption{The O'Neill zero-sum game matrix}
\begin{center}
\begin{tabular}{c|rrrr}
 \hline
&B1&B2&B3&B4\\
 \hline
A1&1&-1&-1&-1\\
 \hline
A2&-1&-1&1&1\\
 \hline
A3&-1&1&-1&1\\
 \hline
A4&-1&1&1&-1\\
 \hline
\end{tabular}
\end{center}
\label{tab:gamemodel}
\end{table}
To investigate dynamics behavior in a laboratory experiment game,
we use the replicator dynamics equation \cite{2011Sandholm}:
\begin{equation}\label{eq:repliequl} \Dot{x}_j=x_j*(U_j - {\overline{U_X}}),
\end{equation}

$x_j$ is the $j$th strategy player's probability
in the population with the $j$th strategy player included, and
$\Dot{x}_j$ is the evolution velocity of the probability;
 $U_j$ the payoff of the $j$th strategy player,
and $\bar{U}_X$ is the average payoff of
the population with the $j$th strategy player included.
The explicitly expression of these nonlinear differential equations
are shown in SI.

One by one, we assign strategy probabilities (A1, A2, A3, A4)
to $(x_1,x_2,x_3,x_4)$, respectively. Similarly,
we assign strategy probabilities (B1, B2, B3, B4)
to $(x_5,x_6,x_7,x_8)$, respectively.
Then, at any time, the system must be
an eight-dimension space, wherein
the Nash equilibrium is
$$x^*:=(x_1^*,x_2^*,...,x_8^*)=(2/5,1/5,1/5, 1/5,2/5,1/5,1/5,1/5).$$
This eight-dimension space has two concentrated
$$ x_1+x_2+x_3+x_4=1 ~ \cap ~ x_5 + x_6 + x_7 + x_8=1 \cap x_k \geq 0 ~ (k \in {1,2,...,8})$$
From the Jacobian matrix (or character matrix, or derivative matrix)
of the velocity vector field
$F:=(\Dot{x}_1,\Dot{x}_2,...,\Dot{x}_8)$ at Nash equilibrium $x^*$ \cite{2011Sandholm},
we can calculate the eigenvalues $\lambda$ and
their related normalized eigenvector $\xi$'s components
$(\eta_1,\eta_2,...\eta_8)$ explicitly.
Here, the components
$(\eta_1,\eta_2,...\eta_8)$ are one-by-one corresponding
to $(x_1,x_2,...,x_8)$.
The explicitly analytic results of
the eigenvalue $\lambda$ and eigenvector $\xi$
are shown in Table \ref{tab:eigenval_vec_cyc}.
From the sign of the eigenvalues
we can read off that the Nash eqilibrium (fixed point)
is linearly along the eigendirection of the real eigenvalues.
Details of the eigen system deduction are shown in appendix section \ref{app:derive_eig}.
\subsection{Eigencycle set}
\begin{flushleft}
\textbf{Definition}
\end{flushleft}
We call the circle constructed by the
two components $(\eta_m,\eta_n)$ within one normalized eigenvector
${\xi}_i = (\eta_1, ..., \eta_m, ..., \eta_n, ... \eta_s)$
the \textbf{eigencycle}, marked as $\sigma^{(mn)}$.  It is calculated as:
\begin{equation}\label{eq:the_ecyc}
 \sigma^{(mn)}=\pi \cdot ||\eta_m|| \cdot ||\eta_n||
               \cdot  \sin\big (\arg(\eta_m)-\arg(\eta_n)\big ),
\end{equation}
where $m$ and $n$ are the abscissa and the ordinate dimension
of the two dimension subspace, respectively.
$||\eta_m||$ and $\arg(\eta_m)$ indicate
the amplitude and the phase angle
of the $\eta_m$, respectively.
$\sigma^{(mn)}$ can determine the direction and amplitude of the eigencycle.
An alternative and equivalent presentations of the eigencycle value are
\begin{eqnarray}\label{eq:the_ecyc}
       \sigma^{(mn)}&=&\pi \cdot \big (\Re(\eta_m)\Im(\eta_n)
                                    - \Re(\eta_n)\Im(\eta_m)\big ),\\
                    &=&\pi \cdot \Re\big(\eta_m^\dag \eta_n \big)
\end{eqnarray}
wherein the $\Re(\eta_m)$ is the real part of
the complex number $\eta_m$ and
the $\Im(\eta_m)$ is the imaginary part;
the superscript $\dag$ indicates the conjugate
of the complex number.

\begin{flushleft}
\textbf{Eigencycle values of the O'Neill game}
\end{flushleft}
According to this formula, for the O'Neill game,
Table \ref{tab:eigenval_vec_cyc} lists the eigencycle values
of the eigenvectors of the replicator dynamics.\\
\begin{flushleft}
\textbf{Interpretation of the definition}
\end{flushleft}
\begin{itemize}
\item \textbf{Invariance of the eigencycle:}
    An eigencycle is constructed by two components in a normalized eigenvector,
    and hence, its value is invariant.
    This is according to Eq.~(\ref{eq:the_ecyc}) and the following points:
    (1) the mode of each component is fixed;  and
    (2) the phase difference between two given components in an eigenvector is fixed;
    Because, they are modulated by the same eigenvalue,
    that is, the relative phase difference between these two components remains fixed.
\item \textbf{Number of eigencycle:}
    There are $N(N-1)/2$ independent eigencycles corresponding to
    a given $N$-component normalized eigenvector,
    as there are $N^2$ pairwise combinations of each component
    in an $N$-dimensional eigenvector.
    Considering that the $N$ self-combination of $(\eta_m,\eta_m)$ is trivial ($\sigma^{(mm)}$ = 0), and $(\eta_n,\eta_m)$ and $(\eta_m,\eta_n)$ are simply reversed ($\sigma^{(mn)}$ = $-\sigma^{(nm)}$), only $N(N\!-\!1)/2$ combinations remain.
\item \textbf{Eigencycle set:} The eigencycle set, as a vector denoted by $\Omega^{(mn)}_{\xi_k}$, is defined to represent a set of $N(N-1)/2$ eigencycle elements. The subscript is the normalized eigenvector $\xi$ indexed by $k$, which generates this eigencycle set. The superscript $(mn)$ is the index of the two-dimensional subspace, where the elements (eigencycles) of the set are located. $(mn)$ is defined as follows: \{$\{m,n\} \in \{1,2,...,n\} \cap (m < n)$\}. In this study, the assignment order is $m$ from 1 to $N$, and then $n$ from 2 to $N$.
\item \textbf{Subspace set:} The subspace set, denoted as $\Omega^{(mn)}$, has $N(N-1)/2$ eigencycle elements. The superscript $(mn)$ is the index of the two-dimensional subspace. Again, $(mn)$ is defined as follows: \{$\{m,n\} \in \{1,2,...,n\} \cap (m < n)$\}. In this study, the assignment order is $m$ from 1 to $N$, and then $n$ from 2 to $N$. Owing to symmetry, there exists a subset of $\Omega^{(mn)}$, wherein the performance of the elements is equal.
\item \textbf{Independence of eigencycle sets:} The eigencycle sets corresponding to eigenvectors are not all independent. Let us take the O'Neill game as an example. Among the eight eigencycle sets, only three (generated by $(\xi_{.8i},\xi_{.4i_1},\xi_{.4i_2})$) are independent. We explain this as follows:  Per the set of eigenvalues, (1) Two eigenvalues (0.2 and -0.2) are real numbers. As only the eigenvectors whose eigenvalues are complex numbers are related to cyclical motion, these two are trivial. (2) The remaining six eigenvectors make three pairs.
Each pair of eigenvectors with complex conjugate eigenvalues are conjugate,
and the generated eigencycle set pair has opposite values to the other pair. Hence, we can omit those related to eigenvalues, that is, $({-0.8i},{-0.4i},{-0.4i})$. Using the eigenvalues as the subscript symbols, the eigenvectors are recorded as $(\xi_{.8i},\xi_{.4i_1},\xi_{.4i_2})$.
In this system,
there two independent eigenvectors
(5-th and 7-th in Table \ref{tab:eigenval_vec_cyc}) belong to the same eigenvalue
${-0.4i}$.
We use the subscripts 1 and 2 to distinguish the eigencycle set from the two degenerate eigenvalues of $0.4i$.
\item \textbf{Geometric presentation:} A geometric presentation of an eigencycle set is shown in figure \ref{fig:Lissajous}.  An eigencycle only depends on the internal components $\eta_m$ and $\eta_n$ of the same normalized eigenvector. As both of the components' amplitude and phase difference is fixed and not arbitrary, so the geometric pattern set is fixed. In geometry, an elements of the eigencycle set is similar to a Lissajous cycle. An explanation of the similarity is shown in appendix section \ref{app:geo_eigc}.
\item \textbf{Relationship between eigencycle value and angular momentum:}
In mathematics, we can strictly prove that, the rate between the angular momentum at any time moment in any two subspace equals to  the rate between eigencycle values of the two subspace. See appendix section \ref{app:sigma_L} for details. Meanwhile, in a time series from a real experiment system, the angular momentum of the two observations in time series is measurable. So the eigencycle concept can be verified in real system.
\item \textbf{Why the O'Neill game?} Since O'Neill is a two-person four-strategy zero-sum game with eight dimensions, it is a high-dimensional problem. It is difficult for us to intuitively express the motion structure of such a high dimension. Using the eigencycle set, the eight-dimension game space can be decomposed into $28$ two-dimension sets. Importantly, \begin{itemize} \item In theory, this game's replicator dynamics equation can be explicitly solved mathematically. The game has high symmetry, so it is easily tractable. \item In experiment, the game matrix has high symmetry and there exist sufficient equivalent observations. The data should provide sufficient opportunity for testing. \item The O'Neill experiment was the first to reveal the reality and accuracy of the Nash Equilibrium in game theory. Revealing these characteristics  of game dynamics using our experiment data is a new challenge. \end{itemize}
\end{itemize}


\begin{figure}[h!]
\centering
\includegraphics[scale=0.7]{Lissa8i.png}
    \caption{Geometric presentation of the eigencycle set. Subfig (mn) illustrate the 28 eigencycles patterns of the eigencycle set $(\sigma_{.8i})$; the value of eigenvecter component ($\eta_m$ and $\eta_n$) and cycles value $\Omega^{(mn)}$ is shown in Table \ref{tab:eigenval_vec_cyc}.
    The red (blue) color indicates the cyclic motion being counter-clockwise (clockwise),
    and its value (called as the signed area value or the angular momentum value), is positive (negative).}
\label{fig:Lissajous}
\end{figure}


\subsection{Explanation of the theoretical results}
Here, we briefly explain the theoretical results in Table \ref{tab:eigenval_vec_cyc}.
Meanwhile, define the fine structure and hypefine structure in the O'Meil game.
\begin{itemize}
\item \textbf{Precision and fine structure} Table \ref{tab:eigenval_vec_cyc} provides a theoretical prediction on cyclic behavior having higher precision than ever known. The eigencycle set $\sigma_{.8i}$ predicts four different values, (0.1964:0.0654:0.0218:0) = (9:3:1:0), that is, a cycle set with four different strengths are expected. This prediction leads to (1) the discovery of the fine structure (see section \ref{R2}); (2) the rising of the test accuracy one order (see section \ref{RE}), which are included in the main results of this study.
\item \textbf{Symmetry and hypefine structure} The contributions of the eigencycles from the different normalized eigenvectors $\xi$ are different, but can be classified as shown in the numerical results in Table \ref{tab:eigenval_vec_cyc}. The classification reflects the symmetry of the game matrix as well as the structure of the $\xi$ inner components. We employ this to identify the contribution of $\xi$ in the data (e.g., see $\sigma_\alpha$ and $\sigma_\beta$ in subsection \ref{R3}), which is one of  the main results of this study too.
\item \textbf{Operable ---} We use the experiment data to validate our eigencycle set decomposition approach. Our results may provide a spectrum analysis tool for game dynamics (see section \ref{R4}).
\end{itemize}

\begin{table}
\caption{The eigenvalues, eigenvectors, and their respective 28 eigencycles}
\begin{center}
\begin{tabular}{crrrrrrrrrr}
 \toprule
 \tabincell{c}{Eigenvalue\\$\lambda_{i}$} ~~& $\lambda_{.8i}$ & $\lambda_{-.8i}$ & $\lambda_{.2}$ & $\lambda_{-.2}$ & $ \lambda_{.4i_1}$ & $\lambda_{-.4i_1}$ & $ \lambda_{.4i_2}$ & $\lambda_{-.4i_2}$ \\
 \hline
 \vspace{0.05cm}
	&$\frac{4}{5}\, \mathrm{i}$ & $-\frac{4}{5}\, \mathrm{i}$
	&$\frac{1}{5}$ & $-\frac{1}{5}$
	&$\frac{2}{5}\, \mathrm{i}_1$ & $-\frac{2}{5}\, \mathrm{i}_1$
	&$\frac{2}{5}\, \mathrm{i}_2$ & $-\frac{2}{5}\, \mathrm{i}_2$ \\
 \hline \vspace{0.1cm}
 \tabincell{c}{Eigenvector\\($\eta_i \in \xi$)} ~~& $\xi_{.8i}$ ~~~& $\xi_{-.8i}$ ~~~ & $\xi_{.2}$~~& $\xi_{-.2}$~~& $ \xi_{.4i_1}$~~ & $\xi_{-.4i_1}$~~~& $ \xi_{.4i_2}$~~ & $\xi_{-.4i_2}$\\
 \hline
$\eta_1$	&$-\frac{1}{4}\, \mathrm{i}$&$\frac{1}{4}\, \mathrm{i}$&$\frac{2}{5}$&0&	$	0 $&$ 0 $&$ 0 $&$ 0$\\
$\eta_2$	&$\frac{1}{12}\, \mathrm{i}$&$-\frac{1}{12}\, \mathrm{i}$&$\frac{1}{5}$&0&	$	\frac{1}{6} $&$ \frac{1}{6} $&$ \frac{1}{6} $&$ \frac{1}{6}$\\
$\eta_3$	&$\frac{1}{12}\, \mathrm{i}$&$-\frac{1}{12}\, \mathrm{i}$&$\frac{1}{5}$&0&	$	\frac{-1+\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{-1-\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{-1-\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{-1+\sqrt{3}\, \mathrm{i}}{12}$\\
$\eta_4$	&$\frac{1}{12}\, \mathrm{i}$&$-\frac{1}{12}\, \mathrm{i}$&$\frac{1}{5}$&0&	$	 \frac{-1 -\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{-1+\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{ -1+\sqrt{3}\, \mathrm{i}}{12} $&$ \frac{-1-\sqrt{3}\, \mathrm{i}}{12}$\\
$\eta_5$	&$\frac{1}{4}$&$\frac{1}{4}$&0&$\frac{2}{5}$&	$	 0 $&$ 0 $&$ 0 $&$ 0$\\
$\eta_6$	&$-\frac{1}{12}$&$-\frac{1}{12}$&0&$\frac{1}{5}$&	$	 -\frac{\mathrm{i}}{6} $&$ \frac{\mathrm{i}}{6} $&$ -\frac{\mathrm{i}}{6} $&$ \frac{\mathrm{i}}{6}$\\
$\eta_7$	&$-\frac{1}{12}$&$-\frac{1}{12}$&0&$\frac{1}{5}$&	$	 \frac{\sqrt{3} + \mathrm{i}}{12} $&$ \frac{\sqrt{3}-\mathrm{i}}{12} $&$ \frac{-\sqrt{3}+\mathrm{i}}{12} $&$ \frac{-\sqrt{3}-\mathrm{i}}{12}$\\
$\eta_8$	&$-\frac{1}{12}$&$-\frac{1}{12}$&0&$\frac{1}{5}$&	$	 \frac{-\sqrt{3}+\mathrm{i}}{12} $&$ \frac{-\sqrt{3}-\mathrm{i}}{12} $&$ \frac{\sqrt{3}+\mathrm{i}}{12} $&$ \frac{\sqrt{3}-\mathrm{i}}{12}$\\
\hline
 \vspace{0.1cm}
\tabincell{c}{Eigencycle\\ ($\Omega^{(mn)}$)} & $\sigma_{.8i}$ ~~~& $\sigma_{-.8i}$ ~~~& $\sigma_{.2}$~~& $\sigma_{-.2}$~~& $ \sigma_{.4i_1}$~~ & $\sigma_{-.4i_1}$~~~& $ \sigma_{.4i_2}$~~ & $\sigma_{-.4i_2}$\\
 \hline
	 12 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 14 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 15 & $-$0.1964 & 0.1964 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 16 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 17 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 18 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 23 & 0 & 0 & 0 & 0 & $-$0.0756 & 0.0756 & 0.0756 & $-$0.0756 \\
	 24 & 0 & 0 & 0 & 0 & 0.0756 & $-$0.0756 & $-$0.0756 & 0.0756 \\
	 25 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 26 & $-$0.0218 & 0.0218 & 0 & 0 & 0.0873 & $-$0.0873 & 0.0873 & $-$0.0873 \\
	 27 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 28 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 34 & 0 & 0 & 0 & 0 & $-$0.0756 & 0.0756 & 0.0756 & $-$0.0756 \\
	 35 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 36 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 37 & $-$0.0218 & 0.0218 & 0 & 0 & 0.0873 & $-$0.0873 & 0.0873 & $-$0.0873 \\
	 38 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 45 & 0.0654 & $-$0.0654 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 46 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 47 & $-$0.0218 & 0.0218 & 0 & 0 & $-$0.0436 & 0.0436 & $-$0.0436 & 0.0436 \\
	 48 & $-$0.0218 & 0.0218 & 0 & 0 & 0.0873 & $-$0.0873 & 0.0873 & $-$0.0873 \\
	 56 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 57 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 58 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
	 67 & 0 & 0 & 0 & 0 & $-$0.0756 & 0.0756 & 0.0756 & $-$0.0756 \\
	 68 & 0 & 0 & 0 & 0 & 0.0756 & $-$0.0756 & $-$0.0756 & 0.0756 \\
	 78 & 0 & 0 & 0 & 0 & $-$0.0756 & 0.0756 & 0.0756 & $-$0.0756 \\
 \hline
\end{tabular}
\end{center}
 \vspace{-0.2cm}
\footnotesize{
Note: $\Omega^{(mn)}$ is the identity of the two-dimensional subspace of an eigencycle.
$m$ ($n$) indicates which among the eight dimensions
is selected as the $x$-axis ($y$-axis) in the eigencycle subspace.
}
\label{tab:eigenval_vec_cyc}
\end{table}


















\section{Experimental observation on subspace cycles}
\subsection{Brief summary of the experiment data}
Table \ref{tab:datasource} summarizes the O?Neill game experiments. The three experiments span 26 years. In total, 358 subjects participated in the experiments, which involved multi-round repeated playing in the game matrix \ref{tab:gamemodel}.
The experiment data help verify the high-dimensional cycling in this study.


\begin{table}[!ht]
\caption{Experiment Data Source}
 \vspace{-0.2cm}
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
 \hline
\tabincell{c}{Experiment\\abbreviate}& \tabincell{c}{Data\\Source}&\tabincell{c}{Experiment\\Summary} &	\tabincell{c}{Published\\Time}&\tabincell{c}{total\\Rounds}&	\tabincell{c}{Matching\\Protocol}\\	\hline
$O$&O'Neill&\tabincell{c}{	The game was played by 50\\ students working in 25 pairs.}	&	1987&	2625&	\tabincell{c}{Fixed\\Paired}\\	\hline
$B$& Binmore et al&\tabincell{c}{		Each experiment session required 12 subjects \\13 experimental sessions in all.\\ Each real game was played 150 times.}&	2001 &	1950&	\tabincell{c}{Randomly\\Matching}\\	\hline
$IT$&	 Okano&\tabincell{c}{20 individuals adopted the player A role\\ against 20 teams adopting the player B role.\\ 20 experimental sessions in all.\\ Each real game was played 150 times.}&	2013&	3000&	\tabincell{c}{Fixed\\Paired}\\	\hline
$TI$&	 Okano&\tabincell{c}{20 teams adopted the player A role \\against 20 individuals adopting \\the player B role, 20 experimental sessions \\in all. Each real game was played 150 times.}&	2013&	3000&	\tabincell{c}{Fixed\\Paired}\\	\hline
$II$&	 Okano&\tabincell{c}{Individuals against individuals, \\18 experimental sessions in all. \\Each real game was played 132 times.}&	2013&	2376&	\tabincell{c}{Fixed\\Paired}\\	\hline
$TT$&	 Okano&\tabincell{c}{Teams against teams, \\18 experimental sessions in all. \\Each real game was played 132 times.}&	2013&	2376&	\tabincell{c}{Fixed\\Paired} \\	\hline
\end{tabular}
\end{center}
\label{tab:datasource}
\end{table}





\subsection{Angular momentum as the measurement}

\begin{flushleft}
\textbf{The measurement for the cycle in the subspace: }
\end{flushleft}


According to the theoretical eigencycle set decomposition approach,
we can measure the cyclic angular momentum
~\footnote{This measurement is the signed area of
the triangle $\Delta_{[O, x(t), x(t+1)]}$ in the ($m,n$) two-dimensional subspace.
For each transition from $x(t)$ to $x(t+1)$, referring to $O$,
the angular momentum is twice the signed area of the triangle.
We suggest using the angular momentum
because it contains
the mass $m$ as a parameter,
which may be compatible with the population size $N$
as the variable in further investigations of game dynamics. }
in each of the two-dimensional subspace,
indicated by the eigencycle $\Omega^{(mn)}$, separately.
The angular momentum $L^{(mn)}_E$\cite{2012Inertia} can be expressed by the following formula: \\
\begin{equation}\label{eq:exp_am} L^{mn}_E=\frac{1}{N-1}\sum_{t=1}^{N-1}\left(x(t)-O\right) \times \left(x(t\!+\!1)-x(t)\right)
\end{equation}



\begin{itemize}
\item $L^{(mn)}_E$ represents the average value of
the accumulated angular momentum over time;
the subscript $mn$ indexes the two-dimensional $(x_m,x_n)$ subspace;
\item $N$ is the length of the experimental time series, that is,
the total number of repetitions of the repeated game experiments;
\item $O$ is the projection of the Nash equilibrium at the subspace $\Omega^{(m,n)}$;
\item $x(t)$ is a two-dimensional vector at time $t$,
which can be expressed as $(x_m(t),x_n(t))$, $x(t+1)$ is a two-dimensional vector at time $t+1$, and
\item $\times$ represents the cross product between two two-dimensional vectors.
\end{itemize}

\begin{flushleft}
\textbf{Interpretation of the measurement: }
\end{flushleft}


\begin{itemize}
\item The sign of the value of $L^{(mn)}_E$ indicates the direction of the cyclical movement (counterclockwise or clockwise): $L^{(mn)}_E >0$ refers to counterclockwise movement and $L^{(mn)}_E < 0$ refers to clockwise movement. $L^{(mn)}_E =0 $ means no cycle was observed. If a system is completely random, as predicted by the mixed strategy Nash equilibrium, the long-term average of its angular momentum is 0.
\item The modulus of the value, $||L^{(mn)}_E||$, indicates the strength of determinate motion. For example, in a one-to-one fixed pairing, there are four states in a two-dimensional subspace, and these four states form a unit square with an area of 1. The accumulated angular momentum of one full cycle is 2, or $||L^{(mn)}_E||=2$.
\item If, in the time series of 1,000 repeated rounds experiment, the observed accumulated $L^{(mn)}_E = 40$, then the total amount of the determinate motion is 20 cycles. This also means there are 80 determinate motions beyond complete stochastic motion, that is, the probability of directional movement is 8\%.
\end{itemize}


\subsection{Experimental results}

We calculate the average angular momentum for the experiments shown in Table \ref{tab:datasource}, and Table \ref{tab:exp_am} shows the results.


\begin{table}
\caption{Experimental angular momentum}
 \vspace{-0.2cm}
\begin{center}
\begin{tabular}{ c r r r r r r }
 \toprule
 \vspace{0.1cm}
$\Omega^{(mn)}$&	$L^{(mn)}_O$~~&	$L^{(mn)}_B$~~&	$L^{(mn)}_{IT}$~~&	$L^{(mn)}_{TI}$~~&	$L^{(mn)}_{II}$~~&	$L^{(mn)}_{TT}~~$\\\hline
12	&	0.003	&	0.002	&	$-0.008$	&	$-0.001$	&	0.002	&	0.008	\\	\hline
13	&	$-0.011$	&	$-0.002$	&	0.003	&	0.000	&	$-0.009$	&	$-0.002$	\\	\hline
14	&	0.008	&	0.000	&	0.004	&	0.002	&	0.007	&	$-0.006$	\\	\hline
15	&	$-0.038$	&	$-0.015$	&	$-0.053$	&	$-0.033$	&	$-0.039$	&	$-0.049$	\\	\hline
16	&	0.009	&	0.010	&	0.018	&	0.019	&	0.008	&	0.020	\\	\hline
17	&	0.021	&	0.002	&	0.020	&	0.006	&	0.015	&	0.014	\\	\hline
18	&	0.007	&	0.003	&	0.015	&	0.008	&	0.016	&	0.014	\\	\hline
23	&	0.011	&	0.003	&	$-0.009$	&	0.001	&	0.008	&	0.007	\\	\hline
24	&	$-0.008$	&	$-0.001$	&	0.001	&	$-0.002$	&	$-0.006$	&	0.001	\\	\hline
25	&	0.010	&	0.003	&	0.021	&	0.005	&	0.015	&	0.018	\\	\hline
26	&	$-0.001$	&	0.000	&	0.004	&	$-0.011$	&	0.002	&	$-0.009$	\\	\hline
27	&	$-0.001$	&	$-0.001$	&	$-0.010$	&	0.004	&	$-0.012$	&$	-0.014$	\\	\hline
28	&	$-0.007	$&	$-0.002$	&	$-0.014	$&	0.002	&	$-0.005$	&	0.005	\\	\hline
34	&	0.000	&	0.001	&	$-0.006$	&	0.000	&	$-0.001$	&	0.005	\\	\hline
35	&	0.011	&	0.008	&	0.015	&	0.011	&	0.014	&	0.013	\\	\hline
36	&	$-0.009$	&	$-0.006$	&$-0.003$	&	$-0.002$	&	$-0.006$	&	$-0.005$	\\	\hline
37	&	$-0.003$	&	0.001	&	$-0.004$	&	$-0.004$	&	0.002	&	0.002	\\	\hline
38	&	0.002	&	$-0.002$	&	$-0.008$	&	$-0.006$	&	$-0.011$	&	$-0.010$	\\	\hline
45	&	0.018	&	0.004	&	0.017	&	0.017	&	0.010	&	0.019	\\	\hline
46	&	0.000	&	$-0.004$	&	$-0.019$	&	$-0.006$	&	$-0.003$	&	$-0.007$	\\	\hline
47	&	$-0.016$	&	$-0.001$	&	$-0.006$	&	$-0.007$	&	$-0.006$	&	$-0.002$	\\	\hline
48	&	$-0.002$	&	0.002	&	0.007	&	$-0.004$	&	$-0.001$	&	$-0.009$	\\	\hline
56	&	$-0.006$	&	0.001	&	$-0.001$	&	0.001	&	$-0.001$	&	0.002	\\	\hline
57	&	0.009	&	0.000	&	0.004	&	0.005	&	$-0.005$	&	$-0.009$	\\	\hline
58	&	$-0.003$	&	$-0.001$	&	$-0.002$	&	$-0.006$	&	0.006	&	0.007	\\	\hline
67	&	$-0.003$	&	0.001	&	$-0.003$	&	0.000	&	0.005	&	0.007	\\	\hline
68	&	$-0.003$	&	0.000	&	0.002	&	0.001	&	$-0.006$	&	$-0.005$	\\	\hline
78	&	0.006	&	0.000	&	0.000	&	0.005	&	0.000	&	$-0.002$	\\	\hline
\end{tabular}
\end{center}
 \vspace{-0.2cm}
\footnotesize{
$\Omega^{(mn)}$ is the $28$ subspaces referring to Table
\ref{tab:eigenval_vec_cyc}; the subscript $L$ refers to the O'Neill and Binmore as well as the AIBT, ATBI, AIBI, and
ATBT experiments in Table \ref{tab:datasource}.
We use the software MATLAB, version R2018a.}\\


\label{tab:exp_am}
\end{table}





\begin{table}
\caption{Consistency of the six experiments}
 \vspace{-0.2cm}
 \begin{center}
\begin{tabular}{crrrrrr}
 \toprule
 	&~~~~$L_O$~~~&~~~~$L_B$~~~&~~~$L_{IT}$~~~	&~~~$L_{TI}$~~~&~~~$L_{II}$~~~&~~~$L_{TT}$~~~\\ \hline
 $L_O$ & 1.000 & & & & & \\ &0.000& & & & & \\\hline
$L_B$ & 0.706 & 1.000 & & & & \\
 & 0.000 &0.000 & & & & \\\hline
$L_{IT}$ & 0.522 & 0.639 & 1.000 & & & \\
 & 0.004 & 0.000 & 0.000 & & & \\\hline
$L_{TI}$ & 0.660 & 0.658 & 0.586 & 1.000 & & \\
 & 0.000 & 0.000 & 0.001 &0.000 & & \\\hline
$L_{II}$ & 0.728 & 0.825 & 0.608 & 0.5306 & 1.000 & \\
 & 0.000 & 0.000 & 0.001 & 0.0037 &0.000 & \\\hline
$L_{TT}$ & 0.474 & 0.756 & 0.470 & 0.5932 & 0.770 & 1.000\\
 & 0.011 & 0.000 & 0.012 & 0.0009 & 0.000 &0.000 \\\hline
\end{tabular}
\end{center}
 \vspace{-0.2cm}
\footnotesize{
Measure: Spearman's rank correlation.
The number of observations is 28.
The variables (observations) are the experimental angular momentum
shown in Table \ref{tab:exp_am}.
The first row reports the correlation coefficients, and
the second row reports the significance level $p$.
We use the software Stata, version:SE 15.1.}
\label{tab:spearman}
\end{table}


\subsection{Explanation of the experimental results:\label{RE}}
The measurement results illustrate the consistency and precision
 of the six experiments, thus validating the inner eigenvector decomposition approach. The cycles in the high-dimension space are real and testable as well, as shown by the following two points:
\begin{itemize}
\item \textbf{Consistency~------}
    In the six experiments, the observed cycle values $L$ in the two-dimensional subspaces show consistency (Spearman rank's test, max($p$)=0.012, mean of the $\rho$ values is 0.6351, Std.~\!Ev = 0.1087, Std.~\!Err = 0.0118).
\item \textbf{Precision~------}
Except for the main cycle found in $\Omega^{(15)}$, we can now test the cycles in the subspace ($\Omega^{(26,27,28,36,37,38,46,47,48)}$). This means that the accuracy increases up to one order of magnitude. We explain this with two numerical examples: \begin{itemize}
\item $L^{(15)}_O$=$-0.038$, where the superscript $O$ indicates the O'Neill game, superscript $(15)$ indicates that the subspace measured is $\Omega^{(15)}$, and the negative sign means the cycle is clockwise. $||L||$ = 0.038, that is, there are ($\frac{105\times 0.038}{2}\simeq$ 2) two cycles obtained in the O'Neill (1987) 105-rounds game, and, in the 25 groups, 2,625 rounds of observations, yielding  50 cycles.
\item $L^{(37)}_O$=$-0.003$, that is, the average angular momentum in subspace 37 is 0.003. Each complete cyclical motion in the four-states subspace of the O'Neill game contributes two units of angular momentum; then, the 105 rounds have 0.2 cycles, and the 2,625 rounds have 4.4 cycles.
\end{itemize}
\end{itemize}

















\section{Eigencycle set verified with experiments}

We have shown the theoretical
eigencycle values $\sigma^{(mn)}$ in Table. \ref{tab:eigenval_vec_cyc} and the experimental average angular momentum $L_{mn}$ in Table \ref{eq:exp_am}.

There is following proposition ---
For various subspace $(m,n)$,
the average accumulated angular momentum
$L_{mn}$ divided by its $\sigma^{(mn)}$ is invariant
(Details of the mathematical proof is provided in appendix \ref{app:sigma_L}).
That is to say, by take $\sigma^{(mn)}$ as predictor (independent variable) and take $L_{mn}$
as the observation (dependent variables),  a linear relation with truncation term being 0 can be expected.
So, we can evaluate the validity of the eigencycle with the experimental $L$ (average angular momentum).

The three subsections can be abstracted as follows:
\begin{enumerate}
\item Report the discovery of the fine structure, because of the contribution of $\sigma_{.8i}$ which is
labeled as $0.8i$ in Figure \ref{fig:fine_structure};
\item Report the finding of the hyperfine structure, because of the contribution of $\sigma_{.4i_1}$ and $\sigma_{.4i_2}$ which is
labeled as $\alpha$ and $\beta$ in Figure \ref{fig:fine_structure}, respectively;
\item Report the multi-OLE results (see Table \ref{tab:multi_reg}) illustrate that
the eigencycle set is an ideal
basis in this study.
The results validate the eigenvector set as a spectrum analysis tool on dynamics system.
\end{enumerate}

\begin{figure}[!ht]
\centering
\includegraphics[scale=0.55]{FineStructure20201202.png}
\caption{Fine and hyperfine structures
of game dynamics in the O'Neill (1987) game experiments.
The labels \{0.0423, 0.0142, and 0.047\} are the observed average angular momentum of the experiments, which linearly match the eigencycle values of $\sigma_{.8i}$.
This figure illustrates how the theoretical eigencycle set
 clearly shows the existence of the
the fine and hyperfine structures, thus enhancing the literature
\cite{Binmore2001Minimax,2012arXiv1203.2591W,dan2014,dan2020}.}
\label{fig:fine_structure}
\end{figure}





\subsection{Fine structure \label{R2}}
\begin{flushleft}
\textbf{Result 1's description: }
\end{flushleft}

The six experiments' cycles
from the 28 subspaces are employed
to test the prediction of the three eigencycle sets
($\sigma_{.8i}$, $\sigma_{.4i_1} $, and $\sigma_{.4i_2}$)
by OLE, respectively.
The results from the six experiments are consistent:
\begin{itemize}
\item Among the 28 samples from the theoretical $\sigma_{.8i}$ (eigencycle set) and
experimental $L$, there exists a significant linear dependence.
The results show that $\sigma_{.8i}$ can independently be the
principle component for the game dynamics.
\item Importantly, we clearly observe
the four cycles values, rated as (9:3:1:0)
and predicted by $\sigma_{.8i}$ (see the red squares in Figure \ref{fig:8iregression}), indicating the discovery of the \textbf{fine structure}
in the O'Neill game.
\item
Neither $\sigma_{.4i_1}$ nor $\sigma_{.4i_2}$ exists in significant linear dependence with experimental $L$.
for the game dynamics. However, they are not invalid, as we later show that they lead to the \textbf{hyperfine structure}.
\end{itemize}
\begin{flushleft}
\textbf{The supporting data of result 1: }
\end{flushleft}
The report of 'the discovery of the fine structure' is based on the
consequence of the $\sigma_{.8i}$
by splitting the $\Omega^{{15}}$ cycle in 5 $\sigma$ significant.
This claim is supported by the data as following:

Figure \ref{fig:8iregression} illustrates the ordinary least squares of the theoretical $\sigma_{.8i}$ and the six experimental $L$. Table \ref{tab:reg8i4i14i2} provides the results of the ordinary least squares between the six experiments and $\{\sigma_{.8i}, \sigma_{.4i_1}, \sigma_{.4i_2}\}$.
\begin{itemize}
\item On the claim of the discovery of the fine structure --- The six subspaces of the six experiments, when observed together, yield 36 samples, and the result is strongly significant ($ttest$, $p$=2.835 $\times 10^{-15}$, $N$=36).
More strictly,  considering the concentration of the 8 dimension,
omitting the 4- and 8-th dimension related subspaces, the result is strongly significant remained ($ttest$, $p$=1.3802 $\times 10^{-10}$, $N$=24).  This stratifies the statistical significance of the five standard deviations (5 $\sigma$, a threshold of $p \leq 2.87 \times 10^{-7}$) above background expectations, at 0. So, we claim the discovery of fine structure.
\item Regarding the fine structure,
the nine subspaces of the six experiments are observed together,
giving us 54 samples. In this case, the result is
strongly significant ($ttest$, $p$=7.689 $\times 10^{-7}$, $N$=54).
We are not suprise by this relatively low significant,
because the theoretical value is closer to background expectations, at 0.
\end{itemize}

\begin{figure}[!ht]
\centering
\includegraphics [width=1.87in]{YfigN7.png}
\includegraphics [width=1.8in]{YfigN8.png}
\includegraphics [width=1.8in]{YfigN9.png}\\
\includegraphics [width=1.85in]{YfigN10.png}
\includegraphics [width=1.8in]{YfigN11.png}
\includegraphics [width=1.8in]{YfigN12.png}
\caption{Linear regression to test the validity of the eigencycle set concept with $\sigma_{.8i}$ through the experiment. The 9:3:1:0 predicted by $\sigma_{.8i}$ is observed, which leads to the discovery of the fine structure of game dynamics. The experiment values for (a)--(f) are from the treatments ordered as in Table \ref{tab:datasource}.}
\label{fig:8iregression}
\end{figure}



\begin{table}
\caption{Ordinary least squares between the six experimental $L$ and $\{\sigma_{.8i}, \sigma_{.4i_1}, \sigma_{.4i_2}\}$} \small
\begin{center}
\begin{tabular}{crccccccc}
 \toprule
& ~~~~~~&$L^{(mn)}_O$&$L^{(mn)}_B$& $L^{(mn)}_{IT}$& $L^{(mn)}_{TI}$& $L^{(mn)}_{II}$& $L^{(mn)}_{TT}$ \\ \hline
$\sigma_{.8i}$&coef.~ $t$&	8.26&	9.31&	12.78&	10.72&	10.64&	12.39 \\	
&$p$&	(0.00)&	(0.00)&	(0.00)&	(0.00)&(0.00)&	(0.00) \\
&cons.~ $t$&0.08&	0.35&	-0.49&	0.25&	-0.03&	0.47 \\	
&$p$&(0.939)&	(0.726)&	(0.630)&	(0.801)&	(0.978)&	(0.640) \\
&$R^2$&	0.724&	0.7692&	0.8627&	0.8155&	0.8131&	0.8552 \\	\hline
$\sigma_{0.4i_1}$&coef.~$t$&	$-0.31$&	0.47&	1.24&	$-0.63$&	0.11&	$-0.36$ \\	
&$p$&	(0.759)&	(0.645)&	(0.227)&	(0.535)&	(0.917)&	(0.719) \\
&cons.~t&	0.01&	0.22&	$-0.05$&	0.04&	0&	0.14 \\	
&$p$&	(0.995)&	(0.827)&	(0.960)&	(0.968)&	(0.999)&	(0.89) \\
&$R^2$&	0.0037&	0.0083&	0.0557&	0.0149&	0.0004&	0.0051 \\	\hline
$\sigma_{.4i_2}$&coef.~$t$&	0.92&	1.11&	0.43&	$-0.2$&	1.39&	0.42 \\
& ~$p$&	(0.367)&	(0.276)&	(0.672)&	(0.843)&	(0.176)&	(0.675) \\
&cons.~$t$&	$-0.06$&	0.05&	$-0.23$&	0.13&	$-0.17$&	0.13 \\	
&p&	(0.952)&(0.960)&(0.822)&	(0.898)&	(0.870)&	(0.896) \\
&$R^2$&	0.0314&	0.0454&	0.007&	0.0015&	0.0692&	0.0069 \\	\hline
$\sigma_{.8i}=-0.0833$&mean&$-0.0042$&	$-$0.0016&	$-$0.0059&	$-0.0037$&	$-$0.0043&	$-$0.0055 \\	
&$\pm SE$&(0.0019)&	(0.0008)&(0.0027)&	(0.0016)&	(0.0017)&	(0.0021) \\	\hline
\end{tabular}
\end{center}
\vspace{-0.2cm}
\footnotesize{Software:Stata, version:SE 15.1}
\label{tab:reg8i4i14i2}
\end{table}

\begin{flushleft}
\textbf{Interpretation of result 1:}
\end{flushleft}
\begin{itemize}
\item \textbf{$\sigma_{.8i}$ as principal component~---~}
The $p$ value of the regression coefficient of $\sigma_{.8i}$ is close to $0.000$, and the constant term cannot reject the $0$ hypothesis. Thus, $\sigma_{.8i}$ is explanatory with respect to the main experimental results. We can regard $\sigma_{.8i}$ as the principal component in this system. $\sigma_{.8i}^{(15)}$ has the largest value, which is consistent with Binmore et al.?s suggestion \cite{Binmore2001Minimax} and confirmed by Wang and Xu \cite{2012arXiv1203.2591W} after converting the $4\times 4$ game into a $2 \times 2$ payoff matrix. To our knowledge, the observed cycles in $\sigma_{.8i}^{(15)}$ are cutting-edge contributions on cycle measurement in the literature.
\item \textbf{Discover the fine structure~---~}
Beyond the $\Omega^{(15)}$ subspaces, $\sigma_{.8i}$ also predicts the cycle
in the six subspaces, $\Omega^{(16,17,18,25,35,45)}$, and
the nine two-dimensional subspaces, $\Omega^{(26,27,28,36,37,38,46,47,48)}$.  These cycles are one-third or one-ninth in strength compared with $\Omega^{(15)}$'s cycle, which cannot be tested by earlier methods. However, the eigencycle set approach shows their significant existence. We call these the 'fine structure'.
\item \textbf{Interpretation of $\sigma_{.4i}$ results~---~} %
The no-zero eigencycles in the $\sigma_{.4i_1}$ and $\sigma_{.4i_2}$ set are all independent of the first or fifth dimension. Hence, the experimental observed cycles in $\Omega^{(15,16,17,18,25,35,45)}$, which are obvious motions by $\sigma_{.8i}$, cannot be captured. As a result, neither $\sigma_{.4i_1}$ nor $\sigma_{.4i_2}$ can globally capture the entire 28-subspace motion reasonably.
\end{itemize}


\subsection{Hyperfine structure \label{R3}}
\begin{flushleft}
\textbf{Result 2's description}
\end{flushleft}
We investigate $\sigma_{.4i_1}$ and $\sigma_{.4i_2}$
using the data.
As per the main results, we find the hyperfine structure.
In $\sigma_{.4i_1}$ and $\sigma_{.4i_2}$, the corresponding eigenvectors ($\xi_{.4i_1}$ and $\xi_{.4i_2}$) and components
$\eta_1$ and $\eta_5$ are all 0. Hence,
$\sigma_{.4i_1}$ and $\sigma_{.4i_2}$
have an effect only in the (2-3-4, 6-7-8) six-dimension subspace,
which includes 15 two-dimensional subspaces
where the eigencycles can exist.
We use the two degenerate eigenvalues' eigencycle set
($\sigma_{.4i_1}$, $\sigma_{.4i_2}$)
to build a pair of orthogonal bases
as follows:
\begin{equation}\label{eq:alphabeta}
\left\{ \begin{array}{lr} \sigma_\alpha= (\sigma_{.4i_2}+\sigma_{.4i_1})/2, \\ \sigma_\beta = (\sigma_{.4i_2}-\sigma_{.4i_1})/2 \end{array}
\right.
\end{equation}
The mathematical explanations for the two bases are as follows:
\begin{itemize}
\item $\sigma_\alpha$ only relates to the (2-3-4, 6-7-8)-dimension outer-cross nine subspaces, $\Omega^{(26,27,28,36,37,38,46,47,48)}$. That is,
$\sigma_\alpha$ has only nine components, but fully covers the nine two-dimensional subspace. Notably, as a strictly mathematical result, there exist only two values, and they are in opposition in $\sigma_\alpha$.
\item $\sigma_\beta$ only relates to the (2-3-4, 6-7-8)-dimension inner-cross six subspaces, $\Omega^{(23,24,34,67,68,78)}$. That is,
$\sigma_\alpha$ has only six components, but fully covers the six two-dimensional subspaces.
Again, there exists only one value, and it is in $\sigma_\beta$.
\end{itemize}
The results of the experimental test are reported as follows:
\begin{itemize}
\item There exists significant linear dependence
between the experimental $L$ and $\sigma_\alpha$. The constant term is also a consequence of
$\sigma_{.8i}$. Hence, we have discovered the hyperfine structure.
\item
There exists significant linear dependence
between the experimental $L$ and $\sigma_\beta$. However, the constant term is zero and cannot be rejected.
\end{itemize}

\begin{flushleft}
\textbf{Supporting data of result 2}
\end{flushleft}
The report of 'find evidence of hyperfine structure' is based on the
statistical significant consequence of $\sigma_\alpha$ and  $\sigma_\beta$ on
the fine structure due to $\sigma_{.8i}$.
The supporting data are as following:
\begin{itemize}
     \item Table \ref{tab:P11_alpha_beta} and Figure \ref{fig:lp11_reg} show the supporting data. As the five treatments (labeled ($O, IT, TI, II, and~TT$) in Table \ref{tab:datasource})
    have the same social state pattern (each two-dimensional subspace has four states), we thus pool the data under the label $P11$-treatment.
    \item By regressing $L_{P11}$ and $\sigma_{.8i}$ as well as $\sigma_{\alpha}$ and $\sigma_{\beta}$, we can obtain the result of the first, second, and third rows in Table \ref{tab:P11_alpha_beta}, respectively, as shown in Figure \ref{fig:lp11_reg}.
        \begin{itemize}
        %
        \item $L_{P11}$ and $\sigma_{.8i}$ have significant positive correlation (linear regression, $t$=27.36, p=0.000, N=28);
        See Figure \ref{fig:lp11_reg}(b).
        %
        \item $L_{P11}$ and $\sigma_{\alpha}$ have significant positive correlation (linear regression, $t$=4.43, $p$=0.003, N=9), wherein the constant item is the influence of $0.8i$ on the nine points ($0.057\times(-0.083)=-0.0047\in\{-0.006,-0.004\}$);
        See Figure \ref{fig:lp11_reg}(b).
        %
        \item $L_{P11}$ and $\sigma_{\beta}$ have significant positive correlation (linear regression, $t$=3.31, $p$=0.030, N=6);
        See Figure \ref{fig:lp11_reg}(c).\\
        %
        \end{itemize}
    \item If sample by the experiments and the subspace simultaneously, we have larger sample size, and we get the same result.
    \begin{itemize}
        \item If the $\sigma_\alpha$'s nine subspaces of the six  experiments are observed together, we obtain 54 samples. These samples can still be interpreted by $\sigma_\alpha$ with significance (Spearman's $\rho$ = 0.3554, $p$(2-tailed) = 0.00836, $N$=54).
    %
        \item Similarly, with the $\sigma_\beta$'s six subspaces under the six experiments, we gain 36 samples, but the result is weakly significant (Spearman's $\rho$ = 0.31767, $p$(2-tailed) = 0.05903, $N$=36).
    \end{itemize}
\end{itemize}
\begin{figure}[!ht]
\centering
\includegraphics [width=1.87in]{Yfig13.png}
\includegraphics [width=1.8in]{figalpha.png}
\includegraphics [width=1.8in]{figbeta.png}\\
\caption{Fine structure and hyperfine structure of the cycles}
\label{fig:lp11_reg}
\end{figure}
 \begin{table}
\caption{Ordinary least squares results of experimental $L_{P11}$ based on the eigencycle set}
 \vspace{-0.2cm}
 \begin{center}
\begin{tabular}{crrrrrc}
 \toprule
\tabincell{c}{eigencycle\\ set}&coef.&\tabincell{c}{coef. \\ $t$}&\tabincell{c}{coef.\\ $p>|t|$}& \tabincell{c}{cons \\ $t$}	& \tabincell{c}{cons \\$p>|t|$} & \tabincell{c}{cons \\ conf.~interval}\\
 	\hline
$\sigma_{.8i}$& 0.057 &27.36& 0.000 &0.11 & 0.910 & [$-$0.001 ~~0.001]\\
\hline
$\sigma_\alpha$& 0.006 &4.43 & 0.003 &-10.94& 0.000&[$-0.006$ $-0.004$] \\
\hline
$\sigma_\beta$ &$ 0.005 $ & $3.31$ &0.030&$-0.77$& 0.487&[$-0.002$ ~~0.001] \\ \hline
\end{tabular}
\end{center}
\vspace{-0.2cm}
\footnotesize{
Note:
By the software Stata, version SE 15.1.
}
\label{tab:P11_alpha_beta}
\end{table}




\begin{flushleft}
\textbf{Interpretation of result 2}
\end{flushleft}
\begin{itemize}
\item \textbf{Finding the hyperfine structure~------~}
$L_{P11}$ and $\sigma_{\alpha}$ show significant correlation, indicating that $\sigma_{\alpha}$
can explain the hyperfine structure in the nine subspaces $\Omega^{(26,27,28,36,37,38,46,47,48)}$.
Beyond $\sigma_{.8i}$, which predicts that the nine points have the same value,
$\sigma_{\alpha}$ can also capture the differences among them.
At the same time, the response mode reflected by $\sigma_\alpha$
is consistent with the best response mode in behavior game theory,
which is also consistent with the win-stay-lose-shift mode in human decision-making \cite{Behavioral2003}.
\item \textbf{Puzzle on $\sigma_\beta$'s result~---~}
$L_{P11}$ and $\sigma_{\beta}$ have significant positive correlation, which cannot be explained by behavioral game theory. We suggest that
this is the geometry factor effect,
which is a natural result of
the eight-dimensional presentation of the dynamics equation \ref{eq:repliequl}.
\end{itemize}



\subsection{Eigencycle spectrum analysis \label{R4}}
\begin{flushleft}
\textbf{Descript of result 3:}
\end{flushleft}


With the eigencycle set as the orthogonal basis, we can explore high-dimensional dynamics through spectrum analysis in the form of
Eq. (\ref{eq:eigdeco1}).
From the two subsections above, we can conclude that $\sigma_{.8i}$, $\sigma_{\alpha}$, and $\sigma_{\beta}$ simultaneously affect $L_{P11}$.


\begin{flushleft}
\textbf{Supporting data of result 3:}
\end{flushleft}


\begin{table}
\caption{Multiple linear regression of experimental $L_{P11}$ by
$\sigma_{.8i}$, $\sigma_{\alpha}$, and $\sigma_{\beta}$}
 \vspace{-0.2cm}
\begin{center}
\begin{tabular}{ c|r r r r r }
 \toprule
 \vspace{0.1cm}
$L_{P11}$&	Coef.~~~~&	Std. Err.~~&	$t$~~~&	$P\!>\!|t|$&~~~~[95\% Conf.	Interval]~~~~~\\\hline
$\sigma_{.8i}$		&	~.0565190	&	~.0015589	&	36.25	&	~.000	&	 ~.0533014~~	~.0597365 ~~	\\	
$\sigma_{\alpha}$		&	~.0057181	&	~.0015589	&	3.67	&	~.001	& ~.0025006~~	~.0089356 ~~	\\	
$\sigma_{\beta}$	&	~.0045221	&	~.0015778	&	2.87	&	~.009	&	 ~.0012656~~	~.0077785 ~~	\\
cons		&	$-$.0000876	&	~.0002982	&	$-$0.29	&	~.772	&	 $-$.000703~~	~.0005279 ~~	\\	\hline
\end{tabular}
\end{center}
\label{tab:multi_reg}
\end{table}


Multiple regression analysis is a powerful technique used for predicting the unknown value of a variable from the known value of two or more variables-  also called the predictors.
Table \ref{tab:multi_reg} shows the supporting data. Multiple linear regression helps us interpret the experimental observation based on three theoretical eigencycle sets:

\begin{equation}\label{eq:empricalfor}
 \underset{SE}{L_{P11}}\underset{=}{=}\underset{(0.002)}{0.057~\!\sigma_{.8i}}+\underset{(0.002)}{0.006\sigma_{\alpha}}+\underset{(0.002)}{0.005\sigma_{\beta}}-0.00009,
\end{equation}
Function~\ref{eq:empricalfor} is evidence that $\sigma_{.8i}$, $\sigma_{\alpha}$, and $\sigma_{\beta}$ have a significantly positive effect on $L_{P11}$. $0.057$ is the partial regression coefficient of $\sigma_{.8i}$, which shows that, with the influence of $\sigma_{\alpha}$ and $\sigma_{\beta}$ held constant, a one-unit increase in $\sigma_{.8i}$ increases $L_{P11}$ by 0.057 units.
Similar significance from $\alpha$ and $\beta$ can be obtained.


\begin{flushleft}
\textbf{ Result 3's interpretation:}
\end{flushleft}
\begin{itemize}
\item
Consistent with the two subsections above, the results in this subsection are also significant. That is,
(1) The experimental observation is mainly determined by $\sigma_{.8i}$.
(2) Compared with $\sigma_{.8i}$,
$\sigma_\alpha$ and $\sigma_\beta$ have a weaker but significant effect.
 \item
 Because of constant cannot reject 0 (linear regression, $t=-0.29$, $p=0.772$), $\sigma_{.8i}$, $\sigma_\alpha$ and $\sigma_\beta$ are ideal basic vector for spectral analysis.
\item
The eigencycle set can be an ideal basis for spectrum analysis,
which is helpful for revealing the motion characteristics of high-dimensional game dynamics.
The results in Table \ref{tab:reg8i4i14i2}are a significant illustration of this fact.
\end{itemize}











\section{Discussion and conclusion}
\begin{itemize}
\item Contributions. This study is unique in its discovery of
        the fine structure, the cycles in high dimension game,
        in the long-existed human behavior O'Neill game experiments.
        To our knowledge, it is the first report of the high dimensional cycle.
        We also find evidence on the existence of the hyperfine structure in the human cycling spectrum (see Figure \ref{fig:fine_structure}).
        Meanwhile we have developed the eigencycle set decomposition approach.
        Through the invariant between the eigenvector components
        in a periodic dynamics system, by which we can
        test out the fine structure spectrum of human social cycling.
\item On the implications of the discovery,
we suggest that it may deeply influence our view on human game behavior.
The conflict between the complex in experiment
and the simply in theory seems visible.
(1) The complex of the experiments game behaviors,
which is high stochastic, discrete time, discrete strategy, non-linear, and high dimensional system, short time series, a extensive studied data which has existed more than 3 decades;
(2) The simply of the theoretical predictors, which is differential, continue time, continue strategy space, and solvable by the tools from the standard concepts of the dynamics system theory. However, referring to the finding of the fine dynamics structure, they are compatible.
It is well known that, the concepts and tools (e.g., stationary concept, Jacobian,  eigensystem concept, Near periodic orbits, Poincare Section, Centre manifold theory) in the dynamics system theory are powerful. Now we see that,
the reality and accuracy of the human behaviour game dynamics, then  the evolutionary game theory, can be well improved along the standard dynamics system theory.
\item   On the implication of the eigencycle approach, we suggest that, it may be an improvement to the standard dynamics system theory.
Not only can bring game dynamics system theory
to experimental game dynamics systematically,
it can be part of the standard analytical approach to where dynamics systems theory can be applied.
This approach is bases on the invariant between two components of a given eigenvector, which, to our knowledge, is novel.
Considering the widely application of the dynamics system theory,
we hope to have more strictly mathematics study
and potential empirical verification on the validation of the approach.
Some details on the comparison of our approach with
existed approach are shown in appendix
section \ref{app:rw_dst}.


\item  We concern the questions raised during this study. Our finding have demonstrated the significant existence of cycles in high dimension game by the eigencycle approach.
Although out of the scope of this study, there are some enigma remained,

\begin{itemize}
    \item The coefficients in Eq.~(\ref{eq:empricalfor}) are not quantitatively explained in a clear manner. These coefficients may have physical meaning, such as the distribution of electrons in their orbit (eigenstates) obeying the Boltzmann distribution. But what governors the coefficients is unclear. On this question,
    some simulation results have shown that
    the coefficients is strongly depends
    on individual decision making methods \cite{2021Qinmei},
    but the answer remains fuzzy.
    \item The eigencycle approach offers new opportunities for formal
    dynamics model analysis, but also has its own limitations.
    %
     First, it is based on linear approximation which is common in eigen system analysis. So, the application
     on large deviation condition needs caution.
     Second,  we ignore the phase interfering between two components from two eigenvectors, but if in a delegation imaginary part of eigenvalue condition (although not common),
     the long-time and sufficient noise condition is necessary (see the proof in appendix section \ref{app:itf_diff_ev}). So, for further strictly application, the boundary of the eigencycle approach needs further investigate.
\end{itemize}






\item  Our final concern is the central concept of game dynamics theory, or evolutionary game theory. Classical game theory has its central concept, Nash equilibrium, which reality, accuracy and applications have been well established. However, game dynamics theory has not such well established central concept.
The ESS (evolution stable strategy) concept,  which is mainly qualitative concept. Since recent experiments quantitative results accumulated \cite{Binmore2001Minimax,dan2014,wang2014,dan2016book},
a quantitative concept is more urgent.
The fixed point theorem leads to the Nash equilibrium,
which has become the central concept
of the classical game statics theory.
Have the finding the high dimensional game eigencycles, 
we calculate the 'invariant manifolds' and compare with eigencycles ant net transit. 
See appendix section \ref{app:im_ec} and \ref{app:net_trans} for details. 
As straightly impress, we suggest that  
'invariant manifolds' could be a candidate as the central concept,
which be used to capture complex dynamics and the long term behavior pattern of game dynamics.
This concept, not only have solid mathematics background, 
but also can be verified in laboratory experiment. 
In the field of game dynamics theory and experiment, on dynamics pattern, there exists a common question \cite{dan2020}. 
\begin{center}
\textit{In the background is a metaphysical question: what is a cycle?}
\end{center}
By the fine structure out of the eigencycles in O'Neill games data, we wish the 'invariant manifolds' can be a concrete answer. 
\end{itemize}

 


~\\
\textbf{Acknowledgement}: We thank O'Neill, Ken Binmore,
and Yoshitaka Okano for providing the data.
Thanks to Yijia Wang for her critical suggestion for Zhijian
to obtain the pair of orthogonal basis ($\alpha, \beta$).
The first author could be changed from Wang to Yao at any time
during the revising and publishing processes of the top five economics journal.
Wang designed the approach and Yao discovered the fine structure.



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