# Eigen mode selection in human subject game experiment

Wang Zhijian and Yao Qinmei and Wang Yijia
Experimental social science laboratory, Zhejiang University, China

###### Abstract

Eigen mode selection ought to be a practical issue in some real game systems,
as it is a practical issue in the dynamics behaviour of a building, bridge, or
molecular, because of the mathematical similarity in theory. However, its
reality and accuracy have not been known in real games. We design a 5-strategy
game which, in the replicator dynamics theory, is predicted to exist two eigen
modes. Further, in behaviour game theory, the game is predicted that the mode
selection should depends on the game parameter. We conduct human subject game
experiments by controlling the parameter. The data confirm that, the
predictions on the mode existence, as well as the mode selection are
significantly supported. This finding suggests that, like the equilibrium
selection concept in classical game theory, eigen mode selection is an issue
in game dynamics theory.

###### Contents

  1. 1 Introduction
    1. 1.1 Eigenmode
    2. 1.2 Research question
  2. 2 Results
    1. 2.1 Theoretical expectation
      1. 2.1.1 Game and its eigensystem
      2. 2.1.2 Parameter specification and theoretical prediction
    2. 2.2 Experimental observation
      1. 2.2.1 Distribution and deviation
      2. 2.2.2 Eigencycle in experiment
    3. 2.3 Verification by experiment
      1. 2.3.1 Eigen mode selection
      2. 2.3.2 Invariant in eigen modes
  3. 3 Discussion
  4. 4 Appendeix
    1. 4.1 Technology Notes
      1. 4.1.1 The consequence of the oppose a
      2. 4.1.2 Details of the eigenvector structure
      3. 4.1.3 The 5 parameter selection
      4. 4.1.4 Measurement of experimental angular momentum
      5. 4.1.5 Theoretical eigencycle set
      6. 4.1.6 Equivalent of eigencycle and angular momentum
      7. 4.1.7 Mode of angular momentum
      8. 4.1.8 Algorithm for the myopic response strength estimation
      9. 4.1.9 An example of the invariant cross the 5 parameters
    2. 4.2 Human subject game experiment protocol
    3. 4.3 Additional results

##  1 Introduction

###  1.1 Eigenmode

We investigate the reality and accuracy of the concept, eigen mode, by theory
and experiment in this study. Here, the the theory indicates the game dynamics
theory (or evolutionary game theory)[1, 2] which is a dynamics version in game
theory [3]; The experiment indicates the laboratory human subjects behaviour
game experiments, which is called also as experimental economics approach [4,
2].

Eigen mode is a normal mode in an oscillating system, being one in which all
parts of the system are oscillating with the same frequency and with a fixed
phase relation. A physical object, such as a building, bridge, or molecule,
has a set of normal modes and their natural frequencies that depend on its
structure, materials and boundary conditions. In music, normal modes of
vibrating instruments (strings, air pipes, drums, etc.) are called ”harmonics”
or ”overtones”. The most general motion of a system is a superposition of its
normal modes. The modes are normal in the sense that they can move
independently, that is to say that an excitation of one mode will never cause
motion of a different mode. In mathematical terms, normal modes are orthogonal
to each other. Further, in mathematical narrative, eigen mode, like the terms
eigenvalue, eigenvector, eigen space, invariant manifold, and so on, belongs
to the concept set ’eigen system’ in dynamics system theory.

Game dynamics theory (evolutionary game theory) mathematically belongs to
dynamics system theory. Game experimenter have applied the eigenvalue to
design experiment. E.g., to predict the stability of a game [5, 6] by the real
part of the eigenvalue, and to interpret the cycle by the imaginary part of
the eigenvalue [7, 8, 9]. The logic chain for eigenvalue judgement is
following

  * •

For a give game, the game dynamics system can be expressed as a velocity
vector field [1][2] as

$\dot{x}=f(x)$ (1)

in which $x\in R^{N}$ and $N$ is the dimension of the strategy space. The Nash
equilibrium is a singular point of the velocity vector field (a point where
$\dot{x}=0$). In linear approximation, near the equilibrium point, the
dynamics can be expressed as $\dot{x}=Jx$ in which $J$ is the Jacobian
(character matrix) at the equilibrium point [1, 2]. An eigenvalue
($\lambda_{i}$) is a root of the characteristic equation $(J-\lambda_{i}I=0)$.
Associated to the eigenvalue, there exists an eigenvector $\xi_{i}$ which
describes the instinct motion shape. These are the basic of the concept set of
eigen system. Suppose that, an initial condition can be expressed as
$x(0)=\sum_{i=1}^{N}a_{i}{\xi}_{i}$ then, the evolution trajectory can be
expressed as

$x(t)=\sum_{i=1}^{N}e^{\lambda_{1}t}a_{i}{\xi}_{i}.$ (2)

Here, each of the eigenvector $\xi_{i}$ describes an eigen mode, which has $N$
components being oscillating with the same frequency $\lambda_{i}$. In motion
shape, the eigenvector $\xi_{i}$ structure play the cruel role. The set
$\\{\xi_{i}\\}$ is called eigen mode set.

  * •

Eigen mode selection should be a practical issue in real game system in
theory. Because, a game dynamics system, in mathematics like a physical
object, can have an eigen modes set; At the same time, the game dynamics
system can be expected as a sum of its eigen modes111This issue has appeared
in quantum physics. In the quantum physics for the electrons of a hydro atom,
by Schrödinger equation, the eigen function (eigen mode) set can be obtained.
But for a given real world condition, how the electron distributes on the
eigen mode is remain an issue. To answer this question, the knowledge from
statistical physics, like Boltzmann constant, is needed. When such question is
answered well, we can, for example, report the temperature of a star far
away.. In a given long run high stochastic dynamics process, statistically,
which of the eigen mode mostly contribute to an observation, this is the
question about the eigen mode selection.

###  1.2 Research question

Game theorists has rare handled the complex eigenvector, let alone eigen mode
concept or the multi eigen mode issue. Although recent experiments [10, 11]
have significantly supported the eigen mode predictions from the logic chain
above, but these game have only unique pair of complex eigenvector. Multi
eigen mode was seemly test out in significant 2 roles and 4 strategy zero sum
game, of which having 3 pairs of complex eigenvectors (eigen modes), in the
existed data ([12, 13, 14]) reported by [15]. But the data does not provide
control variable for eigenvectors, then the cause for multi eigen mode can not
be distinguished.

Naturally, a naive question will be: When a game have multi eigen modes, which
mode will be selected? This question likes that question in game statics
theory — when a system have multi equilibrium, which equilibrium will be
selected — which is the issue namely equilibrium selection. Further question
is why a mode been selected?

We design an ideal controllable 5 strategy game experiment, of which having
two eigen modes. By this exemplified game, we show the reality and accuracy of
the multi eigen mode selection prediction from theory, and then carry out the
verification by real human subject game experiments.

##  2 Results

###  2.1 Theoretical expectation

####  2.1.1 Game and its eigensystem

##### Payoff matrix

The payoff matrix of the one population symmetric game is

$A=\left(\begin{array}[]{rrrrr}0&a&1&-1&-a\\\ -a&0&a&1&-1\\\ -1&-a&0&a&1\\\
1&-1&-a&0&a\\\ a&1&-1&-a&0\end{array}\right)$ (3)

##### Dynamics

To investigate the dynamic behaviors in laboratory experiment game, we begin
with using the replicator dynamics equations [16]:

$\dot{x}_{i}=x_{i}\Big{(}U_{i}-{\overline{U}}\Big{)}$ (4)

in which, $i\in\\{1,2,..,N\\}$. $x_{i}$ is the $i$-th strategy player’s
probability in the population where the $i$-th strategy player included, and
$\dot{x}_{i}$ is the evolution velocity of the probability; $U_{i}$
($:\equiv\sum_{j}A_{ij}x_{j}$) is the payoff of the players using the $i$-th
strategy, and $\overline{U}$ (:$\equiv x_{i}U_{i}$) is the average payoff of
the full population.

##### Equilibrium

Nash equilibrium $(1/5,1/5,1/5,1/5,1/5)^{T}$.

##### Jacobian

This is a time invariant dynamics system. Suppose that, the motion of the
strategy vector $x$ is close the equilibrium and the linear approximation of
dynamical system is validate. Then we can obtain the eigen system from the
Jacobian, which collects all first-order partial derivatives of the
multivariate function (Eq. 4) at the equilibrium as a character matrix [1, 2]:

$J=\frac{1}{5}\left(\begin{array}[]{rrrrr}0&a&1&-1&-a\\\ -a&0&a&1&-1\\\
-1&-a&0&a&1\\\ 1&-1&-a&0&a\\\ a&1&-1&-a&0\end{array}\right)$ (5)

##### Eigenvalue

The eigenvalues are

$\lambda=(0,\chi^{+},~{}\chi^{-},~{}-\chi^{+},~{}-\chi^{-}),$ (6)

in which $\chi^{\pm}=\frac{1}{10}\Big{[}\pm
2\,\sqrt{5}(1-4\,a-a^{2})-10(a^{2}+1)\Big{]}^{1/2}$. Notice that, $\chi^{\pm}$
is full imaginary, and their real parts are the constant 0
($Re(\chi^{\pm})=0).$ The two pairs of conjunction complex eigenvalue depend
on the variable $a$ which can be shown in Fig. 3. The imaginary part of
eigenvalue $\Im(\lambda_{i})$ relates to the angular velocity, the frequency,
and the angular momentum of cycle motion [8]. It is natural to assume that,
when given $a$, a motion having higher $|\Im(\lambda_{i})|$ value has larger
angular momentum, if other factors are same. Moreover, if
$\Im(\lambda_{i})=0$, the angular momentum will be 0. Details of relationship
of $\Im(\lambda)$ and angular momentum can refer to the section 6.2.1
Interfere of two components from the same eigenvector in [15].

##### Eigenvector

There are five eigenvector associated to the five eigenvalues respectively.

  * •

Among the five eigenvectors, There is an the unique pure real eigenvector,
which is $N_{e}=(1,1,1,1,1)^{T}$, the rest point (Nash equilibrium).

  * •

Remained are the two conjunction pairs of complex vectors. The amplitudes of
each eigenvector components are all equal. The phase of each eigenvector
component are fixed as $2\pi/5$ and $4\pi/5$ along strategy-(1,2,3,4,5),
respectively. The pattern of the two eigenvectors in complex plane is shown in
Fig. 1.

  * •

The eigenvector set is orthogonal.

Figure 1: Geometric presentation of the eigenvector (left) $v^{\alpha}$ and
(right) $v^{\beta}$.

##### Eigencycle

An eigencycle is determined by an eigenvector (definition is shown in appendix
4.1.5). The eigencycle of the 10 subspace can be calculated. The results of
eigencycle set from the two eigenvectors ($v^{\alpha}$ and $v^{\beta}$) are
shown in Table 1 as $\sigma_{\alpha}$ and $\sigma_{\beta}$. These two
eigencycle sets, reflecting the two eigenvector then the eigen mode. The
geometric presentation of the two eigencycle set is shown in Fig. 2. Regarded
as two vectors with 10 components, these two vector is orthogonal, because the
inner product (dot product) of the two vectors is zero. So called eigen mode
selection is the projection of cyclic motion to these two eigencycle set
vectors. These two eigencycle set play cruel role of this study.

Figure 2: Geometric presentation of the two eigencycle set ((top)
$\sigma_{\alpha}$ and (bottom) $\sigma_{\beta}$). These two eigencycle set
associate to the two eigenvectors (shown in Fig. 1) respectively. The orbit is
the Lissajous curve in mathematics. Each sub figure represents a 2-d subspace,
whose dimension id is arranged as the first column in Table 1, from left to
right ordered as $mn$=(12, 13, 14, 15, 23, 24, 25, 34, 35, 45). The red (or
blue) indicates the eigencycle value is positive and counter-clockwise (or
negative, clockwise). The eigen mode decomposition refers to these two
eigencycle set ($(\sigma_{\alpha},\sigma_{\beta}$). Meanwhile, the equivalent
among [1,4,5,8,10]-columns, as well as the equivalent among
[2,3,6,7,9]-columns, hint the invariant between the 2-d sub-spaces.

####  2.1.2 Parameter specification and theoretical prediction

The parameter specification, as well as the related theoretical predictions,
plays the cruel role in this experiment study. As the first experiment, to our
knowledge, to study the eigen mode selection, we limit ourselves on following
three research questions:

  1. 1.

(RQ1: Distinguishable) Whether the eigen modes exists and distinguishable?

  2. 2.

(RQ2: Predictable) Whether the eigen modes selection can be predictable?

  3. 3.

(RQ3: Invariant) Whether the parameter selection invariant exists?

##### Parameter specification

In mathematics, the parameter selection mainly base on the solution of the
eigen system, especially on the eigenvalues relationship of the two eigen
mode. This is because, eigen mode (eigenvector) is the companion of its
related eigenvalue in dynamics. If the eigenvalue of an eigenvector close to
0, the related cyclic frequency tends to 0; Then, the cyclic motion referring
to angular momentum measurement is statistically weak and being no observable,
and is regarded as dominated and no selected. This study is based on angular
momentum measurement. So, the mode selection is the contribution weight of an
eigen mode to total observed angular momentum. In this view, special
relationship of the eigenvalues of the two mode, like equal or zero, is cruel.
This is the main reasons to specify the parameter $a$.

In total, we choose the five parameters as
$a=[-4.236,-0.618,0.234,1.618,4.236]$, and call the related game treatment in
this study as Treatment-(1, 2, 3, 4, 5) respectively, which denoted as (Tr1,
Tr2, Tr3, Tr4, Tr5) in short. The reasons of the parameter assignments are to
reply the research questions:

##### Prediction 1

, which is aim at answering the RQ1, we assign the parameter
$a=(-0.618,1.618)$ are chose because at these two value, $\chi^{+}(-0.618)=0$
and $\chi^{-}(1.618)=0$; So, only one mode can be observed in these two
treatments, respectively. This is the impact of the special condition of 0
eigenvalue on angular momentum measurement.

##### Prediction 2

, which is aim at the RQ2, we assign the parameter $a=(-4.236,0.236)$. Because
$\big{|}\chi^{+}(-4.236)\big{|}=\big{|}\chi^{-}(0.236)\big{|}$ at these two
value, So, the two mode observed should be selected equally in these two
treatments, respectively. This is the impact of the special condition of equal
eigenvalue on angular momentum measurement. Meanwhile, for symmetry
consideration, we choose the $a$ which oppose of the $a=-4.236$. At this point
$\big{|}\chi^{+}(4.236)\big{|}>>\big{|}\chi^{-}(4.236)\big{|}$ which is
appreciate for the qualitative analysis.

Details of the deduction is shown in appendix secction 4.1.3.

Table 1: Weight in $mn$-subspace (top), the results of the eigen mode selection of the theory (middle) and the experiment (bottom)  mn | $\sigma_{a}$ | $\sigma_{\beta}$ |  $L$(-4.236) |  $L$(-0.618) |  $L$(0.234) |  $L$(1.618) |  $L$(4.236)
---|---|---|---|---|---|---|---
12 | -2.988 | 1.847 | 2 | 1/2 | -1/2 | -2 | -2
13 | -1.847 | -2.988 | -1 | -1 | -1 | -1 | -1
14 | 1.847 | 2.988 | 1 | 1 | 1 | 1 | 1
15 | 2.988 | -1.847 | -2 | -1/2 | 1/2 | 2 | 2
23 | -2.988 | 1.847 | 2 | 1/2 | -1/2 | -2 | -2
24 | -1.847 | -2.988 | -1 | -1 | -1 | -1 | -1
25 | 1.847 | 2.988 | 1 | 1 | 1 | 1 | 1
34 | -2.988 | 1.847 | 2 | 1/2 | -1/2 | -2 | -2
35 | -1.847 | -2.988 | -1 | -1 | -1 | -1 | -1
45 | -2.988 | 1.847 | 2 | 1/2 | -1/2 | -2 | -2
Theory ($\rho$) |  |  |  |  |  |  |
$\sigma_{\alpha}$ | 1 | 0.0500 | -0.4543 | 0.1307 |  [rgb]1,0.9,0.90.8348 |  [rgb]1,0.9,0.90.9950 |  [rgb]1,0.9,0.90.9950
$\sigma_{\beta}$ | 0.0500 | 1 |  [rgb]1,0.9,0.90.8670 |  [rgb]1,0.9,0.90.9967 | 0.5915 | -0.0498 | -0.0498
Experiment ($\rho$) |  |  |  |  |  |  |
$\sigma_{\alpha}$ |  |  | -0.4630 | 0.4750 |  [rgb]1,0.9,0.9 0.9090 |  [rgb]1,0.9,0.9 0.9750 |  [rgb]1,0.9,0.9 0.9990
($p_{\alpha}$) |  |  | (0.1667) | (0.2924) |  [rgb]1,0.9,0.9(0.0014) |  [rgb]1,0.9,0.9(0.0000) |  [rgb]1,0.9,0.9(0.0000)
$\sigma_{\beta}$ |  |  |  [rgb]1,0.9,0.9 0.8510 |  [rgb]1,0.9,0.9 0.8400 | 0.3330 | 0.2250 | 0.0070
($p_{\beta}$) |  |  |  [rgb]1,0.9,0.9(0.0020) |  [rgb]1,0.9,0.9(0.0013) | (0.1366) | (0.5280) | (0.8563)

##### Parameter dependence predictions

Myopic response is the nature of human being. It play as the central principle
in behaviour game theory [4]. By a simple myopic response strength estimation,
we can estimate which mode should be qualitatively dominate (Details of the
algorithm for estimation is shown in appendix section 4.1.8). Using the
estimation, we can obtain the strength of the cycle, denoted as $L(a)$, for
the 10 sub-spaces. The results are shown in the top panel in Table 1 (a full
version is shown in Table 10 in appendix).

Qualitatively, we can estimate projection of the strength of the response
($L(a)$) on the two mode $(\sigma_{\alpha},\sigma_{\beta})$ respectively by
calculate the correlation coefficients ($\rho$). The results are shown in the
middle panel (labelled as Theory ($\rho$)) in Table 1. The $\rho$ value have
clear splitting between the various $a$. With this estimation, we believe
that, the parameter selection is fit to answer above the research questions.

Quantitatively, for various $a$ in game matrix, the eigenvector set is
identical. The eigenvector set has only two orthogonal complex eigenvectors
with their complex conjunction respectively. So, if a system can be decomposed
by the eigen eigenvector $(\xi_{a},\xi_{b})$, by multi variable linear
regression (MLR), we can learn how the weights depends on the parameter $a$ of
the treatment.

##### Parameter independence predictions

As the components of the eigenvector is of high symmetry, the values of
eigencycle is very limit, which can be $\pm\alpha$ and $\pm\beta$. So, between
the 10 various 2-dimensional spaces, the observed eigencycle values can
frequently be totally equal or totally oppose. This prediction is based on the
eigen mode hypotheses: All parts of the eigenvector are not only oscillating
with the same frequency, but also oscillating with constant phase difference.
An example to interpret the invariant cross the 5 parameters are shown in
appendix 4.1.9

The theoretical prediction of the invariant are shown in the upper triangular
in Table 4, in which $(+)$ indicates the totally equal and $(-)$ indicates the
totally oppose. Details of the algorithm for these theoretical results are
shown in SI. In sum, for the 3rd RQ, we have follow prediction and its check
method:

##### Prediction 3

The motion observed in subspace have the relationship of totally equal or
totally oppose cross $a$; The relationship is predictable shown in Table 4, in
which (+) indicates the equal and ($-$) indicates oppose. There are 20 points
being expected to have the maximum correlation coefficients in significant.

On verification, this prediction can be verified by the correlation
coefficient ($\rho_{mn,m^{\prime}n^{\prime}}$ where the
($mn,m^{\prime}n^{\prime}$ are of the two 2-d subspace index) of the
eigencycles values between the 10 2-dimensional subspace in experimental time
series.

Figure 3: The eigenvalues and observed mode selection. Top panel, the
imaginary part of the eigenvalues as a function of $a$ in the game matrix. The
heavy line indicates the observable eigen mode, referring to the myopic
response estimation. The blue (red) indicates $\sigma_{\alpha}$ and
$\sigma_{\beta}$ associated eigenvalues (imaginary part), respectively. The
arrows indicates the parameter $a$ values used in experiments. Bottom panel,
the experimental weight of the mode selection. Results of the session shown in
small. A Big label indicates the average of the sessions. Red (and Blue)
indicates the $k_{1}$ (and $k_{2}$) value. A big label with fulled (not
fulled) surface indicates the value is deviated from 0 in significant (not
significant) by ttest over the observation from 10 sessions. Numerical results
see Table 8 in appendix.

###  2.2 Experimental observation

To identify the theoretical prediction shown in the logic chain, the test of
of the long run mean and deviation of the strategy distribution, as well as
the eigencycles. There are 5 experiment treatments which depends on the $a$
value. For each treatments, there has 6 sessions repeated. In each session
having 20 minutes and 600 repeated rounds. The experiments were conducted
during March 6 to April 3 at our laboratory.222We thanks San Lixia and Guo
Hangyi for the help of the experiments carrying out. Details of the experiment
protocol see Method section.

####  2.2.1 Distribution and deviation

Results of the mean and standard deviation do not reject the Nash equilibrium
hypotheses. Notice that, the theoretical deviation depends on the population
size. In our experiment, there are 6 human subjects participated, which can be
deduced basing on the Nash equilibrium concept. Details of the results shown
in Table 6 in appendix.

####  2.2.2 Eigencycle in experiment

Referring to the experimental eigencycle measurement (see Method), we can
obtain the angular momentum results from the experiment time series for the
five treatments respectively. The results are shown in Table 2 which labelled
as $L^{\text{Tr}\cdot}$.

In addtion, to identify the strength of the motion, we report the normal of
the eigencycle amplitude, shown in the last row in Table 2.

Table 2: Theoretical eigencycle $\sigma$ and experimental observation $L$ mn | $\sigma_{a}$ | $\sigma_{\beta}$ | $L^{\text{Tr}1}$ | $L^{\text{Tr}2}$ | $L^{\text{Tr}3}$ | $L^{\text{Tr}4}$ | $L^{\text{Tr}5}$
---|---|---|---|---|---|---|---
12 | -0.3804 | 0.2351 | 0.3478 | -0.1060 | -0.2229 | -0.3899 | -0.4050
13 | -0.2351 | -0.3804 | -0.1601 | -0.4085 | -0.3906 | -0.2935 | -0.2057
14 | 0.2351 | 0.3804 | 0.2400 | 0.5420 | 0.2760 | 0.2786 | 0.2349
15 | 0.3804 | -0.2351 | -0.4278 | -0.0275 | 0.3375 | 0.4047 | 0.3758
23 | -0.3804 | 0.2351 | 0.4096 | -0.0319 | -0.1187 | -0.3265 | -0.4006
24 | -0.2351 | -0.3804 | -0.2292 | -0.3648 | -0.4537 | -0.2935 | -0.2081
25 | 0.2351 | 0.3804 | 0.1675 | 0.2907 | 0.3495 | 0.2302 | 0.2038
34 | -0.3804 | 0.2351 | 0.4068 | 0.0591 | -0.1720 | -0.2655 | -0.4086
35 | -0.2351 | -0.3804 | -0.1572 | -0.4995 | -0.3373 | -0.3545 | -0.1977
45 | -0.3804 | 0.2351 | 0.4175 | 0.2363 | -0.3497 | -0.2803 | -0.3818
normal ampl. |  |  | 8.6771 | 4.0952 | 2.6071 | 5.6943 | 11.5525

###  2.3 Verification by experiment

####  2.3.1 Eigen mode selection

##### Weight of the eigen modes

Having the observed angular momentum vector $L^{Tr\cdot}$ from the experiment
shown in Table 2, we can calculate the weight of the $L^{Tr\cdot}$ over the
eigencycle set vectors ($\sigma_{\alpha},\sigma_{\beta}$). The calculation is
carried by multiple linear regression analysis, which is used to assess the
association between two independent variables
($\sigma_{\alpha},\sigma_{\beta}$) and a single dependent variable, the
outcome $L^{Tr\cdot}$. The multiple linear regression equation is as follows:

$L^{Tr\cdot}=c_{0}+k_{\alpha}\sigma_{\alpha}+k_{\beta}\sigma_{\beta}.$ (7)

Herein, $k_{\alpha}$ and $k_{\beta}$ are the adjusted estimated regression
coefficient that quantifies the association between the factors
($\sigma_{\alpha}$ and $\sigma_{\beta}$) and the outcome. The decomposition
model have significant interpretation power when $p$ value close to 0 and
$c_{0}$ does not deviate from 0 in significant.

The results for the five treatments are shown in Table 3. We have also carried
out the same calculate at experiment session level, in which there are 10
sample for each treatment, and the result shown in Table 8. Our interpretation
of the results are following:

  1. 1.

As response to RQ1, the null hypothesis is that the eigen mode does not exist
or is indistinguishable. This null hypothesis can be rejected in significant
by following results.

     * •

Results from the correlation coefficient ($\rho$) as well as the linear
regression ($p$) value between the eigencycle vectors of the theory and the
experiment by treatment shown the bottom panel in Table 1. In the Table, the
[rgb]1,0.9,0.9highlight values enhance the consistence between theory and
experiment for visibility (see also the details in 1st-2nd column in Table
10).

     * •

Results from the $p$ value of the MLR by treatment show in Table 3

     * •

Results from the the $p$ value of the MLR by sessions (6th column in Table 8).

     * •

Results from the statistical ttest results of the $p_{k_{1}}$ and the
$p_{k_{2}}$ values by treatments, in which the samples comes from the $k_{1}$
and the $k_{2}$ of the MLR by sessions (the last two columns in Table 8).

These results illustrate that, the prediction 1 is supported. The dynamics
observation $L^{Tr\cdot}$ from the all five treatment can be decomposed by the
eigen modes in significant.

  2. 2.

For RQ2, the null hypothesis is that the mode selection is unpredictable. This
can be rejected in significant.

     * •

Evidence comes from experiment Tr2 and Tr4 reject the null hypotheses in
significant. Because, the projection of $L^{Tr2}$ appears only in
$\sigma_{\beta}$ ($p$=0.005, ttest on the deviation from 0, $N$=10 session);
And $L^{Tr4}$ appears only in $\sigma_{\beta}$ ($p$=0.000, ttest on the
deviation from 0, $N$=10 session). The statics can be obtained in Table 8 in
the appendix.

     * •

Supporting evidence also comes from experiment Tr1 and Tr3, of which
theoretical expectation is that, the weight $|k_{\alpha}|$ and $|k_{\beta}|$
are same. Both Tr1 ($p$ = 0.090, ttest on the null hypothesis
$|k_{\alpha}|=|k_{\beta}|$, $N$=10 session) and Tr3 ($p$= 0.386, ttest on the
null hypothesis $|k_{\alpha}|=|k_{\beta}|$, $N$=10 session) do not rejected
the theoretical in significant. The statics can be obtained base on the data
shown in Table 9 in the appendix.

Table 3: Multi eigen mode weight Treatment | $a$ | $c_{0}$ | $k_{\alpha}$ | $k_{\beta}$ | $\rho$ | $p$
---|---|---|---|---|---|---
Tr1 |  $-$4.2360 |  $-$0.0000 |  $-$0.5967 | 0.9229 | 0.9884 | 0.0000
Tr2 |  $-$0.6180 | 0.0000 | 0.2214 | 0.4403 | 0.8924 | 0.0004
Tr3 | 0.2360 | 0.0000 | 0.2927 | 0.1345 | 0.9337 | 0.0001
Tr4 | 1.6180 |  $-$0.0000 | 0.7070 | 0.1159 | 0.9716 | 0.0000
Tr5 | 4.2360 |  $-$0.0000 | 1.4666 |  $-$0.0943 | 0.9979 | 0.0000

####  2.3.2 Invariant in eigen modes

In theory, there exist invariant between the cycles in the subspace. The
motion observed in some 2d sub-spaces can be totally equal or totally oppose
independent on $a$; The prediction is predictable shown in the up triangle in
the top panel in Table 4. In the up triangle, the $(+)$ and $(-)$ indicate the
equal and oppose, respectively; If no correlated, the cell remain blank. With
the total 50 sessions data, this prediction can be verified by the correlation
coefficient $\rho$ of the eigencycles values between the 10 2-dimensional
subspace in experimental time series.

As a clean up presentation, Table 4 reports the correlation coefficient $\rho$
of observed eigencycle between the 10 in pairs (Full results is shown in Table
11 in appendix). Notice that, in $\rho_{35,14}$, the low script ($35,14$) are
of the two 2-d subspace index, ($x_{3},x_{5}$) and ($x_{1},x_{4}$),
respectively; and $x_{.}$ is the proposition of the $x_{.}$ strategy in games.
There are totally 45 independent test points. The bottom panel reports the $p$
value of linear regression to verified the dependence of the paired 50
observations. So call clean up means that we select the 20 maximum $|\rho|$
value to fill the top panel, meanwhile the 20 minimum $p$ value to fill the
bottom panel in Table 4 from the Table 11.

As result, referring to the reported $\rho$ and $p$ values, none of the 45
observed check points violates the theoretical prediction. This is the answer
to the RQ3, there exists invariant between the subspace insignificant
($p$=0.0000, Wilcoxon Sign Rank Test, $N$=45 check points).

Table 4: correlation coefficient of observed eigencycle between 10 2-dimensional subspace  | $L^{12}$ | $L^{13}$ | $L^{14}$ | $L^{15}$ | $L^{23}$ | $L^{24}$ | $L^{25}$ | $L^{34}$ | $L^{35}$ | $L^{45}$
---|---|---|---|---|---|---|---|---|---|---
$\rho$ |  |  |  |  |  |  |  |  |  |
$L^{12}$ | 1 |  |  |  ($-$) | (+) |  |  | (+) |  | (+)
$L^{13}$ |  | 1 |  ($-$) |  |  | (+) |  ($-$) |  | (+) |
$L^{14}$ |  | -0.5832 | 1 |  |  |  ($-$) | (+) |  |  ($-$) |
$L^{15}$ | -0.9101 |  |  | 1 |  ($-$) |  |  |  ($-$) |  |  ($-$)
$L^{23}$ | 0.9380 |  |  | -0.8719 | 1 |  |  | (+) |  | (+)
$L^{24}$ |  | 0.5599 | -0.6210 |  |  | 1 |  ($-$) |  | (+) |
$L^{25}$ |  | -0.5855 | 0.5656 |  |  | -0.7145 | 1 |  |  ($-$) |
$L^{34}$ | 0.8823 |  |  | -0.8455 | 0.9427 |  |  | 1 |  | (+)
$L^{35}$ |  | 0.7611 | -0.4853 |  |  | 0.5719 | -0.6138 |  | 1 |
$L^{45}$ | 0.8826 |  |  | -0.9260 | 0.8826 |  |  | 0.9102 |  | 1
$p$ |  |  |  |  |  |  |  |  |  |
$L^{12}$ | 0 |  |  |  |  |  |  |  |  |
$L^{13}$ |  | 0 |  |  |  |  |  |  |  |
$L^{14}$ |  | 0.0000 | 0 |  |  |  |  |  |  |
$L^{15}$ | 0.0000 |  |  | 0 |  |  |  |  |  |
$L^{23}$ | 0.0000 |  |  | 0.0000 | 0 |  |  |  |  |
$L^{24}$ |  | 0.0000 | 0.0000 |  |  | 0 |  |  |  |
$L^{25}$ |  | 0.0000 | 0.0000 |  |  | 0.0000 | 0 |  |  |
$L^{34}$ | 0.0000 |  |  | 0.0000 | 0.0000 |  |  | 0 |  |
$L^{35}$ |  | 0.0000 | 0.0004 |  |  | 0.0000 | 0.0000 |  | 0 |
$L^{45}$ | 0.0000 |  |  | 0.0000 | 0.0000 |  |  | 0.0000 |  | 0

##  3 Discussion

We have illustrated the eigen mode existence as well as the eigen mode
selection in laboratory human subject game experiment. The eigen mode
selection, which identified by eigencycle and angular momentum measurement,
can be qualitatively predicted by two theoretical approaches: the replicator
dynamics and the principle of the myopic best responses of human natural on
decision making.

To obtain the eigen mode selection is not an accident, when we refer to the
cycles in elementary games published near 2014 [7, 9, 8], as well as recent
finding and confirmation in laboratory game experiments:

  1. 1.

By the end of 2020, in the O’Neill game experiment reported in 1987 [12], the
theoretical prediction of the eigensystem has been confirmed by eigencycle
measurement. In the O’Neill game experiment reported in 2001 [13] and in 2013
[14], the dynamics structures in various setting are consistence in high
accuracy in 2020 [15, 10]

  2. 2.

In a five strategy one population game experiment, using the same approach,
the regularity of high dimensional cycle is confirmed, in 2021. [10]

  3. 3.

In a four strategy one population game experiment, using the same approach
again, the regularity of high dimensional cycle is confirmed by two controlled
parameter, too, in 2021. [11]

  4. 4.

It has been notice that, borrowing the pole assignment approach in state
dependent closed-loop feedback control theory in modern control theory, the
game dynamics structure control by design is realisable in laboratory game
experiments in 2022. [17]

Among these, only the O’Neill game (designed by O’Neill 1987 [12]) has 3 pairs
of complex eigenvalue/eigenvector. It has been notice that, among the three
pairs, the low eigenvalue has significant influence in the dynamics structure,
which has already reach the eigen mode selection issue (see Eq. 6.5 in the
master thesis [10]). So, it is not a surprise for us to pay attention on multi
mode selection.

At its root, this study is follow the workflow (logic chain) which has been
existed since 1978 when the game dynamics established [16]. At that time,
Jacobian matrix (character matrix) had been there before 1850. In 2020 [15],
the finding of the dynamics structure (cycling spectrum) in O’Neill game
reported in 1987 is only an application of the long existed workflow. Might
why finding such simple and clear pattern need so long waiting is a puzzle in
the history of studying game dynamics theory.

This study add knowledge to the existed workflow led by [15], by provides the
myopic response estimation to predict the eigen mode selection (see Table 1
and its related explanation as example). Although the estimation is
qualitative approach, but it highlight that, in game dynamics theory, the
eigen mode selection issue is not totally in dark.

On further, the consistency of the theoretical approaches has to be
investigated.333Wang Zhijian thanks Zhao Yuqi from Chu Cochen Honors College
of Zhejiang University, as well as Guo Hangyi from School of Computer Science
and Engineering of Zhejiang University for helpful discussion at this point.
There are two approaches

  * •

The eigenvalues from replicator dynamics in Eq. (6), considering the impact of
eigenvalue on the angular momentum measurement, which was proved in
mathematics in [15] (see Appendix 2: Invariant between eigencycle and the
angular momentum in [15]).

  * •

The myopic response strength estimation approach in section 2.1.2 and shown in
Table 1, which play as the central principle in behaviour game theory [4].

in this report to predict the eigen mode selection. Their predictions are
supported in statistics significant by the experiments data in this study.
However, none of the two approaches straightly predicts the weight of the mode
selected. This is an obvious challenge to game theorist.

## References

  * [1] William H Sandholm.  Population Games and Evolutionary Dynamics.  MIT Press,, 2010.
  * [2] Daniel Friedman and Barry Sinervo.  Evolutionary games in natural, social, and virtual worlds.  Oxford University Press, 2016.
  * [3] Larry Samuelson.  Game theory in economics and beyond.  Journal of Economic Perspectives, 30(4):107–30, 2016.
  * [4] C.F. Camerer.  Behavioral game theory: Experiments in strategic interaction.  Princeton University Press, 2003.
  * [5] Timothy N Cason, Daniel Friedman, and Ed Hopkins.  Testing the tasp: An experimental investigation of learning in games with unstable equilibria.  Journal of Economic Theory, 145(6):2309–2331, 2010.
  * [6] Moshe Hoffman, Sigrid Suetens, Uri Gneezy, and Martin A Nowak.  An experimental investigation of evolutionary dynamics in the rock-paper-scissors game.  Scientific reports, 5(1):1–7, 2015.
  * [7] Timothy N Cason, Friedman Daniel, and E. D. Hopkins.  Cycles and instability in a rock–paper–scissors population game: A continuous time experiment.  Review of Economic Studies, 1:1, 2014.
  * [8] Bin Xu, Shuang Wang, and Zhijian Wang.  Periodic frequencies of the cycles in $2\times 2$ games: evidence from experimental economics.  European Physical Journal B, 87(2):46, 2014.
  * [9] Zhijian Wang, Bin Xu, and Hai-Jun Zhou.  Social cycling and conditional responses in the rock-paper-scissors game.  Scientific reports, 4(1):1–7, 2014.
  * [10] Qinmei Yao.  Theoretical analysis and experiment of dynamic structure of high dimensional game.  https://cdmd.cnki.com.cn/Article/CDMD-10335-1021626407.htm,doi=10.27461/d.cnki.gzjdx.2021.000847, 2021\.
  * [11] Shujie Zhou.  Theory and experiment of dynamic structure in four strategy game.  https://cdmd.cnki.com.cn/Article/CDMD-10335-1021626406.htm, doi=10.27461/d.cnki.gzjdx.2021.000844, 2021.
  * [12] B. O’Neill.  Nonmetric test of the minimax theory of two-person zerosum games.  Proceedings of the National Academy of Sciences, 1987.
  * [13] Ken Binmore, Joe Swierzbinski, and Chris Proulx.  Does minimax work? an experimental study.  Economic Journal, pages 445–464, 2001.
  * [14] Yoshitaka Okano.  Minimax play by team.  Games & Economic Behavior, 2013.
  * [15] Zhijian Wang and Qingmei Yao.  Human social cycling spectrum.  arXiv preprint arXiv:2012.03315, 2020.
  * [16] Peter D Taylor and Leo B Jonker.  Evolutionary stable strategies and game dynamics.  Mathematical biosciences, 40(1-2):145–156, 1978.
  * [17] Zhijian Wang.  Game dynamics structure control by design: an example from experimental economics.  arXiv preprint arXiv:2203.06088, 2022.
  * [18] Yijia Wang, Xiaojie Chen, and Zhijian Wang.  Testability of evolutionary game dynamics based on experimental economics data.  Physica A: Statistical Mechanics and its Applications, 486:455 – 464, 2017.

##  4 Appendeix

###  4.1 Technology Notes

####  4.1.1 The consequence of the oppose a

When oppose $a$ to $-a$, one may think the game well remain same by trans the
s1 strategy with the s5 strategy, the s2 strategy with 3 strategy, and then
the matrix should be:

${A}^{\dagger}=\left(\begin{array}[]{rrrrr}0&-a&-1&1&a\\\ a&0&-a&-1&1\\\
1&a&0&-a&-1\\\ -1&1&a&0&-a\\\ -a&-1&1&a&0\end{array}\right)$ (8)

This matrix appears similar to the original, but actually is not equivalent
because of -1 and 1 elements in the matrix. So, the theoretical results of the
Treatment 1 ($a=-4.236$) can be different from the Treatment 5 ($a=+4.236$).

####  4.1.2 Details of the eigenvector structure

The eigenvector structure of the game play cruel role in this study. Besides
the unique equilibrium (0.2, 0.2, 0.2, 0.2, 0.2) for various $a$, the
eigenvector structure are of high symmetry, which has only two independent
patterns, for various $a$ too. Details of the pattern for various $a$ are
shown in Fig. 4.

It is obviously that, among these pattern, only the two $v^{\alpha}$ and
$v^{\beta}$ shown in Fig. 1 in main text are independent. the remains are of
the complex conjunction of these two vectors.

Figure 4: Geometric presentation of the eigenvector set. From top to bottom
are by the order of $a$, from -4.236 to 4.236. The titles of each sub figure
is the associate eigenvalue of the eigenvector.

####  4.1.3 The 5 parameter selection

It is easy to notice that, the manifold satisfy central manifold theorem
requirement:

$\Re(\lambda_{i})=0~{}~{}~{}~{}i\in\\{1,2,3,4,5\\}$

At the same time, when $a=-0.618$

$\lambda_{2}=\lambda_{4}=0$

and when $a=1.618$

$\lambda_{3}=\lambda_{5}=0$

Meanwhile, when

$a=\left(\begin{array}[]{c}-\sqrt{5}-2\\\
\sqrt{5}-2\end{array}\right)=\left(\begin{array}[]{r}-4.236\\\
0.236\end{array}\right)$

the system has unique imaginary part,

$\begin{array}[]{c}|\Im(\lambda_{i})|\end{array}=\left(\begin{array}[]{c}\sqrt{-2\,\sqrt{5}-1}\\\
\sqrt{2\,\sqrt{5}-1}\end{array}\right)~{}~{}~{}~{}i\in\\{2,3,4,5\\}.$

Having same imaginary part indicates having same frequency of cycles.

For the symmetry consideration, we choose the oppose the maximum value above
is 4.236, which is chose as the last parameter. In sum, the parameters
selected are

$a=\big{[}-4.236,-0.618,0.234,1.618,4.236\big{]}.$

####  4.1.4 Measurement of experimental angular momentum

##### The measurement for the cycle in the subspace:

According to the theoretical eigencycle set decomposition approach, we can
measure the cyclic angular momentum 444This 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
$L^{(mn)}_{E}$ [18] can be expressed by the following formula:

$L^{mn}_{E}=\frac{1}{N-1}\sum_{t=1}^{N-1}\left(x(t)-O\right)\times\left(x(t\\!+\\!1)-x(t)\right)$
(9)

  * •

$L^{(mn)}_{E}$ represents the average value of the accumulated angular
momentum over time; the subscript $mn$ indexes the two-dimensional
$(x_{m},x_{n})$ subspace;

  * •

$N$ is the length of the experimental time series, that is, the total number
of repetitions of the repeated game experiments;

  * •

$O$ is the projection of the Nash equilibrium at the subspace
$\Omega^{(m,n)}$;

  * •

$x(t)$ 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

  * •

$\times$ represents the cross product between two two-dimensional vectors.

Interpretation of the measurement

####  4.1.5 Theoretical eigencycle set

##### Definition

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 eigencycle,
marked as $\sigma^{(mn)}$ and calculated as follows:

$\sigma^{(mn)}=\pi\cdot||\eta_{m}||\cdot||\eta_{n}||\cdot\sin\big{(}\arg(\eta_{m})-\arg(\eta_{n})\big{)},$
(10)

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 presentation of the eigencycle is

$\sigma^{(mn)}=\pi\cdot\big{(}\Re(\eta_{m})\Im(\eta_{n})-\Re(\eta_{n})\Im(\eta_{m})\big{)},$
(11)

wherein the $\Re(\eta_{m})$ is the real part of the complex number $\eta_{m}$
and the $\Im(\eta_{m})$ is the imaginary part.

##### Eigencycle set values of the game

According to this formula, for these game, the column $\sigma^{\alpha}$ and
$\sigma^{\beta}$ in Table 2 lists the eigencycle set values of the two unique
eigenvectors of the replicator dynamics.

Interpretation

  * •

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 $N$.

  * •

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:
{$\\{m,n\\}\in\\{1,2,...,n\\}\cap(m<n)$}.

  * •

Geometric presentation: An eigencycle only depends on the internal components
$\eta_{m}$ and $\eta_{n}$ of the 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 as shown in figure 2.

####  4.1.6 Equivalent of eigencycle 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. In a time series from a real system,
the angular momentum of two observation is measurable, so the eigencycle
concept can be verified in real system.

The invariant (Equivalent) between the eigencycle and the angular momentum is
important in this study. Because, only having this invariant, we can using
eigencycle as predictor for experimental angualr momentum. Here, the
eigencycle as predictor can be obtained by the eigenvector from dynamics
equation, and the angular momentum can he obtained in time series.

Main idea to prove the invariant is to calculate the angular momentum of the
eigencycle explicitly. Details of the deduction are shown in [15]. The main
results applied in this study are following two propositions.

##### Proposition 1

For a given eigenvector $\xi$, in all of the two dimension sub-spaces
$\Omega^{(m,n)}$ at any given $t$, the rate between the instantaneous angular
momentum value and eigencycle value is equal,

$\frac{L_{mn}(t)}{\sigma^{(mn)}}=\textbf{C}(t)~{}~{}~{}~{}\forall(m,n)\in(1,2,...,N)$
(12)

Such that, for various subspace $(m,n)$ and $(m^{\prime},n^{\prime})$, the
average accumulated angular momentum $L_{mn}$ divided by its $\sigma^{(mn)}$
is invariant,

$\frac{1}{t_{1}-t_{0}}\sum_{t=t_{0}}^{t_{1}}\frac{L_{mn}(t)}{\sigma^{(mn)}}=\frac{1}{t_{1}-t_{0}}\sum_{t=t_{0}}^{t_{1}}\frac{L_{m^{\prime}n^{\prime}}(t)}{\sigma^{(m^{\prime}n^{\prime})}}=\textbf{C}~{}~{}~{}~{}\forall(m,n;m^{\prime},n^{\prime})\in(1,2,...,N)$
(13)

So, by take $\sigma^{(mn)}$ as predictor (independent variable) and take
$L_{mn}$ as the observation (dependent variables), the a linear regression
line, with truncation term being 0, is expected.

##### Proposition 2

Interfere of two components from two different eigenvector is vanish, when
following condition hold:

  1. 1.

White noise condition: after a random interrupt, the initial phase of the two
eigenvector distribute between [$-\pi,\pi$] equally;

  2. 2.

Sufficient noise shock: there are sufficient times of the interrupt;

  3. 3.

Long time limit: there are sufficient long time of the time series.

####  4.1.7 Mode of angular momentum

We define mode of angular momentum as,

$\Big{|}L\Big{|}=\Big{(}\sum_{i=1}^{10}L_{mn}^{2}\Big{)}^{1/2}~{}~{}~{}~{}~{}~{}~{}~{}~{}(m,n)\in\\{1,2,3,4,5\\}\cap
m<n$

####  4.1.8 Algorithm for the myopic response strength estimation

Following estimation is not an approach for general game. This is what we hope
to emphasis.

This estimation takes the advantage of the symmetry of the game. So called
symmetry, are of the equilibrium and the eigenvector structure, as well as the
payoff matrix and Jaobian matrix.

At the same time, we assume that, In long run, statistically, the deviation of
the each strategy are symmetrical (equal).

Now, assume that, in a population with huge number of agents system $S$. The
system is exactly at Nash equilibrium, and in which, we add two agents A and
B, and both A and B play $x_{1}$ strategy. This is the initial condition.

Now, only A has opportunity to make a response. Its response will lead to the
change of probability of $x_{2},x_{3},x_{4},x_{5}$, which have consequence on
angular momentum measurement.

When A makes a response, the observed velocity of $\dot{x}_{2}$ in the sub
space $mn=(1,2)$ is close to

$\dot{x}_{2}\simeq J_{21}\cdot\delta(x_{1}).$

then the expected angular momentum measured in in the sub space $(mn)=(1,2)$
is

$L_{1}\simeq\delta(x_{1})\times\dot{x}_{2}\simeq
J_{21}\cdot\delta(x_{1})^{2}.$

Herein, $J$ is the Jacobian (character matrix) of the dynamics as shown in Eq.
5 in main text. Similarly, if the added two agents are playing $x_{2}$, we can
expect that

$L_{2}\simeq-\delta(x_{2})\times\dot{x}_{1}\simeq-
J_{12}\cdot\delta(x_{2})^{2}.$

Here, the symbol $(-)$ is added because of the definition of angular momentum.

As the deviation of $\delta(x_{1})$ and $\delta(x_{2})$ are equal in long run,
statistically, these two factors have same contribution on $L$.

At the same time, in this study case $J_{ij}=-J_{ji}$ for all
$(i,j)=(1,2,3,4,5)\cap(i>j)$, so the contribution of the two terms are equal,
and propitiation to $J_{12}$.

In this study case, there two constant ($\pm 1$) in $J$ which are applied as
invariant. As estimation, we set step function $f$ as

  * •

2, if $J_{ij}>1$

  * •

$-$2, if $J_{ij}<-1$

  * •

1/2, if $0<J_{ij}<1$

  * •

$-$1/2, if $-1<J_{ij}<0$

Similarly we can assume the deviation appear at $x_{2},x_{3},x_{4},x_{5}$
respectively for various game parameter $a$. Then the expected angular
momentum set for each $a$ in Table 1 can be obtained.

Then, by evaluating the projection (correlation coefficient $\rho$) of these
angular momentum set (scale vector) over the eigencycle set (scale vector), we
provide the theoretical predictions of the eigen mode decomposition weight
(eigen mode selection) for various game with the parameter $a$.

####  4.1.9 An example of the invariant cross the 5 parameters

We show an example, using the observations from the subspace (12) and (15) ,
to explain the prediction of the parameter independence. Suppose a motion
$x(t)$ can be decomposed as

$x(t)=C_{a}\xi_{a}+C_{b}\xi_{b}$

in which, $C_{i}$ is the weight (or contribution) of the eigenvector $i$. For
different treatment the weight could differ. So, the theoretical composed
eigencycle $\Omega$ in the 2-d subspace (12) can be expressed as

$\Omega^{12}=C_{a}^{2}\sigma_{a}^{12}+C_{b}^{2}\sigma_{b}^{12};$

At the same time, the composed eigencycle $\Omega$ in the 2-d subspace (15) is

$\Omega^{15}=C_{a}^{2}\sigma_{a}^{15}+C_{b}^{2}\sigma_{b}^{15};$

as $\sigma_{1}^{12}=-\sigma_{1}^{15}$ and $\sigma_{2}^{12}=-\sigma_{2}^{15}$.
So, disregarding the combination of ($C_{1},C_{2}$) for various treatment in
theory, which is instinct dynamics property of the game. we can always have

$\Omega^{12}=-\Omega^{15}.$

Then observations from the subspace (12) and (15) are of the totally oppose
relation, which is of parameter independence.

In the same way, we deduce the theoretical expectation of the parameter
independence relationship for each pair of the 10 subspace.

Table 5: Experiment $L$ in 10 sub space By Treatment (Tr1-Tr5) and Sessions (1-10)  | mn | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---|---|---
Tr1 | 12 | 3.733 | 1.744 | 1.6 | 1.806 | 7.694 | 0.556 | 4.783 | 1.994 | 4.194 | 2.078
| 13 | -1.017 | -3.944 | -2.628 | -0.844 | -0.456 | -0.233 | -2.439 | -1.072 | -1.067 | -0.189
| 14 | -0.539 | 6.089 | 3.95 | 0.967 | 0.072 | 2.394 | 2.783 | 1.311 | 1.567 | 2.228
| 15 | -2.178 | -3.889 | -2.922 | -1.928 | -7.311 | -2.717 | -5.128 | -2.233 | -4.694 | -4.117
| 23 | 2.889 | 1.922 | 4.078 | 3.739 | 7.906 | 1.6 | 3.5 | 3.322 | 4.683 | 1.9
| 24 | -0.3 | -5.628 | -3.594 | -3.061 | -1.206 | -1.828 | -1.639 | 0.233 | -1.906 | -0.961
| 25 | 1.144 | 5.45 | 1.117 | 1.128 | 0.994 | 0.783 | 2.922 | -1.561 | 1.417 | 1.139
| 34 | 3.089 | 2.144 | 3.911 | 5.922 | 7.167 | 1.656 | 1.778 | 3.55 | 4.106 | 1.972
| 35 | -1.217 | -4.167 | -2.461 | -3.028 | 0.283 | -0.289 | -0.717 | -1.3 | -0.489 | -0.261
| 45 | 2.25 | 2.606 | 4.267 | 3.828 | 6.033 | 2.222 | 2.922 | 5.094 | 3.767 | 3.239
Tr2 | 12 | -1.267 | -1.45 | 0.161 | -0.572 | 2.567 | -3.339 | 1.767 | -2.139 | 0.133 | -0.2
| 13 | -1.522 | -3.106 | -3.239 | 0.35 | -4.217 | -0.783 | 1.333 | -2.833 | -1.639 | -1.072
| 14 | 1.417 | 4.039 | 3.644 | 0.878 | 3.367 | 2.383 | 1.45 | 2.283 | 2.017 | 0.717
| 15 | 1.372 | 0.517 | -0.567 | -0.656 | -1.717 | 1.739 | -4.55 | 2.689 | -0.511 | 0.556
| 23 | -1.506 | -1.6 | 0.906 | -1.856 | 3.189 | -2.667 | 1.928 | 0.639 | 0.061 | -0.4
| 24 | -0.372 | -2.983 | -2.739 | 0.2 | -2.561 | -1.972 | 0.683 | -3.883 | -0.506 | -0.806
| 25 | 0.611 | 3.133 | 1.994 | 1.083 | 1.939 | 1.3 | -0.844 | 1.106 | 0.578 | 1.006
| 34 | -1.811 | -0.8 | 0.744 | -0.056 | 3.194 | -1.5 | 2.117 | 0.561 | 0.406 | -0.433
| 35 | -1.217 | -3.906 | -3.078 | -1.45 | -4.222 | -1.95 | 1.144 | -2.756 | -1.983 | -1.039
| 45 | -0.767 | 0.256 | 1.65 | 1.022 | 4 | -1.089 | 4.25 | -1.039 | 1.917 | -0.522
Tr3 | 12 | 0.422 | -1.4 | 0.661 | 0.933 | -1.283 | -2.528 | -1.139 | -0.122 | -0.956 | -0.4
| 13 | -0.611 | -0.8 | -2.422 | -1.556 | -1.894 | -0.544 | 1.144 | -1.778 | 1.128 | -2.85
| 14 | -0.156 | 0.778 | 1.628 | 0.744 | 2.861 | 1.344 | -0.656 | -0.283 | -0.717 | 1.65
| 15 | 0.344 | 1.422 | 0.133 | -0.122 | 0.317 | 1.728 | 0.65 | 2.183 | 0.544 | 1.6
| 23 | 0.578 | -0.956 | -0.417 | 0.872 | 0.306 | -3.1 | -0.467 | 0.122 | -0.361 | 0.328
| 24 | 0.089 | -3.017 | -0.756 | -1.439 | -2.828 | -0.517 | -0.822 | -0.8 | 0.017 | -1.756
| 25 | -0.244 | 2.572 | 1.833 | 1.5 | 1.239 | 1.089 | 0.15 | 0.556 | -0.611 | 1.028
| 34 | -0.35 | -0.783 | -1.394 | 0.717 | -0.828 | -2 | 1.061 | -0.144 | -0.067 | -0.694
| 35 | 0.317 | -0.972 | -1.444 | -1.4 | -0.761 | -1.644 | -0.383 | -1.511 | 0.833 | -1.828
| 45 | -0.417 | -3.022 | -0.522 | 0.022 | -0.794 | -1.172 | -0.417 | -1.228 | -0.767 | -0.8
Tr4 | 12 | -3.817 | -2.472 | -1.628 | -1.161 | -3.106 | -2.922 | 0.494 | -4.067 | -1.172 | -2.35
| 13 | -2.322 | -3.5 | -1.811 | -1.333 | -1.489 | -1.039 | 0.556 | -2.656 | -2 | -1.117
| 14 | 0.883 | 2.739 | 2 | 0.306 | 4.389 | 0.733 | 0.256 | 3.139 | -0.733 | 2.156
| 15 | 5.256 | 3.233 | 1.439 | 2.189 | 0.206 | 3.228 | -1.306 | 3.583 | 3.906 | 1.311
| 23 | -2.922 | -2.444 | -2.767 | -1.089 | -1.744 | -2.672 | 1.2 | -3.478 | -1.272 | -1.406
| 24 | -1.328 | -2.011 | -1.65 | -0.839 | -4.089 | -0.678 | -1.183 | -2.517 | -0.783 | -1.633
| 25 | 0.433 | 1.983 | 2.789 | 0.767 | 2.728 | 0.428 | 0.478 | 1.928 | 0.883 | 0.689
| 34 | -1.767 | -1.956 | -1.272 | -0.778 | -1.733 | -2.033 | 2.394 | -2.9 | -2.956 | -2.117
| 35 | -3.478 | -3.989 | -3.306 | -1.644 | -1.5 | -1.678 | -0.639 | -3.233 | -0.317 | -0.406
| 45 | -2.211 | -1.228 | -0.922 | -1.311 | -1.433 | -1.978 | 1.467 | -2.278 | -4.472 | -1.594
Tr5 | 12 | -5.533 | -4.8 | -1.511 | -5.172 | -2.8 | -6.739 | -0.622 | -4.972 | -6.306 | -8.333
| 13 | -4.344 | -5.461 | -1.517 | -3.333 | -0.211 | -1.2 | -3.006 | -2.722 | -0.467 | -1.506
| 14 | 2.178 | 5.017 | 0.661 | 1.45 | 4.306 | 3.006 | 3.639 | 2.589 | 1.144 | 3.15
| 15 | 7.7 | 5.244 | 2.367 | 7.056 | -1.294 | 4.933 | -0.011 | 5.106 | 5.628 | 6.689
| 23 | -5.394 | -4.383 | -4.367 | -4.606 | -4.022 | -3.822 | -1.661 | -4.083 | -7.3 | -6.644
| 24 | -4.928 | -3.517 | -0.772 | -2.156 | -0.894 | -3.839 | 0.733 | -3.328 | -0.383 | -4.961
| 25 | 4.789 | 3.1 | 3.628 | 1.589 | 2.117 | 0.922 | 0.306 | 2.439 | 1.378 | 3.272
| 34 | -4.339 | -4.05 | -3.1 | -7.717 | -4.956 | -3.739 | -3.133 | -5.217 | -5.583 | -5.372
| 35 | -5.4 | -5.794 | -2.783 | -0.222 | 0.722 | -1.283 | -1.533 | -1.589 | -2.183 | -2.778
| 45 | -7.089 | -2.55 | -3.211 | -8.422 | -1.544 | -4.572 | 1.239 | -5.956 | -4.822 | -7.183
Table 6:  Mean and STD  | $p(x_{1})$ | $p(x_{2})$ | $p(x_{3})$ | $p(x_{4})$ | $p(x_{5})$ | $Std(x_{1})$ | $Std(x_{2})$ | $Std(x_{3})$ | $Std(x_{4})$ | $Std(x_{5})$
---|---|---|---|---|---|---|---|---|---|---
Tr1 | 0.194 | 0.209 | 0.188 | 0.198 | 0.210 | 0.156 | 0.162 | 0.153 | 0.162 | 0.163
Tr2 | 0.213 | 0.215 | 0.185 | 0.188 | 0.199 | 0.158 | 0.162 | 0.157 | 0.158 | 0.158
Tr3 | 0.216 | 0.216 | 0.181 | 0.188 | 0.198 | 0.162 | 0.168 | 0.152 | 0.153 | 0.157
Tr4 | 0.212 | 0.201 | 0.183 | 0.196 | 0.207 | 0.161 | 0.162 | 0.156 | 0.157 | 0.161
Tr5 | 0.209 | 0.196 | 0.195 | 0.194 | 0.207 | 0.163 | 0.158 | 0.159 | 0.157 | 0.157
Table 7: Mean and STD by sessions in experiment  ($a$) | session | $p(x_{1})$ | $p(x_{2})$ | $p(x_{3})$ | $p(x_{4})$ | $p(x_{5})$ | $s(x_{1})$ | $s(x_{2})$ | $s(x_{3})$ | $s(x_{4})$ | $s(x_{5})$
---|---|---|---|---|---|---|---|---|---|---|---
Tr1 | 1 | 0.185 | 0.209 | 0.214 | 0.174 | 0.218 | 0.142 | 0.158 | 0.139 | 0.139 | 0.152
(-4.236) | 2 | 0.221 | 0.227 | 0.159 | 0.199 | 0.194 | 0.170 | 0.176 | 0.149 | 0.178 | 0.163
| 3 | 0.185 | 0.221 | 0.187 | 0.206 | 0.201 | 0.158 | 0.174 | 0.152 | 0.171 | 0.167
| 4 | 0.190 | 0.175 | 0.224 | 0.209 | 0.202 | 0.153 | 0.140 | 0.167 | 0.178 | 0.165
| 5 | 0.190 | 0.210 | 0.207 | 0.191 | 0.202 | 0.165 | 0.175 | 0.175 | 0.159 | 0.173
| 6 | 0.196 | 0.219 | 0.160 | 0.241 | 0.184 | 0.150 | 0.153 | 0.139 | 0.167 | 0.147
| 7 | 0.192 | 0.216 | 0.176 | 0.201 | 0.215 | 0.158 | 0.177 | 0.152 | 0.157 | 0.164
| 8 | 0.199 | 0.212 | 0.193 | 0.196 | 0.200 | 0.145 | 0.153 | 0.151 | 0.158 | 0.157
| 9 | 0.191 | 0.201 | 0.197 | 0.185 | 0.227 | 0.160 | 0.150 | 0.153 | 0.152 | 0.158
| 10 | 0.194 | 0.202 | 0.165 | 0.183 | 0.256 | 0.154 | 0.167 | 0.149 | 0.162 | 0.181
Tr2 | 1 | 0.193 | 0.221 | 0.185 | 0.165 | 0.236 | 0.155 | 0.159 | 0.172 | 0.150 | 0.159
(-0.618) | 2 | 0.224 | 0.209 | 0.157 | 0.195 | 0.215 | 0.151 | 0.158 | 0.144 | 0.162 | 0.163
| 3 | 0.201 | 0.191 | 0.235 | 0.198 | 0.176 | 0.171 | 0.168 | 0.176 | 0.169 | 0.163
| 4 | 0.272 | 0.220 | 0.159 | 0.161 | 0.188 | 0.163 | 0.162 | 0.146 | 0.145 | 0.153
| 5 | 0.153 | 0.184 | 0.220 | 0.228 | 0.215 | 0.147 | 0.163 | 0.178 | 0.178 | 0.173
| 6 | 0.202 | 0.215 | 0.204 | 0.202 | 0.177 | 0.159 | 0.177 | 0.167 | 0.159 | 0.147
| 7 | 0.237 | 0.196 | 0.171 | 0.224 | 0.172 | 0.176 | 0.167 | 0.150 | 0.181 | 0.149
| 8 | 0.210 | 0.286 | 0.155 | 0.158 | 0.190 | 0.159 | 0.158 | 0.146 | 0.137 | 0.154
| 9 | 0.220 | 0.183 | 0.182 | 0.167 | 0.248 | 0.145 | 0.136 | 0.137 | 0.136 | 0.164
| 10 | 0.219 | 0.245 | 0.177 | 0.183 | 0.176 | 0.153 | 0.171 | 0.151 | 0.158 | 0.155
Tr3 | 1 | 0.163 | 0.249 | 0.186 | 0.163 | 0.240 | 0.139 | 0.171 | 0.150 | 0.134 | 0.163
(0.236) | 2 | 0.229 | 0.207 | 0.129 | 0.234 | 0.201 | 0.176 | 0.175 | 0.131 | 0.173 | 0.167
| 3 | 0.239 | 0.196 | 0.198 | 0.167 | 0.199 | 0.162 | 0.152 | 0.165 | 0.146 | 0.151
| 4 | 0.212 | 0.231 | 0.199 | 0.161 | 0.196 | 0.159 | 0.170 | 0.157 | 0.141 | 0.164
| 5 | 0.181 | 0.229 | 0.170 | 0.206 | 0.213 | 0.162 | 0.178 | 0.134 | 0.164 | 0.159
| 6 | 0.244 | 0.208 | 0.204 | 0.177 | 0.168 | 0.176 | 0.168 | 0.164 | 0.149 | 0.144
| 7 | 0.210 | 0.186 | 0.158 | 0.243 | 0.204 | 0.159 | 0.161 | 0.145 | 0.167 | 0.151
| 8 | 0.220 | 0.204 | 0.208 | 0.169 | 0.200 | 0.158 | 0.162 | 0.164 | 0.150 | 0.166
| 9 | 0.205 | 0.211 | 0.183 | 0.173 | 0.228 | 0.151 | 0.162 | 0.148 | 0.151 | 0.166
| 10 | 0.262 | 0.243 | 0.176 | 0.186 | 0.132 | 0.174 | 0.184 | 0.162 | 0.158 | 0.142
Tr4 | 1 | 0.181 | 0.225 | 0.174 | 0.161 | 0.259 | 0.157 | 0.166 | 0.152 | 0.145 | 0.179
(1.618) | 2 | 0.226 | 0.168 | 0.209 | 0.187 | 0.210 | 0.173 | 0.155 | 0.163 | 0.157 | 0.161
| 3 | 0.213 | 0.205 | 0.197 | 0.200 | 0.185 | 0.148 | 0.168 | 0.157 | 0.150 | 0.155
| 4 | 0.252 | 0.170 | 0.196 | 0.193 | 0.189 | 0.164 | 0.147 | 0.157 | 0.151 | 0.150
| 5 | 0.189 | 0.231 | 0.183 | 0.211 | 0.186 | 0.159 | 0.179 | 0.152 | 0.169 | 0.153
| 6 | 0.206 | 0.199 | 0.169 | 0.223 | 0.204 | 0.163 | 0.157 | 0.144 | 0.156 | 0.161
| 7 | 0.227 | 0.202 | 0.164 | 0.225 | 0.181 | 0.162 | 0.168 | 0.160 | 0.176 | 0.166
| 8 | 0.215 | 0.213 | 0.184 | 0.196 | 0.191 | 0.161 | 0.165 | 0.156 | 0.152 | 0.160
| 9 | 0.189 | 0.179 | 0.197 | 0.146 | 0.289 | 0.158 | 0.139 | 0.166 | 0.146 | 0.176
| 10 | 0.221 | 0.222 | 0.161 | 0.223 | 0.173 | 0.164 | 0.175 | 0.154 | 0.169 | 0.150
Tr5 | 1 | 0.200 | 0.208 | 0.197 | 0.199 | 0.195 | 0.150 | 0.155 | 0.159 | 0.151 | 0.156
(4.236) | 2 | 0.206 | 0.193 | 0.183 | 0.190 | 0.229 | 0.171 | 0.165 | 0.168 | 0.163 | 0.175
| 3 | 0.173 | 0.252 | 0.179 | 0.210 | 0.185 | 0.149 | 0.166 | 0.149 | 0.160 | 0.146
| 4 | 0.255 | 0.163 | 0.206 | 0.200 | 0.176 | 0.154 | 0.137 | 0.147 | 0.156 | 0.144
| 5 | 0.209 | 0.181 | 0.224 | 0.168 | 0.218 | 0.164 | 0.147 | 0.174 | 0.148 | 0.154
| 6 | 0.229 | 0.197 | 0.186 | 0.220 | 0.167 | 0.183 | 0.160 | 0.151 | 0.162 | 0.143
| 7 | 0.210 | 0.193 | 0.215 | 0.192 | 0.190 | 0.170 | 0.156 | 0.164 | 0.156 | 0.152
| 8 | 0.198 | 0.182 | 0.189 | 0.170 | 0.262 | 0.157 | 0.148 | 0.166 | 0.154 | 0.162
| 9 | 0.220 | 0.201 | 0.184 | 0.192 | 0.203 | 0.166 | 0.168 | 0.159 | 0.158 | 0.165
| 10 | 0.184 | 0.191 | 0.188 | 0.195 | 0.242 | 0.163 | 0.173 | 0.155 | 0.160 | 0.174

###  4.2 Human subject game experiment protocol

The experiment was approved by the Experimental Social Science Laboratory of
Zhejiang University. The data of controlled treatment
($a=[-4.236,-0.618,0.234,1.618,4.236]$-Treatment) is from the experiment
carried out from March to April in 2022. The authors confirms that this
experiment was performed in accordance with the approved social experiments
guidelines and regulations, which follow the regulation of experimental
economics protocol [4].

A total number of 60 undergraduate and graduate students of Zhejiang
University volunteered to serve as the human subjects of this experiment.
These students were openly recruited through a web registration system.

The 60 human subjects (call also as players) were distributed into 10
populations of equal size $N$ = 6. The six players of each population carried
one experimental sessions. During the game process the players sited
separately in a classroom laboratory, each of which facing a computer screen.
They were not allowed to communicate with each other during the whole
experimental session. Written instructions were handed out to each player and
the rules of the experiment were also orally explained by an experimental
instructor. The rules of the experimental session are as follows:

  1. 1.

Each player plays the 5-strategy game repeatedly with the same other five
players.

  2. 2.

Each player earns virtual points during the experimental session according to
the payoff matrix shown in the written instruction.

  3. 3.

In each game round, each player competes with one player in the other five
players as the opponent.

  4. 4.

Each player has to make a choice among the four candidate actions “x1”, “x2”,
“x3” , “x4”, “x5” . If this time runs out, the player has to make a choice
immediately. After a choice has been made it can not be changed. If a player
does not make a choice in the given 2 second, its strategy of this round will
be set to be the same as its own strategy of its previous round.

Figure 5:  The screen shot of the user interface in human subject game
experiment.

During the experimental session, the computer screen of each player will show
an information window and a decision window. The window on the left of the
computer screen is the information window. The upper panel of this information
window shows the current game round, the time limit of making a choice and the
time left to make a choice. The color of this upper panel turns to green at
the start of each game round. After a round end, the lower panel of the
information window will show the player’s own choice, the opponent strategies,
and own payoff in this game round are shown in the screen. The opponent
strategies is arrange as a list $[P_{1},P_{2},P_{3},P_{4},P_{5}]$ which
present the number of players choice “x1”, “x2”, “x3” , “x4”, “x5”
respectively.

The player’s own accumulated payoff is also shown. The players are asked to
record by handwriting of their choices of the round on the record sheet in
some round for potential checking. Each session last 20 minutes, have more
than 600 periods records. For each ($a=[-4.236,-0.618,0.234,1.618,4.236]$)
treatment, we have 10 sessions repeated by various group subjects. Each group
was asked to play the 5 sessions without repeated parameter $a$, arranging as
$a=[0.234,-0.618,1.618,-4.236,4.236]$ in order. So, we have totally 6000
records in the time series for each treatment.

The window on the right of the computer screen is the decision window. It is
activated only after all the players of the group have made their choices. The
upper panel of this decision window lists the current game round, while the
lower panel lists the five candidate actions (“x1”, “x2”, “x3” , “x4”, “x5”)
horizontally from left to right. The player can make a choice by clicking on
the corresponding action names.

The reward for each player is determined by the rank, which is determined by
the total number of their earning points in experiment sessions participated.
Form the highest to the lowest, each player is payed as 300, 250, 200, 150,
100 and 50 yuan RMB in the sessions participated which cost about 2 hours in
total.

###  4.3 Additional results

This sub section includes several statistical results in details, for the
understanding the main results reported in main text better.

Table 8: MLE Results By Sessions and the statistical ttest By Treatment  ($a$) | session | $c_{0}$ | $k_{\alpha}$ | $k_{\beta}$ | $p$ | $\bar{k_{1}}$ | $\bar{k_{2}}$ | $p_{k_{1}}$ | $p_{k_{2}}$
---|---|---|---|---|---|---|---|---|---
Tr1 | 1 | -0 | -0.591 | 0.575 | 0 | -0.597 | 0.923 | 0.003 | 0
(-4.236) | 2 | -0 | 0.161 | 1.593 | 0 |  |  |  |
| 3 | -0 | -0.401 | 1.168 | 0 |  |  |  |
| 4 | -0 | -0.564 | 0.953 |  0.003 |  |  |  |
| 5 | -0 | -1.676 | 1.199 | 0 |  |  |  |
| 6 | -0 | -0.258 | 0.530 | 0.007 |  |  |  |
| 7 | 0 | -0.563 | 1.051 | 0.001 |  |  |  |
| 8 | -0 | -0.728 | 0.576 | 0.005 |  |  |  |
| 9 | -0 | -0.846 | 0.954 | 0 |  |  |  |
| 10 | -0 | -0.501 | 0.630 | 0.002 |  |  |  |
Tr2 | 1 | -0 | 0.479 | 0.048 | 0.001 | 0.221 | 0.440 | 0.170 | 0.005
(-0.618) | 2 | -0 | 0.713 | 0.708 | 0 |  |  |  |
| 3 | 0 | 0.245 | 0.832 | 0 |  |  |  |
| 4 | -0 | 0.125 | 0.114 | 0.505 |  |  |  |
| 5 | -0 | -0.222 | 1.229 | 0 |  |  |  |
| 6 | 0 | 0.752 | 0.097 | 0.001 |  |  |  |
| 7 | 0 | -0.784 | 0.314 | 0.009 |  |  |  |
| 8 | -0 | 0.611 | 0.483 | 0.011 |  |  |  |
| 9 | 0 | 0.055 | 0.416 | 0.014 |  |  |  |
| 10 | 0 | 0.241 | 0.161 | 0 |  |  |  |
Tr3 | 1 | -0 | -0 | -0.013 | 0.977 | 0.293 | 0.134 | 0.004 | 0.093
(0.236) | 2 | 0 | 0.611 | 0.167 | 0.010 |  |  |  |
| 3 | 0 | 0.329 | 0.337 | 0.005 |  |  |  |
| 4 | 0 | 0.070 | 0.401 | 0 |  |  |  |
| 5 | 0 | 0.428 | 0.377 | 0.004 |  |  |  |
| 6 | -0 | 0.664 | -0.066 | 0.001 |  |  |  |
| 7 | 0 | 0.065 | -0.070 | 0.736 |  |  |  |
| 8 | 0 | 0.303 | 0.105 | 0.118 |  |  |  |
| 9 | 0 | 0.032 | -0.241 | 0.007 |  |  |  |
| 10 | -0 | 0.426 | 0.347 | 0.002 |  |  |  |
Tr4 | 1 | -0 | 1.026 | -0.069 | 0.004 | 0.707 | 0.116 | 0 | 0.155
(1.618) | 2 | -0 | 0.975 | 0.350 | 0 |  |  |  |
| 3 | 0 | 0.735 | 0.319 | 0 |  |  |  |
| 4 | 0 | 0.463 | 0.041 | 0.002 |  |  |  |
| 5 | -0 | 0.823 | 0.441 | 0.004 |  |  |  |
| 6 | -0 | 0.758 | -0.164 | 0 |  |  |  |
| 7 | -0 | -0.272 | 0.302 | 0.012 |  |  |  |
| 8 | -0 | 1.193 | 0.164 | 0 |  |  |  |
| 9 | -0 | 0.765 | -0.255 | 0.013 |  |  |  |
| 10 | -0 | 0.605 | 0.028 | 0.001 |  |  |  |
Tr5 | 1 | -0 | 2.104 | 0.148 | 0 | 1.467 | -0.094 | 0 | 0.409
(4.236) | 2 | 0 | 1.704 | 0.479 | 0 |  |  |  |
| 3 | -0 | 0.985 | 0.018 | 0.003 |  |  |  |
| 4 | -0 | 1.859 | -0.563 | 0 |  |  |  |
| 5 | 0 | 0.786 | -0.030 | 0.129 |  |  |  |
| 6 | -0 | 1.460 | -0.216 | 0 |  |  |  |
| 7 | -0 | 0.434 | 0.251 | 0.206 |  |  |  |
| 8 | -0 | 1.606 | -0.145 | 0 |  |  |  |
| 9 | 0 | 1.602 | -0.618 | 0 |  |  |  |
| 10 | -0 | 2.127 | -0.266 | 0 |  |  |  |
Table 9: MLE Results of $c_{0}$, $k_{\alpha}$ and $k_{\beta}$ By Sessions $(a)$ | session | $c_{0a}$ | $c_{0as}$ | $k_{a}$ | $k_{as}$ | $p_{a}$ | $c_{0b}$ | $c_{0bs}$ | $k_{b}$ | $k_{bs}$ | $p_{b}$
---|---|---|---|---|---|---|---|---|---|---|---
Tr1 | 1 | 0.181 | -1 | -0.559 | -1 | 0.052 | 0.646 | -1 | 0.549 | 1 | 0.03
(-4.2362) | 2 | 0.501 | -1 | 0.249 | -1 | 0.705 | -0.176 | -1 | 1.6 | 1 | 0
| 3 | 0.368 | -1 | -0.337 | -1 | 0.505 | 0.438 | -1 | 1.15 | 1 | 0
| 4 | 0.3 | -1 | -0.511 | -1 | 0.262 | 0.616 | -1 | 0.927 | 1 | 0.007
| 5 | 0.378 | -1 | -1.61 | 1 | 0.01 | 1.831 | -1 | 1.124 | -1 | 0.072
| 6 | 0.167 | -1 | -0.229 | -1 | 0.371 | 0.282 | -1 | 0.518 | 1 | 0.006
| 7 | 0.331 | -1 | -0.505 | -1 | 0.286 | 0.615 | -1 | 1.025 | 1 | 0.002
| 8 | 0.181 | -1 | -0.696 | 1 | 0.041 | 0.795 | -1 | 0.543 | -1 | 0.09
| 9 | 0.3 | -1 | -0.793 | -1 | 0.069 | 0.924 | -1 | 0.916 | 1 | 0.011
| 10 | 0.198 | -1 | -0.467 | -1 | 0.131 | 0.548 | -1 | 0.607 | 1 | 0.015
Tr2 | 1 | 0.015 | -1 | 0.482 | 1 | 0 | -0.524 | -1 | 0.069 | -1 | 0.683
(-0.618) | 2 | 0.223 | -1 | 0.752 | 1 | 0.032 | -0.779 | -1 | 0.741 | 1 | 0.016
| 3 | 0.262 | -1 | 0.291 | -1 | 0.411 | -0.267 | -1 | 0.843 | 1 | 0
| 4 | 0.036 | -1 | 0.131 | -1 | 0.395 | -0.136 | -1 | 0.12 | -1 | 0.388
| 5 | 0.387 | -1 | -0.154 | -1 | 0.764 | 0.243 | -1 | 1.219 | 1 | 0
| 6 | 0.031 | -1 | 0.757 | 1 | 0 | -0.821 | -1 | 0.131 | -1 | 0.629
| 7 | 0.099 | -1 | -0.767 | 1 | 0.007 | 0.857 | -1 | 0.278 | -1 | 0.377
| 8 | 0.152 | -1 | 0.638 | 1 | 0.041 | -0.668 | -1 | 0.511 | -1 | 0.081
| 9 | 0.131 | -1 | 0.078 | -1 | 0.7 | -0.06 | -1 | 0.419 | 1 | 0.003
| 10 | 0.051 | -1 | 0.25 | 1 | 0.006 | -0.263 | -1 | 0.172 | -1 | 0.062
Tr3 | 1 | -0.004 | -1 | -0.001 | -1 | 0.986 | 0 | -1 | -0.013 | -1 | 0.823
(0.236) | 2 | 0.053 | -1 | 0.62 | 1 | 0.004 | -0.667 | -1 | 0.195 | -1 | 0.427
| 3 | 0.106 | -1 | 0.348 | -1 | 0.065 | -0.36 | -1 | 0.352 | 1 | 0.032
| 4 | 0.126 | -1 | 0.092 | -1 | 0.595 | -0.076 | -1 | 0.405 | 1 | 0
| 5 | 0.119 | -1 | 0.449 | 1 | 0.041 | -0.468 | -1 | 0.396 | 1 | 0.048
| 6 | -0.021 | -1 | 0.66 | 1 | 0 | -0.725 | -1 | -0.036 | -1 | 0.877
| 7 | -0.022 | -1 | 0.061 | -1 | 0.62 | -0.071 | -1 | -0.067 | -1 | 0.545
| 8 | 0.033 | -1 | 0.309 | 1 | 0.05 | -0.331 | -1 | 0.118 | -1 | 0.453
| 9 | -0.076 | -1 | 0.018 | -1 | 0.87 | -0.034 | -1 | -0.239 | 1 | 0.001
| 10 | 0.109 | -1 | 0.445 | 1 | 0.028 | -0.466 | -1 | 0.366 | -1 | 0.053
Tr4 | 1 | -0.022 | -1 | 1.023 | 1 | 0.001 | -1.121 | -1 | -0.023 | -1 | 0.952
(1.618) | 2 | 0.11 | -1 | 0.994 | 1 | 0.001 | -1.065 | -1 | 0.394 | -1 | 0.265
| 3 | 0.101 | -1 | 0.752 | 1 | 0.002 | -0.803 | -1 | 0.353 | -1 | 0.198
| 4 | 0.013 | -1 | 0.465 | 1 | 0 | -0.505 | -1 | 0.062 | -1 | 0.711
| 5 | 0.139 | -1 | 0.848 | 1 | 0.008 | -0.899 | -1 | 0.479 | -1 | 0.156
| 6 | -0.051 | -1 | 0.749 | 1 | 0 | -0.828 | -1 | -0.129 | -1 | 0.621
| 7 | 0.095 | -1 | -0.256 | -1 | 0.127 | 0.298 | -1 | 0.29 | 1 | 0.044
| 8 | 0.052 | -1 | 1.202 | 1 | 0 | -1.304 | -1 | 0.218 | -1 | 0.59
| 9 | -0.08 | -1 | 0.751 | 1 | 0.006 | -0.835 | -1 | -0.22 | -1 | 0.474
| 10 | 0.009 | -1 | 0.606 | 1 | 0 | -0.661 | -1 | 0.055 | -1 | 0.797
Tr5 | 1 | 0.047 | -1 | 2.112 | 1 | 0 | -2.298 | -1 | 0.243 | -1 | 0.734
(4.236) | 2 | 0.151 | -1 | 1.73 | 1 | 0 | -1.861 | -1 | 0.556 | -1 | 0.352
| 3 | 0.006 | -1 | 0.986 | 1 | 0 | -1.076 | -1 | 0.062 | -1 | 0.863
| 4 | -0.177 | -1 | 1.828 | 1 | 0 | -2.031 | -1 | -0.479 | -1 | 0.462
| 5 | -0.01 | -1 | 0.785 | 1 | 0.036 | -0.859 | -1 | 0.005 | -1 | 0.99
| 6 | -0.068 | -1 | 1.448 | 1 | 0 | -1.595 | -1 | -0.15 | -1 | 0.769
| 7 | 0.079 | -1 | 0.448 | -1 | 0.13 | -0.474 | -1 | 0.27 | -1 | 0.333
| 8 | -0.046 | -1 | 1.598 | 1 | 0 | -1.755 | -1 | -0.072 | -1 | 0.893
| 9 | -0.195 | -1 | 1.568 | 1 | 0 | -1.75 | -1 | -0.546 | -1 | 0.327
| 10 | -0.084 | -1 | 2.112 | 1 | 0 | -2.323 | -1 | -0.17 | -1 | 0.814
Table 10: Correlation coefficient between experimental $L$ (Change continuously) Theory $\rho$ | $\sigma^{\alpha}$ | $\sigma^{\beta}$ | $M^{1}$ | $M^{2}$ | $M^{3}$ | $M^{4}$ | $M^{5}$
---|---|---|---|---|---|---|---
$\sigma^{\alpha}$ | 1 | 0.0500 | -0.4543 | 0.1307 | 0.8348 | 0.9950 | 0.9950
$\sigma^{\beta}$ | 0.0500 | 1 | 0.8670 | 0.9967 | 0.5915 | -0.0498 | -0.0498
$M^{1}$ | -0.4543 |  [rgb]1,0.9,0.90.8670 | 1 | 0.8238 | 0.1111 | -0.5408 | -0.5408
$M^{2}$ | 0.1307 |  [rgb]1,0.9,0.90.9967 | 0.8238 | 1 | 0.6548 | 0.0313 | 0.0313
$M^{3}$ |  [rgb]1,0.9,0.90.8348 |  [rgb]1,0.9,0.90.5915 | 0.1111 | 0.6548 | 1 | 0.7759 | 0.7759
$M^{4}$ |  [rgb]1,0.9,0.90.9950 | -0.0498 | -0.5408 | 0.0313 | 0.7759 | 1 | 1
$M^{5}$ |  [rgb]1,0.9,0.90.9950 | -0.0498 | -0.5408 | 0.0313 | 0.7759 | 1 | 1
Experiment $\rho$ | $\sigma^{\alpha}$ | $\sigma^{\beta}$ | $L^{1}$ | $L^{2}$ | $L^{3}$ | $L^{4}$ | $L^{5}$
$\sigma^{\alpha}$ | 1 | 0.050 | -0.470 | 0.426 | 0.866 | 0.970 | 0.996
$\sigma^{\beta}$ | 0.050 | 1 | 0.852 | 0.863 | 0.471 | 0.222 | -0.021
$L^{1}$ | -0.470 | 0.852 | 1 | 0.554 | -0.036 | -0.302 | -0.527
$L^{2}$ | 0.426 | 0.863 | 0.554 | 1 | 0.681 | 0.595 | 0.373
$L^{3}$ | 0.866 | 0.471 | -0.036 | 0.681 | 1 | 0.922 | 0.824
$L^{4}$ | 0.970 | 0.222 | -0.302 | 0.595 | 0.922 | 1 | 0.952
$L^{5}$ | 0.996 | -0.021 | -0.527 | 0.373 | 0.824 | 0.952 | 1
Experiment $p$ | $\sigma^{\alpha}$ | $\sigma^{\beta}$ | $L^{1}$ | $L^{2}$ | $L^{3}$ | $L^{4}$ | $L^{5}$
$\sigma^{\alpha}$ | 0 | 0.891 | 0.171 | 0.219 | 0.001 | 0 | 0
$\sigma^{\beta}$ | 0.891 | 0 | 0.002 | 0.001 | 0.169 | 0.537 | 0.954
$L^{1}$ | 0.171 | 0.002 | 0 | 0.096 | 0.921 | 0.396 | 0.118
$L^{2}$ | 0.219 | 0.001 | 0.096 | 0 | 0.030 | 0.070 | 0.289
$L^{3}$ | 0.001 | 0.169 | 0.921 | 0.030 | 0 | 0 | 0.003
$L^{4}$ | 0 | 0.537 | 0.396 | 0.070 | 0 | 0 | 0
$L^{5}$ | 0 | 0.954 | 0.118 | 0.289 | 0.003 | 0 | 0
Table 11: (correlation coefficient of observed eigencycle between 10 2-dimensional subspace  | $L^{12}$ | $L^{13}$ | $L^{14}$ | $L^{15}$ | $L^{23}$ | $L^{24}$ | $L^{25}$ | $L^{34}$ | $L^{35}$ | $L^{45}$
---|---|---|---|---|---|---|---|---|---|---
$\rho$ |  |  |  |  |  |  |  |  |  |
$L^{12}$ | 1 | 0.1977 | -0.2106 | -0.9101 | 0.938 | 0.3283 | -0.2357 | 0.8823 | 0.3181 | 0.8826
$L^{13}$ | 0.1977 | 1 | -0.5832 | -0.3445 | 0.1704 | 0.5599 | -0.5855 | 0.2655 | 0.7611 | 0.23
$L^{14}$ | -0.2106 | -0.5832 | 1 | -0.0211 | -0.1448 | -0.621 | 0.5656 | -0.1799 | -0.4853 | 0.024
$L^{15}$ | -0.9101 | -0.3445 | -0.0211 | 1 | -0.8719 | -0.2554 | 0.207 | -0.8455 | -0.3974 | -0.926
$L^{23}$ | 0.938 | 0.1704 | -0.1448 | -0.8719 | 1 | 0.139 | -0.3082 | 0.9427 | 0.2969 | 0.8826
$L^{24}$ | 0.3283 | 0.5599 | -0.621 | -0.2554 | 0.139 | 1 | -0.7145 | 0.1192 | 0.5719 | 0.2802
$L^{25}$ | -0.2357 | -0.5855 | 0.5656 | 0.207 | -0.3082 | -0.7145 | 1 | -0.2813 | -0.6138 | -0.3258
$L^{34}$ | 0.8823 | 0.2655 | -0.1799 | -0.8455 | 0.9427 | 0.1192 | -0.2813 | 1 | 0.1586 | 0.9102
$L^{35}$ | 0.3181 | 0.7611 | -0.4853 | -0.3974 | 0.2969 | 0.5719 | -0.6138 | 0.1586 | 1 | 0.1824
$L^{45}$ | 0.8826 | 0.23 | 0.024 | -0.926 | 0.8826 | 0.2802 | -0.3258 | 0.9102 | 0.1824 | 1
$p$ |  |  |  |  |  |  |  |  |  |
$L^{12}$ | 0 | 0.1687 | 0.1421 | 0 | 0 | 0.0199 | 0.0994 | 0 | 0.0244 | 0
$L^{13}$ | 0.1687 | 0 | 0 | 0.0143 | 0.2367 | 0 | 0 | 0.0624 | 0 | 0.1082
$L^{14}$ | 0.1421 | 0 | 0 | 0.8842 | 0.3156 | 0 | 0 | 0.2113 | 0.0004 | 0.8687
$L^{15}$ | 0 | 0.0143 | 0.8842 | 0 | 0 | 0.0734 | 0.1491 | 0 | 0.0043 | 0
$L^{23}$ | 0 | 0.2367 | 0.3156 | 0 | 0 | 0.3356 | 0.0294 | 0 | 0.0363 | 0
$L^{24}$ | 0.0199 | 0 | 0 | 0.0734 | 0.3356 | 0 | 0 | 0.4098 | 0 | 0.0487
$L^{25}$ | 0.0994 | 0 | 0 | 0.1491 | 0.0294 | 0 | 0 | 0.0478 | 0 | 0.0210
$L^{34}$ | 0 | 0.0624 | 0.2113 | 0 | 0 | 0.4098 | 0.0478 | 0 | 0.2712 | 0
$L^{35}$ | 0.0244 | 0 | 0.0004 | 0.0043 | 0.0363 | 0 | 0 | 0.2712 | 0 | 0.2050
$L^{45}$ | 0 | 0.1082 | 0.8687 | 0 | 0 | 0.0487 | 0.0210 | 0 | 0.2050 | 0