HHuman Social Cycling Spectrum
Zhijian Wang and Qinmei YaoExperimental Social Science Laboratory, Zhejiang University, Hangzhou 310058, ChinaDecember 8, 2020
Nash equilibrium, its reality and accuracy, is firstly illustrated by the O’Neill (1987) game experiment.In this game experiments (O’Neill 1987, Binmore 2001 and Okano 2013), we discover the fine structurein the human social cycling spectrum. With the eigencycle set from the eigenvectors in game dynamicsequations, we can have the accuracy increased by an order of magnitude on the dynamics testing.
Contents
We study how to bridge theory and experiment in game dynamics theory. In game statics theory,the mixed strategy Nash equilibrium is the central concept which established near 1950. Till 1987,the O’Neill game[8] provided the first illustration that laboratory human strategy behaviour can beaccurately captured by the concept. The experiments involving a long run repeated, two-person, zero-sum game which payoff matrix shown in Table 1. Since then, this game has been extensively repeated invarious setting experiment [6][15] and analysis [1]. Till now, most of the studies focus on the distributionsand time dependence of the individual behaviors, rare on the social dynamics pattern.Intuitively, two reasons bars the dynamics study. First, the game has high dimension strategy spaceand how to handle the dynamical pattern remains a problem now a day [4]; Second, in laboratorygame experiment, the human strategy decision makings are highly stochastic, in which, to test out thedynamical pattern is a hard task [2][5], especially on discrete time game [3].We carry out the task by developing an approach, namely eigencycle set decomposition . This iswell known that, for a given complex harmonic motion system, eigenvalue decomposition (called also asspectrum analysis) is useful for performing mathematical analysis in order to identify the key features.1 a r X i v : . [ ec on . T H ] D ec ny initial probability distribution can be expressed as a linear combination of the eigenvectors ξ , as[7] p (0) = a ξ + a ξ + ... + a n ξ n (1)in which ( λ , λ , ..., λ n ) are the eigenvalues. The probability evolves in time according to p ( t ) = e λ t a ξ + e λ t a ξ + ... + e λ n t a n ξ n (2)namely the coefficients, e.g., ( a , a , ..., a n ). Instated of these coefficients, suppose a normalized eigen-vector having s components and ξ i = ( η , ..., η m , ..., η n , ...η s ), we concern the components ( η ) in ξ i . Thekey of our approach is an invariant with two constants (1) The phase angle difference of any two η is aconstant, and (2) the amplitude of each η is a constant. By this method, we develop a new theoreticalexpected observation, called as eigencycle set, to identify the human dynamics behaviours in high di-mensional game. As illustrated by Figure 2, the results are (1) Finding the fine structure and hypefinestructure in high dimensional game dynamics, with the cycle measurement accuracy increased by anorder of magnitude; (2) providing a tool, namely eigencycle spectrum analysis, for dynamics analysis.In section 2, using O’Neill game as an example, we introduce the theoretical eigencycle set, byapplying inner eigenvector analysis. In section 3, we introduce experimental results. In section 4, weverify whether the theoretical eigencycle set concept validate. The last is section 5. By question andanswer format, we summarize the contribution, the improvement referring to the previous works, andthe research questions newly raised by this work. The O’Neill Game is a zero-sum, 4 ×
4, game. Table 1 shows the payoff matrix: To investigate theTable 1: The O’Neill zero-sum game matrixB1 B2 B3 B4A1 1 -1 -1 -1A2 -1 -1 1 1A3 -1 1 -1 1A4 -1 1 1 -1dynamic behaviors in laboratory experiment game, we use the replicator dynamics equation [9]:˙ x j = x j ∗ ( U j − U X ) (3)in which, x j is the j -th strategy player’s probability in the population where the j -th strategy playerincluded, and ˙ x j is the evolution velocity of the probability; U j the payoff of the j -th strategy player,and ¯ U X is the average payoff of the population where the j -th strategy player included.We assign one by one from the (A1, A2, A3, A4) strategy probability to ( x , x , x , x ). At the sametime, assign (B1, B2, B3, B4) strategy probability to ( x , x , x , x ). Then, at any time, the systemmust be in one point the 8-dimension space. In this 8-dimension space, the Nash equilibrium is at( x , x , ..., x ) = (2 / , / , / , / , / , / , / , / . Naturally, this 8-dimension space have two concentrated x + x + x + x = 1 ∩ x + x + x + x = 1 ∩ x k ≥ k ∈ , , ..., λ and their relatednormalized eigenvector ξ ’s components ( η , η , ...η ) explicitly . The results are shown in Table 2. The components ( η , η , ...η ) is 1-by-1 corresponds to ( x , x , ..., x ) .2 Eigencycle set Definition
We call the circle constructed by two components ( η m , η n ) within one normalized eigenvector ξ i =( η , ..., η m , ..., η n , ...η s ) as the eigencycle, marked as σ ( mn ) , and calculated as follows: σ ( mn ) = π · || η m || · || η n || · sin (arg( η m ) − arg( η n )) (4)in which, the superscript ( mn ) is the index of the 2-d subspace where m and n are the abscissa and theordinate dimension respectively; || η m || and arg( η m ) indicates the amplitude and the phase angle of the η , respectively. σ ( mn ) can determine the direction of the eigencycle and the amplitude of the eigencycle.According to this formula, Table 2 lists the eigencycle values of the eigenvectors. Interpretation • Invariance of eigencycle:
An eigencycle is constructed by two components in a normalizedeigenvector, so its value is invariant. This is according to Eq. (4) and following three points: (1)the phase difference between two given components in an eigenvector is fixed; (2) modulated bythe same eigenvalue, the relative phase difference between these two components remains fixed.and (3) the mode of each component is fixed. • Number of eigencycle:
There are N ( N − / N -component normalized eigenvector. Because there are a total of N pairwise combination ofeach component in an N -dimensional eigenvector. Considering the N self-combination of ( η m , η m )are trivial ( σ ( mm ) = 0), and ( η n , η m ) and ( η m , η n ) is just a simple reverse ( σ ( mn ) = − σ ( nm ) ), sothat only N ( N − / • Eigencycle set:
Eigencycle set, as a vector denoted as Ω ( mn ) ξ k , is defined to represent the setof N ( N − / ξ index by k , who generate this eigencycle set. The superscript ( mn ) is the index of the two-dimensionalsubspace where the elements (eigencycles) of the set are located. ( mn ) is defined as: {{ m, n } ∈{ , , ..., n } ∩ ( m < n ) } . The assignment order is m from 1 to N first and then n from 2 to N inthis paper. We call the method of generating eigencycle set above as eigencycle set approach . • Subspace set:
Subspace set, denoted as Ω ( mn ) , has N ( N − / mn ) is the index of the two-dimensional subspace. Again, ( mn ) is defined as: {{ m, n } ∈ { , , ..., n } ∩ ( m < n ) } . The assignment order is m from 1 to N first and then n from 2 to N in this paper. We will see that, due to symmetry, there exists subset of Ω ( mn ) ; In thesubset, the performance of the elements are equal. • Independence of eigencycle sets:
The eigencycle sets corresponding to eigenvectors are not allindependent. Taking the O’Neill game as an example, among the 8 eigencycle sets, only 3 eigen-cycle set (generated by ( ξ . i , ξ . i , ξ . i )) are independent. Explanation are following. By the set ofeigenvalues, (1) Two eigenvalues (0.2 and -0.2) are real numbers. As only the eigenvectors whoseeigenvalues are complex numbers are related to cyclic motion, these two are trivial. (2) Remaining6 eigenvectors are of three pair. Each pair eigenvectors with complex conjugate eigenvalues areconjugate, and the generated eigencycle set pair have oppose values. So, we can omit those relatedto eigenvalue ( − . i, − . i, − . ξ . i , ξ . i , ξ . i ). We use the subscript 1,2 to distinguish eigencycle set from thetwo degenerate eigenvalue 0 . i . • Geometric presentation:
The geometric presentation of an eigencycle is similar to (1:1)-Lissajous diagrams, but differs. In (1:1)-Lissajous diagram, the amplitude of a component can bearbitrary, but both the components’ amplitude of an eigencycle is fixed and not arbitrary. At thesame time, the eigencycle only depends on the internal components η m and η n of the normalizedeigenvector, and has nothing to do with other eigenvector. We give a counterexample in Figure(d), because its shape and size are not determined by a single eigenvector.3 Why O’Neill game:
Since O’Neill is a two-person four-strategy zero-sum game with 8 dimen-sions, it is a high-dimensional problem. It is difficult for us to intuitively express the motionstructure of such high dimension. But, by eigencycle set, the 8 dimensions game space can bedecompose to 28 2-dimensions set. Importantly, – In theory, this game’s replicator dynamics equation can be solved mathematical explicitly.The game has high symmetry, so could be tractable easily. – In experiment, The game matrix has high symmetry and exists sufficient equivalent observa-tions, so the data should provide sufficient opportunity for testing. – In history, the O’Neill experiment being the firstly reveal the reality and accuracy of NashEquilibrium in the history of Game Theory, to reveal the reality and accuracy of the gamedynamics in the existed experiment data could be a existing task.Figure 1: Eigencycle set. Subfig (a-c) illustrate the 28 eigencycles pattern of the eigencycle set( σ . i , σ . i , σ . i ), the order is as Ω ( mn ) in Table 2. (d) illustrates a counter-example of eigencycle.The Lissajous figure of ξ . i and ξ . i . These curve are not eigenvector, which depend on the weight ofthe two ξ is outside parameter, like the a i in Eq. (1). Here, we explain briefly on the theoretical results shown on Table 2 • Precision —
Table 2 have provide a theoretical prediction on cyclic behavior having higher preci-sion than we ever know. The eigencycle set σ . i predicts 4 different values, (0.1964:0.0654:0.0218:0)=(9:3:1:0),means 4 different strength cycle set expected. This expectation is verified and lead to the discoveryof the fine structure (see section 3.4) • Symmetry —
The contributions of the eigencycles from the different normalized eigenvector ξ are difference and can be classified. This can be seen in the numerical results in Table 2. Theclassification can reflect the symmetry of the game matrix, as well as the structure of the ξ innercomponents. This can be employed to identify the contribution of ξ in data (for example, see the σ α and σ β in subsection 4.2). Using the symmetry property, we will show the the evidence of thesignificant existence of the hypefine structure. • Operable —
We will use the experiment data to verify the validate our eigencycle set decom-position approach. It could provide a spectrum analysis tool for game dynamics (see section4.3). 4able 2: The eigenvalues, eigenvectors and their respective 28 eigencyclesEigenvalue λ i λ . i λ − . i λ . λ − . λ . i λ − . i λ . i λ − . i i − i −
15 25 i − i i − i Eigenvector( η i ∈ ξ ) ξ . i ξ − . i ξ . ξ − . ξ . i ξ − . i ξ . i ξ − . i η − i i η i − i
16 16 16 16 η i − i − √ − −√ − −√ − √ η i − i − −√ − √ − √ − −√ η η − − − i6 i6 − i6 i6 η − − √ √ − i12 −√ −√ − i12 η − − −√ −√ − i12 √ √ − i12 Eigencycle(Ω ( mn ) ) σ . i σ − . i σ . σ − . σ . i σ − . i σ . i σ − . i
12 0 0 0 0 0 0 0 013 0 0 0 0 0 0 0 014 0 0 0 0 0 0 0 015 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − Note: Ω ( mn ) is the identity of the 2-d subspace of an eigencycle. m ( n ) indicates which one in the 8-dimension is selectas the x -axis ( y -axis) in the eigencycle subspace. Experimental observation on subspace cycles
A brief summary of the O’Neill game experiments is shown in Table 3. The experiments, including3 experiments and crossing 26 year, having total 358 subjects participated, are multi-rounds repeatedplaying on the game matrix 1. The experiment data are employed to verify the high dimension cyclingin this paper. Table 3: Experiment Data Source
Experimentabbreviate DataSource ExperimentSummary PublishedTime totleRounds MatchingProtocol O O’Neill The game was played by 50students working in 25 pairs. 1987 2625 FixedPaired B Binmore et al Each experiment session required 12 subjects13 experimental sessions in all.Each real game was played 150 times. 2001 1950 RandomlyMatching IT Okano 20 individuals adopted the player A roleagainst 20 teams adopting the player B role.20 experimental sessions in all.Each real game was played 150 times. 2013 3000 FixedPaired
T I
Okano Twenty teams adopted the player A roleagainst 20 individuals adoptingthe player B role, 20 experimental sessionsin all. Each real game was played 150 times. 2013 3000 FixedPaired II Okano Individuals against individuals,18 experimental sessions in all.Each real game was played 132 times. 2013 2376 FixedPaired
T T
Okano Teams against teams,18 experimental sessions in all.Each real game was played 132 times. 2013 2376 FixedPaired
The measurement for cycle in the subspace:
According to the theoretical the eigencycle set decomposition approach, we can carry out the cyclicangular momentum measurement in each of the two-dimensional subspace, indicated by the eigencycleΩ ( mn ) , separately. The angular momentum L ( mn ) E [13] can be expressed by the following formula: L mnE = 1 N − N − (cid:88) t =1 ( x ( t ) − O ) × ( x ( t +1) − x ( t )) (5) • 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 therepeated game experiments; • O is the projection of the Nash equilibrium at the subspace Ω ( m,n ) ; • x ( t ) is a two-dimensional vector at time t which can be expressed as ( x m ( t ) , x n ( t )), and x ( t + 1)is at time t + 1; • × represents the cross product between two two-dimensional vectors. Interpretation of the measurement: This measurement can be called also as signed area of the triangle ∆ [ O,x ( t ) ,x ( t +1)] in the ( m, n ) 2-d subspace. Foreach transition from x ( t ) to x ( t + 1) referring to O , the angular momentum is twice of the signed area of the triangle. Wesuggest using the concept of the angular momentum, because it contain the mass m as parameter, which may compatiblethe population size N as variable in further game dynamics investigation. ( mn ) L ( mn ) O L ( mn ) B L ( mn ) IT L ( mn ) T I L ( mn ) II L ( mn ) T T
12 0.003 0.002 − . − .
001 0.002 0.00813 − . − .
002 0.003 0.000 − . − . − . − . − . − . − . − . − . − .
009 0.001 0.008 0.00724 − . − .
001 0.001 − . − .
006 0.00125 0.010 0.003 0.021 0.005 0.015 0.01826 − .
001 0.000 0.004 − .
011 0.002 − . − . − . − .
010 0.004 − . − . − . − . − .
014 0.002 − .
005 0.00534 0.000 0.001 − .
006 0.000 − .
001 0.00535 0.011 0.008 0.015 0.011 0.014 0.01336 − . − . − . − . − . − . − .
003 0.001 − . − .
004 0.002 0.00238 0.002 − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − . − .
002 0.002 0.007 − . − . − . − .
006 0.001 − .
001 0.001 − .
001 0.00257 0.009 0.000 0.004 0.005 − . − . − . − . − . − .
006 0.006 0.00767 − .
003 0.001 − .
003 0.000 0.005 0.00768 − .
003 0.000 0.002 0.001 − . − . − . Ω ( mn ) is the 28 subspace id referring to Table 2; The subscript of L relates to the O’Neill, Binmore, and AIBT, ATBI,AIBI and ATBT experiments referring to Table 3. Software MATLAB, version: R2018a. • The sign of the value of L ( mn ) E indicates the direction of cycle movement (counterclockwise orclockwise): L ( mn ) E > L ( mn ) E < L ( mn ) E = 0 meansno cycle obtained. If a system is the completely random as mixed strategy Nash equilibriumprediction, its long-term average of the angular momentum is 0. • 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 4 states in a two-dimensional subspace, and these 4 statesform a unit square with an area of 1. The accumulate angular momentum of one full cycle motionis 2, or || L ( mn ) E || = 2. • If, in the time series of 1000 repeated rounds experiment, the observed accumulated L ( mn ) E = 40,it means the total amount of determinate motion is 20 cycles, also means that there are 80 timesdeterminate motions beyond complete stochastic motion, which also means the probability ofdirectional movement is 8%. We calculate average angular momentum for the experiments shown in Table 3. The results areshown in Table 4. 7able 5: Consisance of the six experiments L O L B L IT L T I L II L T T L O L B L IT L T I L II L T T
Measure: Spearman’s rank correlation. Number of observations are 28. Variables (observations) are the experimentalangular momentum shown in Table 4. The first row in the table cell reports the correlation coefficient, the second rowreports the significance level p . Software: Stata, version:SE 15.1. The measurement results illustrate the consistence and the precision of the six experiment. Thesesuggests that, the inner eigenvector decomposition approach validate, meanwhile the cycles in highdimension space is real and testable. For supporting, see following two points: • Consistences ——
In the six experiments, the observed cycles values L in the 2-d subspacesare consistences (Spearman rank’s test, max( p )=0.012, mean of the ρ values is 0.6351, Std. Ev =0.1087, Std. Err = 0.0118). • Precision ——
Except the main cycle found in Ω (15) , now we can test out the cycles in thesubspace (Ω (26 , , , , , , , , ) . This means that the accuracy increases up to one order ofmagnitudes. Explain with two numerical examples: – L (15) O = − . O indicates the O’Neill game, the superscript (15) indicatesthe subspace measured is Ω (15) , the negative means the cycle is clockwise. || L || = 0.038, itmeans There is ( × . (cid:39) ) 2 cycles obtained in the O’Neill (1987) 105 rounds game, andin the 25 group total 2625 round observation, there 50 cycles obtain. – L (37) O = − . We evaluate the validate of the eigencycle set concept by the experimental L (average angularmomentum). The 3 subsection can be abstract to:1. The discovery of the fine structure, due to the contribution of σ . i labeled as 0 . i in Figure 2.2. The finding of the hypefine structure, due to the contribution of σ . i and σ . i labeled as α and β in Figure 2.3. Multi OLE results (see Table 8) illustrate that, the eigencycle sets is an ideal basis and validatefor the spectrum analysis on game dynamics.8igure 2: Fine structure and hypefine structure of game dynamics in the O’Neill (1987) game exper-iments. The label { } are the observed average angular momentum of the exper-iments, which exactly linearly matching the eigencycle values of σ . i . This figure illustrates how thetheoretical eigencycle set clearly brings out the fine structure discovery and hypefine structure existence,meanwhile the improvements to the existed literatures [6, 11, 3, 4]. Result 1’s description:
The 6 experiments’ cycles from the 28 subspaces are employed to test the three eigencycle set ( σ . i , σ . i and σ . i ) prediction by OLE, respectively. The results from the six experiments are exactly consistent: • Between the 28 samples from theoretical σ . i (eigencycle set) and experimental L , significantlinear dependence exists. The results show that σ . i can be the principle component for the gamedynamics independently. Importantly, the four cycle values rated as (9:3:1:0) expected by σ . i areclearly observed (see the red squares in Figure 3), which means the discovery of the fine structure in the O’Neill game. • Neither σ . i or σ . i exists significant linear dependence with experimental L . That means, thesetwo eigencycle set can not be interpreter for the game dynamics independently. But they are notinvalidate. As we will show latter, they lead to the hypefine structure . Result 1’s supporting data:
Figure 3 illustrate the OLS(ordinary least square) of the theoretical σ . i and the six experimental L .Table 6 is the results of the OLS between the six experiment and { σ . i , σ . i , σ . i } respectively. Result 1’s Interpretation: • σ . i as Principle component — The p value of the regression coefficient of σ . i closes to 0 . σ . i is explanatory tothe experimental main results. Therefore, we can regard σ . i as the Principle Component in thissystem. σ (15) . i has the largest value, which is consistent with Binmore et al suggestion [6] andconfirmation by Wang and Xu [11] after converting the 4 × × σ (15) . i is the cutting edge on cycle measurement in existedpublications. • Discover the fine structure —
Beyond the Ω (15) subspaces, σ . i also predicts the cycle inthe 6 subspace Ω (16 , , , , , , and the 9 two-dimensional subspace Ω (26 , , , , , , , ,
48) 3 In addition, on fine structure, with the 9 subspaces of the 6 experiments as observation together, we have 54 samples,the result is strongly significant ( ttest , p =7.689 × − , N =54). Similarly, With the 6 subspaces of the 6 experimentsas observation together, we have 36 samples, the result is strongly significant ( ttest , p =2.835 × − , N =36). This is astatistical significance of five standard deviations (5 σ , a threshold of p ≤ . × − ) above background expectations 0. σ . i by theexperiment. The 9:3:1:0 predicted by σ . i observed, which means the discovery of the fine structure ofgame dynamics. (a) - (f) experiment values came from the treatments ordered as Table 3.These cycles are 1/3 or 1/9 in strength comparing with the Ω (15) ’s cycle, which is unable to testout by the previous methods. Now, by eigencycle set approach, their existences are significant.We call these as the fine structure. • Interpretation on σ . i results — The no zero eigencycles in σ . i and σ . i eigencycles set are allindependent of 1-st or 5-th dimension. So the experimental observed cycles in Ω (15 , , , , , , ,which are obviously motions by σ . i , can not be captured. As a result, neither σ . i nor σ . i canglobally capture the whole 28 subspace motion is reasonable. Result 2’s description
We investigate σ . i and σ . i by data. As the main result here, we find the hypefine structure. In σ . i and σ . i , their corresponds eigenvector ( ξ . i and ξ . i ) component η and η all are 0. So σ . i and σ . i effect only in (2-3-4, 6-7-8) 6-dimension subspace, which includes 15 2-dimension subspace wherethe eigencycles can existed. We use the two degenerate eigenvalue’s eigencycle set ( σ . i , σ . i ) to builda pair of orthogonal basis as (cid:26) σ α = ( σ . i + σ . i ) / ,σ β = ( σ . i − σ . i ) / σ α only relates to (2-3-4, 6-7-8)-dimension outer cross 9 subspace Ω (26 , , , , , , , , . Thatis, σ α has only 9 components but fully cover the 9 2-d subspace. Notice that, as a strictlymathematical result, there have only two values and their oppose in σ α .2. σ β only relates to (2-3-4, 6-7-8)-dimension inner cross 6 subspace Ω (23 , , , , , .That is, σ α has only 6 components but fully cover the 6 2-d subspace. Again, there has only one value and itsin σ β σ . i , σ α and σ β are mutually orthogonal. 10able 6: OLS between the six experiment L and { σ . i , σ . i , σ . i } respectively L ( mn ) O L ( mn ) B L ( mn ) IT L ( mn ) TI L ( mn ) II L ( mn ) TT σ . i coef. t p (0.00) (0.00) (0.00) (0.00) (0.00) (0.00)cons. t p (0.939) (0.726) (0.630) (0.801) (0.978) (0.640) R σ . i coef. t − .
31 0.47 1.24 − .
63 0.11 − . p (0.759) (0.645) (0.227) (0.535) (0.917) (0.719)cons. t 0.01 0.22 − .
05 0.04 0 0.14 p (0.995) (0.827) (0.960) (0.968) (0.999) (0.89) R σ . i coef. t − . p (0.367) (0.276) (0.672) (0.843) (0.176) (0.675)cons. t − .
06 0.05 − .
23 0.13 − .
17 0.13p (0.952) (0.960) (0.822) (0.898) (0.870) (0.896) R σ . i = − . − . − − − . − − ± SE (0.0019) (0.0008) (0.0027) (0.0016) (0.0017) (0.0021) Software:Stata, version:SE 15.1
Table 7: OLS results of experimental L P by the eigencycle seteigencycleset coef. coef. t coef. p > | t | cons t cons p > | t | consconf. interval σ . i − σ α − . − . σ β .
005 3 .
31 0.030 − .
77 0.487 [ − .
002 0.001]
Note: Software Stata, version: SE 15.1.
Experimental testing results are as following: • There exists significant linear dependence between experimental L and σ α . Meanwhile, the con-stant term is consequence of σ . i . This is why we claim the finding of the hypefine structure. • There exists significan linear dependence between experimental L and σ β . Meanwhile, the constantterm is zero can not be rejected. Result 2’s supporting data • Supporting data refers to Table 7. As the 5 treatments (labelled as (
O, IT, T I, II, T T ) in Ta-ble 3) have the same social state pattern (each 2-d subspace is 4 states), meanwhile, largersample size is necessary for game dynamics testing, we pool the data and label the pooled as P L P and σ . i , σ α and σ β , we can get the result of the first, sec-ond and third row in Table 7, respectively. L P and σ . i have significant positive correlation(linear regression, t =27.36, p=0.000), L P and σ α have significant positive correlation (linearregression, t =4.43, p =0.003), in which the constant item is the influence of 0 . i on 9 points(0 . × ( − . − . ∈ {− . , − . } ). L P and σ β have significant positive correla-tion (linear regression, t =3.31, p =0.030). Result 2’s interpretation • Finding hypefine structure —— L P and σ α have significant correlation, which indicatesthat σ α can explain the hyperfine structure in the 9 subspaces Ω (26 , , , , , , , , . Beyond σ . i who predicts the 9 point having same value, σ α can capture the differences between them.11able 8: Multiple linear regression of experimental L P by σ . i , σ α and σ β L P Coef. Std. Err. t P > | t | [95% Conf. Interval] σ . i .0565190 .0015589 36.25 .000 .0533014 .0597365 σ α .0057181 .0015589 3.67 .001 .0025006 .0089356 σ β .0045221 .0015778 2.87 .009 .0012656 .0077785cons − .0000876 .0002982 − .29 .772 − .000703 .0005279 Meanwhile, the response mode reflected by σ α is consistent with the best response mode inbehavior game theory, which is also consistent with win-stay-lose-shift mode in human decision-making. • Puzzle on σ β ’s result — L P and σ β have significant positive correlation. This can not beexplained by behavior game theory. We suggest this is the geometry factor effect, which is anatural result of the 8-dimesion presentation of the dynamics equation 3. Result 3’s description:
Having the eigencycle set as the orthogonal basis, it is possible to explore the high dimension dynam-ics by spectrum analysis in the form as Eq. (1). From the two subsections above, it can be concludedthat L P is affected by σ . i , σ α and σ β simultaneously. Further, through multiple linear regression,the significance of σ . i , σ α and σ β over L P can be obtained. Moreover, the impact of σ . i is moresignificant, which is about 10 times of σ α and 12 times of σ β in magnitude. Result 3’s supporting data:
The supporting data are shown in Table 8. The multiple linear-regression analysis is used forinterpretation the experimental observation by the three theoretical eigencycle sets. In formula form isas L P SE = = . σ . i (0 . + 0 . σ α (0 . + 0 . σ β (0 . − . σ . i , σ α and σ β have significantly positive impact on L P . The 0 .
057 is the partial regression coefficientof σ . i , and tells us that with the influence of σ α and σ β held constant, as σ . i increases a unit, L P goes up by 0.057 units. Similar significance from α and β can be obtained. The R value of about0.9824 means that about 98.24 percent of the L P is explained by σ . i , σ α and σ β . Result 3’s interpretation: • Consist with the two subsections above, the results in this subsection is also significant. (1)Experimental observation is determined mainly by σ . i . (2) Compared with σ . i , σ α and σ β haveless effect, but their effects are significant. • The eigencycle set can be an ideal basis for the spectrum analysis, which is helpful to revealthe motion characteristics of high-dimensional game dynamics. The results in the Table 6 is asignificant illustration. In addition, on hypefine structure, if using the 9 subspaces of the 6 experiments as observation together, we have 54samples. These sample can still interpreted by σ α in significant (Spearman’s ρ = 0.3554, p (2-tailed) = 0.00836, N =54.Similarly, With the 6 subspaces related to σ β of the 6 experiments as observation together, we have 36 samples, the resultremain weakly significant (Spearman’s ρ = 0.31767, p (2-tailed) = 0.05903, N =36). .4 Summery of the verification result • Find the fine structure and the hypefine structure in the cycling spectrum of gamedynamics.
The cycles exists in significant and in high precision as expected by the eigencycleset. The principle component is the σ . i . Further, we can test out the cycles in the subspace 2-3-4and 6-7-8 dimension. • Eigencycle concept is validate:
Basing on the high precision finding and clear classificationof the cycles in the game, we suggest the eigencycle concept is validate. • Two raised questions:–
The coefficients in Eq. (7), which like the coefficients ( a i ) in Eq. (1) can not be interpreted byReplicator dynamics equation. We do not know which theoretical approach being validate.This is the first and obvious question remained; – We construct σ α and σ β by the σ . i and σ . i in Eq. (6). We hope can have better interpre-tation for such constructing. This is the second question remained. • What is the main contribution of this work? – Discovery of the fine structure of the game dynamics in the historic O’Neill game. At thesame time, finding the evidence on the existence of the hypefine structure in human cyclingspectrum. Figure 2 – Develop the eigencycle set decomposition approach. By the invariant of an eigenvector per-formance in a periodic dynamics system, we build the eigencycle set to test out the finestructure spectrum of the human social cycling. • Why use 2-person fixed-paired discrete time game experiment to test the dynamical pattern? – In previous work [14], we have noticed that, in long run fix paired 2 × – As the myopic is the natural of human behaviour, the cyclic pattern can be expected. Al-though the time series is highly stochastic, using the time reserve asymmetry measurement,like velocity field or angular momentum measurement [10], the dynamic behaviour can betest out. • Why using the replicator dynamics and the perturbation expansion to model? – The replicator dynamics is the earliest developed model. Evidences show that the model, aswell as the perturbation expansion, can capture the main character of the dynamics behaviour[12]. – The replicator dynamics and the perturbation expansion can be analysis with standard ap-proach in mathematics, from which, the eigen system have clear pratical picture. • What is the fact to show the advantage of the inner eigenvector decomposition method for gamedynamics? – Subfigure (1987) — This is the game matrix designed by O’Neill. This game is historic byproviding the first evidence on the reality and accuracy of the mixed strategy Nash equilib-rium. – Subfigure (1978) — The evolutionary character, when transited to a 2 × – Subfigure (2001) — This figure comes from the figure in [6]. On cyclic behaviour in geometryway, were the the firstly presented. The authors conducted experiment of 150 rounds repeatedper session and totally 13 sessions. But, by their measurement, the results have not reportedin significant confidently from the 13 sessions. They have reported a manual selected 150rounds likely-cycle trajectory only. 13
Subfigure (2012) — Borrow the method to concentrated the 4 × × – Subfigure (2020), by applying the eigencycle set analysis, to analysis the O’Neill game again,the combination 2-3-4 and 6-7-8 strategy tech is not necessary. So, it is possible to test allof the cycles in all of the 28 2-d subspaces. Further, we can archive the discovery of the finestricture. • What is the further expectation on evolutionary game dynamics basing on this finding? – The coefficients in Eq. (7), are not clearly explained quantitatively. The coefficients musthave their physical meaning, like the electron distribution in their orbit (eigen states) obeyingBoltzmann distribution. At the same time, to our knowledge, no theory has established forthis issue. This need further study. – The degeneracy of the eigenvalue come from the symmetry of the payoff matrix of the O’Neillgame. Any linear combination of the two eigenvectors can a new basis, but, which one isphysical fact is not known. We can do better in this issue. – How to distinguish the second order interaction consequence? In O’Neill game case, thephysical mean of α ’s consequence is unclear. Although the eigenvector belongs to the λ = . i coupling with the eigenvector belongs to λ = . i has been obtained at the subspace set (the9 subspace, 2-3-4 and 5-6-7 cross) in significant. • What area the eigencycle set approach may validate? – In our previous work, we have notice in the velocity field based the Markovian, the sec-ond largest eigenvector inner structure, in complex plain, the components performance likesstationary wave. – In testing and forecasting a time series, when having potential equilibrium and potentialdynamics equation, this approach could be a way out, especially those puzzled by the highdimension dynamics pattern [4].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 ( α, β ).The first author could be transited from Wang to Yao at any time during the top 5 economics journalrevising and publishing processes. Wang designed the approach and Yao discovered the fine structure.
References [1] James N Brown and Robert W Rosenthal. Testing the Minimax Hypothesis: A Re-examination ofO’Neill’s Game Experiment. Econometrica, 58(5):1065–1081, September 1990.[2] C.F. Camerer. Behavioral game theory: Experiments in strategic interaction. Princeton UniversityPress, 2003.[3] Timothy N Cason, Friedman Daniel, and E. D. Hopkins. Cycles and instability in arock–paper–scissors population game: A continuous time experiment. Review of Economic Studies,1:1, 2014.[4] Timothy N. Cason, Daniel Friedman, and Ed Hopkins. An experimental investigation of pricedispersion and cycles. Journal of Political Economy, 0(ja):null, 0.[5] John Van Huyck, Frederick Rankin, and Raymond Battalio. What does it take to eliminate theuse of a strategy strictly dominated by a mixture? Experimental Economics, 2(2):129–150, 1999.14igure 4: (a) The O’Neill 4 × × imes ××