#  Non-Hermitian topology in evolutionary game theory: Exceptional points and
skin effects in rock-paper-scissors cycles

Tsuneya Yoshida  Tomonari Mizoguchi  Yasuhiro Hatsugai  Department of Physics,
University of Tsukuba, Ibaraki 305-8571, Japan

###### Abstract

Non-Hermitian topology is a recent hot topic in condensed matters. In this
paper, we propose a novel platform drawing interdisciplinary attention: rock-
paper-scissors (RPS) cycles described by the evolutionary game theory.
Specifically, we demonstrate the emergence of an exceptional point and a skin
effect by analyzing topological properties of their payoff matrix.
Furthermore, we discover striking dynamical properties in an RPS chain: the
directive propagation of the population density in the bulk and the
enhancement of the population density only around the right edge. Our results
open new avenues of the non-Hermitian topology and the evolutionary game
theory.

In these decades, the notion of the topology has played a central role in
condensed matter physics Kane and Mele (2005); Bernevig  _et al._ (2006);
König  _et al._ (2007); Hasan and Kane (2010); Qi and Zhang (2011). Analysis
of topological aspects of condensed matters dates back to integer quantum Hall
systems Klitzing  _et al._ (1980) which exhibit robust chiral edge modes
induced by the topology of the Hermitian Hamiltonian in the bulk Thouless  _et
al._ (1982); Halperin (1982); Hatsugai (1993). While these robust edge modes
are originally reported for quantum systems, they have been extended to
various disciplines of science Haldane and Raghu (2008); Wang  _et al._
(2009); Fu  _et al._ (2010); Ozawa  _et al._ (2019); Prodan and Prodan (2009);
Kane and Lubensky (2013); Kariyado and Hatsugai (2015); Süsstrunk and Huber
(2015); Delplace  _et al._ (2017); Sone and Ashida (2019); Yoshida and
Hatsugai (2021). In particular, the notion of topology has been extended to an
interdisciplinary field Knebel  _et al._ (2020); Yoshida  _et al._ (2021); the
emergence of topological edge modes has been predicted Yoshida  _et al._
(2021) for networks of rock-paper-scissors (RPS) cycles which are described by
the evolutionary game theory Weibull (1997); Sigmund (2011); Kirkup and Riley
(2004); Rosas (2010); Wang  _et al._ (2013); Perc  _et al._ (2017).

Among the variety of topological phenomena, non-Hermitian topology Hatano and
Nelson (1996); Bender and Boettcher (1998); Gong  _et al._ (2018); Kawabata
_et al._ (2019); Bergholtz  _et al._ (2021); Ashida  _et al._ (2020) has
become one of the recent hot topics because it results in novel phenomena
which do not have Hermitian counterparts. A representative example is the
emergence of exceptional points (EPs) Rotter (2009); Shen  _et al._ (2018);
Kozii and Fu (2017); Yoshida  _et al._ (2018) (and their symmetry-protected
variants Budich  _et al._ (2019); Okugawa and Yokoyama (2019); Yoshida  _et
al._ (2019); Zhou  _et al._ (2019); Kimura  _et al._ (2019); Yoshida and
Hatsugai (2019); Delplace  _et al._ (2021)) on which eigenvalues of the non-
Hermitian Hamiltonian touch both for the real- and imaginary-parts. Another
typical example is a non-Hermitian skin effect Yao and Wang (2018); Lee and
Thomale (2019); Zhang  _et al._ (2020); Okuma  _et al._ (2020); Lee  _et al._
(2019); Yoshida  _et al._ (2020) which is extreme sensitivity of the
eigenvalues and eigenstates to the presence/absence of boundaries. As is the
case of the Hermitian topology, the above non-Hermitian phenomena have also
been reported in a variety of systems Guo  _et al._ (2009); Zhen  _et al._
(2015); Hassan  _et al._ (2017); Zhou  _et al._ (2018); Zyuzin and Zyuzin
(2018); Papaj  _et al._ (2019); Shen and Fu (2018); Matsushita  _et al._
(2019); Hofmann  _et al._ (2020); Helbig  _et al._ (2020) (e.g., photonic
systems Guo  _et al._ (2009); Zhen  _et al._ (2015); Hassan  _et al._ (2017);
Zhou  _et al._ (2018) and quantum systems Zyuzin and Zyuzin (2018); Papaj  _et
al._ (2019); Shen and Fu (2018); Matsushita  _et al._ (2019, 2021)).

Despite the above significant progress, topological phenomena of the
evolutionary game theory, which attract interdisciplinary attention, are
restricted to the Hermitian topology. Highlighting non-Hermitian topology of
such systems is significant as it may provide a new insights and may open a
new avenue of the evolutionary game theory.

In this paper, we report non-Hermitian topological phenomena in the
evolutionary game theory: an EP and a skin effect in RPS cycles. The EP in the
single RPS cycle is protected by the realness of the payoffs which is
mathematically equivalent to parity-time ($PT$) symmetry. Our linearized
replicator equation elucidates that the EP governs dynamics of the RPS cycle.
Furthermore, we discover striking dynamical phenomena in an RPS chain induced
by the skin effect: the directive propagation of the population density in the
bulk and the enhancement of the population density only around the right edge.
These dynamical properties are in sharp contrast to those in the Hermitian
systems. The above results open new avenues of the non-Hermitian topology and
the evolutionary game theory.

Results
EP in a single RPS cycle. Firstly, we demonstrate the emergence of an EP at
which two eigenvalues touch both for the real- and imaginary-parts.

Consider two players play the RPS game [see Fig. 1(a)] whose payoff matrix is
given by

$\displaystyle A(\lambda)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&-1&1\\\ 1&0&-1\\\
-1&1&0\end{array}\right)+\lambda\left(\begin{array}[]{ccc}0&0&0\\\ 0&1&-1\\\
0&-1&1\end{array}\right),$ (7)

with a real number $\lambda$. Here, players choose one of the strategies
$(s_{1},s_{2},s_{3})=(``\mathrm{R}",``\mathrm{P}",``\mathrm{S}")$. The payoff
of a player is $A_{IJ}$ when the player chooses the strategy $s_{I}$ and the
other player chooses $s_{J}$ ($I,J=1,2,3$). For $\lambda=0$, this game is
reduced to the standard zero-sum RPS game where the sum of all players’ payoff
is zero for an arbitrary set of strategies.

The above payoff matrix exhibits an EP, which can be deduced by noting the
following two facts. (i) The first (second) term is anti-Hermitian
(Hermitian). (ii) For an arbitrary $\lambda$, eigenvalues $\epsilon_{n}$
($n=1,2,3$) form a pair $\epsilon_{n}=\epsilon^{*}_{n^{\prime}}$ ($n\neq
n^{\prime}$) or are real numbers $\epsilon_{n}\in\mathbb{R}$. This constraint
arises from the realness of the payoffs

$\displaystyle\mathcal{K}A(\lambda)\mathcal{K}$ $\displaystyle=$
$\displaystyle A(\lambda),$ (8)

where the operator $\mathcal{K}$ takes complex conjugate. Equation (8) can be
recognized as $PT$ symmetry by regarding $\lambda$ as a momentum (for more
details, see Sec. S1 of Supplemental Material sup, ). For $\lambda=0$, the
energy eigenvalues are aligned along the imaginary axis due to the anti-
Hermiticity of $A(\lambda=0)$. Increasing $\lambda$, two eigenvalues touch at
a critical value $\lambda_{\mathrm{c}}$ so that they become real numbers when
the second term is dominant. At $\lambda=\lambda_{\mathrm{c}}$, the EP
emerges.

The emergence of the EP is supported by Fig. 1(b). For $\lambda=0$, the energy
eigenvalues are pure imaginary due to anti-Hermiticity of $A(\lambda=0)$. As
$\lambda$ is turned on, two eigenvalues approach each other, and the EP emerge
at $\lambda=\lambda_{c}=\sqrt{3}$. One can also characterize the topology of
this EP by computing the $\mathbb{Z}_{2}$-invariant $\nu$ Delplace  _et al._
(2021)

$\displaystyle\nu(\lambda)$ $\displaystyle=$
$\displaystyle\mathrm{sgn}\left[\mathrm{Disc}_{E}P(E,\lambda)\right],$ (9)

with $P(E,\lambda)=\mathrm{det}[A(\lambda)-E\mbox{1}\mbox{l}]$. Here,
$\mathrm{sgn}(x)$ takes $1$ ($-1$) for $x>0$ ($x<0$), and
$\mathrm{Disc}_{E}P(E,\lambda)$ denotes discriminant of $P(E,\lambda)$. For
the $3\times 3$-matrix $A(\lambda)$,
$\mathrm{Disc}_{E}P(E,\lambda)=(\epsilon_{1}-\epsilon_{2})^{2}(\epsilon_{1}-\epsilon_{3})^{2}(\epsilon_{2}-\epsilon_{3})^{2}$
holds. Figure 1(c) plots the $\mathbb{Z}_{2}$-invariant as a function of
$\lambda$. Corresponding to the emergence of the EP, the
$\mathbb{Z}_{2}$-invariant jumps at $\lambda=\lambda_{\mathrm{c}}$,
elucidating the topological protection of the EP.

Figure 1:  (a): Sketch of the RPS cycle. The arrows denote dominance
relationship between the strategies for $\lambda=0$; “R” beats “S”; “S” beats
“P”; “P” beats “R”. The second terms proportional to $\lambda$ are introduced
between sites connected by the double line. (b): The real- and imaginary-parts
of eigenvalues as functions of $\lambda$. An eigenvalue of $A(\lambda)$ is
zero for an arbitrary $\lambda$ (see horizontal dashed lines). (c): The
$\mathbb{Z}_{2}$-invariant characterizing the EP. The dashed vertical lines in
panels (b) and (c) denote $\lambda=\lambda_{\mathrm{c}}$.

Dynamical properties and the EP.  Suppose that a large number of players
repeat the above game whose dynamics is described by the replicator equation
[see Eq. (10)]. In this case, the EP governs the dynamical behaviors when the
population density slightly deviates from a fixed point; the population
density shows an oscillatory behavior for $0\leq\lambda<\lambda_{\mathrm{c}}$,
while such a behavior disappear for $\lambda_{\mathrm{c}}<\lambda$ due to the
emergence of the EP.

Firstly, we linearize the replicator equation. When a large number of players
repeat the game, the time-evolution is described by the replicator equation

$\displaystyle\partial_{t}\bm{e}^{T}_{I}\cdot\bm{x}$ $\displaystyle=$
$\displaystyle\bm{e}^{T}_{I}\cdot\bm{x}\left(\bm{e}^{T}_{I}A\bm{x}-\bm{x}^{T}A\bm{x}\right),$
(10)

where $\bm{x}$ denotes the population density ($\sum_{I}x_{I}=1$), and
$\bm{e}_{I}$ is a unit vector whose $I$-th element is unity. The second term
vanishes when the payoff matrix $A$ is anti-symmetric. However, in order to
access the non-Hermitian topology, the second term is inevitable which makes
the argument in Ref. Yoshida  _et al._ , 2021 unavailable.

Nevertheless, we can still obtain the following linearized equation

$\displaystyle\partial_{t}\delta\bm{x}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{0}}A\delta\bm{x},$ (11)

which is mathematically equivalent to the Schrödinger equation. Here,
$\delta\bm{x}$ is defined as $\delta\bm{x}=\bm{x}-\bm{c}$ with
$\bm{c}=(1,1,\ldots,1)^{T}/N_{0}$, and $N_{0}$ denotes dimensions of the
matrix $A$. Key ingredients are the following relations:

$\displaystyle\sum_{J}A_{IJ}=0\quad$ $\displaystyle\mathrm{and}$
$\displaystyle\quad\sum_{J}A_{JI}=0,$ (12)

for an arbitrary $I$. This equation guarantees that $\bm{c}$ denotes a fixed
point. Linearizing the replicator equation around this fixed point, we obtain
Eq. (11) as detailed in the “Methods”.

Figure 2:  Dynamical properties of the RPS game (7). Panels (a), (c), and (e)
display the time-evolution of the population $\bm{x}$ for $\lambda=0$, $0.5$,
and $2$, respectively. The data are obtained by employing a fourth order
Runge-Kutta method Süli and Mayers (2003) with discretized time $t_{n}=n\Delta
t_{\mathrm{RK}}$ with $\Delta t_{\mathrm{RK}}=0.05$. We set the initial
condition as $\bm{x}(t=0)=(1-\delta_{0},1+\delta_{0}/2,1+\delta_{0}/2)/3$ with
$\delta_{0}=0.1$. The time-evolution is computed up to $t_{n}=120$. Panels
(b), (d), and (f) are phase plots for $\bm{x}$. The arrows denote the
direction of the dynamics $(\Delta x_{1},\Delta x_{2},\Delta x_{3})$ which is
computed by $\Delta x_{I}=x_{I}(\bm{e}^{T}_{I}A\bm{x}-\bm{x}^{T}A\bm{x})\Delta
t$ with $\Delta t=0.1$. The black crosses denote the fixed point specified by
$\bm{c}=(1,1,1)/3$. In panels (b), (d), and (f), the blue line denotes the
population density $\bm{x}(t)$ plotted in panels (a), (c), and (e),
respectively.

The linearized replicator equation indicates that the dynamics is governed by
the spectrum of $A$. In particular, the imaginary-part of the eigenvalues
results in oscillatory behaviors.

To verify the above statement, we numerically solve the replicator equation
(10). Figure 2 plots the time-evolution of the population density for several
values of $\lambda$. For $\lambda=0$, the matrix $A(\lambda)$ is reduced to
the payoff matrix of the standard zero-sum RPS game. In this case, the
eigenvalues of $A$ are pure imaginary, which results in the oscillatory
behavior [see Fig. 2(a)]. This oscillatory behavior is also observed in the
phase plot [see Fig. 2(b)], where the orbit forms a closed loop. The above
oscillatory behavior is also observed for $\lambda=0.5$ [see Figs. 2(c) and
2(d)], which is due to the imaginary-part of the eigenvalues. We also note
that the deviation $\delta\bm{x}$ is enhanced as $t$ increases, which is
because the real-part of the eigenvalues is positive [see Fig. 1(b)]. Further
increasing $\lambda$ induces the EP [see Fig. 1(b)], and the eigenvalues
become real. Correspondingly, the above oscillatory behavior is not observed
for $\lambda=2$ [see Figs. 2(e) and 2(f)], The above numerical data verify
that the EP governs the dynamics around the fixed point $\bm{c}$; the
oscillatory behavior of the RPS cycle disappears as the EP emerges. We note
that for $\lambda<0$, the fixed point corresponds to evolutionary stable
strategy. A similar EP emerges also in this case which governs the dynamics.

We close this part with three remarks. Firstly, so far, we have seen the
emergence of the EP in the RPS game by changing $\lambda$. We note that
symmetry-protected exceptional rings are also observed by introducing an
additional parameter (see Sec. S2 of Supplemental Material sup, ), whose
topology is also characterized by the $\mathbb{Z}_{2}$-invariant $\nu$.
Secondly, although Ref. Szolnoki  _et al._ , 2014 discusses topology of
interaction networks among strategies, it differs from the topology discussed
in this paper. Thirdly, we note that EPs are also reported for active matters
Sone  _et al._ (2020); Tang  _et al._ (2021). We would like to stress,
however, that significance of this paper is to reveal the emergence of the EPs
and how they affect the dynamics in systems of the evolutionary game theory
which describes the population density of biological systems and human
societies.

Skin effect in an RPS chain.  Now, we discuss a one-dimensional system showing
the skin effect whose origin is the non-trivial topology characterized by the
winding number Zhang  _et al._ (2020); Okuma  _et al._ (2020). Because of this
non-trivial topology, switching from the periodic boundary condition (PBC) to
the open boundary condition (OBC) significantly changes the spectrum.
Correspondingly, almost of all right eigenstates are localized around the
right edge which are called skin modes.

Figure 3:  (a): Sketch of the RPS chain for $L_{x}=4$ with an additional site
at $I=9$. The arrows describes payoffs. The terms proportional to $\lambda$
are introduced between sites connected by double lines. As shown in this
panel, $I$ takes $I=1,2,3,\ldots$. For the PBC, $I+2L_{x}=I$ holds. Dashed
line denotes the unit cell. For $J=1,2,\ldots$, $R_{2I-1}=R_{2I}$ holds. (b):
Spectrum of the RPS chain for $\lambda=-0.5$. The data colored with blue are
obtained under the PBC. Here, the shade of color denotes $k/\pi$. The data
denoted by red dots are obtained for $L_{x}=15$ and the OBC. (c): Amplitude of
the right eigenvectors $\phi^{\mathrm{R}}_{In}$ ($n=1,2,\ldots,2L_{x}$) as
functions of $I$ for $L_{x}=15$. Red (gray) lines denote data for the OBC
(PBC).

The above skin effect can be observed in an RPS chain illustrated in Fig.
3(a). Applying the Fourier transformation, the payoff matrix is written as

$\displaystyle A(k)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}0&1-e^{-ik}\\\ -1+e^{ik}&-2i\sin
k\end{array}\right)+\lambda\left(\begin{array}[]{cc}-2&1+e^{-ik}\\\
1+e^{ik}&-2\end{array}\right),$ (17)

under the PBC. Here, $\bm{x}_{k}$ is defined as
$\bm{x}^{T}_{k}=(x_{k\mathrm{A}},x_{k\mathrm{B}})$ with
$x_{k\alpha}=\frac{1}{L_{x}}\sum_{R_{I}}e^{ikR_{I}}x_{R_{I}\alpha}$ and
$\alpha=\mathrm{A},\mathrm{B}$. Sets of $R_{I}$ and $\alpha_{I}$ are specified
by $I$ ($x_{I}=x_{R_{I}\alpha_{I}}$). For the explicit form of the payoff
matrix in the real-space, see Sec. S3 of Supplemental Material sup, .

Figure 3(b) plots the spectrum of the payoff matrix for $\lambda=-0.5$. When
the PBC is imposed, eigenvalues form a loop structure as denoted by blue lines
in Fig. 3(b). Accordingly, the winding number Gong  _et al._ (2018); Zhang
_et al._ (2020); Okuma  _et al._ (2020) defined as

$\displaystyle W$ $\displaystyle=$ $\displaystyle\oint\frac{dk}{2\pi
i}\partial_{k}\log\mathrm{det}[A(k)-\epsilon_{\mathrm{ref}}\mbox{1}\mbox{l}],$
(19)

takes $-1$ for $\epsilon_{\mathrm{ref}}=1$, which implies the skin effect.
Indeed, imposing the OBC significantly changes the spectrum [see red dots in
Fig. 3(b)]. Correspondingly, almost all of the right eigenvectors are
localized around the edges, meaning the emergence of skin modes [see Fig.
3(c)].

Figure 4:  Time-evolution of the population density
$\delta\bm{x}(t)=\bm{x}(t)-\bm{c}$ for $\lambda=-0.5$ and $L_{x}=15$. The
horizontal axis denotes $I$. (a)-(d) [(e)-(f)]: The time-evolution under the
PBC [OBC’]. The data in panels (a) and (e) are multiplied by 5. The data are
obtained by employing the fourth order Runge-Kutta method Süli and Mayers
(2003) with discretized time $t_{n}=n\Delta t_{\mathrm{RK}}$ with $\Delta
t_{\mathrm{RK}}=0.05$. We set the initial condition so that $N_{0}x_{I}(0)$
takes $1+\delta_{0}$ and $1-\delta_{0}$ with $\delta_{0}=0.1$ for $I=25$ and
$I=26$, respectively. For the other $I$, $N_{0}x_{I}(0)$ takes 1. Here,
$N_{0}$ is chosen so that $\sum_{I}x_{I}(0)=1$ holds.

The above data [Figs. 3(b) and 3(c)] indicate that the skin effect is observed
in the RPS chain. Here, we note that the spectrum and skin modes are almost
unchanged under the following perturbation to the edges (see Sec. S3 of
Supplemental Material sup, ): attaching site at $I=2L_{x}+1$ and tuning
diagonal elements to $A_{II}=\lambda$ only for $I=1,2L_{x}+1$ so that Eq. (12)
holds [see Fig. 3(a)]. We refer to this boundary condition as OBC’.

The above skin effect results in striking dynamical properties: the directive
propagation of the population density in the bulk and the enhancement of the
population density only around the right edge. The directive propagation is
observed by imposing the PBC. Figures 4(a)-4(d) indicate that the propagation
only to the right direction is enhanced. This phenomenon can be understood by
noting the following facts as well as linearized approximation: (i) as $t$
increases, each mode is enhanced corresponding to $\mathrm{Re}\epsilon_{n}$;
(ii) the group velocity of each mode is proportional to
$-\partial_{k}\mathrm{Im}\epsilon_{n}$. Because of the loop structure
resulting in $W=-1$, the modes propagating to the right are more enhanced than
the ones propagating to the left.

The enhancement of the population density at the right edge is observed by
imposing OBC’. We note that Eq. (12) is satisfied for OBC’ while it is not for
OBC. As shown in Figs. 4(e)-4(g), the population density around the right edge
is enhanced as $t$ increases. This phenomenon is due to the skin modes whose
eigenvalues satisfy $\mathrm{Re}\epsilon_{n}>0$. The above dynamical behaviors
are unique to non-Hermitian systems; turning off $\lambda$, $iA(\lambda=0)$
becomes Hermitian and the above behaviors disappear (see Sec. S3 of
Supplemental Material sup, ).

We close this part with two remarks. Firstly, we note that a similar behaviors
can be observed for another RPS chain, implying ubiquity of the skin effect
(see Sec. S4 of Supplemental Material sup, ). Secondly, Ref. Knebel  _et al._
, 2020 has analyzed an RPS chain whose payoff matrix is Hermitian up to the
imaginary unit $i$. We would like to stress, however, that our aim is to
observe non-Hermitian topological phenomena which are not accessible with the
model in Ref. Knebel  _et al._ , 2020. In this non-Hermitian case, we need to
take into account the second term of Eq. (10).

Discussion  We have proposed a new platform of non-Hermitian topology in an
interdisciplinary field: RPS cycles described by the evolutionary game theory.
Specifically, by analyzing the payoff matrix, we have demonstrated the
emergence of the EP and the skin effect which are representative non-Hermitian
topological phenomena. In addition, our linearized replicator equation has
revealed that the EP governs the dynamics of the population density around the
fixed point. Furthermore, we have discovered the striking dynamical phenomenon
in the RPS chain: the directive propagation of the population density in the
bulk and the enhancement of the population density only around the right edge
which are induced by the skin effect.

Our results not only provide novel platforms but also open new avenues of the
evolutionary game theory. Here, we mention several future directions. The
experimental observation is one of the central issues. In particular, our
results provide a first step toward the observation of the non-Hermitian
topology beyond natural science because the game theory describes a wide
variety of systems from biological systems Kirkup and Riley (2004); Sinervo
and Lively (1996) to human societies Rosas (2010); Wang  _et al._ (2013); Perc
_et al._ (2017). The experimental observation in such a system is considered
to be a significant step to the application of topological phenomena beyond
natural science. In addition, as is the case of equatorial wave Delplace  _et
al._ (2017), our result may provide a novel perspective of well-known
phenomena for biological systems such as bacteria Kirkup and Riley (2004) and
side-blotched lizards Sinervo and Lively (1996). We also note that the
topological classification for systems described by the game theory is also a
crucial issue to be addressed.

Methods
Derivation of the linearized replicator equation. Here, we derive Eq. (11). We
start with noting Eq. (12) results in the relations $A\bm{c}=0$ and
$\bm{c}^{T}A=0$ which mean that $\bm{c}=(1,1,1,\ldots,1)^{T}/N_{0}$ is a fixed
point. By making use of the above relations we have

$\displaystyle\partial_{t}\bm{e}^{T}_{I}\cdot\delta\bm{x}$
$\displaystyle=\bm{e}^{T}_{I}\cdot(\bm{c}+\delta\bm{x})\left[\bm{e}^{T}_{I}A(\bm{c}+\delta\bm{x})-(\bm{c}+\delta\bm{x})^{T}A(\bm{c}+\delta\bm{x})\right]$
$\displaystyle=\bm{e}^{T}_{I}\cdot(\bm{c}+\delta\bm{x})\left(\bm{e}^{T}_{I}A\delta\bm{x}-\delta\bm{x}^{T}A\delta\bm{x}\right)$
$\displaystyle\sim(\bm{e}^{T}_{I}\cdot\bm{c})\bm{e}^{T}_{I}A\delta\bm{x}.$
(20)

From the second to the third line, we have used the relations $A\bm{c}=0$ and
$\bm{c}A=0$. In the last line, we have discarded the second and third order
terms of $\delta\bm{x}$.

Because $(\bm{e}^{T}_{I}\cdot\bm{c})=1/N_{0}$ for an arbitrary $I$, we have

$\displaystyle\partial_{t}\delta\bm{e}^{T}_{I}\cdot\bm{x}$ $\displaystyle\sim$
$\displaystyle\frac{1}{N_{0}}\bm{e}^{T}_{I}A\delta\bm{x},$ (21)

which is equivalent to Eq. (11).

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Acknowledgements. This work is supported by JSPS Grant-in-Aid for Scientific
Research on Innovative Areas “Discrete Geometric Analysis for Materials
Design”: Grant No. JP20H04627. This work is also supported by JSPS KAKENHI
Grants No. JP17H06138, No. JP20K14371, and No. JP21K13850.

Author contributions T.Y., T. M., and Y. H. planned the project. T. Y.
performed the calculations. T.Y., and T. M., and Y. H. were involved in the
discussion of the materials and the preparation of the manuscript.

Competing financial interests The authors declare no competing interests.

Supplemental Materials:
Non-Hermitian topology in evolutionary game theory: Exceptional points and
skin effects in rock-paper-scissors cycles

##  S1  $PT$ symmetry

Here, we briefly review topological properties under $PT$ symmetry (i.e.,
symmetry described by the product of the time-reversal operation and
inversion). When the system is $PT$ symmetric, the Bloch Hamiltonian
$h(\bm{k})$ satisfies

$\displaystyle PTh^{*}(\bm{k})PT^{-1}$ $\displaystyle=$ $\displaystyle
h(\bm{k}),$ (S1)

with an anti-unitary operator $PT=U_{PT}\mathcal{K}$, a unitary matrix
$U_{PT}$, and $\bm{k}$ denoting the momentum. Here, each of the time-reversal
operation and the spatial inversion flips the momentum ($\bm{k}\to-\bm{k}$),
and thus, their product does not flip $\bm{k}$.

Equation (S1) results in the following constraint on the eigenvalue
$\epsilon_{n}$ ($n=1,2,\ldots$)

$\displaystyle\epsilon_{n}=\epsilon^{*}_{n^{\prime}}\ $
$\displaystyle\mathrm{or}$ $\displaystyle\ \epsilon_{n}\in\mathbb{R}.$ (S2)

The above equation is obtained by a straightforward calculation. Suppose that
$|\psi_{n}(\bm{k})\rangle_{R}$ and $\epsilon_{n}(\bm{k})$ are a right
eigenvector and an eigenvalue of $h(\bm{k})$,
$h(\bm{k})|\psi_{n}(\bm{k})\rangle_{R}=\epsilon_{n}(\bm{k})|\psi_{n}(\bm{k})\rangle_{R}$.
Then, $PT|\psi_{n}(\bm{k})\rangle_{R}$ is an eigenstate with eigenvalue
$\epsilon^{*}_{n}$ because

$\displaystyle h(\bm{k})PT|\psi_{n}(\bm{k})\rangle_{R}$ $\displaystyle=$
$\displaystyle PTh(\bm{k})|\psi_{n}(\bm{k})\rangle_{R}$ (S3) $\displaystyle=$
$\displaystyle\epsilon^{*}_{n}(\bm{k})PT|\psi_{n}(\bm{k})\rangle_{R},$

holds. Therefore, we obtain Eq. (S2).

Now, let us discuss the topology for $PT^{2}=1$. When a point-gap opens at
$\epsilon_{\mathrm{ref}}\in\mathbb{R}$ [i.e.,
$\mathrm{det}(h-\epsilon_{\mathrm{ref}})\neq 0$ holds], the system may possess
topological properties which are characterized by the following zero-
dimensional $\mathbb{Z}_{2}$-invariant $\nu^{\prime}$

$\displaystyle\nu^{\prime}$ $\displaystyle=$
$\displaystyle-\mathrm{sgn}[\mathrm{det}(h-\epsilon_{\mathrm{ref}})],$ (S4)

with $\mathrm{sgn}(x)$ takes $1$ ($-1$) for $x>0$ ($x<0$). For
$\epsilon_{\mathrm{ref}}=\epsilon_{\mathrm{EP}}$
($\epsilon_{\mathrm{EP}}=\sqrt{3}$), $\nu^{\prime}$ and $\nu$ defined in Eq.
(9) characterizes the same topology of Eq. (7); because
$\nu^{\prime}=\prod_{n}(\epsilon_{n}-\epsilon_{\mathrm{\mathrm{ref}}})$ holds,
we have $\nu^{\prime}$ taking $1$ ($-1$) for $\lambda>\lambda_{\mathrm{c}}$
($\lambda<\lambda_{\mathrm{c}}$).

##  S2  Symmetry-protected exceptional rings in an RPS cycle

We demonstrate the emergence of a symmetry-protected exceptional rings in
evolutionary game theory. Consider an extended RPS cycle whose payoff matrix
is written as

$\displaystyle A(\lambda,\kappa)$ $\displaystyle=$ $\displaystyle
A(\lambda)+\kappa\left(\begin{array}[]{ccc}1&0&-1\\\ 0&0&0\\\
-1&0&1\end{array}\right),$ (S8)

with $\lambda,\kappa\in\mathbb{R}$.

We can block-diagonalize the matrix $A$. Applying the unitary transformation
with

$\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{6}}\\\
\frac{1}{\sqrt{3}}&-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{6}}\\\
\frac{1}{\sqrt{3}}&0&\frac{-\sqrt{2}}{\sqrt{3}}\end{array}\right),$ (S12)

we have

$\displaystyle V^{T}A(\lambda,\kappa)V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-\sqrt{3}\\\
0&\sqrt{3}&0\end{array}\right)+\lambda\left(\begin{array}[]{ccc}0&0&0\\\
0&\frac{1}{2}&-\frac{\sqrt{3}}{2}\\\
0&-\frac{\sqrt{3}}{2}&\frac{3}{2}\end{array}\right)+\kappa\left(\begin{array}[]{ccc}0&0&0\\\
0&\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
0&\frac{\sqrt{3}}{2}&\frac{3}{2}\end{array}\right).$ (S22)

Diagonalizing the above matrix, we have eigenvalues $0$ and

$\displaystyle\epsilon_{\pm}$ $\displaystyle=$
$\displaystyle(\lambda+\kappa)\pm\sqrt{\frac{3}{4}(\lambda-\kappa)^{2}+\frac{1}{4}(\lambda+\kappa)^{2}-3}.$
(S23)

We note that for $\kappa=0$, the EP emerges at
$\lambda=\lambda_{\mathrm{c}}=\sqrt{3}$.

Figure S1(a) [S1(b)] plots the real- [imaginary] part of $\epsilon_{\pm}$.
These figures indicate the emergence of the symmetry-protected exceptional
rings; EPs, where the two bands touch both for the real- and the imaginary-
parts, form a ring. We also note that $\nu$ takes $-1$ ($1$) inside (outside)
of the ring, elucidating the topological protection.

Figure S1:  (a) and (b): The real- and imaginary-parts of $\epsilon_{\pm}$.
The data colored with blue (dark blue) represents $\epsilon_{+}$
($\epsilon_{-}$).

The above results verify the emergence of the symmetry-protected exceptional
rings in the extended RPS cycle.

##  S3  Details of the RPS chain

We discuss details of the RPS chain defined in the main text. Under the OBC’,
the payoff matrix is written as

$\displaystyle A(\lambda)$ $\displaystyle=$ $\displaystyle A_{0}+\lambda B.$
(S24)

We have introduced an additional site at $I=9$ [see Fig. 3(a)] for $L_{x}=4$.
Hence, the vector $\bm{x}$ consists of nine components,
$\bm{x}=(x_{1},x_{2},x_{3},\cdots,x_{9})^{T}$. Here, $A_{0}$ and $B^{\prime}$
are defined as

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}A_{0\mathrm{d}}&A^{T}_{0\mathrm{c}}&0&0&0\\\
A_{0\mathrm{c}}&A_{0\mathrm{d}}&A^{T}_{0\mathrm{c}}&0&0\\\
0&A_{0\mathrm{c}}&A_{0\mathrm{d}}&A^{T}_{0\mathrm{c}}&0\\\
0&0&A_{0\mathrm{c}}&A_{0\mathrm{d}}&A^{T}_{0\mathrm{cR}}\\\
0&0&0&A_{0\mathrm{cR}}&0\end{array}\right),$ (S25f) with $\displaystyle
A_{0\mathrm{d}}=\left(\begin{array}[]{cc}0&1\\\ -1&0\end{array}\right),\quad
A_{0\mathrm{c}}=\left(\begin{array}[]{cc}-1&0\\\ 1&0\end{array}\right),\quad
A_{0\mathrm{cR}}=\left(\begin{array}[]{cc}1&-1\end{array}\right),$ (S25l)

and

$\displaystyle B$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}B_{\mathrm{dL}}&B^{T}_{\mathrm{c}}&0&0&0\\\
B_{\mathrm{c}}&B_{\mathrm{d}}&B^{T}_{\mathrm{c}}&0&0\\\
0&B_{\mathrm{c}}&B_{\mathrm{d}}&B^{T}_{\mathrm{c}}&0\\\
0&0&B_{\mathrm{c}}&B^{T}_{\mathrm{d}}&B^{T}_{\mathrm{cR}}\\\
0&0&0&B^{T}_{\mathrm{cR}}&-1\end{array}\right),$ (S26f) with $\displaystyle
B_{\mathrm{d}}=\left(\begin{array}[]{cc}-2&1\\\ 1&-2\end{array}\right),\quad
B_{\mathrm{c}}=\left(\begin{array}[]{cc}0&1\\\ 0&0\end{array}\right),\quad
B_{\mathrm{dL}}=\left(\begin{array}[]{cc}-1&1\\\ 1&-2\end{array}\right),\quad
B_{\mathrm{cR}}=\left(\begin{array}[]{cc}0&1\end{array}\right),$ (S26n)

respectively. In addition, we have imposed $B_{11}=B_{99}=-1$ to satisfy Eq.
(12).

In Fig. S2, we can see that the spectrum and amplitude of the right
eigenvectors for OBC’ are similar to those for OBC, which indicates the
robustness of the skin effect against perturbations. This is because the non-
Hermitian topology induces the skin effect. We also note that
$\bm{c}=(1,1,\ldots,1)/31$ is the right eigenvector for $L_{x}=15$ [see Fig.
S2(b)] because Eq. (12) holds under OBC’.

Figure S2:  Spectrum and amplitude of right eigenvectors under the OBC’. (a):
The spectrum for $\lambda=-0.5$ and $L_{x}=15$. (b):Amplitude of the right
eigenvectors $\phi^{\mathrm{R}}_{In}$ ($n=1,2,\ldots,2L_{x}+1$) as functions
of $I$ for $L_{x}=15$. The data are plotted in a similar way to Fig. 3.

When Eq. (12) is satisfied, the replicator equation (10) can be linearized as
discussed in the main text. In this case, the dynamics is governed by

$\displaystyle i\partial_{t}\delta\bm{x}(t)$ $\displaystyle=$ $\displaystyle
H\delta\bm{x}(t),$ (S27)

with $H=iA/N_{0}$ [see also Eq. (11)] which is mathematically equivalent to
the Schrödinger equation. In quantum systems, the group velocity and the
lifetime are written as $\partial_{k}\mathrm{Re}E_{n}(k)$ and
$1/\mathrm{Im}E_{n}(k)$ ($\mathrm{Im}E_{n}(k)>0$), respectively. Here,
$E_{n}(k)$ $(n=1,2,\ldots)$ denotes eigenvalues of $H$.

Figure S3:  Time-evolution of the population density
$\delta\bm{x}(t)=\bm{x}(t)-\bm{c}$ for $\lambda=0$ and $L_{x}=15$. The
horizontal axis denotes $I$. (a)-(d) [(e)-(f)]: The time-evolution under the
PBC [OBC’]. The data in all panels are multiplied by 10. These data are
obtained in the same way as those of Fig. 4.

Figure S3 displays the dynamical properties for $\lambda=0$ where the payoff
matrix is anti-Hermitian. Because all eigenvalues $\epsilon$’s are pure
imaginary, the directive perpetration which arises from the real-part of the
eigenvalues $\epsilon$’s is not observed. In addition, no localized mode is
enhanced due to the absence of skin modes.

##  S4  Another RPS chain

We introduce another RPS chain [see Fig.S4(a)] exhibiting the skin effect. For
$L_{x}=5$, the payoff matrix is written as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccccc}A_{0}&0&0&0&A^{\prime}_{\mathrm{c}}\\\
A_{\mathrm{c}}&A_{0}&0&0&0\\\ 0&A_{\mathrm{c}}&A_{0}&0&0\\\
0&0&A_{\mathrm{c}}&A_{0}&0\\\ 0&0&0&A_{\mathrm{c}}&A_{0}\end{array}\right),$
(S28f) with $\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}d&0&r_{2}\\\ r_{3}&d&0\\\
0&r_{1}&d\end{array}\right),$ (S28j) $\displaystyle A_{\mathrm{c}}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}0&0&r_{4}\\\ 0&0&0\\\
0&0&0\end{array}\right),$ (S28n) and the vector $\bm{x}$ consists of fifteen
components, $\displaystyle\bm{x}$ $\displaystyle=$
$\displaystyle(x_{1},x_{2},x_{3},\cdots,x_{15})^{T}.$ (S28o)

Here, $A^{\prime}_{\mathrm{c}}$ is equal to $A_{\mathrm{c}}$ (the zero matrix)
under the PBC (OBC).

Figure S4:  (a): Sketch of the RPS chain for $L_{x}=3$. The arrows and $r$’s
describe payoffs. Here, $A_{II}=d$ for an arbitrary $I$. As shown in this
panel, $I$ takes $I=1,2,\ldots,3L_{x}$. For the PBC, $I+3L_{x}=I$ holds.
Dashed line denotes the unit cell. For $J=1,2,\ldots,L_{x}$,
$R_{3J-2}=R_{3J-1}=R_{3J}=J$ holds. (b): Spectrum of the RPS chain for
$(r_{1},r_{2},r_{3},r_{4},d)=(1,1/2,1,1/2,-1)$. The data colored with blue are
obtained under the PBC. Here, the shade of color denotes $k/\pi$. The data
colored with red are obtained for $L_{x}=10$ and the OBC. (c): Amplitude of
the right eigenvectors $|\phi^{\mathrm{R}}_{In}|$ $n=1,2,\ldots,3L_{x}$ as
functions of $I$ for $L_{x}=10$. Red (gray) lines denote data for the OBC
(PBC). For the OBC, all of the eigenvectors are localized around the right
edge.

Applying the Fourier transformation, the payoff matrix is written as

$\displaystyle A(k)$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}d&0&r_{2}+r_{4}e^{ik}\\\ r_{3}&d&0\\\
0&r_{1}&d\end{array}\right),$ (S32)

under the PBC. Here, $\bm{x}_{k}$ is defined as
$\bm{x}^{T}_{k}=(x_{k\mathrm{S}},x_{k\mathrm{R}},x_{k\mathrm{P}})$ with
$x_{ks}=\frac{1}{L_{x}}\sum_{R_{I}}e^{ikR_{I}}x_{R_{I}s}$ and
$s=\mathrm{R},\mathrm{P},\mathrm{S}$. Sets of $R_{I}$ and $s_{I}$ are
specified by $I$ ($x_{I}=x_{R_{I}s_{I}}$).

Figure S4(b) plots the spectrum of the payoff matrix for
$(r_{1},r_{2},r_{3},r_{4},d)=(1,1/2,1,1/2,-1)$. When the PBC is imposed,
eigenvalues form a loop structure as denoted by blue lines in Fig. S4(b).
Accordingly, the winding number [Eq. (19)] takes $1$ for
$\epsilon_{\mathrm{ref}}=0.5$, which implies the skin effect. Indeed, imposing
the OBC significantly changes the spectrum [see red dots in Fig. S4(b)].
Correspondingly, all of the right eigenvectors are localized around the edges,
meaning the emergence of skin modes [see Fig. S4(c)]. The above data [Figs.
S4(b) and S4(c)] indicate that the skin effect is observed in this RPS chain
as well. The essentially same results of eigenvalues and eigenvectors are also
obtained for $(r_{1},r_{2},r_{3},r_{4},d)=(2,2,2,2,1)$. For this parameter
set, the chain is composed of an ordinary RPS cycles Wang  _et al._ (2014)
whose payoff matrix is written as $\left(\begin{array}[]{ccc}1&0&2\\\ 2&1&0\\\
0&2&1\end{array}\right)$.

Figure S5:  The time-evolution of the population density
$\delta\bm{x}(t)=\bm{x}(t)-\bm{c}$ for
$(r_{1},r_{2},r_{3},r_{4},d)=(1,1/2,1,1/2,-1)$ and $L_{x}=10$. The horizontal
axis denotes $I$. (a)-(d) [(e)-(f)]: The time-evolution under the PBC [OBC].
The data in panels (b)-(d) [(a) and (e)-(h)] are multiplied by $100$ [$3$].
The data are obtained by employing the fourth order Runge-Kutta method Süli
and Mayers (2003) with discretized time $t_{n}=n\Delta t_{\mathrm{RK}}$ with
$\Delta t_{\mathrm{RK}}=0.05$. We set the initial condition as
$\delta\bm{x}(0)=\frac{\delta_{0}}{(3L_{x}-1)3L_{x}}(3L_{x}-1,-1,-1,\ldots,-1)$
with $\delta_{0}=2$.

As discussed in main text, the non-Hermitian topology characterized by $W=1$
induces the directive propagation under the PBC [see Fig. S5(a)-S5(d)] where
Eq. (12) is satisfied for this parameter set. We also note that under the OBC,
the players prefer to be localized around the right edge for large $t$ [see
Fig. S5(e)-S5(f)]. However, introducing the edges violates Eq. (12) which
prevents us from mapping Eq. (10) to the Schrödinger equation; introducing
perturbations around the edge does not help to recover Eq. (12) for this
model.