Physica A 392 (2013) 2652–2659

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Physica A

journal homepage: www.elsevier.com/locate/physa

A golden point rule in rock–paper–scissors–lizard–spockgameYibin Kang a, Qiuhui Pan a,b, Xueting Wang a, Mingfeng He a,b,∗

a School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Chinab School of Innovation Experiment, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

Article history:Received 14 January 2012Received in revised form 8 October 2012Available online 16 October 2012

Keywords:Monte-Carlo simulationODEsMean Field Theory

a b s t r a c t

We study a novel five-species system on two-dimensional lattices when each species havetwo superior and two inferior partners. Here we simplify the huge parameter space ofpredation probability to only two parameters. Both of Monte Carlo simulation and MeanField Theory reveal that two of strategiesmay die out when the ratio of the two parametersis close to the golden point 0.618, and the remaining three strategies are provided a cyclicdominance system.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

In the study of population dynamics of biological systems, it is usually to start with computational simulation or differ-ence equations that model the time evolution of populations [1–8]. Considering the spatial patterns of the system and theevolution rules for a given individual, the features upon the population and spatial distribution evolving through time is ob-tained by the computational simulationmethod (such asMonte Carlo simulation and Cellular Automaton). These simulationworks have be proven valuable [9–13]. It is easily to find the characteristic of the system evolving through time for somesimple simulation systems, such asMoran process andWright–Fisher process [14,15]. For the usual complex system, the fea-tures of the system are hard to prove. If the various constituents of the system are well mixed and spatial is absent, the term‘‘Mean field theory’’ (MFT) is most appropriate. In addition, it yields significant insights into many interesting phenomenaassociated with the nonlinear dynamics (such as a rich structure of bifurcation and chaos) by studying the MFT [16].

In our paper, we focus on a simple model involving five-species competing cyclically by both Monte Carlo and MFT. Thestudy of cyclically competing system is numerous [17–44].

As we know, the rock–paper–scissors game exhibits the closed 3-cyclic relationship of ‘‘rock blunts scissor cuts pa-per wrap rock’’. Recently, a new game which invented by Sam Kass and Karen Bryla was brought to popular attentionthrough its appearance on an episode of the television show ‘‘The Big Bang Theory’’. The game was called rock–paper–scissors–lizard–Spock (RPSLS), and it is an extension of the rock–paper–scissors game. The rules are as follows:

‘‘Scissors cut paper;Paper covers rock;Rock crushes lizard;Lizard poisons Spock;Spock smashes scissors;Scissors decapitate lizard;

∗ Corresponding author at: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China.E-mail address:mfhe@dlut.edu.cn (M. He).

0378-4371/$ – see front matter© 2012 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2012.10.011

Y. Kang et al. / Physica A 392 (2013) 2652–2659 2653

Fig. 1. (Color online) A diagram explaining the outcomes of RPSLS.

Lizard eats paper;Paper disproves Spock;Spock vaporizes rock;Rock crushes scissors’’.

A more detailed illustration of these relations can be seen in Fig. 1 [31,45].Because of the huge parameter space for the five-species system (5ss), it is hard to show the faithfully system. In 2009,

Laird and Schamp [8] present the characteristic of the 5SS system evolving through time by fixing the predation probabilityequal to one. In our paper, we assume that the predation probability ‘‘Scissors cut paper; Paper covers rock; Rock crusheslizard; Lizard poisons Spock; Spock smashes scissors’’ is p1 and the predation probability ‘‘Scissors decapitate lizard; Lizardeats paper; Paper disproves Spock; Spock vaporizes rock; Rock crushes scissors’’ is p2. We focus on the features of the systemevolving through time under the different ratio of p1 and p2.

In the Monte Carlo simulation, the density of every strategy fluctuates above and below that average 0.2. The period andamplitude of the waves are related to the ratio of p1 to p2 and the size of the lattice L. When the ratio of p1 to p2 is close to0.618, the amplitude and the period of the waves are bigger than the situation that the ratio is not close to 0.618. The periodand the amplitude of the waves become small along with the increasing size of lattice. When the amplitude is big enough,one of the strategies become extinct sooner or later due to the fluctuations. Within a short time its stronger predator diesout too, and survival of the remaining three strategies is provided by the cyclic dominance on the analogy of a spatial rock-scissors-paper game. We define the time that the first one strategy became extinct the extinction time. The first extinctiontime proportional to the population of the species, and this conclusion is similar to theMoran Process with finite population.In the part of Mean Field Theory, we find when the value p1/p2 is not equal to 0.618, the phase space has a finite number ofequilibrium points and the interior equilibrium point is stable and unique. When the value of p1/p2 is equal to 0.618, thereare an infinite number of equilibrium points in phase space. These points link up a curved surface, all of the equilibriumpoints in this surface are stable. We introduce the simulation rules in the part of Model andwe simulate the system by usingMonte Carlo simulation and discuss the results. In the part ofMFT, we transform this simulation rules to ordinary differentialequations and we discuss the existence and stability of the equilibrium point under different parameters especially p1/p2and L. In the part of conclusion, we relate our results to the golden point. When the ratio of the two parameters p1 and p2is close to 0.618, one of the strategies and its stronger predator become extinct soon, and survival of the remaining threestrategies is provided by the cyclic dominance on the analogy of a spatial rock-scissors-paper game.

2. Model

Weuse the rock–paper–scissors–lizard–Spock game tomodel the cyclic interaction among fivemobile strategies. Agentswith different strategies populate a square lattice with periodic boundary conditions. Each site is occupied by one agent.Interactions occur among the four nearest neighboring agents, as follows:

ABp1→AA, BC

p1→ BB, CD

p1→ CC,DE

p1→DD, EA

p1→ EE, (1)

ADp2→AA, BE

p2→ BB, CA

p2→ CC,DB

p2→DD, EC

p2→ EE, (2)

A ⃝pm→ ⃝A, B ⃝

pm→ ⃝B, C ⃝

pm→ ⃝C,D ⃝

pm→ ⃝D, E ⃝

pm→ ⃝E (3)

where five populations A, B, C, D and E cyclically dominate each other, ⃝ represents any agents. A outperforms B through‘‘replacing’’, symbolized by the reaction AB → AA. In the sameway, B outperforms C, C outperforms D, D outperforms E, andE beats A in turn. We denote the corresponding rate of these processes by p1. Then, A outperforms D through ‘‘replacing’’,

2654 Y. Kang et al. / Physica A 392 (2013) 2652–2659

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.20.4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.20.4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.20.4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.20.4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.2

0.4

ABCDE

p1 =0 p2 =0.5 pm =0.5

p1 =0.3 p2 =0.5 pm =0.5

p1 =0.5 p2 =0.5 pm =0.5

p1 =0.5 p2 =0.3 pm =0.5

p1 =0.5 p2 =0 pm =0.5

(a)

(b)

(c)

(d)

(e)

Den

sity

Den

sity

Den

sity

Den

sity

Den

sity

MCS

Fig. 2. (Color online) The density of each strategy under different values of p1 and p2 .

50 100 150 200

200

180

160

140

120

100

80

60

40

20

50 100 150 200

200

180

160

140

120

100

80

60

40

20

(a) p1 = 0.3, p2 = 0.5, pm = 0.5. (b) p1 = 0.5, p2 = 0.3, pm = 0.5.

Fig. 3. (Color online) Spatial patterns of each strategy under different values of p1 and p2 after 5000 MCS.

symbolized by the reaction AD → AA. In the same way, D outperforms B, B outperforms E, E outperforms C, and C beats Ain turn. We denote the corresponding rate of these processes by p2. Relation Eq. (3) defines the migration rate pm.

We consider a square lattice of linear size Lwith periodic boundary conditions. Each site of the lattice contains only oneagent. And the initial strategy of each agent is chosen randomly.

The process mentioned above is a Monte-Carlo step, each time step contains L × LMonte-Carlo steps.The initial proportion of each strategy is equals to 0.2, which is same to the initial value in Laird and Schamp’s work [8].

And we discuss the evolution of each strategy with L = 200 in Fig. 2.We set the parameter pm equal to 0.5, and under the different values of p1/p2, the population densities of different

strategies are shown in Fig. 2. We can see the spatial distribution in Fig. 3. We find that the density of every strategyfluctuates around average 0.2. The period and the amplitude of the density fluctuation is changing follow the values ofp1/p2. The fluctuation in Fig. 2(b) and (c) is stochastic fluctuations, the period and the amplitude are short. The period of thesituation (d) is waver from 0.1 to 0.3 and the amplitude is bigger than (b) and (c). In Fig. 3, we use five colors to representfive strategies separately. We see that the same strategies have the trend to gather in (a) and (b), especially in Fig. 3(b).

Y. Kang et al. / Physica A 392 (2013) 2652–2659 2655

ABCDE

1L=500

MCS

0

0.5

0

Den

sity

0.5

0

Den

sity

0.5

0

Den

sity

0.5

0

Den

sity

0.5

0

Den

sity

0.6

0.4

0.2

0D

ensi

ty

0

0

0

0

0.5L=200

1 × 105

1 × 105

2 × 106

2 × 106

2 × 106

2 × 106

0.5L=300

0

1L =200

1L=300

1L=400

Fig. 4. (Color online) The density of each strategy under different values of L from 1 to 2000,000 MCS.

In order to find the influence to the system of the size of the lattice, we set the parameters p1 = 0.5, p2 = 0.3 andpm = 0.5 and the results are under a single simulation run. It is shown in Fig. 4 thatwhen L < 300, the random two strategiesmay disappear and the remaining three strategies are stable in coexistence; when L > 400, all of the five strategies can bein coexistence. In fact, we have simulated under the parameter L = 500 for 107 MCS, and the five strategies can still remainin coexistence. In addition to Fig. 4, we find that the size of the lattice is seriously influence the population densities. AsL is increased, the frequency and the amplitude of the fluctuation are going down. When L is small enough, the randomtwo strategies may disappear and the remaining three strategies are stable in coexistence. Fig. 5 shows the different spatialdistribution when L is different. All of these individuals with same strategy are trend to gather, but this phenomenon is notobvious.

Then we fix the parameters L = 200, pm = 0.5. We traversed the parameters p1 and p2 from 0 to 1 by the step 0.01.Focus on the coexistence of the strategies after 105 MCS, we get Fig. 6. From Fig. 6, we can see that when the value of p1/p2is close to 0.6, only three of the strategies will coexist in the final. But in all other cases, five strategies will coexist in thesystem. The system is the same to the research by Laird and Schamp on the case of p1 = p2 = 1 [8]. They got the resultsthat all strategies would coexist in the system. From this figure we find that in some cases (the value of p1/p2 is not close to0.6), all the strategies will coexist. But if the value of p1/p2 is close to 0.6, a stable ‘Rock-Scissors-paper’ system with threestrategies will emerge and other two strategies will extinct.

Now we discuss the relationship between the extinction time of the first strategy and the lattice size.We define extinction time as the average time of the first strategy becoming extinct. The extinction time of the first

strategy as the function of lattice size with p1 = 0.618, p2 = 1 and pm = 0.5 is shown in Fig. 4.

2656 Y. Kang et al. / Physica A 392 (2013) 2652–2659

3.5

3

2.5

2

1.5

100 200 300 400 500

500

400

300

200

100

300

250

200

150

100

50

30025020015010050L =300

L =500

100 200 300 400

L =400

400

350

300

250

200

150

100

50

200

150

100

50

50 100 150 200L =200

Fig. 5. (Color online) Spatial patterns of each strategy under different values of p1 and p2 after 5000,000 MCS.

Three remaining strategies

Five remaining strategies

Five remaining strategies

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

p 2

p1

Fig. 6. The existence of the strategies in the system with different p1 and p2 .

Based on the data in Fig. 7, we got:

T ∝ L2α,

where

α ≈ 1.

This result is similar to the result in Moran process in finite population [14].

3. Mean field theory

Let x1, x2, x3, x4, x5 denote the densities of subpopulations A, B, C, D and E respectively. We denote the correspondingrate of processes ‘‘AB → AA, BC → BB, CD → CC, DE → DD, EA → EE’’ by p1 and the corresponding rate of processes

Y. Kang et al. / Physica A 392 (2013) 2652–2659 2657

2 2.5 3 3.5 4 4.5 5 5.50

5

10

15

20

25

30

2ln(L)

ln(T

)

Fig. 7. The extinction time of the first strategy in the system with different value of L. Each point is the average of 100 times simulations.

‘‘AD → AA, DB → DD, BE → BB, EC → EE, CA → CC’’ by p2. To summarize these reactions, we define the equations asfollows,

x′

1 = x1(p1x2 − p2x3 + p2x4 − p1x5)x′

2 = x2(−p1x1 + p1x3 − p2x4 + p2x5)x′

3 = x3(p2x1 − p1x2 + p1x4 − p2x5)x′

4 = x4(−p2x1 + p2x2 − p1x3 + p1x5)x′

5 = x5(p1x1 − p2x2 + p2x3 − p1x4).

(4)

The overall density is given by5

i=1

xi = 1. (5)

To illustrate the evolutionary stability of equations Eq. (5), we divided p1 and p2 into two kinds of cases.Case 1, p1/p2 = (

√5 − 1)/2.

The equilibrium points in the equation and their existence and stabilities can be list in Table 1 [46].Case 2, p1/p2 = (

√5 − 1)/2.

The equilibrium points in the equation and their existence and stabilities can be list in Table 2.x∗ in Table 2 is

x∗=

x∗

1

x∗

2

x∗

3

x∗

4

x∗

5

=

(5 +

√5)/10

−√5/5

(5 +√5)/10

00

+ k1

−(3 +

√5)/2

√5 + 1

−(3 +√5)/2

11

+ k2

(√5 − 1)/2

0−(

√5 − 1)/21

−1

. (6)

The value of k1 and k2 are in the internal of the pentagon in Fig. 8.When p1/p2 = (

√5 − 1)/2, there is only one stable equilibrium point (1/5, 1/5, 1/5, 1/5, 1/5), and all boundary

equilibrium points are unstable. Thus, all the strategies in the system are in coexistence. These results are similar withthe works in Section 2 by Monte Carlo simulation. When p1/p2 = (

√5 − 1)/2, there is a stable surface in the system, and

all the points on the surface are stable but not local approximately stable. These stable points include the cases where three,four, or five strategies coexist. However, results in Monte Carlo simulation are three strategies remaining when L is smalland five strategies remaining when L is bigger.

4. Conclusion

In this paper, we proposed a RPSLS game with five strategies. Based on the relationships of the strategies, we built asymmetrical system with two parameters p1 and p2 and discussed the properties of the system by Monte Carlo simulation

2658 Y. Kang et al. / Physica A 392 (2013) 2652–2659

A

B

C D

E

-0.25 -0.2 -0.15 -0.1 0.05 0.05 0.1 0.15 0.2 0.250

0.28

0.26

0.24

0.22

0.2

0.18

0.16

0.14

k2

k 1

Fig. 8. The range of free variable k1 , k2 is the pentagon ABCDE. In the internal of the pentagon, x∗

i > 0, i ∈ I . Thus, all the equilibrium points in the internalof the pentagon are internal equilibrium point. At the edge AB, x∗

1 = 0; At the edge BC, x∗

4 = 0; At the edge CD, x∗

2 = 0; At the edge DE, x∗

5 = 0; At the edgeEA, x∗

3 = 0. At the point A, x∗

1 = x∗

3 = 0; At the point B, x∗

1 = x∗

4 = 0; At the point C, x∗

2 = x∗

4 = 0; At the point D, x∗

2 = x∗

5 = 0; At the point E, x∗

3 = x∗

5 = 0.

Table 1The existence and stability of each equilibrium point in case 1.

Equilibrium point Existence Stability

(1, 0, 0, 0, 0) Always Unstable(0, 1, 0, 0, 0) Always Unstable(0, 0, 1, 0, 0) Always Unstable(0, 0, 0, 1, 0) Always Unstable(0, 0, 0, 0, 1) Always Unstable(p1/(2p1 +p2), p2/(2p1 +p2), p1/(2p1 +p2), 0, 0) Always Unstable(0, p1/(2p1 +p2), p2/(2p1 +p2), p1/(2p1 +p2), 0) Always Unstable(0, 0, p1/(2p1 +p2), p2/(2p1 +p2), p1/(2p1 +p2)) Always Unstable(p1/(2p1 +p2), 0, 0, p1/(2p1 +p2), p2/(2p1 +p2)) Always Unstable(p2/(2p1 +p2), p1/(2p1 +p2), 0, 0, p1/(2p1 +p2)) Always Unstable(1/5, 1/5, 1/5, 1/5, 1/5) Always Stable

Table 2The existence and stability of each equilibrium point in case 2.

Equilibrium point Existence Stability

(1, 0, 0, 0, 0) Always Unstable(0, 1, 0, 0, 0) Always Unstable(0, 0, 1, 0, 0) Always Unstable(0, 0, 0, 1, 0) Always Unstable(0, 0, 0, 0, 1) Always Unstablex∗ Always Stable but not locally approximately stable

and mean field theory. We got a result about the Golden Point. When p1/p2 = (√5 − 1)/2, there is only one stable

equilibrium point (1/5, 1/5, 1/5, 1/5, 1/5), and all boundary equilibrium points are unstable. Thus, all the strategies in thesystem are in coexistence.When p1/p2 = (

√5−1)/2, there is a stable surface in the system, and all the points in the surface

are stable but not local approximately stable. ‘‘The latter stable points include the cases where three, four, or five strategiescoexist. In the simulations the density of every strategy fluctuates around the average value. The period and amplitude ofthe fluctuations depend on the ratio of p1 to p2 and also on the lattice size L. More precisely, both the amplitude and period ofthe density fluctuations increase when the ratio p1/p2 approaches to 0.618 from both directions meanwhile the amplitudedecreases when increasing the lattice size. For intensive fluctuations one of the species becomes extinct sooner or later.The extinction of the first species is followed by a second one and the remaining three strategies coexist in a way describedby the spatial rock–paper–scissors games. According to our simulations the average extinction time of the first species isproportional to the initial number of the given species and also to the lattice size L in agreement with the prediction of theMoran process.

In cyclic competitive systems the final stationary composition (fraction of species) depends on the ratio of invasion rates[17–44]. For large numbers of species, however, the systematic analysis is time-consuming. Laird and Schamp [8] have

Y. Kang et al. / Physica A 392 (2013) 2652–2659 2659

studied five species predator–prey systems when varying only the sign of the equal strength invasion rates. In the presentpaperwe have studied only the caseswhere each species has two prey and two predators with only two values (p1 and p2) ofinvasion rates. According to our MC simulations only three strategies remain alive in sufficiently small finite systems whenp1/p2 is close to the golden ratio. The properties of similar five-species systems will be discussed in the future for largernumbers of parameters’’.1

Acknowledgments

The authors thank anonymous referees for their helpful comments.

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1 The authors thank an anonymous referee for corrections in this part.