Non-Hermitian topology in rock–paper–scissors games

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.


Non-Hermitian topology in rock-paper-scissors games Tsuneya Yoshida * , Tomonari Mizoguchi & Yasuhiro Hatsugai
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.
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. 1a) whose payoff matrix is given by [64][65][66][67] with a real number . Here, players choose one of the strategies (s 1 , s 2 , s 3 ) = ("R", "P", "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 = 0 , this game

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Department of Physics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan. * email: yoshida@rhodia. ph.tsukuba.ac.jp 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. (1) The first (second) term is anti-Hermitian (Hermitian). (2) For an arbitrary , eigenvalues ǫ n ( n = 1, 2, 3 ) form a pair ǫ n = ǫ * n ′ ( n = n ′ ) or are real numbers ǫ n ∈ R . This constraint arises from the realness of the payoffs where the operator K takes complex conjugate. Equation (2) can be recognized as PT symmetry by regarding as a momentum (for more details, see Sect. S1 of Supplemental Material [68]). For = 0 , the energy eigenvalues are aligned along the imaginary axis due to the anti-Hermiticity of A( = 0) . Increasing , two eigenvalues touch at a critical value c so that they become real numbers when the second term is dominant. At = c , the EP emerges. The emergence of the EP is supported by Fig. 1b. For = 0 , the energy eigenvalues are pure imaginary due to anti-Hermiticity of A( = 0) . As is turned on, two eigenvalues approach each other, and the EP emerges at = c = √ 3 . One can also characterize the topology of this EP by computing the Figure 1c plots the Z 2 -invariant as a function of . Corresponding to the emergence of the EP, the Z 2 -invariant jumps at = c , elucidating the topological protection of the EP.

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. (4)]. 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 ≤ < c , while such a behavior disappears for c < due to the emergence of the EP. Firstly, we linearize the replicator equation. When a large number of players repeat the game, the timeevolution is described by the replicator equation where x denotes the population density ( I x I = 1 ), and 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. 22 unavailable.
Nevertheless, we can still obtain the following linearized equation which is mathematically equivalent to the Schrödinger equation. This mathematical equivalence reveals that the dynamics of the RPS cycle (i.e., a classical system) can be understood in terms of quantum physics. Here, δx is defined as δx = x − c with c = (1, 1, . . . , 1) T /N 0 , and N 0 denotes dimensions of the matrix A. Key ingredients are the following relations: To verify the above statement, we numerically solve the replicator Eq. (4). Figure 2 plots the time-evolution of the population density for several values of . For = 0 , the matrix A( ) 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. 2a). This oscillatory behavior is also observed in the phase plot (see Fig. 2b), where the orbit forms a closed loop. The above oscillatory behavior is also observed for = 0.5 (see Fig. 2c,d), which is due to the imaginary-part of the eigenvalues. We also note that the deviation δx is enhanced as t increases, which is because the real-part of the eigenvalues is positive (see Fig. 1b). Further increasing induces the EP (see Fig. 1b), and the eigenvalues become real. Correspondingly, the above oscillatory behavior is not observed for = 2 (see Fig. 2e,f), The above numerical data verify that the EP governs the dynamics around the fixed point c ; the oscillatory behavior of the RPS cycle disappears as the EP emerges. We note that for < 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 . We note that symmetry-protected exceptional rings are also observed by introducing an additional parameter (see Sect. S2 of Supplemental Material [68]), whose topology is also characterized by the Z 2 -invariant ν . Secondly, although Refs. 69-72 discuss topology of interaction networks among strategies (i.e., interaction topology), it differs from the topology discussed in this paper. Thirdly, we note that EPs are also reported for active matters 73,74 . 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 48,49 . 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.
The above skin effect can be observed in an RPS chain illustrated in Fig. 3a. Applying the Fourier transformation, the payoff matrix is written as  [68]. Figure 3b plots the spectrum of the payoff matrix for = −0.5 . When the PBC is imposed, eigenvalues form a loop structure as denoted by blue lines in Fig. 3b. Accordingly, the winding number 31,48,49 defined as takes −1 for ǫ ref = 1 , which implies the skin effect. Indeed, imposing the OBC significantly changes the spectrum (see red dots in Fig. 3b). Correspondingly, almost all of the right eigenvectors are localized around the edges, meaning the emergence of skin modes (see Fig. 3c). The above data (Fig. 3b,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 Sect. S3 of Supplemental Material [68]): attaching site at I = 2L x + 1 and tuning diagonal elements to A II = only for I = 1, 2L x + 1 so that Eq. (6) holds (see Fig. 3a). 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. Figure 4a-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: (1) as t increases, each mode is enhanced corresponding to Reǫ n ; (2) the group velocity of each mode is proportional to −∂ k Imǫ 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. (6) is satisfied for OBC' while it is not for OBC. As shown in Fig. 4e-h, the population density around the right edge is enhanced as t increases. This phenomenon is due to the skin modes whose eigenvalues satisfy Reǫ n > 0 . The above dynamical behaviors are unique to non-Hermitian systems; turning off , iA( = 0) becomes Hermitian and the above behaviors disappear (see Sect. S3 of Supplemental Material [68]).
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 Sect. S4 of Supplemental Material [68]). Secondly, Ref. 21 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. 21 . In this non-Hermitian case, we need to take into account the second term of Eq. (4).

Discussion.
We have proposed a new platform of non-Hermitian topology in an interdisciplinary field, i.e., 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 phenom- Our results pose several future directions which we discuss below. 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 25,75 to human societies [26][27][28]76,77 ; for instance, dynamics of human cooperation has been discussed in Refs. 76,77 . 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 18 , our result may provide a novel perspective of well-known phenomena for biological systems such as bacteria 25 and side-blotched lizards 75 ; dynamics of such systems may be understood in terms of exceptional points. 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. (5). We start with noting Eq. (6) results in the relations Ac = 0 and c T A = 0 which mean that c = (1, 1, 1, . . . , 1) T /N 0 is a fixed point. By making use of the above relations we have From the second to the third line, we have used the relations Ac = 0 and cA = 0 . In the last line, we have discarded the second and third order terms of δx.
Because (e T I · c) = 1/N 0 for an arbitrary I, we have which is equivalent to Eq. (5).   www.nature.com/scientificreports/