# Stable and Historic Behavior in Replicator Equations Generated by Similar-
Order Preserving Mappings

Mansoor Saburov  Mansoor Saburov: College of Engineering and Technology,
American University of the Middle East, Kuwait<EMAIL_ADDRESS>
###### Abstract.

One could observe drastically different dynamics of zero-sum and non-zero-sum
games under replicator equations. In zero-sum games, heteroclinic cycles
naturally occur whenever the species of the population supersede each other in
cyclic fashion (like for the Rock-Paper-Scissors game). In this case, the
highly erratic oscillations may cause the divergence of the time averages. In
contrast, it is a common belief that the most “reasonable” replicator
equations of non-zero-sum games satisfy “The Folk Theorem of Evolutionary Game
Theory” which asserts that $(i)$ a Nash equilibrium is a rest point; $(ii)$ a
stable rest point is a Nash equilibrium; $(iii)$ a strictly Nash equilibrium
is asymptotically stable; $(iv)$ any interior convergent orbit evolves to a
Nash equilibrium. In this paper, we propose two distinct vast classes of
replicator equations generated by similar-order preserving mappings which
exhibit stable as well as mean historic behavior. In the latter case, the time
averages of the orbit will slowly oscillate during the evolution of the system
and do not converge to any limit. This will eventually cause the divergence of
higher-order repeated time averages.

###### Key words and phrases:

Replicator equation; Nash equilibrium; Schur-convex function; stable and
historic behavior

###### 2010 Mathematics Subject Classification:

37C75, 37D05, 47H25, 37N25, 91A22

##  1\. Introduction

The ergodic theory studies the long term average behavior of systems evolving
in time. Two of the most important theorems are those of Birkhoff and von
Neumann which assert the convergence of time averages along almost each orbit.
For the special class of ergodic systems, these time averages have the same
limit for almost all initial points: statistically speaking, the system that
evolves for a long time forgets its initial state, i.e., it does not have
history. The set of all initial points for which the time averages do not
converge has measure zero with respect to any invariant measure. This however
does not imply that it has Lebesgue measure zero, in general. Furthermore,
even if the set has Lebesgue measure zero, it is possible for the set to be
topologically large [6, 43, 44]. This motivates to introduce a notion of
historic behavior in a dynamical system. Roughly speaking, we say that the
forward orbit has historic behavior if its time averages do not converge in
the embedded topology. The terminology historic behavior was coined by Ruelle
[23]. The problem of describing the persistent family of dynamical systems
with historic behavior was popularized by Takens [39]. Recently, this problem
under the name of Takens’ Last Problem was studied in the papers [2, 3, 5, 16,
17, 18, 19, 47].

These two extreme behaviors, stable and historic behavior, of dynamical
systems can be frequently observed in evolutionary game theory. The main aim
of this paper is to provide two vast classes of replicator equations which
exhibit stable as well as historic behavior.

Evolutionary game theory is the study of frequency-dependent natural selection
in which the fitness of individuals is not constant, but depends on
frequencies of the different phenotypes in population [37]. Evolutionary game
theory differs from classical non-cooperative game theory in focusing more on
the dynamics of strategy change. Moreover, it does not rely on rationality,
i.e., the players do not choose their strategy and cannot change it: they are
born with a strategy and their offspring inherit that same strategy.
Evolutionary game theory interprets payoff as biological fitness or
reproductive success. The success of a strategy is determined by how good the
strategy is in the presence of competing strategies. The strategies that do
well reproduce faster and the strategies that do poorly are out-competed.
There has been a veritable explosion of interest by economists and social
scientists in evolutionary game theory [7, 22, 33, 36, 38].

The Nash equilibrium is a strategy profile in which no player can do better by
unilaterally changing their strategy to another strategy [21]. An
evolutionarily stable strategy (ESS) is akin to the Nash equilibrium which is
“evolutionarily” stable, i.e., once it is fixed in a population, natural
selection alone is sufficient to prevent mutant strategies from invading
successfully [41, 42]. Therefore, the ESS is effective against any mutant
strategy when it is initially rare and successful when it is eventually
abundant.

The replicator equation is the cornerstone of evolutionary game theory [40]. A
replicator can be a gene in population genetics, a molecule in biochemical
evolution, an organism in population ecology, a strategy in evolutionary game.
Several evolutionary models in distinct biological fields – population
genetics, biochemical evolution, population ecology, and evolutionary game –
lead independently to the same class of replicator equations [34, 40]. The
general idea is that replicators whose fitness is larger (smaller) than the
average fitness of population will increase (decrease) in numbers. In
replicator equation, the relative growth rate of players using a certain
strategy is equal to the difference between the average payoff of that
strategy and the average payoff of the population as a whole. Hence, the
static approach of evolutionary game theory has been complemented by the
dynamic stability analysis of rest (stationary) solutions of the replicator
equations.

On the one hand, it is a common belief [4, 11, 12] that the most “reasonable”
replicator equations satisfy the “Folk Theorem of Evolutionary Game Theory”
which asserts that $(i)$ a Nash equilibrium is a rest point; $(ii)$ a stable
rest point is a Nash equilibrium; $(iii)$ a strictly Nash equilibrium is
asymptotically stable; $(iv)$ any interior convergent orbit evolves to a Nash
equilibrium.

On the other hand, in the zero-sum game [9, 10, 35], heteroclinic cycles
naturally occur whenever the species of the population supersede each other in
cyclic fashion (like for the Rock-Paper-Scissors game). The dynamics near the
stable heteroclinic cycle is characterized by intermittency: an orbit
remaining near the heteroclinic cycle, spends long periods of time close to
each saddle and makes fast transitions from one saddle to the next one. Since
the orbit converges to the heteroclinic cycle, it comes closer and closer to
the saddles and consequently remains there for longer and longer times and the
time spent near each saddle increases exponentially. Such highly erratic
oscillations cause the divergence of the time averages which is known as
“historic behavior” in literature. Namely, an orbit keeps having new ideas
about what it wants to do and keeps changing considerably so that the values
of the time averages give information about the epoch to which the sojourn
time belongs: it has a history. Hence, the orbit witnesses the history of the
dynamical system and records the fluctuations.

In this paper, we propose two vast classes of replicator equations generated
by similar-order preserving mappings which exhibit stable as well as mean
historic behavior. In the latter case, the time averages of the orbit will
slowly oscillate during the evolution of the system and do not converge to any
limit. This will eventually cause the divergence of higher-order repeated time
averages.

##  2\. The Main Results

Let $\mathbf{I}_{m}:=\\{1,\cdots,m\\}$ be a finite set and
$\\{\mathbf{e}_{k}\\}_{k=1}^{m}$ be the standard basis of $\mathbb{R}^{m}$.
Suppose that $\mathbb{R}^{m}$ is equipped with the $l_{1}-$norm
$\|\mathbf{x}\|_{1}:=\sum_{k=1}^{m}|x_{k}|$ where
$\mathbf{x}:=(x_{1},\cdots,x_{m})\in\mathbb{R}^{m}$. We also set
$\langle\mathbf{x},\mathbf{y}\rangle:=\sum_{k=1}^{m}x_{k}y_{k}$ for any two
vectors $\mathbf{x},\mathbf{y}\in\mathbb{R}^{m}$. We say that $\mathbf{x}\geq
0$ if $x_{k}\geq 0$ for all $k\in\mathbf{I}_{m}$. Moreover, we say that
$\mathbf{x}\geq\mathbf{y}$ if one has $\mathbf{x}-\mathbf{y}\geq 0$, i.e.,
$x_{k}\geq y_{k}$ for all $k\in\mathbf{I}_{m}$. Let
$\mathbb{R}_{+}^{m}:=\\{\mathbf{x}\in\mathbb{R}^{m}:\mathbf{x}\geq 0\\}$ be a
positive orthant,
$\mathbb{B}_{+}^{m}:=\\{\mathbf{x}\in\mathbb{R}_{+}^{m}:\|\mathbf{x}\|_{1}\leq
1\\}$ be a part of unit ball located in the positive orthant, and
$\mathbb{S}^{m-1}:=\\{\mathbf{x}\in\mathbb{R}_{+}^{m}:\|\mathbf{x}\|_{1}=1\\}$
be the standard simplex. For a vector $\mathbf{x}\in\mathbb{R}_{+}^{m}$, we
set

$\textup{{supp}}(\mathbf{x})=\\{k\in\mathbf{I}_{m}:x_{k}>0\\}\quad\textup{and}\quad\textup{{null}}(\mathbf{x})=\\{k\in\mathbf{I}_{m}:x_{k}=0\\}.$

Two vectors $\mathbf{x},\mathbf{y}\in\mathbb{R}_{+}^{m}$ are called similarly
ordered, denoted by $\mathbf{x}\approx\mathbf{y}$, if one has that
$x_{i}\left(\gtreqqless\right)x_{j}$ if and only if
$y_{i}\left(\gtreqqless\right)y_{j}$ for all $i,j\in\mathbf{I}_{m}$.
Obviously, for a vector $\mathbf{x}\in\mathbb{R}_{+}^{m}$, a set
$\textsc{{SOC}}[\mathbf{x}]=\\{\mathbf{y}\in\mathbb{R}_{+}^{m}:\mathbf{x}\approx\mathbf{y}\\}$
of all vectors $\mathbf{y}\in\mathbb{R}_{+}^{m}$ for which $\mathbf{x}$ and
$\mathbf{y}$ are similarly ordered is a convex cone.

######  Definition 2.1.

A continuous mapping $\mathbf{F}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ is
called similar-order preserving if one has
$\mathbf{F}(\mathbf{x})\in\textsc{{SOC}}[\mathbf{x}]$, i.e,
$\mathbf{F}(\mathbf{x})\approx\mathbf{x}$ for all
$\mathbf{x}\in\mathbb{R}_{+}^{m}$.

In the next section, we provide some examples for similar-order preserving
mappings.

Let $\mathbf{F}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$,
$\mathbf{F}(\mathbf{x}):=\left(f_{1}(\mathbf{x}),\cdots,f_{m}(\mathbf{x})\right)$
be a continuously differentiable similar-order preserving mapping. Throughout
this paper, without loss of generality, we may assume that
$0<\mathbf{F}(\mathbf{x})\leq\mathbbm{1}$ for all
$\mathbf{x}\in\mathbb{B}_{+}^{m}$ where $\mathbbm{1}=(1,1,\cdots,1)$.
Otherwise, for small $\varepsilon>0$, we consider a continuous mapping
$\widehat{\mathbf{F}}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$,

$\widehat{\mathbf{F}}(\mathbf{x}):=\frac{1}{M_{1}}\left(\mathbf{F}(\mathbf{x})+\varepsilon\mathbbm{1}\right)$

which is similar-order preserving and
$0<\widehat{\mathbf{F}}(\mathbf{x})\leq\mathbbm{1}$ for all
$\mathbf{x}\in\mathbb{B}_{+}^{m}$ where

$M_{1}:=\max\\{M+\varepsilon,1\\}\quad\textup{and}\quad
M:=\max\limits_{k\in\mathbf{I}_{m}}\max\limits_{\mathbf{x}\in\mathbb{B}_{+}^{m}}f_{k}(\mathbf{x}).$

We consider a discrete-time replicator equation
$\mathcal{R}_{S}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$ generated by a
continuously differentiable similar-order preserving mapping
$\mathbf{F}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$,
$\mathbf{F}(\mathbf{x}):=\left(f_{1}(\mathbf{x}),\cdots,f_{m}(\mathbf{x})\right)$

$\left(\mathcal{R}_{S}(\mathbf{x})\right)_{k}=x_{k}\left(1+f_{k}(\mathbf{x})-\sum\limits_{i=1}^{m}x_{i}f_{i}(\mathbf{x})\right),\quad\forall\
k\in\mathbf{I}_{m}.$ (2.1)

Obviously, in order the simplex $\mathbb{S}^{m-1}$ to be an invariant set for
the replicator equation given by (2.1), we must have some constraints on a
continuous similar-order preserving mapping
$\mathbf{F}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$,
$\mathbf{F}(\mathbf{x}):=\left(f_{1}(\mathbf{x}),\cdots,f_{m}(\mathbf{x})\right)$.
For example, the following constraint

$0\leq\max_{k\in\mathbf{I}_{m}}\max_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})-\min_{k\in\mathbf{I}_{m}}\min_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})\leq
1$

is sufficient for the simplex $\mathbb{S}^{m-1}$ to be invariant. Indeed,
since

$\min_{k\in\mathbf{I}_{m}}\min_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})\leq\sum\limits_{i=1}^{m}x_{i}f_{i}(\mathbf{x})\leq\max_{k\in\mathbf{I}_{m}}\max_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x}),\quad\forall\
\mathbf{x}\in\mathbb{S}^{m-1},$

we obtain for all $\mathbf{x}\in\mathbb{S}^{m-1}$ and $k\in\mathbf{I}_{m}$
that

$1+f_{k}(\mathbf{x})-\sum\limits_{i=1}^{m}x_{i}f_{i}(\mathbf{x})\geq
1+f_{k}(\mathbf{x})-\max_{k\in\mathbf{I}_{m}}\max_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})\geq\\\
f_{k}(\mathbf{x})-\min_{k\in\mathbf{I}_{m}}\min_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})\geq
0.$

Hence, we have
$\sum_{k=1}^{m}\left(\mathcal{R}_{S}(\mathbf{x})\right)_{k}=\sum_{k=1}^{m}x_{k}=1$
and $\left(\mathcal{R}_{S}(\mathbf{x})\right)_{k}\geq 0$ for all
$\mathbf{x}\in\mathbb{S}^{m-1}$ and $k\in\mathbf{I}_{m}$. Since, throughout
this paper, we always assume that $0<\mathbf{F}(\mathbf{x})\leq\mathbbm{1}$
for all $\mathbf{x}\in\mathbb{B}_{+}^{m}$ where $\mathbbm{1}=(1,1,\cdots,1)$,
it is needless to say that the constraint considered above

$0\leq\max_{k\in\mathbf{I}_{m}}\max_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})-\min_{k\in\mathbf{I}_{m}}\min_{\mathbf{x}\in\mathbb{S}^{m-1}}f_{k}(\mathbf{x})\leq
1$

is always satisfied.

Recall (see [4, 12, 33, 36]) that $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a
Nash equilibrium of the discrete-time replicator equation given by (2.1) if
$\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle\geq\langle\mathbf{y},\mathbf{F}(\mathbf{x})\rangle$
for all $\mathbf{y}\in\mathbb{S}^{m-1}$. Moreover,
$\mathbf{x}\in\mathbb{S}^{m-1}$ is called a strictly Nash equilibrium if
$\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle>\langle\mathbf{y},\mathbf{F}(\mathbf{x})\rangle$
for all $\mathbf{y}\in\mathbb{S}^{m-1}$ but $\mathbf{y}\neq\mathbf{x}$. A
point $\mathbf{x}\in\mathbb{S}^{m-1}$ is called a rest (fixed) point if
$\mathcal{R}_{S}(\mathbf{x})=\mathbf{x}$. A rest (fixed) point
$\mathbf{x}\in\mathbb{S}^{m-1}$ is called stable if for every neighborhood
$U(\mathbf{x})\subset\mathbb{S}^{m-1}$ of $\mathbf{x}$ there exists another
neighborhood $V(\mathbf{x})\subset U(\mathbf{x})\subset\mathbb{S}^{m-1}$ of
$\mathbf{x}$ such that an orbit
$\\{\mathbf{y},\mathcal{R}_{S}(\mathbf{y}),\cdots,\mathcal{R}_{S}^{(n)}(\mathbf{y}),\cdots\\}$
of any initial point $\mathbf{y}\in V(\mathbf{x})$ remains inside of the
neighborhood $U(\mathbf{x})$. A rest (fixed) point
$\mathbf{x}\in\mathbb{S}^{m-1}$ is called attracting if there exists a
neighborhood $V(\mathbf{x})\subset\mathbb{S}^{m-1}$ of $\mathbf{x}$ such that
an orbit
$\\{\mathbf{y},\mathcal{R}_{S}(\mathbf{y}),\cdots,\mathcal{R}_{S}^{(n)}(\mathbf{y}),\cdots\\}$
of any initial point $\mathbf{y}\in V(\mathbf{x})$ converges to $\mathbf{x}$.
A rest (fixed) point $\mathbf{x}\in\mathbb{S}^{m-1}$ is called asymptotically
stable if it is both stable and attracting.

The following result shows that a vast class of discrete-time replicator
equations satisfies “The Folk Theorem of Evolutionary Game Theory”.

######  Theorem A.

The following statements are true for the discrete-time replicator equation
$\mathcal{R}_{S}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$ given by (2.1):

  * $(i)$

A Nash equilibrium is a rest point;

  * $(ii)$

A stable rest point is a Nash equilibrium;

  * $(iii)$

A strictly Nash equilibrium is asymptotically stable;

  * $(iv)$

Any interior convergent orbit evolves to a Nash equilibrium.

It is worth mentioning that some particular cases of the discrete-time
replicator equations $\mathcal{R}_{S}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$
given by (2.1) were studied in the literature (see [30, 32]).

We now present a discrete-time replicator equation of a zero-sum game that
exhibits mean historic behavior. We consider either one of the following
discrete-time replicator equations
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ of a zero-sum game defined
by a continuously differentiable similar-order preserving mapping
$\mathbf{F}:\mathbb{B}^{3}_{+}\to\mathbb{R}^{3}_{+}$,
$\mathbf{F}(\mathbf{x}):=\left(f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),f_{3}(\mathbf{x})\right)$

$\displaystyle\mathcal{R}_{H}:\left\\{\begin{matrix}(\mathcal{R}_{H}(\mathbf{x}))_{1}=x_{1}\left(1+x_{2}f_{1}(\mathbf{x})-x_{3}f_{3}(\mathbf{x})\right)\\\
(\mathcal{R}_{H}(\mathbf{x}))_{2}=x_{2}\left(1+x_{3}f_{2}(\mathbf{x})-x_{1}f_{1}(\mathbf{x})\right)\\\
(\mathcal{R}_{H}(\mathbf{x}))_{3}=x_{3}\left(1+x_{1}f_{3}(\mathbf{x})-x_{2}f_{2}(\mathbf{x})\right)\end{matrix}\right.$
(2.2)

or

$\displaystyle\mathcal{R}_{H}:\left\\{\begin{matrix}(\mathcal{R}_{H}(\mathbf{x}))_{1}=x_{1}\left(1+x_{3}f_{1}(\mathbf{x})-x_{2}f_{2}(\mathbf{x})\right)\\\
(\mathcal{R}_{H}(\mathbf{x}))_{2}=x_{2}\left(1+x_{1}f_{2}(\mathbf{x})-x_{3}f_{3}(\mathbf{x})\right)\\\
(\mathcal{R}_{H}(\mathbf{x}))_{3}=x_{3}\left(1+x_{2}f_{3}(\mathbf{x})-x_{1}f_{1}(\mathbf{x})\right)\end{matrix}\right.$
(2.3)

It is wroth mentioning that a density-dependent payoff matrix of a zero-sum
game is given by

$\displaystyle\left(\begin{matrix}0&f_{1}(\mathbf{x})&-f_{3}(\mathbf{x})\\\
-f_{1}(\mathbf{x})&0&f_{2}(\mathbf{x})\\\
f_{3}(\mathbf{x})&-f_{2}(\mathbf{x})&0\end{matrix}\right)\quad\textup{or}\quad\left(\begin{matrix}0&-f_{2}(\mathbf{x})&f_{1}(\mathbf{x})\\\
f_{2}(\mathbf{x})&0&-f_{3}(\mathbf{x})\\\
-f_{1}(\mathbf{x})&f_{3}(\mathbf{x})&0\end{matrix}\right).$

In the language of evolutionary game theory, the discrete-time replicator
equation $\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) or
(2.3) represents the interactions of three competing species where the species
1 beats (loses to) the species 2, the species 2 beats (loses to) the species
3, and the species 3 beats (loses to) the species 1 (like the Rock-Paper-
Scissors game). Some particular cases of the discrete-time replicator
equations $\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ were studied in
the literature (see [1, 8, 13, 14, 15, 24, 25, 26, 27, 28, 29, 31, 45, 46,
48]).

We say that the discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) or (2.3) has
mean historic behavior if the set of initial points
$\mathbf{x}\in\mathbb{S}^{2}$ which give rise to orbits with divergent time
averages $\frac{1}{n}\sum_{k=0}^{n-1}\mathcal{R}_{H}^{(k)}(\mathbf{x})$ has
positive Lebesgue measure (see [31]). We define the $s^{th}-$order repeated
time averages
$\left\\{\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right)\right\\}_{n=1}^{\infty}$
for all $s\in\mathbb{N}$ as follows

$\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right):=\frac{1}{n}\sum_{k=1}^{n}\mathcal{A}^{(s-1)}_{k}\left(\mathbf{x}\right),\quad\mathcal{A}^{(1)}_{n}\left(\mathbf{x}\right):=\frac{1}{n}\sum_{k=0}^{n-1}\mathcal{R}_{H}^{(k)}(\mathbf{x}).$

######  Theorem B.

The discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) or (2.3) has
mean historic behavior. Moreover, for any interior initial point, the
$s^{th}-$order repeated time averages
$\left\\{\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right)\right\\}_{n=1}^{\infty}$
do not converge for any $s\in\mathbb{N}$.

##  3\. Some Examples for Similar-Order Preserving Mappings

Let $\mathbf{F},\mathbf{G}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$,
$\mathbf{F}(\mathbf{x}):=\left(f_{1}(\mathbf{x}),\cdots,f_{m}(\mathbf{x})\right)$,
$\mathbf{G}(\mathbf{x}):=\left(g_{1}(\mathbf{x}),\cdots,g_{m}(\mathbf{x})\right)$
be continuous similar-order preserving mappings and
$\varphi,\psi:\mathbb{R}^{m}_{+}\to(0,\infty)$ be continuous functions. Let
$h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ be a strictly increasing continuous
function and $\mathbf{H}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$,
$\mathbf{H}(\mathbf{x}):=\left(h(x_{1}),\cdots,h(x_{m})\right)$. It is easy to
check that the following mappings

  * $(i)$

$\mathbf{F}_{\varphi,\psi}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ where
$\mathbf{F}_{\varphi,\psi}(\mathbf{x}):=\bigl{(}\varphi(\mathbf{x})f_{1}(\mathbf{x})+\psi(\mathbf{x}),\cdots,\varphi(\mathbf{x})f_{m}(\mathbf{x})+\psi(\mathbf{x})\bigr{)}$;

  * $(ii)$

$\varphi\mathbf{F}+\psi\mathbf{G}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$
where
$\left(\varphi\mathbf{F}+\psi\mathbf{G}\right)(\mathbf{x}):=\varphi(\mathbf{x})\mathbf{F}(\mathbf{x})+\psi(\mathbf{x})\mathbf{G}(\mathbf{x})$;

  * $(iii)$

$\mathbf{H}\circ\mathbf{F}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ where
$\left(\mathbf{H}\circ\mathbf{F}\right)(\mathbf{x}):=\mathbf{H}\left(\mathbf{F}(\mathbf{x})\right)=\bigl{(}h(f_{1}(\mathbf{x})),\cdots,h(f_{m}(\mathbf{x}))\bigr{)}$;

  * $(iv)$

$\mathbf{F}\circ\mathbf{H}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ where
$\left(\mathbf{F}\circ\mathbf{H}\right)(\mathbf{x}):=\mathbf{F}\left(\mathbf{H}(\mathbf{x})\right)=\bigl{(}f_{1}(\mathbf{H}(\mathbf{x})),\cdots,f_{m}(\mathbf{H}(\mathbf{x}))\bigr{)}$;

  * $(v)$

$\mathbf{F}\circ\mathbf{G}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ where
$\left(\mathbf{F}\circ\mathbf{G}\right)(\mathbf{x}):=\mathbf{F}\left(\mathbf{G}(\mathbf{x})\right)=\bigl{(}f_{1}(\mathbf{G}(\mathbf{x})),\cdots,f_{m}(\mathbf{G}(\mathbf{x}))\bigr{)}$

are also similar-order preserving. Moreover, if $\mathbf{F}(\mathbf{x})>0$ and
$\mathbf{G}(\mathbf{x})>0$ for all $\mathbf{x}\in\mathbb{R}_{+}^{m}$ then the
mapping

  * $(vi)$

$\mathbf{F}\odot\mathbf{G}:\mathbb{R}^{m}_{+}\to\mathbb{R}^{m}_{+}$ where
$\left(\mathbf{F}\odot\mathbf{G}\right)(\mathbf{x}):=\bigl{(}f_{1}(\mathbf{x})\cdot
g_{1}(\mathbf{x}),\cdots,f_{m}(\mathbf{x})\cdot g_{m}(\mathbf{x})\bigr{)}$

is also similar-order preserving.

These properties allow to generate various kinds of similar-order preserving
mappings. Let us now provide some concrete examples for similar-order
preserving mappings.

Let $x_{[1]}\geq\cdots\geq x_{[m]}$ denote the components of
$\textbf{x}=(x_{1},\cdots,x_{m})\in\mathbb{R}^{m}$ in a non-increasing order.
We say that $\mathbf{x}$ majorizes $\mathbf{y}$ (or $\mathbf{y}$ is majorized
by $\mathbf{x}$), written $\mathbf{x}\succ\mathbf{y}$, if one has
$\sum_{i=1}^{k}x_{[i]}\geq\sum_{i=1}^{k}y_{[i]}$ for all
$k\in\mathbf{I}_{m-1}$ and $\sum_{i=1}^{m}x_{i}=\sum_{i=1}^{m}y_{i}$.

Recall (see [20]) that a function $\varphi:\mathbb{B}_{+}^{m}\to\mathbb{R}$ is
said to be (strictly) increasing if $\mathbf{x}\geq\mathbf{y}$ (resp.
$\mathbf{x}\geq\mathbf{y}$ but $\mathbf{x}\neq\mathbf{y}$) implies
$\varphi(\mathbf{x})\geq\varphi(\mathbf{y})$ (resp.
$\varphi(\mathbf{x})>\varphi(\mathbf{y})$). A function
$\varphi:\mathbb{B}_{+}^{m}\to\mathbb{R}$ is said to be (strictly) Schur-
convex if $\mathbf{x}\succ\mathbf{y}$ (resp. $\mathbf{x}\succ\mathbf{y}$ but
$\mathbf{y}$ is not a permutation of $\mathbf{x}$) implies
$\varphi(\mathbf{x})\geq\varphi(\mathbf{y})$ (resp.
$\varphi(\mathbf{x})>\varphi(\mathbf{y})$).

It is worth mentioning (see [20]) that, since
$\varphi:\mathbb{B}_{+}^{m}\to\mathbb{R}$ is a strictly Schur-convex function,
we have that $\frac{\partial\varphi}{\partial
x_{i}}(\mathbf{x})\left(\gtrless\right)\frac{\partial\varphi}{\partial
x_{j}}(\mathbf{x})$ if and only if $x_{i}\left(\gtrless\right)x_{j}$ for
$i,j\in\mathbf{I}_{m}$. Moreover, since
$\varphi:\mathbb{B}_{+}^{m}\to\mathbb{R}$ is symmetric, we also have that
$\frac{\partial\varphi}{\partial
x_{i}}(\mathbf{x})=\frac{\partial\varphi}{\partial x_{j}}(\mathbf{x})$ if and
only if $x_{i}=x_{j}$ for $i,j\in\mathbf{I}_{m}$.

Hence, for any symmetric, continuously differentiable, strictly increasing and
strictly Schur-convex function $\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,
a gradient vector field
$\nabla\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}^{m}$ where

$\nabla\varphi(\mathbf{x}):=\left(\frac{\partial\varphi}{\partial
x_{1}}(\mathbf{x}),\cdots,\frac{\partial\varphi}{\partial
x_{m}}(\mathbf{x})\right)$

is always similar-order preserving. Particularly, the gradient vector fields
of the following functions are also similar-order preserving (the reader may
refer to the monograph [20]):

  * $(i)$

Complete Symmetric Functions:
$\varphi_{k}:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,

$\varphi_{k}(\mathbf{x})=\sum\limits_{i_{1}+i_{2}+\cdots+i_{m}=k}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots
x_{m}^{i_{m}},\qquad\forall\ k\in\mathbf{I}_{m};$

  * $(ii)$

Symmetric Gauge Functions: $\varphi_{p}:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,

$\varphi_{p}(\mathbf{x})=\|\mathbf{x}\|_{p}=\sqrt[p]{\sum_{k=1}^{m}x^{p}_{k}},\qquad
p>1;$

  * $(iii)$

Gamma Functions: $\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,

$\varphi(\mathbf{x})=\prod_{k=1}^{m}\Gamma(x_{k}+a),\qquad a\geq 1,$

where $\Gamma:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a gamma function;

  * $(iv)$

Convex Separate Variables Functions:
$\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,

$\varphi(\mathbf{x})=\sum_{k=1}^{m}f(x_{k}),$

where $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuously differentiable,
strictly increasing, and strictly convex function;

  * $(v)$

Symmetric Composite Functions:
$\varphi_{k}:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,

$\varphi_{k}(\mathbf{x})=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq
m}f(x_{i_{1}})\cdots f(x_{i_{k}}),\qquad\forall\ k\in\mathbf{I}_{m},$

where $f:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuously differentiable,
strictly increasing, and strictly log-convex function;

  * $(vi)$

General Convex Symmetric Functions:
$\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$ is a continuously
differentiable, symmetric, strictly increasing, and strictly convex function.

  * $(vii)$

Composition Functions: If $\psi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$ is a
continuously differentiable, strictly increasing, and strictly Schur-convex
function and $h:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is a continuously
differentiable, strictly increasing and strictly convex function then
$\varphi:\mathbb{R}_{+}^{m}\to\mathbb{R}_{+}$,
$\varphi(\mathbf{x})=\psi(h(x_{1}),\cdots,h(x_{m}))$ is also a continuously
differentiable, strictly increasing, and strictly Schur-convex function.

##  4\. The Proof of Theorem A

In order to study the stability of rest (fixed) points of the discrete-time
replicator equations $\mathcal{R}_{S}:\mathbb{S}^{m-1}\to\mathbb{S}^{m-1}$
given by (2.1), we employ a Lyapunov function. Recall (see [4, 11, 33, 36])
that a continuous function $\varphi:\mathbb{S}^{m-1}\to\mathbb{R}$ is called a
Lyapunov function if the number sequence
$\\{\varphi(\mathbf{x}),\varphi(\mathcal{R}_{S}(\mathbf{x})),\cdots,\varphi(\mathcal{R}_{S}^{(n)}(\mathbf{x})),\cdots\\}$
is a bounded monotone sequence for any initial point
$\mathbf{x}\in\mathbb{S}^{m-1}$.

Part $(i)$:

Let us first show that the center
$\mathbf{c}_{\alpha}:=\frac{1}{|\alpha|}\sum_{k\in\alpha}\mathbf{e}_{k}$ of
any face

$\mathbb{S}_{\alpha}:=\\{\mathbf{x}\in\mathbb{S}^{m-1}:\textup{{supp}}(\mathbf{x})\subset\alpha\\}$

of the simplex $\mathbb{S}^{m-1}$ is a rest (fixed) point for all
$\alpha\subset\mathbf{I}_{m}$. Indeed, it is clear that if $k\not\in\alpha$
then $(\mathcal{R}_{S}(\mathbf{c}_{\alpha}))_{k}=(\mathbf{c}_{\alpha})_{k}=0$
and if $k\in\alpha$ then, since
$\mathbf{F}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$ is a similar-order
preserving mapping, we have
$f_{k}(\mathbf{c}_{\alpha})=\sum\limits_{i\in\alpha}(\mathbf{c}_{\alpha})_{i}f_{i}(\mathbf{c}_{\alpha})$
and
$(\mathcal{R}_{S}(\mathbf{c}_{\alpha}))_{k}=(\mathbf{c}_{\alpha})_{k}=\frac{1}{|\alpha|}$.

Moreover, there is no any other rest (fixed) point except the center of every
face of the simplex. Indeed, let $\mathbf{x}\in\mathbb{S}^{m-1}$ be any rest
(fixed) point. We then obtain $(\mathcal{R}_{S}(\mathbf{x}))_{k}=x_{k}$ for
all $k\in\textup{{supp}}(\mathbf{x})$. This is equivalent saying that
$f_{k_{1}}(\mathbf{x})=\sum\limits_{i\in\alpha}x_{i}f_{i}(\mathbf{x})=f_{k_{2}}(\mathbf{x})$
for any $k_{1},k_{2}\in\textup{{supp}}(\mathbf{x})$ and $k_{1}\neq k_{2}$.
Since $\mathbf{F}:\mathbb{B}^{m}_{+}\to\mathbb{R}^{m}_{+}$ is a similar-order
preserving mapping, we obtain $x_{k_{1}}=x_{k_{2}}$ for any
$k_{1},k_{2}\in\textup{{supp}}(\mathbf{x})$ and $k_{1}\neq k_{2}$. This means
that $\mathbf{x}$ is the center of the face $\mathbb{S}_{\alpha}$ where
$\alpha=\textup{{supp}}(\mathbf{x})$.

We now describe all Nash equilibria. Since

$\langle\mathbf{c}_{\alpha},\mathbf{F}(\mathbf{c}_{\alpha})\rangle=\max\limits_{k\in\mathbf{I}_{m}}f_{k}(\mathbf{c}_{\alpha})\geq\langle\mathbf{y},\mathbf{F}(\mathbf{c}_{\alpha})\rangle$

for all $\mathbf{y}\in\mathbb{S}^{m-1},$ the center of every face of the
simplex is a Nash equilibrium.

Moreover, if $\mathbf{x}\in\mathbb{S}^{m-1}$ is a Nash equilibrium then

$\max\limits_{k\in\mathbf{I}_{m}}f_{k}(\mathbf{x})\geq\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle\geq\langle\mathbf{e}_{j},\mathbf{F}(\mathbf{x})\rangle=\max\limits_{k\in\mathbf{I}_{m}}f_{k}(\mathbf{x})$

where $j\in\mathbf{I}_{m}$ is an index such that
$f_{j}(\mathbf{x})=\max\limits_{k\in\mathbf{I}_{m}}f_{k}(\mathbf{x})$. Since

$f_{i}(\mathbf{x})\left(\gtreqqless\right)f_{j}(\mathbf{x})\quad\textup{if and
only if}\quad x_{i}\left(\gtreqqless\right)x_{j}\quad\textup{for all}\quad
i,j\in\mathbf{I}_{m},$

we then obtain $\mathbf{x}=\mathbf{c}_{\alpha}$ for a subset

$\alpha:=\left\\{j\in\mathbf{I}_{m}:f_{j}(\mathbf{x})=\max\limits_{k\in\mathbf{I}_{m}}f_{k}(\mathbf{x})\right\\}\subset\mathbf{I}_{m}.$

This shows that every Nash equilibrium is a rest (fixed) point.

Part $(ii)$:

Since every face $\mathbb{S}_{\alpha}$ is invariant, we first show that

$\mathcal{M}_{\alpha,k}(\mathbf{x}):=\max\limits_{i\in\alpha}x_{i}-x_{k}$

is an increasing Lyapunov function over the interior

$\textup{int}\mathbb{S}_{\alpha}:=\\{\mathbf{x}\in\mathbb{S}_{\alpha}:\textup{{supp}}(\mathbf{x})=\alpha\\}$

of the face $\mathbb{S}_{\alpha}$ for every $\alpha\subset\mathbf{I}_{m}$ and
for every $k\in\alpha$. Indeed, for any $k,t\in\alpha$, if $x_{t}=x_{k}$ then
$(\mathcal{R}_{S}(\mathbf{x}))_{t}=(\mathcal{R}_{S}(\mathbf{x}))_{k}$ and if
$x_{t}\neq x_{k}$ then

$\frac{(\mathcal{R}_{S}(\mathbf{x}))_{t}-(\mathcal{R}_{S}(\mathbf{x}))_{k}}{x_{t}-x_{k}}=\left[\frac{x_{t}f_{t}(\mathbf{x})-x_{k}f_{k}(\mathbf{x})}{x_{t}-x_{k}}+\sum\limits_{i\in\alpha}x_{i}\left(1-f_{i}(\mathbf{x})\right)\right].$

Since $0<\mathbf{F}(\mathbf{x})\leq\mathbbm{1}$ and
$x_{t}f_{t}(\mathbf{x})\left(\gtreqqless\right)x_{k}f_{k}(\mathbf{x})$ if and
only if $x_{t}\left(\gtreqqless\right)x_{k}$ for all $k,t\in\alpha$, we obtain
that
$(\mathcal{R}_{S}(\mathbf{x}))_{t}\left(\gtreqqless\right)(\mathcal{R}_{S}(\mathbf{x}))_{k}$
if and only if $x_{t}\left(\gtreqqless\right)x_{k}$ for all $k,t\in\alpha$.
This yields

$\mathsf{\textbf{MaxInd}}_{\alpha}\left(\mathcal{R}_{S}(\mathbf{x})\right)=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$

for any $\mathbf{x}\in\textup{int}\mathbb{S}_{\alpha}$ where
$\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x}):=\\{k\in\alpha:x_{k}=\max\limits_{i\in\alpha}x_{i}\\}$.

Moreover, if
$t\in\mathsf{\textbf{MaxInd}}_{\alpha}\left(\mathcal{R}_{S}(\mathbf{x})\right)=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$
then for all $k\in\alpha$ with $x_{t}\neq x_{k}$ we have that

$\frac{(\mathcal{R}_{S}(\mathbf{x}))_{t}-(\mathcal{R}_{S}(\mathbf{x}))_{k}}{x_{t}-x_{k}}=1+\left[\frac{x_{k}\left(f_{t}(\mathbf{x})-f_{k}(\mathbf{x})\right)}{x_{t}-x_{k}}+\sum\limits_{i\in\alpha}x_{i}\left(f_{t}(\mathbf{x})-f_{i}(\mathbf{x})\right)\right].$

This yields
$\mathcal{M}_{\alpha,k}(\mathcal{R}_{S}(\mathbf{x}))\geq\mathcal{M}_{\alpha,k}(\mathbf{x})$
for all $\mathbf{x}\in\textup{int}\mathbb{S}_{\alpha}$ and for all
$k\in\alpha$. By repeating this process, we obtain

$\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n+1)}_{S}(\mathbf{x}))\geq\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n)}_{S}(\mathbf{x})),\
\mathsf{\textbf{MaxInd}}_{\alpha}\left(\mathcal{R}^{(n)}_{S}(\mathbf{x})\right)=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$

for all $\mathbf{x}\in\textup{int}\mathbb{S}_{\alpha}$, $k\in\alpha$ and
$n\in\mathbb{N}$. This shows that $\mathcal{M}_{\alpha,k}(\mathbf{x})$ is an
increasing Lyapunov function.

We now show that an orbit starting from any initial point
$\mathbf{x}\in\textup{int}\mathbb{S}_{\alpha}$ converges to the center
$\mathbf{c}_{\beta}$ of the face
$\mathbb{S}_{\beta}\subset\mathbb{S}_{\alpha}$ where
$\beta=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$. Indeed, for every
$k\in\alpha$, we have

$(\mathcal{R}^{(n)}_{S}(\mathbf{x}))_{k}=\frac{1}{|\alpha|}\left[1+\sum_{i\in\alpha}\left((\mathcal{R}^{(n)}_{S}(\mathbf{x}))_{k}-(\mathcal{R}^{(n)}_{S}(\mathbf{x}))_{i}\right)\right]=\\\
\frac{1}{|\alpha|}\left[1+\sum_{i\in\alpha}\left(\mathcal{M}_{\alpha,i}(\mathcal{R}^{(n)}_{S}(\mathbf{x}))-\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n)}_{S}(\mathbf{x}))\right)\right].$

Since the sequence
$\left\\{\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n)}_{S}(\mathbf{x}))\right\\}_{n=0}^{\infty}$
is convergent for every $k\in\alpha$, so is the sequence
$\left\\{(\mathcal{R}^{(n)}_{S}(\mathbf{x}))_{k}\right\\}_{n=0}^{\infty}$ for
every $k\in\alpha$. Consequently, an orbit
$\left\\{\mathcal{R}^{(n)}_{S}(\mathbf{x})\right\\}_{n=0}^{\infty}$ is also
convergent and its limit is a rest point that is the center
$\mathbf{c}_{\beta}$ of a face $\mathbb{S}_{\beta}\subset\mathbb{S}_{\alpha}$
for some $\beta\subset\alpha$. Since
$\mathsf{\textbf{MaxInd}}_{\alpha}\left(\mathcal{R}^{(n)}_{S}(\mathbf{x})\right)=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$,
we have
$\beta=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{c}_{\beta})\supset\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$.
Since
$\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n+1)}_{S}(\mathbf{x}))\geq\mathcal{M}_{\alpha,k}(\mathcal{R}^{(n)}_{S}(\mathbf{x}))$,
we have
$\beta=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{c}_{\beta})\subset\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$.
Hence, we obtain $\beta=\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})$.

We are now ready to show that the stable rest points are only vertices of the
simplex which are Nash equilibria.

On the one hand, the center $\mathbf{c}_{\alpha}$ of the face
$\mathbb{S}_{\alpha}$ is not stable for all $\alpha\subset\mathbf{I}_{m}$ with
$|\alpha|\geq 2$. Indeed, for any small neighborhood $U(\mathbf{c}_{\alpha})$
of the center $\mathbf{c}_{\alpha}$ there are points $\mathbf{x},\mathbf{y}\in
U(\mathbf{c}_{\alpha})$ such that
$|\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})|=|\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{y})|=1$
and
$\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{x})\neq\mathsf{\textbf{MaxInd}}_{\alpha}(\mathbf{y})$
for which their orbits converge to two different vertices of the simplex.

On the other hand, the vertex $\mathbf{e}_{k}$ of the simplex is stable for
all $k\in\mathbf{I}_{m}$. Indeed, for any small neighborhood
$U(\mathbf{e}_{k})$ of the vertex $\mathbf{e}_{k}$ one has

$\mathsf{\textbf{MaxInd}}\left(\mathcal{R}_{S}(\mathbf{x})\right)=\mathsf{\textbf{MaxInd}}(\mathbf{x})=\\{k\\}$

for all $\mathbf{x}\in U(\mathbf{e}_{k})$ and

$\left(\mathcal{R}_{S}(\mathbf{x})\right)_{k}=x_{k}\left[1+\sum\limits_{i=1}^{m}x_{i}\left(f_{k}(\mathbf{x})-f_{i}(\mathbf{x})\right)\right]\geq
x_{k}.$

Hence, we obtain
$\|\mathcal{R}_{S}(\mathbf{x})-\mathbf{e}_{k}\|_{1}=2(1-\left(\mathcal{R}_{S}(\mathbf{x})\right)_{k})\leq
2(1-x_{k})=\|\mathbf{x}-\mathbf{e}_{k}\|_{1}$ which implies
$\mathcal{R}_{S}(U(\mathbf{e}_{k}))\subset U(\mathbf{e}_{k})$ and consequently
$\mathcal{R}_{S}^{(n)}(U(\mathbf{e}_{k}))\subset U(\mathbf{e}_{k})$. This
means that the stable rest points are only vertices of the simplex which are
Nash equilibria.

Part $(iii)$:

We first show that strictly Nash equilibria are only vertices of the simplex.

On the one hand, since $f_{k}(\mathbf{e}_{k})>f_{i}(\mathbf{e}_{k})$ for all
$i\neq k$, we have that

$\langle\mathbf{e}_{k},\mathbf{F}(\mathbf{e}_{k})\rangle=f_{k}(\mathbf{e}_{k})=\max\limits_{i\in\mathbf{I}_{m}}f_{i}(\mathbf{e}_{k})>\langle\mathbf{x},\mathbf{F}(\mathbf{e}_{k})\rangle=\sum_{i\in\textup{{supp}}(\mathbf{x})}x_{i}f_{i}(\mathbf{e}_{k})$

for all $\mathbf{x}\in{\mathbb{S}}^{m-1}\setminus\\{\mathbf{e}_{k}\\}$. This
shows that every vertex $\mathbf{e}_{k}$ of the simplex is a strictly Nash
equilibrium for all $k\in\mathbf{I}_{m}$.

On the other hand, for any $\alpha\subset\mathbf{I}_{m}$ with $|\alpha|\geq 2$
and $k\in\alpha$, we have that

$\langle\mathbf{c}_{\alpha},\mathbf{F}(\mathbf{c}_{\alpha})\rangle=\left(\sum_{i\in\alpha}\frac{1}{|\alpha|}\right)\max\limits_{i\in\alpha}f_{i}(\mathbf{c}_{\alpha})=\max\limits_{i\in\alpha}f_{i}(\mathbf{c}_{\alpha})=\langle\mathbf{e}_{k},\mathbf{F}(\mathbf{c}_{\alpha})\rangle.$

This means that the Nash equilibrium $\mathbf{c}_{\alpha}$ is not the strictly
Nash equilibrium for any $\alpha\subset\mathbf{I}_{m}$ with $|\alpha|\geq 2$.
As we already showed in Part $(ii)$ that the vertex $\mathbf{e}_{k}$ of the
simplex is both stable and attracting for all $k\in\mathbf{I}_{m}$. Hence, a
strictly Nash equilibrium is asymptotically stable.

Part $(iv)$:

Due to Part $(ii)$, an interior orbit starting from any initial point
$\mathbf{x}\in\textup{int}\mathbb{S}^{m-1}$ converges to the center
$\mathbf{c}_{\beta}$ of the face $\mathbb{S}_{\beta}$ where
$\beta=\mathsf{\textbf{MaxInd}}(\mathbf{x})$ which is a Nash equilibrium. This
completes the proof of Theorem A.

##  5\. The Proof of Theorem B

It is easy to check that two discrete-time replicator equations of a zero-sum
game given by (2.2) and (2.3) resemble the similar dynamics. Hence, we discuss
only the first discrete-time replicator equation given by (2.2). We are aiming
to show that the discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) has uniformly
historic behavior. Recall (see [31]) that the discrete-time replicator
equation $\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) is
said to have uniformly historic behavior if there exist at least two disjoint
convex compact subsets $U_{1},U_{2}\subset\mathbb{S}^{2}$ such that a
uniformly historic initial set $\mathcal{HIS}_{\mathcal{R}_{H}}(U_{1},U_{2})$
has positive Lebesgue measure where
$\mathcal{HIS}_{\mathcal{R}_{H}}(U_{1},U_{2})$ is a set of all initial points
$\mathbf{x}\in\mathbb{S}^{2}$ whose orbits have uniformly
$(U_{1},U_{2})-$historic behavior, i.e., there exist $\lambda_{i},\mu_{i}>0$
(depending on $\mathbf{x}$) for $i=1,2$ such that the trapping-escaping
sequence

$\left\\{\chi_{{U}_{i}}\left(\mathcal{R}_{H}^{(n)}(\mathbf{x})\right)\right\\}_{n=0}^{\infty}=\left\\{1^{p^{(i)}_{n}}0^{q^{(i)}_{n}}\right\\}_{n=1}^{\infty}:=\\\
\left\\{\underbrace{1,\cdots,1}_{p^{(i)}_{1}},\
\underbrace{0,\cdots,0}_{q^{(i)}_{1}},\
\underbrace{1,\cdots,1}_{p^{(i)}_{2}},\
\underbrace{0,\cdots,0}_{q^{(i)}_{2}},\ \cdots\
\underbrace{1,\cdots,1}_{p^{(i)}_{n}},\
\underbrace{0,\cdots,0}_{q^{(i)}_{n}},\cdots\right\\}$

of the orbit over the set $U_{i}$ has $(\lambda_{i},\mu_{i})-$gap for all
$i=1,2$, i.e.,

$\displaystyle\liminf\limits_{n\to\infty}\frac{p^{(i)}_{n+1}}{\sum\limits_{k=1}^{n}(p^{(i)}_{k}+q^{(i)}_{k})}\geq\lambda_{i}\quad\textup{and}\quad\liminf\limits_{n\to\infty}\frac{q^{(i)}_{n+1}}{\sum\limits_{k=1}^{n}(p^{(i)}_{k}+q^{(i)}_{k})+p^{(i)}_{n+1}}\geq\mu_{i},\quad
i=1,2$

where $\chi_{U}:U\to\mathbb{R}$ is a characteristic function of a set $U$,
i.e., $\chi_{U}(\mathbf{x})=1$ if $\mathbf{x}\in U$ and
$\chi_{U}(\mathbf{x})=0$ if $\mathbf{x}\not\in U$.

One of the main results of the paper [31] (see Theorem B, [31]) states that if
the discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) has uniformly
historic behavior then for any
$\mathbf{x}\in\mathcal{HIS}_{\mathcal{R}_{H}}(U_{1},U_{2})$, the
$s^{th}-$order repeated time averages
$\left\\{\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right)\right\\}_{n=1}^{\infty}$
of the discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ do not converge for any
$s\in\mathbb{N}$. That’s why, in order to prove Theorem B, it is enough to
show that the discrete-time replicator equation
$\mathcal{R}_{H}:\mathbb{S}^{2}\to\mathbb{S}^{2}$ given by (2.2) has uniformly
historic behavior.

Step-1:

Since $f_{1}(\mathbf{x}),f_{2}(\mathbf{x}),f_{3}(\mathbf{x})>0$ for any
$\mathbf{x}\in\mathbb{S}^{2}$, it is easy to check that the rest (fixed)
points of the discrete-time replicator equation (2.2) which lie on the
boundary
$\partial\mathbb{S}^{2}=\\{\mathbf{x}\in\mathbb{S}^{2}:x_{1}x_{2}x_{3}=0\\}$
of 2D simplex $\mathbb{S}^{2}$ are only three vertices
$\mathbf{e}_{1}=(1,0,0)$, $\mathbf{e}_{2}=(0,1,0)$, and
$\mathbf{e}_{3}=(0,0,1)$. Moreover, since
$f_{1}(\mathbf{c})=f_{2}(\mathbf{c})=f_{3}(\mathbf{c})$, the center
$\mathbf{c}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ of 2D simplex is an
interior rest (fixed) point. We now show that there is no any other interior
rest point except the center
$\mathbf{c}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ of the simplex
$\mathbf{x}\in\mathbb{S}^{2}$. Let $\mathbf{x}\in\textup{int}\mathbb{S}^{2}$
be any interior rest (fixed) point. We must then have
$x_{1}f_{1}(\mathbf{x})=x_{3}f_{2}(\mathbf{x})$,
$x_{2}f_{2}(\mathbf{x})=x_{1}f_{3}(\mathbf{x})$, and
$x_{3}f_{3}(\mathbf{x})=x_{2}f_{1}(\mathbf{x})$. Consequently, we derive

$x_{1}f_{1}(\mathbf{x})+x_{2}f_{2}(\mathbf{x})+x_{3}f_{3}(\mathbf{x})=x_{3}f_{2}(\mathbf{x})+x_{1}f_{3}(\mathbf{x})+x_{2}f_{1}(\mathbf{x}).$

Since $f_{i}(\mathbf{x})\left(\gtreqqless\right)f_{j}(\mathbf{x})$ if and only
if $x_{i}\left(\gtreqqless\right)x_{j}$ for all $i,j\in\mathbf{I}_{3}$, it
follows from the rearrangement inequality (see [20]) that
$\mathbf{x}=\mathbf{c}$.

Three vertices of the simplex are the saddle points which are attractive only
along the boundaries of the simplex in the clockwise direction. Moreover,
since a similar-order preserving mapping is continuously differentiable, the
center of the simplex is a repelling rest (fixed) point with complex
eigenvalues which induce a spiral behavior towards the boundary of the
simplex. We are aiming to show that the orbit of any interior initial point
(except the center of the simplex) alternatively spends exponentially
increasing amount of times close to the consecutive vertices (by clockwise
direction) of the simplex.

Step-2:

We now show that $\xi(\mathbf{x}):=x_{1}x_{2}x_{3}$ is a decreasing Lyapunov
function. Indeed, it follows from the arithmetic-geometric mean and
rearrangement inequalities (see [20]) that

$\displaystyle\xi(\mathcal{R}_{H}(\mathbf{x}))\leq\xi(\mathbf{x})\left(1+\frac{\langle\pi(\mathbf{x}),\mathbf{F}(\mathbf{x})\rangle-\langle\mathbf{x},\mathbf{F}(\mathbf{x})\rangle}{3}\right)^{\frac{1}{3}}\leq\xi(\mathbf{x}).$

where $\pi(\mathbf{x}):=(x_{2},x_{3},x_{1})$. Hence,
$\left\\{\xi\left(\mathcal{R}_{H}^{(n)}(\mathbf{x})\right)\right\\}_{n=0}^{\infty}$
is a decreasing sequence and converges to some limit $\lambda$. Since
$\mathbf{x}\neq\mathbf{c}$, we have

$0\leq\lambda<\xi(\mathbf{x})<\xi(\mathbf{c}).$

If $\lambda>0$ then

$1=\lim\frac{\xi\big{(}\mathcal{R}_{H}^{(n+1)}(\mathbf{x})\big{)}}{\xi\big{(}\mathcal{R}_{H}^{(n)}(\mathbf{x})\big{)}}=\lim\zeta\big{(}\mathcal{R}_{H}^{(n)}(\mathbf{x})\big{)}\leq
1,$

where

$\zeta(\mathbf{x})=(1+x_{2}f_{1}(\mathbf{x})-x_{3}f_{3}(\mathbf{x}))(1+x_{3}f_{2}(\mathbf{x})-x_{1}f_{1}(\mathbf{x}))(1+x_{1}f_{3}(\mathbf{x})-x_{2}f_{2}(\mathbf{x})).$

Consequently, for any $\mathbf{x}^{*}\in\omega(\mathbf{x})$ (where
$\omega(\mathbf{x})$ is an omega-limit set of an orbit starting from an
initial point $\mathbf{x}$), one has
$1=\zeta(\mathbf{x}^{*})=\max\limits_{\mathbf{y}\in\mathbb{S}^{2}}\zeta(\mathbf{y})=1$,
i.e, $\mathbf{x}^{*}=\mathbf{c}$. However, this contradicts to
$\xi(\mathbf{x}^{*})=\lambda<\xi(\mathbf{c})$. It shows that $\lambda=~{}0$,
i.e., $\omega(\mathbf{x})\subset\partial\mathbb{S}^{2}$.

Obviously, we have
$\omega(\mathbf{x})\neq\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\\}$.
Since $\mathcal{R}_{H}(\omega(\mathbf{x}))=\omega(\mathbf{x})$, the orbit
$\\{\mathcal{R}_{H}^{(n)}(\mathbf{x}^{*})\\}_{n=0}^{\infty}\subset\omega(\mathbf{x})$
is infinite for any
$\mathbf{x}^{*}\in\omega(\mathbf{x})\setminus\\{\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}\\}$.
It means that an omega-limit set $\omega(\mathbf{x})$ of an orbit starting
from an initial point
$\mathbf{x}\in{\textup{int}}\mathbb{S}^{2}\setminus\\{\mathbf{c}\\}$ is
infinite and lies on the boundary $\partial\mathbb{S}^{2}$ of the simplex
$\mathbb{S}^{2}$.

Step-3:

An orbit starting from any interior initial point eventually moves along the
periodic itinerary

$G_{1}\hookrightarrow G_{2}\hookrightarrow G_{3}\hookrightarrow
G_{4}\hookrightarrow G_{5}\hookrightarrow G_{6}\hookrightarrow G_{1}$

where $G_{i}\hookrightarrow G_{j}$ stands for $\mathcal{R}_{H}(G_{i})\subset
G_{i}\cup G_{j}$ and the set $G_{i}$ is defined as follows for $i=1,2\cdots,6$

$\displaystyle G_{1}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{1}\geq x_{2}\geq
x_{3}\Big{\\}},\quad G_{2}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{1}\geq
x_{3}\geq x_{2}\Big{\\}},$ $\displaystyle
G_{3}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{3}\geq x_{1}\geq
x_{2}\Big{\\}},\quad G_{4}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{3}\geq
x_{2}\geq x_{1}\Big{\\}},$ $\displaystyle
G_{5}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{2}\geq x_{3}\geq
x_{1}\Big{\\}},\quad G_{6}=\Big{\\{}\mathbf{x}\in\mathbb{S}^{2}:x_{2}\geq
x_{1}\geq x_{3}\Big{\\}}.$

Let us show that $G_{1}\hookrightarrow G_{2}$. The rest is similar to this
case. Let
$\partial\mathbb{S}^{2}_{\varepsilon}=\\{\mathbf{x}\in\mathbb{S}^{2}:{\textup{dist}}(\mathbf{x},\partial\mathbb{S}^{2})<\varepsilon\\}$.
Since $\omega(\mathbf{x})\subset\partial\mathbb{S}^{2}$, one has
$\mathcal{R}_{H}^{(n)}(\mathbf{x})\in\partial\mathbb{S}^{2}_{\varepsilon}$ for
sufficiently small $\varepsilon>0$ and $n\geq n_{0}$. We show
$\mathcal{R}_{H}(G_{1}\cap\partial\mathbb{S}^{2}_{\varepsilon})\subset
G_{1}\cup G_{2}$.

Let $\mathbf{x}=(x_{1},x_{2},x_{3})\in
G_{1}\cap\partial\mathbb{S}^{2}_{\varepsilon}$ and
$\mathbf{x}^{(1)}=(x_{1}^{(1)},x_{2}^{(1)},x_{3}^{(1)})=\mathcal{R}_{H}(\mathbf{x})$.
Since $\mathbf{x}\in G_{1}$, we have $x_{1}\geq x_{2}\geq x_{3}$. Moreover,
since $f_{i}(\mathbf{x})\left(\gtreqqless\right)f_{j}(\mathbf{x})$ if and only
if $x_{i}\left(\gtreqqless\right)x_{j}$ for all $i,j\in\mathbf{I}_{3}$, we
obtain

$x_{2}f_{1}(\mathbf{x})-x_{3}f_{3}(\mathbf{x})\geq 0\quad\textup{and}\quad
x_{3}f_{2}(\mathbf{x})-x_{1}f_{1}(\mathbf{x})\leq 0.$

Consequently, we have ${{x}^{(1)}_{1}}\geq{x_{1}}\geq{x}_{2}\geq x^{(1)}_{2}.$
Obviously, for $\mathbf{x}\in G_{1}\cap\partial\mathbb{S}^{2}_{\varepsilon}$,
one has $x_{3}<\varepsilon$ for sufficiently small $\varepsilon$. We then
obtain ${x^{(1)}_{3}}\leq 2x_{3}\leq
2\varepsilon\leq{x_{1}}\leq{x^{(1)}_{1}}.$ Therefore, we obtain
$\mathcal{R}_{H}(G_{1}\cap\partial\mathbb{S}^{2}_{\varepsilon})\subset
G_{1}\cup G_{2}$.

Step-4:

We choose a neighborhood $\mathbb{U}_{0}$ of the repelling rest point
$\mathbf{c}$ such that $\mathbb{U}_{0}\subset\textup{int}\mathbb{S}^{2}$ and
$\mathbb{U}_{1}=(G_{1}\cup G_{2})\setminus\mathbb{U}_{0}$,
$\mathbb{U}_{2}=(G_{3}\cup G_{4})\setminus\mathbb{U}_{0}$,
$\mathbb{U}_{3}=(G_{5}\cup G_{6})\setminus\mathbb{U}_{0}$ are convex compact
sets which satisfy
$\mathbb{U}_{1}\cap\mathbb{U}_{2}\cap\mathbb{U}_{3}=\emptyset$

Let $\mathbb{U}$ be one of the sets $\mathbb{U}_{1}$, $\mathbb{U}_{2}$, and
$\mathbb{U}_{3}$ defined above and let
$\mathbf{x}\notin\mathbb{U}\cup\mathbb{U}_{0}$,
$\mathcal{R}_{H}^{(k)}(\mathbf{x})\in\mathbb{U}$ for all $1\leq k\leq n$, and
$\mathcal{R}_{H}^{(n+1)}(\mathbf{x})\notin\mathbb{U}$. Then we have
$n>A\log\frac{B}{\xi(\mathcal{R}_{H}(\mathbf{x}))}$ where $A$ and $B$ are some
positive constants.

Indeed, let $\mathbf{x}=(x_{1},x_{2},x_{3})$ and
$\mathbf{x}^{(k)}=(x^{(k)}_{1},x_{2}^{(k)},x_{3}^{(k)})=\mathcal{R}_{H}^{(k)}(\mathbf{x})$.
We also assume $\mathbb{U}:=\mathbb{U}_{1}$. Since
$\mathbf{x}\notin\mathbb{U}_{1}$,
$\mathcal{R}_{H}^{(k)}(\mathbf{x})\in\mathbb{U}_{1}$ for all $1\leq k\leq n$,
and $\mathcal{R}_{H}^{(n+1)}(\mathbf{x})\notin\mathbb{U}_{1}$, we then obtain
$\mathbf{x}\in G_{6}$, $\mathcal{R}_{H}(\mathbf{x})\in G_{1}$, and
$\mathcal{R}_{H}^{(n+1)}(\mathbf{x})\in G_{3}$. Therefore,
$x_{2}\geq\frac{1}{3}$, $x^{(1)}_{1}\geq\frac{1}{3}$, and
$x^{(n+1)}_{3}\geq\frac{1}{3}$. Hence, we have that

$x_{2}^{(1)}=x_{2}\left(1+x_{3}f_{2}(\mathbf{x})-x_{1}f_{1}(\mathbf{x})\right)\geq
x_{2}^{2}\geq\frac{1}{9}$

and

$\displaystyle\frac{1}{81\xi(\mathcal{R}_{H}(\mathbf{x}))}\leq\frac{x_{1}^{(1)}x_{2}^{(1)}}{3\xi(\mathcal{R}_{H}(\mathbf{x}))}=\frac{1}{3x_{3}^{(1)}}\leq\frac{x_{3}^{(n+1)}}{x_{3}^{(1)}}=\prod\limits_{k=1}^{n}\frac{x_{3}^{(k+1)}}{x_{3}^{(k)}}\leq
2^{n},$

which yields $n>A\log\frac{B}{\xi(\mathcal{R}_{H}(\mathbf{x}))}$ for some
constants $A>0$ and $B>0$.

Step-5:

Let $\mathbb{U}$ be one of the sets $\mathbb{U}_{1}$, $\mathbb{U}_{2}$, and
$\mathbb{U}_{3}$ given above and let $(n_{i},m_{i})_{i=1}^{\infty}$ be a
sequence of natural numbers such that
$\mathcal{R}_{H}^{(n_{i})}(\mathbf{x})\notin\mathbb{U}$,
$\mathcal{R}_{H}^{(n_{i}+k)}(\mathbf{x})\in\mathbb{U}$ for $1\leq k\leq
m_{i}$, and $\mathcal{R}_{H}^{(n_{i}+m_{i}+1)}(\mathbf{x})\notin\mathbb{U}$.

It follows from Step-2 and Step-3 that $n_{i}\to\infty$. Let
$\rho:=\max\limits_{\mathbf{x}\in\mathbb{S}^{2}\setminus\mathbb{U}_{0}}\zeta(\mathbf{x})<1$
(see Step-2). Then, we obtain that

$\xi(\mathcal{R}_{H}(\mathbf{x}))=\xi(\mathbf{x})\zeta(\mathbf{x})<\rho\xi(\mathbf{x})$

for any $\mathbf{x}\in\mathbb{S}^{2}\setminus\mathbb{U}_{0}$. It follows from
Step-4 that

$m_{i}>A\log\frac{B}{\xi(\mathcal{R}_{H}^{(n_{i})}(\mathbf{x}))}>A\log\frac{B}{\rho^{n_{i}}\xi(\mathbf{c})}>Cn_{i}$

for some constant $C>0$.

Step-6:

It follows from Step-3 that the forward orbit eventually moves along the
periodic itinerary

$\underbrace{{\mathbb{U}_{1}}\ {{}^{\curvearrowright}}\
{\mathbb{U}_{1}}}_{p_{1}}\rightarrow\underbrace{{\mathbb{U}_{2}}\
{{}^{\curvearrowright}}\
{\mathbb{U}_{2}}}_{q_{1}}\rightarrow\underbrace{{\mathbb{U}_{3}}\
{{}^{\curvearrowright}}\ {\mathbb{U}_{3}}}_{r_{1}}\rightarrow\\\
\underbrace{{\mathbb{U}_{1}}\ {{}^{\curvearrowright}}\
{\mathbb{U}_{1}}}_{p_{2}}\rightarrow\underbrace{{\mathbb{U}_{2}}\
{{}^{\curvearrowright}}\
{\mathbb{U}_{2}}}_{q_{2}}\rightarrow\underbrace{{\mathbb{U}_{3}}\
{{}^{\curvearrowright}}\ {\mathbb{U}_{3}}}_{r_{2}}\rightarrow\\\
\cdots\rightarrow\underbrace{{\mathbb{U}_{1}}\ {{}^{\curvearrowright}}\
{\mathbb{U}_{1}}}_{p_{n}}\rightarrow\underbrace{{\mathbb{U}_{2}}\
{{}^{\curvearrowright}}\
{\mathbb{U}_{2}}}_{q_{n}}\rightarrow\underbrace{{\mathbb{U}_{3}}\
{{}^{\curvearrowright}}\ {\mathbb{U}_{3}}}_{r_{n}}\rightarrow\cdots$

where $\\{p_{n}\\}$, $\\{q_{n}\\}$, and $\\{r_{n}\\}$ are sequences of
trapping times of the forward orbit over the sets $\mathbb{U}_{1}$,
$\mathbb{U}_{2}$, and $\mathbb{U}_{3}$, respectively. It follows from Step-5
that there exists a constant $C>0$ such that

$p_{n+1}>C\sum_{i=1}^{n}(p_{i}+q_{i}+r_{i}),\\\
q_{n+1}>C\left(\sum_{i=1}^{n}(p_{i}+q_{i}+r_{i})+p_{n+1}\right),\\\
r_{n+1}>C\left(\sum_{i=1}^{n}(p_{i}+q_{i}+r_{i})+p_{n+1}+q_{n+1}\right).$

Hence, the trapping-escaping sequence
$\left\\{\chi_{\mathbb{U}_{i}}(\mathcal{R}_{H}^{(n)}\mathbf{x})\right\\}_{n=0}^{\infty}$
has $(C;C(C+2))-$ gap (see [31]) for $i=1,2,3$ where
$\chi_{\mathbb{U}}(\cdot)$ is a characteristic function of a set $\mathbb{U}$,
i.e., $\chi_{\mathbb{U}}(\mathbf{x})=1$ if $\mathbf{x}\in\mathbb{U}$ and
$\chi_{\mathbb{U}}(\mathbf{x})=0$ if $\mathbf{x}\not\in\mathbb{U}$.
Consequently, the discrete-time replicator equation of a zero-sum game given
by (2.2) has uniformly historic behavior which eventually causes the
divergence of higher-order repeated time averages
$\left\\{\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right)\right\\}_{n=1}^{\infty}$
for any $s\in\mathbb{N}$ where
$\mathcal{A}^{(s)}_{n}\left(\mathbf{x}\right):=\frac{1}{n}\sum_{k=1}^{n}\mathcal{A}^{(s-1)}_{k}\left(\mathbf{x}\right)$
and
$\mathcal{A}^{(1)}_{n}\left(\mathbf{x}\right):=\frac{1}{n}\sum_{k=0}^{n-1}\mathcal{R}_{H}^{(k)}(\mathbf{x})$
(see Theorem B, [31]). It completes the proof of Theorem B.

## References

  * [1] K. Baranski and M. Misiurewicz, _Omega-limit sets for the Stein-Ulam spiral map_ , Top. Proc. 36 (2010), 145–172.
  * [2] P.G. Barrientos, S. Kiriki, Y. Nakano, A. Raibekas, and T. Soma, _Historic behavior in nonhyperbolic homoclinic classes_ , Proc. Amer. Math. Soc. 148 (2020), 1195–1206.
  * [3] M. Carvalho and P. Varandas, _Genericity of historic behavior for maps and flows_ , Nonlinearity 34 (2021), no. 10, 7030–7044.
  * [4] R. Cressman, _Evolutionary dynamics and extensive form games_ , MIT Press, Cambridge, 2003.
  * [5] H. L. de Santana, _Historic behavior for flows with the gluing orbit property_ , J. Korean Math. Soc. 59 (2022), no. 2, 337–352.
  * [6] Y. Dowker, _The mean and transitive points of homeomorphisms_ , Ann. Math. 58 (1953), no. 1, 123–133.
  * [7] D. Friedman, _On economic applications of evolutionary game theory_ , J. Evol. Econ. 8 (1998), no. 1, 15–43.
  * [8] N. Ganikhodzhaev and D. Zanin, _On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex_ , Russ. Math. Surv. 59 (2004), no. 3, 571–572.
  * [9] A. Gaunersdorfer, _Time averages for heteroclinic attractors_ , SIAM J. Math. Anal. 52 (1992), 1476–1489.
  * [10] J. Hofbauer, _Heteroclinic cycles in ecological differential equations_ , Tatra Mount. Math. Publ. 4 (1994), 105–116.
  * [11] J. Hofbauer and K. Sigmund, _Evolutionary dynamics and extensive form games_ , Cambridge University Press, Cambridge, 1998.
  * [12] by same author, _Evolutionary game dynamics_ , Bull. Amer. Math. Soc. 40 (2003), no. 4, 479–519.
  * [13] U. Jamilov and F. Mukhamedov, _A class of Lotka-Volterra operators with historical behavior_ , Results Math 77 (2022), no. 4, 169.
  * [14] by same author, _Historical behavior for a class of Lotka-Volterra systems_ , Math Meth Appl Sci. (2022), 1–10.
  * [15] U. Jamilov, M. Scheutzow, and I.Vorkastner, _A prey-predator model with three interacting species_ , (2019).
  * [16] S. Kiriki, M. Li, and T. Soma, _Geometric lorenz flows with historic behavior_ , Disc. Cont. Dyn. Sys. 36 (2016), no. 12, 7021–7028.
  * [17] S. Kiriki, Y. Nakano, and T. Soma, _Historic behaviour for nonautonomous contraction mappings_ , Disc. Cont. Dyn. Sys. 32 (2019), no. 3, 1111–1124.
  * [18] by same author, _Emergence via non-existence of averages_ , Adv. Math. 400 (2022), 108254.
  * [19] S. Kiriki and T. Soma, _Takens’ last problem and existence of non-trivial wandering domains_ , Adv. Math. 306 (2017), 524–588.
  * [20] A. Marshall, I. Olkin, and B. Arnold, _Inequalities: Theory of majorization and its applications_ , Springer, New York, 2011.
  * [21] J.F. Nash, _Non-cooperative games_ , Ann. Math 54 (1951), no. 2, 287–295.
  * [22] B. Perthame, M. Strugarek, and C. Taing, _Selection–mutation dynamics with asymmetrical reproduction kernels_ , Nonlinear Analysis 222 (2022), 112947.
  * [23] D. Ruelle, _Historic behavior in smooth dynamical systems_ , Global Analysis of Dynamical Systems (London) (H. W. Broer et al, ed.), IOP Publishing, 2001, pp. 63–66.
  * [24] M. Saburov, _A class of nonergodic Lotka–Volterra operators_ , Math. Notes 97 (2015), no. 5–6, 759–763.
  * [25] by same author, _On divergence of any order Cesaro mean of Lotka–Volterra operators_ , Ann. Fun. Anal. 6 (2015), no. 4, 247–254.
  * [26] by same author, _Dichotomy of iterated means for nonlinear operators_ , Funct. Anal. its Appl. 52 (2018), no. 1, 74–76.
  * [27] by same author, _Nonergodic quadratic stochastic operators_ , Math Notes 106 (2019), no. 1, 142–145.
  * [28] by same author, _Iterated means dichotomy for discrete dynamical systems_ , Qual. Theory Dyn. Syst. 19 (2020), 25.
  * [29] by same author, _The discrete-time kolmogorov systems with historic behavior_ , Math Meth Appl Sci. 44 (2021), no. 1, 813–819.
  * [30] by same author, _On replicator equations with nonlinear payoff functions defined by the Ricker models_ , Advances Pure Applied Mathematics 12 (2021), 139–156.
  * [31] by same author, _Uniformly historic behaviour in compact dynamical systems_ , J. Differ. Equ. Appl. 27 (2021), no. 7, 1006–1023.
  * [32] by same author, _On discrete-time replicator equations with nonlinear payoff functions_ , Dyn. Games Appl. 12 (2022), no. 2, 643–661.
  * [33] W.H. Sandholm, _Population games and evolutionary dynamics_ , MIT Press, Cambridge, 2010.
  * [34] P. Schuster and K. Sigmund, _Replicator dynamics_ , J Theor Biol 100 (1983), no. 3, 533–538.
  * [35] K. Sigmund, _Time averages for unpredictable orbits of deterministic systems_ , Ann. Oper. Res. 37 (1992), 217–228.
  * [36] by same author, _Evolutionary game dynamics: American mathematical society short course_ , Providence, Rhode Island,, 2010.
  * [37] J. Maynard Smith and G.R. Price, _The logic of animal conflict_ , Nature 246 (1973), no. 5427, 15–18.
  * [38] P. Sun, D. Hou, and H. Sun, _The shapley value for cooperative games with restricted worths_ , J. Math. Anal. Appl. 495 (2021), no. 2, 124762.
  * [39] F. Takens, _Orbits with historic behavior, or non-existence of averages – Open Problem_ , Nonlinearity 21 (2008), no. 3, 33–36.
  * [40] P.D. Taylor and L. Jonker, _Evolutionarily stable strategies and game dynamics_ , Math. Biosci. 40 (1978), no. 1-2, 145–156.
  * [41] B. Thomas, _Evolutionary stability: states and strategies_ , Theor. Popul. Biol. 26 (1984), no. 1, 49–67.
  * [42] by same author, _On evolutionarily stable sets_ , J. Math. Biol. 22 (1985), 105–115.
  * [43] D. Thompson, _The irregular set for maps with the specification property has full topological pressure_ , Dynamical Systems 25 (2010), no. 1, 25–51.
  * [44] by same author, _Irregular sets, the -transformation and the almost specification property_ , Trans. Amer. Math. Soc. 364 (2012), no. 10, 5395–5414.
  * [45] S. S. Vallander, _The limiting behavior of the sequences of iterates of certain quadratic transformations_ , Soviet Math. Dokl. 13 (1972), 123–126.
  * [46] by same author, _One family of cubic stochastic transformations_ , OP–PM Surveys on Applied and Industrial Mathematics 14 (2007), no. 6, 1074.
  * [47] D. Yang, _On the historical behavior of singular hyperbolic attractors_ , Proc. Amer. Math. Soc. 148 (2020), 1641–1644.
  * [48] M. Zakharevich, _On the behaviour of trajectories and the ergodic hypothesis for quadratic mappings of a simplex_ , Russ. Math. Surv. 33 (1978), no. 6, 265–266.