tcb@breakable

# Pulse in collapse: a game dynamics experiment

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

(January 2023)

## Abstract

The collapse process is a constitutional sub-process in the full finding Nash
equilibrium process. We conducted laboratory game experiments with human
subjects to study this process. We observed significant pulse signals in the
collapse process. The observations from the data support the completeness and
the consistency of the game dynamics paradigm.

###### Contents

  1. 1 Introduction
    1. 1.1 Collapse
    2. 1.2 Research questions
    3. 1.3 Outline
  2. 2 Game and the prediction
    1. 2.1 Game design
    2. 2.2 Classical game theory predictions
    3. 2.3 Game dynamics theory prediction
    4. 2.4 Game experiment
  3. 3 Results
    1. 3.1 Distribution
    2. 3.2 Eigencycle
    3. 3.3 Collapse
    4. 3.4 Model comparison
  4. 4 Discussion and conclusions
    1. 4.1 Implications
    2. 4.2 Related works
    3. 4.3 Conclusion
  5. A
    1. A.1 Abbreviations
    2. A.2 Distribution
    3. A.3 Cycle
      1. A.3.1 Verify the statics
      2. A.3.2 Verification of the dynamics theory
    4. A.4 Collapse
      1. A.4.1 Measurement definition
      2. A.4.2 Data
    5. A.5 The game dynamics paradigm and its workflow

##  1 Introduction

Game equilibrium finding is a process [18, 25, 8, 13]. This process is also
referred to as evolution, or adjustment process, or deviation from equilibrium
process, or the convergence process in game theory and experiment. It is a
non-linear spontaneous evolution process driven by game participates’
strategic interactions.

In the study of game theory, terms such as adjustment dynamics [18], learning
theory [12], population game dynamics, evolutionary game theory [10, 26] or
market dynamics [23], are related to this the process. These terms are
included in the game dynamics paradigm.

As a scientific paradigm that moves beyond equilibrium theory (classical game
theory, or game statics theory), the game dynamics paradigm is expected to be
able to describe the regularity of the game dynamic process in terms of
completeness and consistency, as well as reality and accuracy [25, 8, 13].

Instead of using the statics theory to study the predictable equilibrium
(stable statistical relationships among strategy), game dynamics theory
studies the predictable motion (temporary deviations from equilibrium). For
example, in the rock-paper-scissors game [33, 6, 39, 38], the persistent cycle
is a style motion pattern of predictable temporary deviation from equilibrium
(PTDE). Such a pattern can be exploited [11]. In real life, most high-
frequency trading strategies which are not fraudulent, but instead exploit the
PTDE [36].

Following this paradigm, we investigate an unignorable sub-process of the full
finding Nash equilibrium process. In this section, we introduce the sub-
process, namely collapse process; then, we introduce the research questions
and offer a summary of the contents of this paper.

###  1.1 Collapse

A real game system can involve a large number of strategies, most of which
would be dominated during the process of finding Nash equilibrium [3, 22]. As
illustrated in Figure 1(a), the full equilibrium finding process can be
classified into three stages:

  1. 1.

In the initial stages, having no information about the game, agents randomly
choose a strategy vector from the strategy space for optimal payoff. During
this period, the game’s social evolution trajectory is highly stochastic,
smoothly distributed throughout the full game space.

  2. 2.

Later, in the period of infancy, each agent may start to identify their own
nonprofit strategy by learning from strategy interactions. As a result, the
dimensions of the game strategy space may collapse. In theory, ‘a dominated
strategy can dominate a domination before being dominated’, meaning that a
phenomenon known as a pulse could exist. A conceptual rendering of a pulse in
the collapse process is shown in Figure 1(b).

  3. 3.

Finally, when there is no dominant strategy, the evolution trajectory will
converge to a fixed pattern (a persistent cycle or fixed equilibrium point) of
the game. When rendered as an image, the pattern is the same as a condensed
structures in astrophysics.

(a)

(b)

(c)

Figure 1: Conceptual figure. (a) The three periods in the full Nash-
equilibrium-finding process, moving from random to collapse to fixed pattern.
This study focus on the collapse process. (b) An illustration of a pulse in a
collapse.(c) The workflow of this report: The Game, Theory (Static and
Dynamic) vs. Experiment by Observations (1-distribution; 2-cycle; 3-collapse).
For the abbreviations, see Table A1.

The collapse appears in the second stage. The term ‘collapse’, introduced to
game dynamics here, is borrowed from the concept of ‘gravitational collapse’
in astrophysics [35] which is a fundamental mechanism and a legitimate
constituent part of the formation of structures in the universe. This study
benefits from the physical pictures in astrophysics.

###  1.2 Research questions

We investigate the collapse process by asking the following two questions:

  * •

In controlled laboratory game experiments with human subjects, can significant
new observations be obtained during a collapse process?

  * •

Can the existing game theory paradigm consistently and completely capture the
collapse process?

Answering these questions is not trivial, for the following reasons:

  * •

From a scientific perspective, without evidence from the collapse process, our
knowledge of the full equilibrium-finding process is obviously incomplete.

  * •

Within a scientific paradigm, without conducting experiments on collapses,
there is no basis for the reality of the theory [16, 15, 21].

These issues relate to reality and accuracy, as well as to the completeness
and consistency of the game dynamics paradigm, and are therefore not trivial.

###  1.3 Outline

To answer our research questions, following the experimental game theory
paradigm [23, 4, 6, 39, 29, 2], we carried out the workflow shown in Figure
1(c):

  * 1

We designed a parameterised three card poker game for use in a laboratory
study of the the statics theory and the dynamics theory (see section 2.1).

  * 2

We derived the predictions for three variables (distribution, cycle, and
pulse) by the statics theory and the dynamics theory (see sections 2.2 and
2.3).

  * 3

We conducted laboratory experiments which include 3-treatments with 72 human
subjects, fixed-paired, and 1000 repeated games (see section 2.4).

  * 4

We outlined the experimental observations and their theoretical verification
of the three variables (distribution, cycle, and pulse in sections 3.1, 3.2
and 3.3, respectively). Additionally, we we report the results of comparing
the models (see section 3.4).

In the discussion section, we offer a summary, discuss the related literature,
address the implications and applications, and introduce questions for future
research.

##  2 Game and the prediction

This section explains the design of the game and the experiment, as well as
the predictions made using the statics theory and the dynamics theory. A
summary is shown in Table 1.

###  2.1 Game design

The original game is a simplified 3-card porker game. The game is introduced
by Binmore, by replacing Von Neumann’s numerical cards by a deck with only the
1, 2, 3 (page 93 in [3]). This game is a dynamic game with incomplete
information. We generalised this game to study the dynamics process.

The extensive form of the generalised $(m,n)$ game is shown in Figure 2. When
($m,n$) = (2,1), the game is exactly the same as the original game (page 93
[3]). Following [3], the normal form could be represented as the $8\times 8$
bi-matrix shown in Figure 2(b). That is, each player (X or Y) has eight
strategies. This study mainly makes use of the bi-matrix form.

(a) The $(m,n)$-game

(b) The payoff matrix of the (2,1)-game: Treatment A $(\textbf{m,n})\\!=\\!(2,1)$ | $Y_{1}$ | $Y_{2}$ | $Y_{3}$ | $Y_{4}$ | $Y_{5}$ | $Y_{6}$ | $Y_{7}$ | $Y_{8}$
---|---|---|---|---|---|---|---|---
$X_{1}$ | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0
$X_{2}$ | 0, 0 | 0, 0 | 1, -1 | 1, -1 | 1, -1 | 1, -1 | 2, -2 | 2, -2
$X_{3}$ | 2, -2 | -1, 1 | 2, -2 | -1, 1 | 3, -3 | 0, 0 | 3, -3 | 0, 0
$X_{4}$ | 2, -2 | -1, 1 | 3, -3 | 0, 0 | 4, -4 | 1, -1 | 5, -5 | 2, -2
$X_{5}$ | 4, -4 | 1, -1 | 1, -1 | -2, 2 | 4, -4 | 1, -1 | 1, -1 | -2, 2
$X_{6}$ | 4, -4 | 1, -1 | 2, -2 | -1, 1 | 5, -5 | 2, -2 | 3, -3 | 0, 0
$X_{7}$ | 6, -6 | 0, 0 | 3, -3 | -3, 3 | 7, -7 | 1, -1 | 4, -4 | -2, 2
$X_{8}$ | 6, -6 | 0, 0 | 4, -4 | -2, 2 |  8, -8 | 2, -2 |  6, -6 | 0, 0

(c) The payoff matrix of the (3,2)-game: Treatment B $(\textbf{m,n})\\!=\\!(3,2)$ | $Y_{1}$ | $Y_{2}$ | $Y_{3}$ | $Y_{4}$ | $Y_{5}$ | $Y_{6}$ | $Y_{7}$ | $Y_{8}$
---|---|---|---|---|---|---|---|---
$X_{1}$ | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0
$X_{2}$ | -2, 2 | -2, 2 | 0, 0 | 0, 0 | 0, 0 | 0, 0 | 2, -2 | 2, -2
$X_{3}$ | 2, -2 | -2, 2 | 2, -2 | -2, 2 | 4, -4 | 0, 0 | 4, -4 | 0, 0
$X_{4}$ | 0, 0 | -4, 4 | 2, -2 | -2, 2 | 4, -4 | 0, 0 | 6, -6 | 2, -2
$X_{5}$ | 6, -6 | 2, -2 | 2, -2 | -2, 2 | 6, -6 | 2, -2 | 2, -2 | -2, 2
$X_{6}$ | 4, -4 | 0, 0 | 2, -2 | -2, 2 | 6, -6 | 2, -2 | 4, -4 | 0, 0
$X_{7}$ | 8, -8 | 0, 0 | 4, -4 | -4, 4 | 10, -10 | 2, -2 | 6, -6 | -2, 2
$X_{8}$ | 6, -6 | -2, 2 | 4, -4 | -4, 4 | 10, -10 | 2, -2 |  8, -8 | 0, 0

(d) The payoff matrix of the (4,2)-game: Treatment C $(\textbf{m,n})\\!=\\!(4,2)$ | $Y_{1}$ | $Y_{2}$ | $Y_{3}$ | $Y_{4}$ | $Y_{5}$ | $Y_{6}$ | $Y_{7}$ | $Y_{8}$
---|---|---|---|---|---|---|---|---
$X_{1}$ | 0, 0 | *0, 0 | 0, 0 | *0, 0 | 0, 0 | 0, 0 | 0, 0 | 0, 0
$X_{2}$ | -2, 2 | *-2, 2 | 1, -1 | 1, -1 | 1, -1 | 1, -1 | 4, -4 | 4, -4
$X_{3}$ | 2, -2 | -3, 3 | 2, -2 | -3, 3 | 5, -5 | 0, 0 | 5, -5 | 0, 0
$X_{4}$ | 0, 0 | -5, 5 | 3, -3 | -2, 2 | 6, -6 | 1, -1 | 9, -9 | 4, -4
$X_{5}$ | 6, -6 | *1, -1 | 1, -1 | -4, 4 | 6, -6 | 1, -1 | 1, -1 | -4, 4
$X_{6}$ | 4, -4 | -1, 1 | 2, -2 | -3, 3 | 7, -7 | 2, -2 | 5, -5 | 0, 0
$X_{7}$ | 8, -8 | -2, 2 | 3, -3 | -7, 7 | 11, -11 | 1, -1 | 6, -6 | -4, 4
$X_{8}$ | 6, -6 | -4, 4 | 4, -4 | -6, 6 | 12, -12 | 2, -2 | 10, -10 | 0, 0

Figure 2: The parameterized (m,n) game. (a) The extensive form [3]. (b,c,d)
The normal form of the game (the matrix elements are multiplied by 6).

The parameters are assigned as $(m,n)$ = (2, 1), (3, 2) and (4, 2), which are
referred to as treatment A, B, and C, respectively. Details on the game design
and the motivations are provided in SI – Game.

###  2.2 Classical game theory predictions

##### Method:

The concepts of static equilibrium and the iterated eliminating dominated
strategy (IEDS) are useful when describing the full Nash-equilibrium-finding
process, according to the classical game theory paradigm. For details of the
method, see SI – Statics.

  * •

For the three treatments, to solve the Nash equilibrium of the three bimatrix
zero sum two person game, we can use quadratic programming method [20].

  * •

For the three treatments, the order of the eliminating dominated strategies,
as well as the surviving strategies, can be obtained by IEDS method.

##### Predictions:

The predictions and their verification methods are as follows:

  * TS1

Distribution. The equilibrium distribution results are shown in figure 2a.
Following the methods of conventional experimental research [4, 28, 6], the
predictions can be verified using the data, denoted as $\rho^{E}$ and shown in
Table A2.

  * TS2

Cycle. IEDS provides the surviving strategy for a game, which is shown in
Figure 3 and in the row titled ‘Surviving Strategy’ in Table 1. According to
best-response analysis, the cycle’s existence and relative strength in each
treatment (A, B, and C) can be predicted as (Yes, No, Yes), referring to
Figure 3. These points can be verified using the experimental cycle
($\vartheta^{E}$) in section 3.2.

  * TS3

Collapse. IEDS provides the eliminating order of a game. The results is shown
in the row ’IEDS round’ in Table 1. Mathematical analysis of the best response
analysis show that, among the dominated strategies, the strategy that is
eliminated last could provide the pulse signal during the collapse. These
points can be verified by the pulse ($\psi^{E}$) and the crossover points
($\chi^{E}$) using the experimental observations in section 3.3.

In additions, these three predictions can be applied to compare with those
from dynamics theory. The results will be reported in section 3.4.

(a) A | $Y_{2}$ | $Y_{4}$
---|---|---
$X_{2}$ | 0, 0 | 1, -1
$X_{6}$ | 1, -1 | -1, 1

(b) B | $Y_{4}$
---|---
$X_{1}$ | 0, 0

(c) C | $Y_{2}$ | $Y_{4}$
---|---|---
$X_{1}$ | 0, 0 | 0, 0
$X_{2}$ | -2, 2 | 1, -1
$X_{5}$ | 1, -1 | -4, 4

Figure 3: The surviving strategy of the (A, B, C) game. The expected fixed
patterns of the (A, B, C) game are a fixed orbital cycle, pure Nash
equilibrium (a fixed point or signal star), and a mixed pattern, respectively.
Table 1: Equilibrium theory predictions and experiment protocol.

ID | Treatment ID | A | B | C
---|---|---|---|---
Game-$(m,n)$ | (2,1) | (3,2) | (4,2)
Player ID | X | Y | X | Y | X | Y
Equili- brium Theory | IEDS Round 1 | 1,3,5,7 | $\alpha^{*}$ | 3 | $\alpha$ | 3 | $\alpha$
IEDS Round 2 | 4,8 | - | 2,4,6,7,8 | - | 4,6,7,8 | -
IEDS Round 3 |  |  | - | 2 |  |
IEDS Round 4 |  |  | 5 | - |  |
Survive | 2,6 | 2,4 | 1 | 4 | 1,2,5 | 2,4
strategy
Experi- ment Protocal | Session number | 12 | 12 | 12
Player in session | 2 | 2 | 2
Repeated round | 1000 | 1000 | 1000
Matching |  Fix paired |  Fix paired |  Fix paired

${}^{*}\alpha$=1,3,5,6,7,8 means the strategies are eliminated simultaneously.

###  2.3 Game dynamics theory prediction

The game dynamics paradigm provides a set of concepts and methods on game
evolution process [10, 8, 30, 17, 33]; This study adheres to this paradigm.

##### Method:

We utilize the logic dynamics equations system [26]; At the same time,
referring to [6, 39, 17, 10], this study makes use of the noise parameter
($\lambda$ = 50) and the adjustment time step ($\triangle t$ = 0.02). Then,
with random initial conditions, we can generate the time series of 1000 rounds
data as long as the 1000 round experiment conducted in this research. Then,
using the generated time series, we can calculate the predictions of the
observations (distribution, cycle and collapse) simultaneously.

It is worth noting that, based on the experiences of previous studies [6, 39],
we take the two parameters $(\lambda,\triangle t)$ simply by scanning two-
dimensional parametric space, aiming for
$\rho^{T}_{\textbf{Dyn}(\lambda,\triangle t)}$ to be approximately close to
$\rho^{T}_{\textbf{QP}}$. We do not fix the two parameters by the experiment
data.

##### Predictions:

Following the methods, the results are predictions that can be experimentally
verified. These results are as follows:

  * TD1

Distribution. The distribution of the treatments is shown in Figure 4(a) and
Table A2 as $\rho^{T}_{\text{Dyn}}$, which is verifiable by the experiment.

  * TD2

Cycle. The eigencycle spectra of the treatments are shown in Figure 4(b) as
$\vartheta^{T}$. For further details about eigencycle spectra see Appendix
A.3. The refined predictions are that, for treatments (A,B,C), there exist
(strong, no, weak) cycles, respectively.

  * TD3

Collapse. The theoretical accumulated curves ($\varrho^{T}$) are shown in
Figure 4(c)-4(h). The numerical result of the crossover points are shown in
Table A8. The statistically significant theoretical pulse signals are shown in
Table A6. These predictions can be applied as the criterion to evaluate the
consistency between the theory and the experiment.

The crossover points ($\chi_{-}$) with crossover time $\tau>100$ are labelled
with red arrows. According to the numerical results shown in Table A8, there
are (4,0,2) $\chi_{-}$ red arrows in treatments (A,B,C).

###  2.4 Game experiment

The experiment was performed at Zhejiang University between August and
September 2018. A total of 72 undergraduate students from Zhejiang University
volunteered to serve as the human subjects of this experiment.

The main parameters and protocol are shown in the ’Experiment’ rows in Table
1. There were three treatments, and each was assigned 24 human subjects. The
24 subjects were divided into 12 sessions of 2 participants. The two players
played a game with 1000 repeated in this fixed pairing. The reasoning behind
the protocol design are explained in the Appendix—Experiment: Section 3.3.1
Design consideration).

In a session, a player can see observe sides strategy id
($X_{1},X_{2},...,X_{8}$) for player $X$ or ($Y_{1},Y_{2},...,Y_{8}$) for
player $Y$. But a player can only observe her/his own realised payoffs and the
strategy used of its own opponent. We do not provide the subject the payoff
matrix, and no discussion allowed.

The experimental sessions lasted for an average of two hours. The payment of a
subject, including CNY 20 for attending the session, as well as a payment
based on the player’s rank, averaged CNY 120. The rank fee was determined by
the rank of a participant’s total earning score in the 1000-round game, based
on a comparison of players in the same role (player X or player Y) in the same
experimental session. For details on the design, procedures and data, see
SI—Experiment.

##  3 Results

In this section, we report the results of the experiment and the theoretical
verification of the observations (distribution, cycle, and pulse), as well as
the results of the model comparison.

###  3.1 Distribution

In order to illustrate the completeness and consistency, we report the
distribution in brief. For the method and details relating to the data, see
Appendix—A.3.

##### Measurement.

The strategy distribution and its convergence are reported. The convergence is
measured according to the Euclidean distance $\delta$ from the experimental
distribution $\rho^{E}$ in respect to two theoretical predictions (1) the
static (Nash) equilibrium $\rho^{T}_{\text{S}}$, and (2) the time series
produced by the dynamics model $\rho^{T}_{\text{D}}$.

##### Results.

For the experimental distribution ($\rho^{E}$) and its evolution, which is
determined by the distribution values at various time intervals, as shown in
Table A2, the main results are as follows:

  1. EO1.1

As expected, for all (A,B,C) treatments, over time, the observed distribution
($\rho^{E}$) moves closer to the static equilibrium ($\rho^{T}_{\text{S}}$)
and to the dynamics prediction ($\rho^{T}_{\text{D}}$). This is because the
Euclidean distances decrease along time ($\triangle\delta_{\text{S}}<0$ and
$\delta_{\text{D}}<0$) for all of the treatments, as shown in Table A3.

  2. EO1.2

In all (A,B,C) treatments, the dynamics prediction $\rho^{T}_{\text{D}}$
performs better than the statics prediction $\rho^{T}_{\text{S}}$. This
finding is supported by comparing the Euclidean distance during the last
rounds, $\triangle\delta<0$, for all of the treatments, as shown in Table A3.

To summarize, for the distributions, the experimental observations are
consistent with the theoretical predictions and with the existing literature
[4, 23, 6, 5, 28, 17]. In other words, the full process shown in Figure 1(a)
can be observed in our 1000 rounds of data.

###  3.2 Eigencycle

In order to illustrate illustrate the completeness and consistency, we report
the cycle in brief.

##### Measurement

Referring to [34, 41, 40], we used the eigencycle spectrum $\vartheta^{E}$ to
present the cycle in the experiments. We measured the eigencycle value of the
120 subspace of the game state space for each treatment, respectively; then
normalise $\vartheta^{E}_{A},\vartheta^{E}_{B},\vartheta^{E}_{C}$ for
treatments (A,B,C) to attain the maximum $|\vartheta^{E}|=1$. For more details
on the measurements and the methods used for data presentation, see SI –
Cycle.

##### Results

The experimental eigencycle spectrum $\vartheta^{E}$ is shown in Figure 5(b).
The main results are as follows:

  1. EO2.1

Observations on the (A,B,C) treatments,
$\vartheta^{E}_{A},\vartheta^{E}_{B},\vartheta^{E}_{C}$ are consistent with
the predictions made based on dynamics theory,
$\vartheta^{T}_{A},\vartheta^{T}_{B},\vartheta^{T}_{C}$, respectively, as
shown in Figure 4(b) and Figure 5(b). This is because the linear regression
between $\vartheta^{E}$ and $\vartheta^{T}$ has $p<0.000$ for each treatment.

  2. EO2.2

Our observations were consistent with the predictions made based on the
statics theory. According to the eigencycle spectra for the three treatments,
the cycle $(X_{2},X_{6})\times(Y_{2},Y_{4})$ in treatment A is the strongest.
This consists the result of the best-response analysis, as shown in Figure 3.
These finding are supported by the numerical analysis, as shown in Appendix
A.3.1

In summary, the cycles observed in the experiment are consistent with the
theoretical predictions. The consistency of the results is in keeping with the
existing literature on cycles [34, 6, 31, 40, 41].

(a) Theoretical distributions ($\rho^{T}$)

(b) Theoretical eigencycle spectrum ($\vartheta^{T}$)

(c) $\varrho^{T}_{\text{(X-A)}}$

(d) $\varrho^{T}_{\text{(X-B)}}$

(e) $\varrho^{T}_{\text{(X-C)}}$

(f) $\varrho^{T}_{\text{(Y-A)}}$

(g) $\varrho^{T}_{\text{(Y-B)}}$

(h) $\varrho^{T}_{\text{(Y-C)}}$

Figure 4: Theoretical results of treatments A, B, and C. (a) Distributions
$\rho^{T}$ from the two models; (b) eigencycle spectrum $\vartheta^{T}$; (c-h)
accumulated curve ($\varrho^{T}$) of the player (X or Y). The crossover points
($\chi$) are labelled with arrows referring to the data in Table A8.

(a) Experimental distributions ($\rho^{E}$)

(b) Experimental eigencycle spectrum($\vartheta^{E}$)

(c) $\varrho^{E}_{\text{(X-A)}}$

(d) $\varrho^{E}_{\text{(X-B)}}$

(e) $\varrho^{E}_{\text{(X-C)}}$

(f) $\varrho^{E}_{\text{(Y-A)}}$

(g) $\varrho^{E}_{\text{(Y-B)}}$

(h) $\varrho^{E}_{\text{(Y-C)}}$

Figure 5: Experiment results of treatments A, B, and C. (a)Average time of
distribution for the 1-200 period and the 800-1000 period; (b) eigencycle
spectrum $\vartheta^{E}$; (c-h) accumulated curve ($\varrho^{E}$) of the (X,Y)
player over time. The crossover points ($\chi$) are labelled with arrows
referring to the data in Table A7.

###  3.3 Collapse

The first research question requires us to identify the pulse in the collapse.
To evaluate the consistency is to validate the dynamics paradigm during the
collapse. Here, we offer the main results obtained regarding the collapse.

##### Measurement.

For the collapse, there are three relevant measurements – the accumulated
curve ($\varrho$), the pulse signal ($\psi$) and the crossover points
($\chi$). For the definition of the measurements, see Appendix—A.4.1.

##### Results and explanation.

The main experimental results are as follows:

  1. EO3.1

Accumulated curves ($\varrho^{E}$):  The experimental accumulated curves
$\varrho^{E}_{\text{j - i}}$ indicates the $i\in$(A,B,C) treatment and $j\in$
(X,Y) player, respectively, and there are six conditions. These are shown in
Figure 5(c) \- 5(h), which can be individually compared to the dynamics
predictions $\varrho^{T}_{\text{j-i}}$, as shown in Figure 4(c) \- 4(h). The
results of the visual comparison support the consistency between the dynamics
theory and the experimental results.

  2. EO3.2

Crossover point ($\chi^{E}$): Following the dynamics prediction $\chi^{T}$ in
Table A8, in the accumulated curve ($\varrho^{E}$), the $\chi^{E}_{-}$ are
labelled using red arrows between $\textbf{D}^{-}/\textbf{D}^{+}$; meanwhile
the $\chi^{E}_{+}$ are labelled using blue arrows between the
$\textbf{D}^{+}/\textbf{D}^{+}$. These are shown in figure 5(c)-5(h). The
consistency between theory and experiment is in visible.

  3. EO3.2.1

The numerical results in Table A7 and Table A8 support the consistency between
the theory and the experiment. For the dynamics predictions shown in Table A8,
if $\tau^{T}\geq 89$ of $\psi([Y_{2},Y_{4}],\tau)$ is taken as the benchmark,
there are have nine predicted samples. The experiment showed that, following
the nine that were theoretically predicted, all eight of the $\chi^{E}$
labelled arrows in experiment have the largest $\tau$ values in terms of the
related role (X,Y) and the treatment (A,B,C).

  4. EO3.2.2

The crossover point is a consequence of the pulse in collapse. Referring to
the relation between the pulse and the crossover time shown in section A.4.1,
the observed crossover points are used to support the existence of pulse in
[EO3.3.6] and [EO3.3.7].

  5. EO3.3

Pulse signals ($\psi^{E}$): A pulse is a distinct observation made during the
collapse. For the definition of a pulse, see Equation A13 in Appendix A.4.1.
By examining the data, we observed pulses in significant. The main results
regarding the pulses are as follows:

     * (a)

Existence of the pulse:

     * EO3.3.1

There are a total of nine pulses in significant ($p<0.05$), as shown in Table
A5.

     * EO3.3.2

Among these nine pulses, three pulses are of the strongest statistical
significance ($p<0.010$), as shown in Table 2.

     * (b)

The pattern of the pulses’ existence is consistent with the theoretical
pattern:

     * EO3.3.3

The number of pulses in the treatments does not deviate the expectation by
dynamics theory (see Table A6). More pulses are observed in treatment A than
in B, and in B than in C.

     * EO3.3.4

All three of the pulses with the strongest significance ($p<0.010$) are
included in category of the most expected pulses, according to dynamics theory
(for details, see Table A6).

     * EO3.3.5

All four pulses with the strongest significance ($p\leq 0.011$) have the
maximum surplus $\psi$ value among the nine pulse signals shown in Table A5.

     * (c)

The pattern of the pulses’ existence is consistent with the pattern of the
crossover points.

     * EO3.3.6

Of the nine significant pulses ($p<0.05$), six are from treatment A and three
are from C. More pluses are observed in treatment A than in C, as shown in
Table A5. This is consistent with the $\chi^{E}$, in that there more
significant crossover points observed in treatment A than in C, according to
the $\chi^{E}_{-}$ with the red arrow in Table A7.

     * EO3.3.7

As predicted in Table A8, there are (4,0,2) red arrows in treatment (A,B,C),
respectively. This is supported by the number of pulses and their significance
in the experiment, as shown in Table A5.

     * EO3.3.8

On the other hand, there is no significant pulse signal ($p<0.05$) from the Y
player in treatment A, B, or C. This is consistent with the dynamics
predictions for pulses, as shown in Table A6.

According to the pulse measurements, all of the significant results for pulses
are consistent with the predictions made based on dynamics. No statistically
significant evidence against the dynamics model was obtained.

Treat- | Domin- | Domin- | Time | paired- | Surplus | Sample
---|---|---|---|---|---|---
ments | ated | ation | block |  ttest ($p$) |  | size
| $s_{j}$ | $s_{i}$ | $[t_{0},t_{1}]$ |  | $\psi$ | N
A | $X_{8}$ | $X_{2}$ | 11-20 | 0.007 | 17 | 120
A | $X_{8}$ | $X_{6}$ | 11-20 | 0.002 | 19 | 120
A | $X_{4}$ | $X_{6}$ | 81-90 | 0.007 | 15 | 120
Table 2: All of the observed pulses $\psi^{E}$ were highly significant
($p<0.010$). The symbols are explained in Equation A13 in Appendix A.4.1

The main result from the collapse in experiment is that, the significant pulse
signals are observed, which supported by the observed crossover points. In
addition to existed literature [33, 6, 5, 37, 17, 29, 2], in the collapse
process, the consistency between the theory and the experiment empower by new
empirical evidence.

###  3.4 Model comparison

Comparing the performances of the two theories under consideration is a key
component of experiment research. Our main results are as follows:

  1. ET1

In terms of distribution, the dynamics prediction ($\rho^{T}_{D}$)
outperformed statics prediction ($\rho^{T}_{S}$). This is supported by data
($\rho^{E}$) in section 3.1.

  2. ET2

For the cycle, the dynamics prediction outperformed the statics prediction.
This is supported by comparing the results shown in EO2.1 and EO2.2 in section
3.2. Further analysis on cycle spectra can provide additional evidence to
enhance the results, see section A.3.

  3. ET3

For the collapse, dynamics theory can quantitatively predict the pulse
($\psi$) as well as the crossover point ($\chi^{E}$). The predictions made
using IEDS, a component of statics theory in statics theory (TS3 in section
2.2), offer only limited explanation of the pulsing signals and the crossover
point in terms of quantity. As a result, the dynamics model outperforms the
statics.

In summary, game dynamics theory performs better than the static equilibrium
theory, according to our data. This finding is consistent with the existed
literature [6, 5, 17, 37, 33, 29, 2]. The summary of this section, see the
first paragraph of the Discussion.

##  4 Discussion and conclusions

The main results of this study are the answers to the two key research
questions posed above.

  1. 1.

Regarding the first question, we observed that the pulse signal, which is a
distinct observation of the collapse predicted using the dynamic model, is
significant.

  2. 2.

Regarding the second question, in the full process of finding equilibrium, we
showed that all of the significant observations were those most clearly
predicted by the dynamic model. The completeness and the consistency of the
dynamics paradigm are supported by our data.

We posit that these results will assist the development of the game dynamics
paradigm.

###  4.1 Implications

To explain the implications of our results, we can refer to the scientific
paradigm of game dynamics theory [8, 9, 25, 18, 10, 26]. This paradigm has a
self-reinforcing closed feedback workflow loop (for a brief introduction, see
section A.5), which enhances its distinct set of concepts (e.g., natural
selection, the evolutionary stable strategy, and, potentially, cycle [25, 10,
17, 6, 38, 37, 33, 29, 2]).

Since 1990 [8, 25],it has been expected that this paradigm would establish its
own narrative of the game dynamics process, bringing about a paradigm shift
with regard to the classical statics (Nash equilibrium) theory. However, the
paradigm shift has not come to fruition. Those who are skeptical about using
dynamics research to solve social and economic problems mainly focus on
empirical observations [18, 25, 4].

Since 2010, the convenience of this method has been improved by empirical
observations. The cycle is one example of this [7, 6, 40, 37, 39, 38, 32, 29,
2], and it may become a constitute element in the paradigm’s distinct set of
concepts. Because a cycle generally exists in mixed-strategy Nash equilibrium
behaviours, it is a consequence of the engagement of strategy interactions;
moreover, it demonstrates game evolution, and is an exploitable PTDE.

We posit that the pulse in collapse can be constantly developed, similarly to
the cycle, because of the following:

  1. 1.

The pulse fulfils the experimental observation gap in the collapse, which is a
legitimate constituent part of the full equilibrium-finding process (as shown
in Figure 1).

  2. 2.

The study on the pulse in collapse process can be helpful to the completeness
and consistency of the game dynamics paradigm.

###  4.2 Related works

##### On the elimination of a dominated strategy

This study is not the unique to note the behaviours of the dominated strategy.
For examples:

  * •

Theoretically speaking, ‘A dominated strategy can dominate a domination before
being dominated’; this phenomenon, namely, a pulse, is evident (e.g., in
Gintis’ book [14], Figure 5.3, in section 5.8, the red mark during the infancy
stage is the pulse signal). In finding the full Nash equilibrium process, the
dominated strategies that are eliminated or survive have been extensively
studied [16, 15, 21], but research on the pulse phenomenon is rare.

  * •

In experimental terms, research into dominated strategies is not uncommon [24,
30, 27]. But these reports mainly focus on whether the dominated strategies
are eliminated or attain equilibrium, but do not address what can be observed
during the collapse.

The unique of this study in existed literature is that, it firstly provides
the quantitative observation, in terms of the pulse, of the collapse process.
By examining the collapse process, which has not been investigated before, the
consistency and the completeness of the game dynamics paradigm are
demonstrated.

##### Further research

Further research on the collapse and the pulse is needed. Although the
existence of the pulse in the collapse has been explored and pulse experiment
is consistent with the theory, it is far from being considered a legitimate
constituent part of the paradigm.

The open question on how to select a dynamics model and its parameters when
facing a real-life system is remained. In this study, following [1, 6, 39], we
utilised logit dynamics, and fix its parameter grossly by the distribution
closing to the equilibrium distribution by QR. Although we can capture all of
the various observations in various treatments without changing the
parameters, admittedly, this work can not answer the open question.

The collapse stage in games does not only move beyond the stationary concept
of static equilibrium [28, 37],but also departs from the concept of non-
equilibrium stationary states (e.g., cycle) [19, 39, 37]. By developing a
better understanding of collapse, and of how to obtain the hierarchy of
condensed structures or various dynamic equilibrium structures from a
primordial soup or an existing state, we can expect solutions to appear in the
coming decades.

##### Potential applications

Exploiting the PTDE can be profitable, e.g., in the arbitrage by high
frequency trading (HFT) [36]. The arbitrage of HFT comes from predictable non-
equilibrium in the short-term, rather than equilibrium in the long-term. The
collapse is part of a process that is distant from equilibrium. It could
therefore provide a helpful framework for understanding sudden shocks in a
social system. During such periods, many unconventional strategy behaviors
might appear to dominate. The findings related to the pulse support the
statement: ’A dominated can dominate a domination before dominated’. This
means that non-equilibrium behaviors can be profitable in the short term.

This study has real life applications in fields including artificial
intelligence [22] and and the social systems addressed in [9, 25]. the
collapse is a legitimate constituent part of the process of game evolution.
The Nash equilibrium is not necessarily the best strategy during the infancy
period when playing a complex game, e.g., Stratego [22].

Regarding general decision making in real-life social systems,
philosophically, linear thinking may be harmful. In periods of turmoil, the
power of myopia, or the power of the best response, must be properly
acknowledged.

###  4.3 Conclusion

Based on a dynamic game with incomplete information, this investigation
illustrates the full equilibrium finding process shown in figure 1, from
random play to collapse to a fixed pattern (cycling and equilibrium). As a
PTDE, the collapse constitutes fertile ground for the science of game
dynamics, especially as it relates to the system of human social interactions.

## Appendix A

###  A.1 Abbreviations

The abbreviations and mathematical symbols, shown in Table A1, are used in the
main text, this appendix and the supplementary information.

PTDE |  | Predictable temporary deviation from equilibrium
---|---|---
N |  | Sample size for statistical analysis
$\text{Dyn}(\lambda,\Delta)$ |  | Logit Dynamics with noise $\lambda$ and time step $\Delta$,
|  | a model in game dynamics theory
IEDS |  | Iterated eliminating dominated strategy,
|  | a method to find the Nash equilibrium
QP |  | Quadratic programming, an algorithm
|  | for Nash equilibrium distribution
$\textbf{D}^{\pm}$ |  |  $+$, domination strategy, survival strategy;
EO |  | Experimental observation
TS |  | Predictions from static equilibrium theory
TD |  | Predictions from dynamics theory
ET |  | Experiment data, used to compare the models’
Obs. |  | observations
SI |  | Supplementary information
$s_{i}$ |  | The $i$-th optional strategy for a player
$\rho$ |  | The proportion vector of strategies used
| $\rho(s_{i})$ | The $i$-th component of $\rho$,
| $\rho(s_{i},t)$ | The $\rho(s_{i})$ at time (round) $t$
| $\rho_{([t_{1},t_{2}])}$ | The time average of $\rho$ between $[t_{1},t_{2}]$
| $\rho^{+}_{*}([s_{i},s_{j}])$ | A surplus sample set
$\delta$ |  | The Euclidean distance between two distribution
|  | vectors,used to evaluate their difference
$o$ |  | A fixed point, rest point, or zero velocity point, equilibrium
$\lambda$ |  | Eigenvalue
$\xi$ |  | Eigenvector
$\eta_{i}$ |  | The $i$-th component of a given eigenvector $\xi$
$\vartheta$ |  | The eigencycle set in the 2D subspace set of a game
|  | space, describing the cyclic motion strength
| $\vartheta(m,n)$ | The $\vartheta$ at dimension (1,2):=($\eta_{m},\eta_{n}$)
| $\vartheta(x)$ | Eigencycle spectrum, where $x$ is a natural number
$\varrho$ |  | Time-accumulated value curve of $\rho(t)$
| $\varrho(s_{i})$ | the $\varrho$ of the $s_{i}$ strategy
$\psi$ |  | A pulse signal
| $\psi([s_{i},s_{j}],[t_{1},t_{2}])$ | A pulse signal of strategy $j$ over $i$
|  | during time interval $[t_{1},t_{2}]$
$\chi$ |  | A crossover point, having two results $t$ and $\varrho$
| $\chi([s_{i},s_{j}],\tau)$ | A crossover point of $\varrho(s_{i})$ and $\varrho(s_{j})$ at time
|  |  $\tau$, and when $t=\tau-0^{+}$, $s_{j}>s_{i}$
| $\chi_{+}(s_{i},s_{j})$ |  $\chi$ if $s_{i}\in\textbf{D}^{+}\cap s_{j}\in\textbf{D}^{+}$
| $\chi_{-}(s_{i},s_{j})$ |  $\chi$ if $s_{i}\in\textbf{D}^{+}\cap s_{j}\in\textbf{D}^{-}$
Table A1: Abbreviations and mathematical symbols.

###  A.2 Distribution

This section reports the results on the distribution and its evolution,
including the following elements:

  * •

Theoretical and experimental numerical results of the average time
distribution are shown Table A2.

  * •

Numerical results relating to Euclidean distance, illustrating the evolution
of the distribution, are shown in A3.

For more details of the measurements and data, see SI – Distribution.

##### Explanation of support [EO1] and [EO2].

In Table A3, the numerical Euclidean distance and its evolution are reported
as follow:

  1. 1.

The (1,2,4,5)-th row is the Euclidean distance of the given time interval.

  2. 2.

The (3, 6)-th row is the difference in the Euclidean distance

$\triangle\delta_{\textbf{QP}}=\delta_{\textbf{QP[801,1000]}}-\delta_{\textbf{QP[1,200]}}<0$
(A1)
$\triangle\delta_{\textbf{Dyn}}=\delta_{\textbf{Dyn[801,1000]}}-\delta_{\textbf{Dyn[1,200]}}<0$
(A2)

This means that $\rho^{E}$ is close to the predictions during the 1000 round
game in all of the treatments (A,B,C).

  3. 3.

The (7)-th row is the difference between the two models.

$\triangle\delta=\delta_{\textbf{Dyn[801,1000]}}-\delta_{\textbf{QP[801,1000]}}<0$
(A3)

This means that the $\rho^{E}_{800,1000}$ is closer to the expectation of the
dynamics model than the Nash equilibrium according to the QP algorithm in all
of the treatments (A,B,C).

[.5]0 Treatment | Period | id | $\rho^{E}_{(1,1000)}$ | $\rho^{E}_{(1,500)}$ | $\rho^{E}_{(500,1000)}$ | $\rho^{E}_{(800,1000)}$ | $\rho^{T}_{QP}$ | $\rho^{T}_{Dyn(0.02,50)}$
---|---|---|---|---|---|---|---|---
[.5]0 | Game | X | Y | X | Y | X | Y | X | Y | X | Y | X | Y
A | (2,1) | 1 | 0.040 | 0.050 | 0.055 | 0.042 | 0.025 | 0.058 | 0.021 | 0.057 | 0.000 | 0.000 | 0.034 | 0.001
A | (2,1) | 2 | 0.580 | 0.497 | 0.486 | 0.431 | 0.673 | 0.563 | 0.691 | 0.595 | 0.667 | 0.667 | 0.670 | 0.629
A | (2,1) | 3 | 0.019 | 0.017 | 0.029 | 0.028 | 0.009 | 0.007 | 0.005 | 0.007 | 0.000 | 0.000 | 0.000 | 0.000
A | (2,1) | 4 | 0.045 | 0.346 | 0.076 | 0.367 | 0.014 | 0.324 | 0.011 | 0.307 | 0.000 | 0.333 | 0.000 | 0.370
A | (2,1) | 5 | 0.066 | 0.015 | 0.087 | 0.023 | 0.045 | 0.006 | 0.045 | 0.005 | 0.000 | 0.000 | 0.014 | 0.000
A | (2,1) | 6 | 0.185 | 0.019 | 0.163 | 0.030 | 0.207 | 0.007 | 0.205 | 0.005 | 0.333 | 0.000 | 0.280 | 0.000
A | (2,1) | 7 | 0.026 | 0.011 | 0.038 | 0.018 | 0.014 | 0.005 | 0.011 | 0.002 | 0.000 | 0.000 | 0.000 | 0.000
A | (2,1) | 8 | 0.039 | 0.045 | 0.066 | 0.061 | 0.012 | 0.029 | 0.010 | 0.022 | 0.000 | 0.000 | 0.000 | 0.000
B | (3,2) | 1 | 0.624 | 0.061 | 0.540 | 0.071 | 0.707 | 0.051 | 0.712 | 0.044 | 1.00 | 0.033 | 0.842 | 0.025
B | (3,2) | 2 | 0.165 | 0.188 | 0.160 | 0.189 | 0.170 | 0.186 | 0.167 | 0.186 | 0.000 | 0.186 | 0.087 | 0.239
B | (3,2) | 3 | 0.019 | 0.030 | 0.026 | 0.040 | 0.012 | 0.020 | 0.010 | 0.017 | 0.000 | 0.041 | 0.000 | 0.053
B | (3,2) | 4 | 0.029 | 0.579 | 0.046 | 0.521 | 0.011 | 0.636 | 0.012 | 0.650 | 0.000 | 0.573 | 0.000 | 0.501
B | (3,2) | 5 | 0.058 | 0.030 | 0.077 | 0.034 | 0.039 | 0.026 | 0.040 | 0.022 | 0.000 | 0.021 | 0.063 | 0.006
B | (3,2) | 6 | 0.029 | 0.029 | 0.041 | 0.034 | 0.016 | 0.025 | 0.015 | 0.027 | 0.000 | 0.049 | 0.007 | 0.053
B | (3,2) | 7 | 0.043 | 0.020 | 0.062 | 0.030 | 0.025 | 0.011 | 0.020 | 0.009 | 0.000 | 0.023 | 0.000 | 0.012
B | (3,2) | 8 | 0.033 | 0.063 | 0.047 | 0.081 | 0.019 | 0.045 | 0.025 | 0.045 | 0.000 | 0.074 | 0.000 | 0.111
C | (4,2) | 1 | 0.750 | 0.051 | 0.588 | 0.064 | 0.912 | 0.039 | 0.943 | 0.038 | 1.000 | 0.058 | 0.882 | 0.078
C | (4,2) | 2 | 0.092 | 0.386 | 0.135 | 0.362 | 0.049 | 0.411 | 0.034 | 0.426 | 0.000 | 0.422 | 0.077 | 0.413
C | (4,2) | 3 | 0.015 | 0.027 | 0.029 | 0.033 | 0.000 | 0.022 | 0.000 | 0.014 | 0.000 | 0.043 | 0.000 | 0.061
C | (4,2) | 4 | 0.024 | 0.363 | 0.046 | 0.325 | 0.002 | 0.400 | 0.001 | 0.425 | 0.000 | 0.330 | 0.000 | 0.321
C | (4,2) | 5 | 0.061 | 0.043 | 0.092 | 0.044 | 0.031 | 0.042 | 0.016 | 0.040 | 0.000 | 0.025 | 0.038 | 0.011
C | (4,2) | 6 | 0.016 | 0.040 | 0.030 | 0.051 | 0.001 | 0.030 | 0.002 | 0.004 | 0.000 | 0.055 | 0.003 | 0.061
C | (4,2) | 7 | 0.019 | 0.018 | 0.036 | 0.025 | 0.001 | 0.010 | 0.002 | 0.009 | 0.000 | 0.021 | 0.000 | 0.009
C | (4,2) | 8 | 0.024 | 0.071 | 0.044 | 0.096 | 0.004 | 0.047 | 0.003 | 0.044 | 0.000 | 0.046 | 0.000 | 0.047
Table A2: Distribution. Columns 1-3 show the treatment, game parameter, and
strategy, respectively. The blocks $\rho^{E}(t_{0},t_{1})$ are the
experimental (X,Y) player’s time average of $\rho$ between $(t_{0},t_{1})$.
$\rho^{T}_{QP}$ and $\rho^{T}_{Dyn(0.02,50)}$ are the prediction from statics
and dynamics theory, respectively.

| Treatment
---|---
| A | B | C
$\delta_{\textbf{QP[1,200]}}$ |  0.4298 |  0.5551 |  0.5745
$\delta_{\textbf{QP[801,1000]}}$ |  0.3766 |  0.4047 |  0.4436
$\triangle\delta_{\textbf{QP}}$ |  $-$0.0532 |  $-$0.1504 |  $-$0.1309
$\delta_{\textbf{Dyn[1-200]}}$ |  0.0781 |  0.2088 |  0.1442
$\delta_{\textbf{Dyn[801-1000]}}$ |  0.0059 |  0.0097 |  0.0072
$\triangle\delta_{\textbf{Dyn}}$ |  $-$0.0722 |  $-$0.1991 |  $-$0.1370
$\triangle\delta$ |  $-$0.3707 |  $-$0.3950 |  $-$0.4364
Table A3:  The Euclidean distance and its evolution. For details, ′Explanation
of support [EO1] and [EO2]′.

###  A.3 Cycle

As the main focus of this study is the collapse, we report the cycle only
briefly. The definition of the eigencycle spectrum, as well as the details of
the numerical results of the eigencycle spectrum, are shown in SI – Cycle.
This section includes the following contents:

  1. 1.

Results on the maximum eigencycles in the eigencycle spectrum, in the
experiment and in theory, respectively, are shown in Table A4.

  2. 2.

The verification of the theories by experiment.

$\begin{array}[]{|c|c|cc|r|}\hline\cr\addstackgap[.5]{\vphantom{0}}\text{Treatment}&x&m&n&\vartheta^{E}{}{}{}\\\
\hline\cr\text{A}&25&2&12&-1.000\\\ \text{A}&23&2&10&0.830\\\
\text{A}&71&6&12&0.700\\\ \text{A}&69&6&10&-0.660\\\
\text{A}&101&10&12&-0.189\\\ \text{A}&59&5&10&-0.110\\\
\text{A}&9&1&10&-0.104\\\ \text{A}&11&1&12&0.104\\\ \text{A}&61&5&12&0.081\\\
\text{A}&22&2&9&0.078\\\ \text{A}&55&5&6&0.070\\\ \text{A}&93&9&10&-0.067\\\
\hline\cr\text{B}&11&1&12&-0.269\\\ \text{B}&101&10&12&-0.148\\\
\text{B}&59&5&10&-0.143\\\ \text{B}&23&2&10&0.126\\\
\text{B}&111&12&13&-0.098\\\ \text{B}&80&7&12&0.081\\\
\text{B}&29&2&16&-0.081\\\ \text{B}&102&10&13&0.075\\\
\text{B}&13&1&14&0.074\\\ \text{B}&65&5&16&0.067\\\ \text{B}&12&1&13&0.063\\\
\text{B}&71&6&12&0.061\\\ \hline\cr\text{C}&23&2&10&0.272\\\
\text{C}&9&1&10&-0.257\\\ \text{C}&25&2&12&-0.153\\\
\text{C}&101&10&12&-0.094\\\ \text{C}&11&1&12&0.086\\\
\text{C}&93&9&10&-0.077\\\ \text{C}&26&2&13&-0.066\\\
\text{C}&12&1&13&0.066\\\ \text{C}&61&5&12&0.056\\\
\text{C}&111&12&13&-0.051\\\ \text{C}&59&5&10&-0.045\\\
\text{C}&104&10&15&0.044\\\
\hline\cr\end{array}~{}~{}~{}~{}\begin{array}[]{|c|c|cc|r|}\hline\cr\addstackgap[.5]{\vphantom{0}}\text{Treatment}&x&m&n&\vartheta^{T}{}{}{}\\\
\hline\cr\text{A}&25&2&12&-1.000\\\ \text{A}&23&2&10&0.995\\\
\text{A}&71&6&12&0.928\\\ \text{A}&69&6&10&-0.923\\\ \text{A}&61&5&12&0.068\\\
\text{A}&59&5&10&-0.068\\\ \text{A}&9&1&10&-0.021\\\ \text{A}&11&1&12&0.021\\\
\text{A}&1&1&2&0.011\\\ \text{A}&5&1&6&-0.011\\\ \text{A}&86&8&10&0.008\\\
\text{A}&88&8&12&-0.008\\\ \hline\cr\text{B}&23&2&10&0.084\\\
\text{B}&9&1&10&-0.058\\\ \text{B}&61&5&12&0.056\\\ \text{B}&11&1&12&-0.046\\\
\text{B}&10&1&11&0.044\\\ \text{B}&29&2&16&-0.037\\\
\text{B}&59&5&10&-0.030\\\ \text{B}&24&2&11&-0.028\\\
\text{B}&15&1&16&0.025\\\ \text{B}&101&10&12&-0.024\\\
\text{B}&106&11&12&0.018\\\ \text{B}&8&1&9&0.017\\\
\hline\cr\text{C}&61&5&12&0.256\\\ \text{C}&11&1&12&-0.168\\\
\text{C}&23&2&10&0.150\\\ \text{C}&59&5&10&-0.117\\\
\text{C}&25&2&12&-0.107\\\ \text{C}&58&5&9&-0.084\\\ \text{C}&13&1&14&0.067\\\
\text{C}&101&10&12&-0.060\\\ \text{C}&29&2&16&-0.051\\\
\text{C}&1&1&2&-0.049\\\ \text{C}&22&2&9&0.048\\\ \text{C}&8&1&9&0.042\\\
\hline\cr\end{array}$

Table A4: Experimental cycle $\vartheta^{E}$ and the theoretical cycle
$\vartheta^{T}$. The top 12 maximum strengths in the eigencycles of the
spectrum from treatments (A,B,C). $x$ is the $x$-axis value in the eigencycle
spectrum, and ($m,n$) are the strategy IDs in the unified dimension of the
game, which are defined in SI—Cycle.

####  A.3.1 Verify the statics

In the main text section 2.2 [TS2], we state that the cycle’s existence in
treatment-(A, B, C) can be predicted as (Yes, No, Yes), respectively. This
prediction is based on the IEDS approach in the static equilibrium theory.

From Table A4, we obtain the following results.

  * •

In treatment A, the closed loops are

$2\rightarrow 10\rightarrow 6\rightarrow 12\rightarrow 2$ (A4)

  * •

In treatment C, the closed loops are

$2\rightarrow 10\rightarrow 5\rightarrow 12\rightarrow 2$ (A5) $2\rightarrow
10\rightarrow 1\rightarrow 12\rightarrow 2$ (A6)

the IEDS predictions for the cycle are supported. We observed a weak cycle in
treatment B, which is not regarded as evidence against IDES because of its
weakness, referring to equation A7.

####  A.3.2 Verification of the dynamics theory

The predictions given in the main text section 2.3 [TD2] are verified here. In
treatments A and C, the predictions match the experimental data:

  * •

In the cycle loop, there exist cycles for equation A4 in treatment A, and for
equation A5 and A6 in treatment C;

  * •

For the strength of the cycle loop, referring to Table A4, comparing the
average of the strength of $\vartheta^{E}_{\text{Treatment}}$ in the loops,
the result is

$\vartheta^{E}_{\text{A}}>\vartheta^{E}_{\text{C}}>\vartheta^{E}_{\text{B}.}$
(A7)

For treatment B, the dynamics predicts little cycle. Considering equation A7,
we do not regard the loop of $\vartheta_{B}^{E}$ as significant evidence
against the prediction. As result, the consistency between the dynamics and
the experiment is supported.

###  A.4 Collapse

This section includes the measurement definition for collapse, as well as the
experimental data. For more details on this subsection, see SI – Collapse.

####  A.4.1 Measurement definition

##### Conceptual Example

Figure A1 offers an example of the pulse signal, accumulated curve and the
crossover points. It is derived from the theoretical results of the strategy
evolution over time of the X population in treatment A.

Figure A1: Conceptual figure for the pulse signal and the crossover point
during a space collapse process. (a) Illustrates a pulse of strategy 8 in the
strategy proportion time series $\rho(x_{8},t)$, which dominates when
$t\in[0,30]$ and is eliminated (at $t\approx 130$). (b) Illustrates a
crossover point $\chi([x_{2},x_{8}],70)$ (see the red arrows) in the
accumulated curves $\varrho(x_{2},t)$ and $\varrho(x_{8},t)$.

##### Accumulated curve ($\varrho$)

We denote the accumulated $s_{k}$ strategy used in the time interval
$[t_{0},t_{1}]$, which is: $\varrho(x_{k},t)$, which is

$\varrho(s_{k},t_{0},t_{1})=\int_{t_{0}}^{t_{1}}\rho(s_{k},t)dt,$ (A8)

where $\rho(x_{k},t)$ is the proportion of the observed $s_{k}$ at time $t$.
In this study, we ake use of discrete time version and set $dt=1$. Without
additional specification, in this study we set $t_{0}=0$, and the discrete
version of the Equation A8 can be rewrites as

$\varrho(s_{k},t)=\sum_{t^{\prime}=0}^{t}\rho(s_{k},t^{\prime}),$ (A9)

which is the accumulated curve in this study.

##### Pulse signal

A pulse signal, or pulse, can be defined following these steps:

  1. 1.

A sample of surplus in time series, denoted as
$\rho^{+}\big{(}[s_{i},s_{j}],t\big{)}$, is the proportion difference between
two strategies $[s_{i},s_{j}]$ at $t$,

$\rho^{+}\big{(}[s_{i},s_{j}],t\big{)}=\rho(s_{j},t)-\rho(s_{i},t).$ (A10)

  2. 2.

A surplus sample set $\rho^{+}_{*}$ refers to the surplus samples from a given
successive time interval $[t_{0},t_{1}]$ (time block),

$\rho^{+}_{*}\big{(}[s_{i},s_{j}],[t_{0},t_{1}]\big{)}=\\{\rho^{+}\big{(}[s_{i},s_{j}],t_{k}\big{)},~{}\forall
t_{k}\in[t_{0}\\!\\!+\\!\\!1,t_{0}\\!\\!+\\!\\!2,...,t_{1}]\\}.$ (A11)

  3. 3.

Sample size N. For statics analysis, the sample set size is

$\textbf{N}=(t_{1}-t_{0})\times\text{Number of experiment session}$ (A12)

For example, assuming that a time block is set as 10 rounds, from the
81${}^{s}t$ round to 90${}^{t}h$ round in a 1000 round repeated game session,
the element of the set is 10. We have 12 sessions for a given treatment, so
the sample size N is 120.

  4. 4.

Pulse ($\psi$) is the total surplus of $s_{j}$ over $s_{i}$ in a time block
$[t_{0},t_{1}]$:

$\psi\big{(}[s_{i},s_{j}],[t_{0},t_{1}]\big{)}=\sum_{t_{k}=t_{0}+1}^{t_{1}}\rho^{+}\big{(}[s_{i},s_{j}],t_{k}\big{)};$
(A13)

For example, in the top line in Table A5, the $\psi=17$. The explanation for
this is that:

     * •

$X_{8}\in\textbf{D}^{-}$ is a dominated, and $X_{2}\in\textbf{D}^{+}$ is a
domination.

     * •

There are 120 samples from the X population of treatment A and in the
$t\in[11-20]$ rounds of the $\text{ses}\in[1,12]$ sessions.

     * •

The surplus observed is the sum over the sample set shown in equation A11.
That is

$\sum_{\text{ses}=1}^{12}\sum_{t=11}^{20}\big{(}\rho(X_{8},t,\text{ses})-\rho(X_{2},t,\text{ses})\big{)}=17$
(A14)

  5. 5.

The statistical significance of a pulse $\psi$ is reported by the ttest over
the surplus sample set $\rho^{+}_{*}$ shown in Equation A11. On the sample
size N for ttest, see the example following the Equation A12.

  6. 6.

Strong significant When $p<0.010$, the statistic result is reported to be ‘in
strongly significant’. When $0.010\leq p<0.050$, the statistic result is
reported as ‘in significant‘ but not as ‘in strongly significant‘.

##### Crossover point: $\chi(x_{i},x_{j},\tau)$

The definition of the crossover point depends on the accumulated curve
($\varrho$) shown in Equation A9. Assuming that:

$\varrho(s_{i},t)-\varrho(s_{j},t)=0$ (A15)

the crossover point is defined as $\chi(s_{i},s_{j},\tau)$ in which

  * •

$\tau$ is the solution of the $t$

  * •

$\chi(s_{i},s_{j},\tau)$ := $\varrho(s_{i},t)=\varrho(s_{j},t)$

##### The two classes of crossover points

  * •

$\chi_{+}$ — when the paired comparison is between two ($\textbf{D}^{+}$), for
example, the blue arrows in Figure 5(c) \- 5(h) are the crossover points
$\chi_{+}$.

  * •

$\chi_{-}$ — when the paired comparison is between the ($\textbf{D}^{-}$) and
($\textbf{D}^{+}$), for example, the red arrows in Figure 5(c) \- 5(h) are the
crossover points $\chi_{-}$.

##### The relation between the pulse and the crossover time.

Based on the results reported in [E32.6] in section 3.3, we hypothesize that,
if a crossover $\chi_{-}([x_{i},x_{j}],\tau)$ ( $s_{j}\in\textbf{D}^{-}$ and
$s_{i}\in\textbf{D}^{+}$) has larger $\tau$, $s_{j}$ is more strongly expected
to provide the pulse signal.

Assuming that $\rho(s_{i})$ and $\rho(s_{j})$ are equal at $t=0$, if a long
run dominated ($s_{j}\in\textbf{D}^{-}$) has a pulse signal referring to a
long run domination ($s_{i}\in\textbf{D}^{+}$), that is

$\rho(x_{j},t)>\rho(x_{i},t)~{}~{}~{}~{}~{}~{}\forall t\in[0,T],$ (A16)

As a mathematical result, there must be a crossover point in its accumulated
curve $\varrho(x_{j},\tau)$ referring to $\varrho(x_{i},\tau)$ in which
$\tau>T$. This is one explanation for the hypothesis.

For example, in experimental pulse $\psi^{E}$ (Table 2), all the strongest
significant pulses belong to the largest $\tau$ in $\chi^{E}$ (Table A7). The
result reported in [E32.6] refers to this mathematical property.

####  A.4.2 Data

Here, we present the data that support the results shown in section 3.3 in
main text.

  1. 1.

1\. The experimental pulse signal observed is statistically significant
(p¡0.05): see Table A5

  2. 2.

The theoretical pulse signal in statistical significant; Listed are the top 10
samples ordered by the time block surplus values of treatments (A,B,C): see
Table A6. Note: the time series from the dynamics model are smooth, meaning
that more significant pulse signals can be obtained from this model, when
compared with those from the highly stochastic human subject experimental time
series..

  3. 3.

Experimental crossover point: see Table A7

  4. 4.

Theoretical crossover point: see Table A8

Treat- | Domin- | Domin- | time | paired- | Surplus | Sample
---|---|---|---|---|---|---
ments | ated | ation | block | ttest($p$) |  | size
| $s_{j}$ | $s_{i}$ | $[t_{0},t_{1}]$ |  | $\psi^{E}$ | N
A | $X_{8}$ | $X_{2}$ | 11-20 | 0.007 | 17 | 120
A | $X_{8}$ | $X_{6}$ | 11-20 | 0.002 | 19 | 120
A | $X_{8}$ | $X_{2}$ | 21-30 | 0.011 | 15 | 120
A | $X_{4}$ | $X_{6}$ | 41-50 | 0.027 | 13 | 120
A | $X_{1}$ | $X_{6}$ | 81-90 | 0.023 | 12 | 120
A | $X_{4}$ | $X_{6}$ | 81-90 | 0.007 | 15 | 120
C | $X_{8}$ | $X_{5}$ | 1-10 | 0.027 | 11 | 120
C | $X_{7}$ | $X_{5}$ | 11-20 | 0.049 | 10 | 120
C | $X_{8}$ | $X_{5}$ | 11-20 | 0.049 | 10 | 120
Table A5: Statistically significant experimental pulses ($\psi^{E}$) ($p$¡0.05). Treat- | Domin- | Domin- | time | paired- | Surplus | Sample
---|---|---|---|---|---|---
ments | ated | ation | block | ttest($p$) |  | size
| $s_{j}$ | $s_{i}$ | $[t_{0},t_{1}]$ |  | $\psi^{T}$ | N
A | $X_{8}$ | $X_{6}$ | 21-31 | 0.000 | 35.04 | 120
A | $X_{8}$ | $X_{2}$ | 21-31 | 0.000 | 34.75 | 120
A | $X_{8}$ | $X_{6}$ | 31-41 | 0.000 | 29.97 | 120
A | $X_{8}$ | $X_{2}$ | 11-21 | 0.000 | 29.20 | 120
A | $X_{8}$ | $X_{6}$ | 11-21 | 0.000 | 29.08 | 120
A | $X_{8}$ | $X_{6}$ | 41-51 | 0.000 | 24.47 | 120
A | $X_{4}$ | $X_{6}$ | 41-51 | 0.000 | 23.87 | 120
A | $X_{8}$ | $X_{2}$ | 31-41 | 0.000 | 23.84 | 120
A | $X_{4}$ | $X_{6}$ | 51-61 | 0.000 | 22.01 | 120
A | $X_{4}$ | $X_{6}$ | 31-41 | 0.000 | 20.47 | 120
A | $X_{8}$ | $X_{6}$ | 51-61 | 0.000 | 19.98 | 120
A | $X_{4}$ | $X_{6}$ | 61-71 | 0.000 | 18.84 | 120
B | $X_{7}$ | $X_{1}$ | 21-31 | 0.000 | 18.05 | 120
C | $X_{8}$ | $X_{5}$ | 11-21 | 0.000 | 21.64 | 120
C | $X_{8}$ | $X_{1}$ | 11-21 | 0.000 | 21.64 | 120
C | $X_{8}$ | $X_{2}$ | 11-21 | 0.000 | 18.75 | 120
C | $X_{8}$ | $X_{5}$ | 21-31 | 0.000 | 18.39 | 120
C | $X_{8}$ | $X_{1}$ | 21-31 | 0.000 | 18.39 | 120
Table A6:  Theoretical pulse ($\psi^{T}$). The most strongly expected top 18
samples (twice the $\psi^{E}$ is significant) ordered by the $\psi^{T}$ values
of all the treatments (A,B,C). Note: the time series from the dynamics model
are smooth, meaning that more significant pulse signals can be obtained from
this model, when compared with those from the highly stochastic experimental
time series.

Treat- | Domin- | Domin- | Crossover | Arrow
---|---|---|---|---
[.5]0 ment | ation | ated | Time ($\tau$) | Color | ID | Fig ID
$A$ | $X_{2}$ | $X_{1}$ | 47 | - | - | Fig 5(c)
$A$ | $X_{2}$ | $X_{6}$ | 47 | - | - |
$A$ | $X_{2}$ | $X_{7}$ | 52 | - | - |
$A$ | $X_{2}$ | $X_{3}$ | 55 | - | - |
$A$ | $X_{2}$ | $X_{4}$ | 58 | - | - |
$A$ | $X_{2}$ | $X_{5}$ | 58 | - | - |
$A$ | $X_{2}$ | $X_{8}$ | 67 | - | - |
$A$ | $X_{6}$ | $X_{1}$ | 44 | - | - |
$A$ | $X_{6}$ | $X_{3}$ | 74 | - | - |
$A$ | $X_{6}$ | $X_{3}$ | 79 | - | - |
$A$ | $X_{6}$ | $X_{7}$ | 92 | red | 4 |
$A$ | $X_{6}$ | $X_{1}$ | 106 | red | 3 |
$A$ | $X_{6}$ | $X_{5}$ | 139 | - | - |
$A$ | $X_{6}$ | $X_{4}$ | 169 | red | 2 |
$A$ | $X_{6}$ | $X_{8}$ | 186 | red | 1 |
$A$ | $Y_{2}$ | $Y_{4}$ | 297 | blue | 1 | Fig 5(f)
$C$ | $X_{2}$ | $X_{8}$ | 45 | - | - | Fig 5(e)
$C$ | $X_{5}$ | $X_{6}$ | 49 | - | - |
$C$ | $X_{5}$ | $X_{6}$ | 54 | - | - |
$C$ | $X_{5}$ | $X_{3}$ | 56 | - | - |
$C$ | $X_{5}$ | $X_{4}$ | 100 | red | 3 |
$C$ | $X_{5}$ | $X_{8}$ | 138 | red | 1 |
$C$ | $Y_{2}$ | $Y_{4}$ | 74 | blue | 1 | Fig 5(h)
Table A7: Experimental crossover point $\chi^{E}$ at $\tau^{E}>40$. The arrow columns refer to the arrows in Figure 5(c)-5(h), based on the theoretical arrows in Table A8. For the definition of $\tau$, see Equation A15. Treat- | Domin- | Domin- | Crossover | Arrow
---|---|---|---|---
[.5]0 ment | ation | ated | Time ($\tau$) | Color | ID | Fig ID
$A$ | $X_{2}$ | $X_{4}$ | 62 | - | - | Fig 4(c)
$A$ | $X_{2}$ | $X_{8}$ | 86 | - | - |
$A$ | $X_{6}$ | $X_{5}$ | 65 | - | - |
$A$ | $X_{6}$ | $X_{7}$ | 116 | red | 4 |
$A$ | $X_{6}$ | $X_{1}$ | 117 | red | 3 |
$A$ | $X_{6}$ | $X_{4}$ | 177 | red | 2 |
$A$ | $X_{6}$ | $X_{8}$ | 209 | red | 1 |
$A$ | $Y_{2}$ | $Y_{1}$ | 71 | - | - | Fig 4(f)
$A$ | $Y_{2}$ | $Y_{5}$ | 74 | - | - |
$A$ | $Y_{2}$ | $Y_{4}$ | 277 | blue | 1 |
$B$ | $X_{1}$ | $X_{3}$ | 42 | - | - |
$B$ | $X_{1}$ | $X_{6}$ | 56 | - | - |
$B$ | $X_{1}$ | $X_{8}$ | 69 | - | - |
$B$ | $X_{1}$ | $X_{7}$ | 72 | - | - |
$B$ | $X_{1}$ | $X_{5}$ | 75 | - | - |
$B$ | $X_{1}$ | $X_{2}$ | 84 | - | - |
$C$ | $X_{1}$ | $X_{7}$ | 52 | - | - | Fig 4(e)
$C$ | $X_{1}$ | $X_{6}$ | 66 | - | - |
$C$ | $X_{1}$ | $X_{4}$ | 77 | - | - |
$C$ | $X_{1}$ | $X_{8}$ | 82 | - | - |
$C$ | $X_{1}$ | $X_{2}$ | 126 | blue | 2 |
$C$ | $X_{5}$ | $X_{7}$ | 61 | - | - |
$C$ | $X_{5}$ | $X_{6}$ | 82 | - | - |
$C$ | $X_{5}$ | $X_{4}$ | 112 | red | 3 |
$C$ | $X_{5}$ | $X_{8}$ | 144 | red | 1 |
$C$ | $Y_{2}$ | $Y_{4}$ | 89 | blue | 1 | Fig 4(h)
Table A8: Theoretical crossover point $\chi^{T}$ at $\tau^{T}>40$. The arrow
columns relate to the arrows ($\tau>89$) in Figure 4(c)-4(h). For the
definition of $\tau$, see Equation A15.

###  A.5 The game dynamics paradigm and its workflow

Similarly to the paradigms found in natural science, the game dynamics
paradigm is also a distinct set of concepts, including theories, research
methods, postulates, and standards for what constitutes legitimate
contributions to a field. The reality and accuracy, as well as the
completeness and consistency, of this distinct set of concepts must be proven.
Key to developing a paradigm is a closed-loop of workflow. The workflow
includes the following steps:

  1. 1.

Describe a game (G) and its playing protocol (P);

  2. 2.

Take a proper dynamics equation system T (e.g., replicator dynamics or logit
dynamics) and specify the parameters of T by (G,P). Then, derive the
theoretical observation OT(G,P) by solving T(G,P), e.g., the reset point, the
eigen system, or by time series analysis among others technologies;

  3. 3.

Conduct experiments or collect empirical data to measure the observation
OE(G,P)

  4. 4.

Evaluate the OT(G,P) by empirical observation OE(G,P).

  5. 5.

Iterate Step 1 to Step 4 and feedback to step 1 as a loop, by trial and error,
to find the best fit between the theory and the empirical system in general.

This iterated closed-loop workflow is widely applied in various scientific
fields, and can constantly improve the establishment of a paradigm. In
addition to the reality and the accuracy, the paradigm is endorsed by the
completeness and the consistency between:

$O^{T}(P,G)~{}~{}\text{vs}~{}~{}O^{E}(P,G)$

endorse the paradigm. Social science follows this workflow, as does natural
science[10, 4, 23].

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# Universality class of alternative phase space and Van der Waals criticality

Amin Dehyadegaria<EMAIL_ADDRESS>Bibhas Ranjan Majhic
<EMAIL_ADDRESS>Ahmad Sheykhia,b<EMAIL_ADDRESS>Afshin
Montakhaba<EMAIL_ADDRESS>a Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
b Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran
c Department of Physics, Indian Institute of Technology Guwahati, Guwahati
781039, Assam, India

###### Abstract

A new perspective toward thermodynamic phase space of Reisser-Nordstrom (RN)
black holes in an anti-de-Sitter (AdS) spaces was recently proposed AAA ,
where the square of the electric charge $(Q^{2})$ of black hole was regarded
as a thermodynamic variable and the cosmological constant (pressure) as a
fixed quantity. In this paper, we address the universality class and critical
properties of any AdS black hole in this alternative phase space. We disclose
the critical behavior of AdS black hole in the alternative phase space in
which a continuous phase transition happens and in a very general framework,
independent of the spacetime metric. Based on the expansion of the equation of
state and Landau thermodynamic potential in the neighborhood of a critical
point in the alternative phase space, we confirm that the set of values for
critical exponents for generic black hole is analogous to the Van der Waals
fluid system. Finally, we reveal that the scalar curvature in geometry
thermodynamic diverges at the critical point of black hole. Our study shows
that the approach here is powerful enough to investigate the critical behavior
of any black holes and further supports the viability of the alternative
viewpoint toward phase space of black holes suggested in AAA .

##  I Introduction

Thermodynamic phase transitions are one of the most intriguing phenomena in
black hole physics which can provide insight into underlying structure of
spacetime geometry. In particular, phase transition of anti-de-Sitter (AdS)
black holes has received much attention since the discovery of the
correspondence between gravity in an AdS spacetime and the conformal field
theory (CFT) living on its boundary. The pioneering work in this regard was
performed by Hawking and Page HP , who demonstrated the existence of a first-
order phase transition in the Schwarzschild AdS black hole. According to the
AdS/CFT duality, this phase transition can be interpreted as a confinement-
deconfinement transition in the quark gluon plasma Witten . Later, the analogy
between the small-large black hole phase transition and the liquid-gas Van der
Waals phase transition was reported in VDW1 , where RN black hole was
considered in asymptotic AdS spacetime. Later on, it was revealed that this
similarity happens in the extended phase space of the RN-AdS black hole in
which the cosmological constant ($\Lambda$) is regarded as a thermodynamic
variable corresponding to the thermodynamic pressure with a black hole’s
volume as a conjugate quantity Dolan ; PV . In the extended phase space, the
cosmological constant as a dynamical quantity can take on arbitrary values in
the first law of black hole thermodynamics where the mass of AdS black hole is
interpreted as the enthalpy enthalpy . In recent years, various studies on
black holes phase transition in an extended phase space have been carried out
such as zeroth-order phase transition NAAA , reentrant phase transition BIRPT
as well as superfluid-like phase transition superfluidBH  as well as study of
triple points MRPT . For more details, we refer to BHCh  and references
therein.

Criticality is an interesting topic in phase transition context because
thermodynamic quantities of the system show non-analytic behavior as one
approaches the critical point, where a continuous phase transition occurs.
This non-analytic behavior is expressed in the terms of power law functions
which are governed by the critical exponents. The set of critical exponents
define the universality class of a system, which are thought to obey the same
symmetry principles. In case of charged AdS black hole, a continuous (second
order) phase transition occurs between small-large black hole in an extended
phase space PV . Furthermore, the critical exponents associated with this
transition are the same as those in the Van der Waals liquid-gas transition,
i.e., both systems belong to the same universality class PV . Critical
behavior and universality class of AdS black holes in the extended phase space
have been investigated in various spacetimes variousBH1 ; variousBH2 .
Recently, authors of Ref. Majhi1  have shown that if there is a critical point
in an extended phase space of general AdS black hole, the small-large black
hole transition is in the Van der Waals universality class. Another
interesting approach towards critical phenomena of AdS black hole is through
variation of the electric charge of black hole in a fixed AdS background
geometry, i.e. keeping the cosmological constant (pressure) as a fixed
parameter. In this perspective, the charge of black hole is treated as a
natural thermodynamic variable that can take on arbitrary values in the
thermodynamic process. In this view, the thermodynamic behavior of AdS black
hole is analyzed in a thermodynamics phase space. The critical point and
associated critical exponents in a phase space of black hole have been studied
in a general way Majhi2 . It was found that the values of critical exponents
differ from those of the Van der Waals phase transition Majhi2 . All these
phase transitions are also discussed from the thermogemetrical point of view
Majhi3  where the phase transition is identified as the divergence of the
Ricci scalar of the thermodynamic geometry. In additional, an interesting
phenomenon of black hole reentrant phase transition has also been investigated
in four-dimensional Born-Infeld AdS black in which the charge of system can
vary and the cosmological constant is fixed AA .

On the other hand, an alternative view of phase space has been proposed more
recently where various thermodynamic quantities find more natural and physical
meaning AAA . In this view, the _square_ of the electric charge of black hole
is considered as a thermodynamic variable and $\Psi=1/2r_{+}$ is subsequently
considered as its conjugate, where $r_{+}$ is the horizon radius. It was
indeed shown that in this alternative phase space, phase transition and
critical behavior of RN-AdS black hole, in four dimensions, occur in the
$Q^{2}$-$\Psi$ plane, where relevant response function clearly signifies
stable and unstable region AAA . Remarkably, in this viewpoint, the small-
large black hole phase transition is quite similar to the Van der Waals
liquid-gas system and belongs to the same universality class in contrast with
previous study of Majhi2 . Additionally, it would be interesting to study the
universality class and critical properties for any AdS black hole in an
alternative phase space where the cosmological constant ($\Lambda$) is taken
to be constant. Hence, in this paper, we provide a general framework, i.e. a
metric independent way, for investigating the critical behavior of AdS black
hole in the above-mentioned alternative phase space approach in which a
continuous phase transition happens. In this analysis, the exsistance of the
phase transition with respect to the alternative phase space variables is
assumed to be there. Based on the expansion of the equation of state and
Landau thermodynamic potential in the neighborhood of a critical point in the
alternative phase space, we find that a set value of critical exponents for
generic black hole is analogous to usual Van der Waals system. Also, we show
that the scalar curvature in thermogeometric picture diverges at the critical
point of black hole in an alternative phase space.

The present Article is structured as follows: In the next section, we obtain
the critical exponents by using the equation of state of black hole in a
general scheme. In section III, by considering the Landau thermodynamic
potential, we study universality properties of black hole at phase transition.
In section IV, we investigate thermodynamic geometry of the system at critical
point in alternative phase space. The last section is devoted to concluding
remarks.

##  II Equation of state: General approach

Here, we intend to prove that the critical exponents of a continuous (second-
order) phase transition are independent of the metric function (black hole) in
an alternative phase space where $Q^{2}$ is treated as a thermodynamic
quantity and its conjugate is $\Psi$ as proposed in AAA . For this purpose,
the first law of black hole thermodynamics, for constant pressure, is written
as

$dM=TdS+\Psi dq,$ (1)

where $q=Q^{2}$. Here $M$, $S$ and $T$ are the total mass, entropy and Hawking
temperature of black hole, respectively. In general, the entropy only depends
on the event horizon, $r_{+}$, i.e. $S=S(r_{+})$. The Gibbs free energy is a
thermodynamic potential that can be calculated by Legendre transform of the
above equation

$dG=-SdT+\Psi dq,$ (2)

where $G=G(T,q)$. Thermodynamics of black hole may be described by equation of
state $q=q(\Psi,T)$ where $q$ depends on $\Psi$ and $T$. As we know, the
critical point in $q-\Psi$ plane is characterized by AAA

$\frac{\partial
q}{\partial\Psi}\Big{|}_{T_{c}}=0,\quad\quad\frac{\partial^{2}q}{\partial\Psi^{2}}\Big{|}_{T_{c}}=0,$
(3)

where the subscript $c$ refers to the critical point and critical quantities
are $T_{c}$, $\Psi_{c}$ and $q_{c}$. The behavior of thermodynamic functions
near the critical point is identified by the critical exponents which are
defined for a Van der Waals system as Callen

$\displaystyle C\sim\left|T-T_{c}\right|^{-\alpha},$
$\displaystyle\left|v_{l}-v_{g}\right|\sim\left|T-T_{c}\right|^{\beta},$
$\displaystyle P-P_{c}\sim\left|v_{l}-v_{g}\right|^{\delta},$
$\displaystyle\chi_{{}_{T}}$ $\displaystyle=$
$\displaystyle-\frac{1}{v}\frac{\partial v}{\partial
P}\Big{|}_{T}\sim\left|T-T_{c}\right|^{-\gamma}.$

The first equation defines the exponent for thermal response function $C$
(heat capacity), the second equation characterizes the non-analytic behavior
of order parameter at the critical point, the third defines the critical
isotherm, and the last equation defines the singularity in (mechanical)
response function, $\chi_{{}_{T}}$ (isothermal compressibility). Here, $v$ and
$P$ are specific volume and pressure, respectively, which define the
thermodynamic phase space. Now, in order to find the critical exponents for a
black hole, we expand $q$ around the critical point

$\displaystyle q(\Psi,T)$ $\displaystyle=$ $\displaystyle
a_{00}+a_{01}\left(T-T_{c}\right)+a_{02}\left(T-T_{c}\right)^{2}$ (4)
$\displaystyle+a_{11}\left(T-T_{c}\right)\left(\Psi-\Psi_{c}\right)+a_{12}\left(T-T_{c}\right)^{2}$
$\displaystyle\times\left(\Psi-\Psi_{c}\right)+a_{30}\left(\Psi-\Psi_{c}\right)^{3}+\ldots,$

where we have used the convention
$a_{ij}\equiv\left(i!j!\right)^{-1}\partial^{i+j}q/\partial\Psi^{i}\partial
T^{j}\Big{|}_{\Psi_{c},T_{c}}$, and use has been made of Eq.(3). For
simplicity, we rewrite thermodynamic variables in dimensionless form as

$\displaystyle q$ $\displaystyle=$ $\displaystyle q_{c}(1+\varrho),$
$\displaystyle T$ $\displaystyle=$ $\displaystyle T_{c}(1+t),$
$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\Psi_{c}(1+\psi),$ (5)

where $\varrho$, $t$ and $\psi$ are deviation from critical point.
Substituting above expressions into Eq.(4), we have

$\varrho=a_{01}^{\prime}t+a_{02}^{\prime}t^{2}+a_{11}^{\prime}t\psi+a_{12}^{\prime}t^{2}\psi+a_{30}^{\prime}\psi^{3},$
(6)

in which the prime quantities are equal to the rescaled coefficients in
Eq.(4). Since phase transition occurs between small and large black hole under
constant charge, one writes

$\displaystyle\varrho$ $\displaystyle=$ $\displaystyle
a_{01}^{\prime}t+a_{02}^{\prime}t^{2}+a_{11}^{\prime}t\psi_{s}+a_{12}^{\prime}t^{2}\psi_{s}+a_{30}^{\prime}\psi_{s}^{3}$
(7) $\displaystyle=$ $\displaystyle
a_{01}^{\prime}t+a_{02}^{\prime}t^{2}+a_{11}^{\prime}t\psi_{l}+a_{12}^{\prime}t^{2}\psi_{l}+a_{30}^{\prime}\psi_{l}^{3},$

here, $\psi_{s}$ ($\psi_{l}$) stands for $\psi$ at small (large) horizon.
Also, applying the Maxwell’s equal area law, $\oint\Psi dq=0$, which is
obtained from Gibbs free energy AAA ; Callen , and using Eq.(6) one can arrive
at

$\int_{\psi_{l}}^{\psi_{s}}\psi\Big{(}a_{11}^{\prime}t+a_{12}^{\prime}t^{2}+3a_{30}^{\prime}\psi^{2}\Big{)}d\psi=0.$
(8)

It is a matter of calculations to show that Eqs.(7) and (8) have the following
solution

$\psi_{l}=-\psi_{s}=\sqrt{-\frac{a_{11}^{\prime}t+a_{12}^{\prime}t^{2}}{a_{30}}}.$
(9)

Therefore, the order parameter near the critical point behaves as

$\left|\psi_{s}-\psi_{l}\right|=2\psi_{s}\sim
t^{1/2}\Longrightarrow\beta=1/2.$ (10)

In the vicinity of critical point, the shape of the critical isotherm, $t=0$,
is obtained by

$\varrho=a_{30}^{\prime}\psi^{3}\Longrightarrow\delta=3.$ (11)

According to equation (6), the behavior of response function,
$\chi_{{}_{T}}=\partial\Psi/\partial q\Big{|}_{T}$, is given by

$\chi_{{}_{T}}\sim\frac{1}{a_{11}^{\prime}t}\Longrightarrow\gamma=1.$ (12)

To denote the specific heat at fixed $\Psi$ close to the critical point, we
perform the expansion of entropy around the critical point

$\displaystyle S(\Psi,T)$ $\displaystyle=$ $\displaystyle
s_{00}+s_{01}(T-T_{c})+s_{10}\left(\Psi-\Psi_{c}\right)$ (13)
$\displaystyle+s_{11}\left(T-T_{c}\right)\left(\Psi-\Psi_{c}\right)+\ldots.$

where
$s_{ij}\equiv\left(i!j!\right)^{-1}\partial^{i+j}S/\partial\Psi^{i}\partial
T^{j}\Big{|}_{\Psi_{c},T_{c}}$. With $S$ at hand, we can extract the critical
exponent $\alpha$, as follows

$C_{\Psi}=T\frac{\partial S}{\partial
T}\Big{|}_{\Psi}=T_{c}s_{01}\Longrightarrow\alpha=0.$

It is remarkable to note that $S$ is only a function of $\Psi$ for Maxwell
electrodynamics, i.e. $s_{ij}=0$ for $j\neq 0$ AAA . In this way we calculate
the critical exponents of a black hole in a general framework without
specifying the form of the metric, which coincide with those obtained for Van
der Waals fluid system.

##  III Phenomenological aspect: Landau function

Let us define the thermodynamic potential in this case as

$K=M-TS-\Psi q~{}.$ (14)

Then use of first law Eq.(1) yields

$dK=-SdT-qd\Psi~{}.$ (15)

The above states that $K$ is function of both $T$ and $\Psi$ and

$S=-\Big{(}\frac{\partial K}{\partial T}\Big{)}_{\Psi};\,\,\,\
q=-\Big{(}\frac{\partial K}{\partial\Psi}\Big{)}_{T}~{}.$ (16)

Consequently, the conditions Eq.(3) at the critical point take the following
forms:

$\frac{\partial^{2}K}{\partial\Psi^{2}}\Big{|}_{T_{c}}=0,\quad\quad\frac{\partial^{3}K}{\partial\Psi^{3}}\Big{|}_{T_{c}}=0.$
(17)

Now since $K=K(T,\Psi)$, Taylor expansion of it near the critical point is
given by

$\displaystyle
K(T,\Psi)=b_{00}+b_{10}(T-T_{c})+b_{11}(T-T_{c})(\Psi-\Psi_{c})$
$\displaystyle+b_{01}(\Psi-\Psi_{c})+b_{20}(T-T_{c})^{2}+b_{21}(T-T_{c})^{2}(\Psi-\Psi_{c})$
$\displaystyle+b_{22}(T-T_{c})^{2}(\Psi-\Psi_{c})^{2}+b_{12}(T-T_{c})(\Psi-\Psi_{c})^{2}$
$\displaystyle+b_{13}(T-T_{c})(\Psi-\Psi_{c})^{3}+b_{04}(\Psi-\Psi_{c})^{4}+\dots~{}.$
(18)

In the above we have used
$b_{ij}=\left(i!j!\right)^{-1}(\partial^{i+j}K)/(\partial^{i}T\partial^{j}\Psi)\Big{|}_{\Psi_{c},T_{c}}$and
the condition Eq.(17). Using Eq.(5), we obtain the near critical point $K$ as

$\displaystyle
K(t,\psi)=b_{00}+b_{10}^{\prime}t+b_{11}^{\prime}t\psi+b_{01}^{\prime}\psi+b_{20}^{\prime}t^{2}$
$\displaystyle+b_{12}^{\prime}t\psi^{2}+b_{21}^{\prime}t^{2}\psi+b_{22}^{\prime}t^{2}\psi^{2}+b_{13}^{\prime}t\psi^{3}+b_{04}^{\prime}\psi^{4}~{}.$
(19)

Here the prime quantities are the rescaled coefficients which appear in
Eq.(18). Since their explicit forms are not needed, we do not mention their
values. The same logic will be followed again and again. Using the first and
second relations of Eq.(16), we obtain

$\displaystyle
S=b_{10}^{\prime\prime}+b_{11}^{\prime\prime}\psi+b_{20}^{\prime\prime}t+b_{21}^{\prime\prime}t\psi+b_{12}^{\prime\prime}\psi^{2}$
$\displaystyle+b_{22}^{\prime\prime}t\psi^{2}+b_{13}^{\prime\prime}\psi^{3}~{},$
(20) $\displaystyle
q=a_{01}^{\prime\prime}+a_{11}^{\prime\prime}t+a_{12}^{\prime\prime}t\psi+a_{21}^{\prime\prime}t^{2}+a_{22}^{\prime\prime}t^{2}\psi$
$\displaystyle+a_{13}^{\prime\prime}t\psi^{2}+a_{04}^{\prime\prime}\psi^{3}~{},$
(21)

respectively. In the case of Maxwell electrodynamics, the coefficients of $t$
in Eq.(20) is zero i.e. $b_{2i}^{\prime\prime}=0$ where $i=0,1,3$.

Using the Maxwell’s equal area law, one obtains

$\int_{\psi_{s}}^{\psi_{l}}\psi
dq=\int_{\psi_{s}}^{\psi_{l}}\psi(a_{12}^{\prime\prime}t+a_{22}^{\prime\prime}t^{2}+2a_{13}^{\prime\prime}t\psi+3a_{04}^{\prime\prime}\psi^{2})d\psi=0~{}.$
(22)

Near the critical point, $q$ is assumed to be the same for $\psi_{s}$ and
$\psi_{l}$, i.e.:

$\displaystyle
q=a_{01}^{\prime\prime}+a_{11}^{\prime\prime}t+a_{12}^{\prime\prime}t\psi_{s}+a_{21}^{\prime\prime}t^{2}+a_{22}^{\prime\prime}t^{2}\psi_{s}$
$\displaystyle+a_{13}^{\prime\prime}t\psi_{s}^{2}+a_{04}^{\prime\prime}\psi_{s}^{3},$
$\displaystyle
q=a_{01}^{\prime\prime}+a_{11}^{\prime\prime}t+a_{12}^{\prime\prime}t\psi_{l}+a_{21}^{\prime\prime}t^{2}+a_{22}^{\prime\prime}t^{2}\psi_{l}$
$\displaystyle+a_{13}^{\prime\prime}t\psi_{l}^{2}+a_{04}^{\prime\prime}\psi_{l}^{3}.$
(23)

Equating the above two and using Eq.(22), one obtains

$\psi_{l,s}=\frac{-a_{13}^{\prime\prime}t\pm\sqrt{3t\left[a_{13}^{\prime\prime
2}t-3a_{04}^{\prime\prime}\left(a_{12}^{\prime\prime}+a_{22}^{\prime\prime}t\right)\right]}}{3a_{04}^{\prime\prime}}~{},$
(24)

which leads to

$\left|\psi_{s}-\psi_{l}\right|=\frac{2\sqrt{3t\left[a_{13}^{\prime\prime
2}t-3a_{04}^{\prime\prime}\left(a_{12}^{\prime\prime}+a_{22}^{\prime\prime}t\right)\right]}}{3a_{04}^{\prime\prime}}~{},$
(25)

and to the lowest order in $t$

$\left|\psi_{s}-\psi_{l}\right|\sim(T-T_{c})^{1/2}~{},$ (26)

which yields $\beta=1/2$. Next, Eq.(21) at $T=T_{c}$ reduces to

$q\sim\psi^{3}\sim(\Psi-\Psi_{c})^{3}~{},$ (27)

and so $\delta=3$.

The isothermal compressibility is defined as $\chi_{T}=(\partial\Psi/\partial
q)_{T}$. Therefore differntiating Eq.(21) we find (to the lowest order in
$t$), $\chi_{T}\sim t^{-1}$ which yields $\gamma=1$. Similarly, the specific
heat is given by $C_{\Psi}=T(\partial S/\partial T)_{\Psi}$ and can be
calculated using Eq.(20)

$C_{\Psi}=(1+t)\Big{(}\frac{\partial S}{\partial
t}\Big{)}_{\psi}=(1+t)\left(b_{20}^{\prime\prime}+b_{21}^{\prime\prime}\psi+b_{22}^{\prime\prime}\psi^{2}\right)~{},$
(28)

which to the lowest order is just a constant. Therefore we find the critical
exponent $\alpha=0$.

##  IV Thermogeometric description

The approach toward phase transition by considering thermodynamic geometry
(thermogeometric) was first introduced by Weinhold Weinhold  and Ruppeinner
Ruppeiner1 . In this case the Ricci scalar of the metric diverges at the
critical point or in other words divergence of Ricci scalar is the signature
of the phase transition Ruppeiner2 ; DCP . Both of these approaches are
conformally related to each other by the temperature $T$. For recent studies
on the Ruppeiner thermodynamic geometry, see Mirza1 ; Mirza2 ; 1502.00386 ;
1602.03711 ; 1610.06352  and references therein. This approach has different
properties due to the different thermodynamic potentials, i.e. is not
invariant under Legendre transformation Mrugala ; Quevedo . This distinct
problem was later remedied by Quevedo Quevedo , who presented a Legendre
invariant set of metrics in the phase space. In particular, a Legendre
invariant metric has been investigated for black holes in Alvarez  (see also
Majhi4 ). Below, we follow procedure in Alvarez , to construct the Legendre
invariant metric in an alternative phase space.

The idea is as follows: First construct a thermodynamic phase space
$\mathcal{T}$ on which the coordinates are
$\mathcal{Z}^{A}=(\mathcal{F},\mathcal{X}^{a},\mathcal{P}^{a})$ where
$\mathcal{F}$ is the thermodynamic potential and $\mathcal{P}^{a}$ are the
conjugate variables of the thermodynamic variables $\mathcal{X}^{a}$. In this
representation the fundamental one form is given by

$\theta_{\mathcal{F}}=d\mathcal{F}-\sum_{a,b}\delta_{ab}\mathcal{P}^{a}d\mathcal{X}^{b}~{},$
(29)

where $\delta_{ab}$ is the Kronecker delta. With the present setup, one can
choose the thermodynamic geometry on $\mathcal{T}$ as

$\displaystyle\mathcal{G}=\Big{(}d\mathcal{F}-\sum_{a,b}\delta_{ab}\mathcal{P}^{a}d\mathcal{X}^{b}\Big{)}^{2}$
$\displaystyle+\lambda\Big{(}\sum_{a,b}\xi_{ab}\mathcal{P}^{a}\mathcal{X}^{b}\Big{)}\Big{(}\sum_{c,d}\eta_{cd}d\mathcal{P}^{c}d\mathcal{X}^{b}\Big{)}~{},$
(30)

which is invariant under the following set of Legendre transformations:

$\displaystyle\mathcal{F}_{old}=\mathcal{F}_{new}-\delta_{ab}\mathcal{X}_{new}^{a}\mathcal{P}_{new}^{b}~{},$
$\displaystyle\mathcal{X}_{old}^{a}=-\mathcal{P}_{new}^{a};\,\,\,\,\
\mathcal{P}_{old}^{a}=\mathcal{X}_{new}^{a}~{}.$ (31)

Here $\eta_{ab}=diag(-1,1,1,\dots)$ and $\lambda$ is an arbitrary Legendre
invariant function of $\mathcal{X}^{a}$ while $\xi_{ab}$ is an arbitrary
diagonal constant matrix. The simplest choice is chosen as $\lambda=1$ and
$\xi_{ab}=diag(1,1,\dots)$. Therefore the general form of the simplest
Legendre invariant metric is

$\mathcal{G}=\theta_{\mathcal{F}}^{2}+\Big{(}\sum_{a,b}\xi_{ab}\mathcal{P}^{a}\mathcal{X}^{b}\Big{)}\Big{(}\sum_{c,d}\eta_{cd}d\mathcal{P}^{c}d\mathcal{X}^{b}\Big{)}~{}.$
(32)

We shall work with this particular form.

For the present case, we start with $K$, Eq.(14), as a thermodynamic
potential. So according to Eq.(15), the coordinates of the thermodynamic phase
space are $\mathcal{Z}^{A}=(K,\mathcal{X}^{a},\mathcal{P}^{a})$ with
$\mathcal{X}^{a}=(\Psi,T)$ and $\mathcal{P}^{a}=(-q,-S)$. So the metric
Eq.(32) takes the form:

$\mathcal{G}_{1}=\theta_{K}^{2}+(-ST-q\Psi)(-dSdT+dqd\Psi)~{},$ (33)

with $\theta_{K}=dK+qd\Psi+SdT$. Use of Eqs.(15) and (16) leads to the
following form:

$\mathcal{G}_{1}=(TK_{T}+\Psi
K_{\Psi})(-K_{\Psi\Psi}d\Psi^{2}+K_{TT}dT^{2})~{},$ (34)

where we use the conventions $X_{Y}=\partial X/\partial Y$ and
$X_{YY}=\partial^{2}X/\partial Y^{2}$. The above metric is two dimensional
which has the general form:

$ds^{2}=-f(x,y)dx^{2}+g(x,y)dy^{2}~{}.$ (35)

The Ricci scalar of this metric is given by

$R=\frac{1}{2f^{2}g^{2}}\Big{[}f\Big{(}f_{y}g_{y}-g_{x}^{2}\Big{)}+g\Big{\\{}f_{y}^{2}-f_{x}g_{x}-2f\Big{(}f_{yy}-g_{xx}\Big{)}\Big{\\}}\Big{]}~{}.$
(36)

Here $f=(TK_{T}+\Psi K_{\Psi})K_{\Psi\Psi}$ and $g=(TK_{T}+\Psi
K_{\Psi})K_{TT}$. Therefore the Ricci scalar $R$ diverges when
$K_{\Psi\Psi}=0$ provided the numerator is finite. If both vanish, then one
needs to be careful and use L’Hospital’s rule to arrive at the same
conclusion. This shows that the first condition (see Eq.(17)) at the critical
point implies the divergence of Ricci tensor of the metric Eq.(34) at the
critical point, i.e. $K_{\Psi\Psi}=2b_{02}=0$.

For the other condition we consider the original situation where $q$ is
expressed as function of $\Psi$ and $T$. In this case
$dq=q_{\Psi}d\Psi+q_{T}dT$. Therefore the Legendre invariant metric can be
chosen as

$\mathcal{G}_{2}=\theta_{q}^{2}+(q_{\Psi}\Psi+q_{T}dT)(-d\Psi
dq_{\Psi}+dTdq_{T})~{},$ (37)

where we have considered the thermodynamic phase space as
$\mathcal{Z}^{A}=(q,\mathcal{X}^{a},\mathcal{P}^{a})$ with
$\mathcal{X}^{a}=(\Psi,T)$ and $\mathcal{P}^{a}=(q_{\Psi},q_{T})$. Here the
fundamental one form is $\theta_{q}=dq-q_{\Psi}d\Psi-q_{T}dT$. Now since
$q_{\Psi}$ and $q_{T}$ are functions of both $\Psi$ and $T$, proceeding as
before, the above reduces to the following form:

$\mathcal{G}_{2}=(q_{\Psi}\Psi+q_{T}dT)(-q_{\Psi\Psi}d\Psi^{2}+q_{TT}dT^{2})~{}.$
(38)

This is again of the form Eq.(35) whose Ricci scalar is given by Eq.(36). Here
$f=(q_{\Psi}\Psi+q_{T}dT)q_{\Psi\Psi}$ and $(q_{\Psi}\Psi+q_{T}dT)q_{TT}$.
Therefore the Ricci scalar for the present metric diverges at the critical
point as $q_{\Psi\Psi}$ vanishes at this point, i.e. $q_{\Psi\Psi}=2a_{20}=0$.

##  V concluding remarks

Choosing the correct independent thermodynamic variables is a key starting
point in any thermodynamic treatment. The convenient choice of independent
thermodynamic variables can lead to easier solutions, as is clearly
demonstrated by usefulness of Legendre transform in thermodynamics. However,
the “wrong” set of independent variables could lead to nonphysical results. In
Ref.AAA , an alternative thermodynamic phase space was proposed for RN black
holes in AdS space. In this alternative view, the square of black hole
electric charge (instead of the usual charge) was considered to be the
independent thermodynamic variable. It was shown that the thermodynamic
behavior in such an alternative view makes more physical sense and that the
the critical behavior resembled strongly with Van der Waals fluid, belonging
to the same universality class. In the present work, we have approached the
same problem by (i) generalizing to _any_ AdS black hole independent of
spacetime metric, and (ii) solving the problem both from the equation of state
as well as thermodynamic potential approach. Both approaches lead clearly to
the set of four critical exponents which are the same as the Van der Waals
fluid system. This provides further evidence for the generality of Van der
Waals universality class for AdS black holes on one hand, as well as
indicating the validity of the alternative phase space proposed in AAA .
Furthermore, we have also provided a thermogeometric approach where Ricci
scaler has been calculated and shown to diverge at the critical point within
the general alternative phase space.

###### Acknowledgements.

We thank Shiraz University Research Council. The work of AS has been supported
financially by Research Institute for Astronomy and Astrophysics of Maragha,
Iran. The work of BRM is supported by a START-UP RESEARCH GRANT (No.
SG/PHY/P/BRM/01) from Indian Institute of Technology Guwahati, India.

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