Fixation probabilities in evolutionary games with the Moran and Fermi processes

Journal of Theoretical Biology 364 (2015) 242–248

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Fixation probabilities in evolutionary games with the Moran and Fermi processes Xuesong Liu a, Qiuhui Pan a,b, Yibin Kang a, Mingfeng He a,n a b

School of Mathematical Science, Dalian University of Technology, Dalian 116024, China School of Innovation Experiment, Dalian University of Technology, Dalian 116024, China

H I G H L I G H T S


A Moran and Fermi mixed process is proposed.
Fixation probability of a single co-operator with update mechanism of the Fermi process is higher than neutral evolution's.
More co-operators with update mechanism of the Fermi process lead to higher fixation probabilities when co-operators' quantity is the same.

art ic l e i nf o

a b s t r a c t

Article history: Received 20 May 2014 Received in revised form 19 August 2014 Accepted 27 August 2014 Available online 11 September 2014

An evolutionary dynamic model of 2  2 games with finite population of size N þM was built. Among these individuals, N individuals have the same update mechanism as that of the Moran process, while the other M individuals have the same update mechanism as that of the Fermi process. We obtain the balance equations of the fixation probability and analyze some concrete cases. In contrast with the results of neutral evolution, the fixation probability of a single co-operator with the same update mechanism as that of the Fermi process is higher. Besides, more co-operators with the update mechanism of the Fermi process lead to higher fixation probabilities when co-operators' quantity is the same. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Evolutionary game dynamics Evolution of co-operation Stochastic dynamics Fixation events

1. Introduction The Evolutionary game theory is an approach to study frequency dependent selection (Smith, 1982; Weibull, 1997; Hofbauer and Sigmund, 1998; Gintis, 2000; Nowak and Sigmund, 2004; Nowak, 2006, 2013). It describes the evolutionary dynamics of a well-mixed population, which is consisting of interacting individuals taking different strategies. The fitness of individuals depends on the outcome of their interplay. Strategies can spread by imitation or natural selection. Due to differences in payoff, different strategies spread with different rates under natural selection (Wu et al., 2010). From 1920s to early 1940s, Ronald Fisher、J.B. Haldane and Sewall Wright established Mathematical biology combining Mendelian genetics with Darwinian evolution, and studied on the fixation of a mutant gene (Nowak, 2006). The success of a mutant gene is dependent not only on selection but also on chance. The probability that the mutant gene will eventually fix within a population was first

n

Corresponding author. E-mail address: [email protected] (M. He).

http://dx.doi.org/10.1016/j.jtbi.2014.08.047 0022-5193/& 2014 Elsevier Ltd. All rights reserved.

addressed quantitatively by Fisher (1922), who worked out the fixation probability for the case of genetic selection (i.e. no dominance) (Fisher, 1930). Similar results have been obtained by Haldane (1927) and Wright (1931). In 1950s, neutral theory of evolution was put forward by Motoo Kimura, who extended the formula of fixation probability to include any level of dominance (Nowak, 2006). His formula gives the fixation probability in terms of the initial frequency, the selection coefficients and the effective population number (Kimura, 1957). This function was used by Robertson (1960) in his theory of selection limits in plant and animal breeding. Moran (1962) worked out another method in obtaining the fixation probability for a special case. At almost the same time, Kimura (1962) also obtained a quite general formula for the probability of fixation. The Game theory was first brought to biology by Hamilton (1967) and Trivers (1971). The Evolutionary game theory was put forward by John Maynard Smith (1973). For infinitely populations, such systems are provided by the deterministic replicator dynamics (Taylor and Jonker, 1978; Hofbauer et al., 1979; Schuster and Sigmund, 1983; Hofbauer and Sigmund, 1998, 2003; Nowak et al., 2004; Zeeman, 1980). For finite populations, the outcome of the evolutionary game, which is

X. Liu et al. / Journal of Theoretical Biology 364 (2015) 242–248

described by stochastic, is determined by the interaction of random drift and frequency dependent selection (Antal and Scheuring, 2006; Nowak, 2006). In the last few years, many research studies have been carried out on the stochastic dynamics of evolutionary processes and all kinds of phenomena induced by stochasticity have now been explored (Goel and Richter-Dyn, 1974; Traulsen et al., 2005; Traulsen et al., 2006a; Traulsen et al., 2006b; Traulsen and Hauert, 2009; Mobilia, 2012). Once the population has been full of one type individuals, the extinct species are never re-introduced. Naturally, the system without mutations can't be driven into other states, but ends up in such an absorbing state. There are different ways to introduce weak selection under frequency dependent selection (Traulsen et al., 2006a; Wu et al., 2010). In the context of evolutionary game dynamics with infinite population, weak selection plays an important role (Nowak, 2006). The payoff from an evolutionary game has a weak or strong effect on fitness of individuals. When the effect is weak to a certain degree, corresponding selection is called weak selection (Nowak, 2006; Altrock et al., 2012). Normally, under frequency dependent selection the probability that one strategy replaces another can be fairly complicated. Under weak selection, some significant insights can be obtained analytically (Traulsen et al., 2007; Wu et al., 2010; Wu et al., 2013). In recent work, a major concern in cooperation is to continue to improve (Doebeli and Hauert, 2005; Ohtsuki et al., 2006; Nowak and Highfield, 2011; Masuda, 2012; Nowak, 2012). In this context, we focus on the stochastic evolutionary dynamics of two strategies, C (cooperation) and D (defection), in a finite, well-mixed population of size N þM. The concept of the Moran process was investigated quite sufficiently, and much work has been reported in the Fermi process in recent years. Moreover, a comprehensive study on the contrast analysis of the two processes has been undertaken. However, combining the two processes to a mixed process has not been established. The mixed process can make sense under the following condition. There are two kinds of people in a population. Among them, some people get information by surfing the Internet, while others get the information only by communicating with neighbors. This leads to a system with information asymmetry. In our model, the individuals with the update mechanism of the Moran process know everyone's strategies and payoffs. They get global information, which is equivalent to people who get information by surfing the Internet. In our model, the individuals with the update mechanism of the Fermi process only know the chosen one's strategy and payoff. They get local information, which is equivalent to people who get the information only by communicating with neighbors. Our motivation to study a combination of the Moran Fermi processes is to explore whether or not the system with information asymmetry is in favor of the formation of co-operation. Our goal here is to extend observations to the process, specifically we provide theoretical expressions of fixation probabilities (Van Kampen, 1997). We investigate what characteristics of the system contribute most to the formation of cooperation. The paper is organized as follows. In Section 2, we introduce a particular evolutionary process for our analysis. We obtain the balance equations of fixation probabilities in Section 3. We discuss the relationship between the Moran and Fermi processes in Section 4. Some concrete examples can be found in Section 5. Section 6 offers a conclusion of our results.

2. Model In a well-mixed population of size N þM, with two strategies C (cooperation) and D (defection), an individual plays a symmetric 2  2

243

game with each other. Among these N þ M individuals, N individuals have the same update mechanism as that of the Moran process (Altrock et al., 2012), while the other M individuals have update mechanism the same as that of Fermi process (Altrock and Traulsen, 2009). At each Monte Carlo step, a randomly chosen individual's strategy is updated. If the individual X's update mechanism is the same as that of the Moran process, one of individuals including itself is chosen proportional to fitness to produce one offspring with the same strategy. To keep the population size N þ M constant, X is removed from the population before the offspring is added (Traulsen and Hauert, 2009; Wu et al., 2010). It's noteworthy that the update mechanism of the offspring is the same as that of X. If the update mechanism of X is the same as that of the Fermi process, the strategy of a second randomly chosen individual will be imitated by X or not. Let the two randomly chosen individual X and Y have payoffs π X and π Y . Following some research (Altrock and Traulsen, 2009; Traulsen et al., 2007; Wu et al., 2010), we choose X imitates Y's strategy with probability pX-Y ¼ 1=ð1 þ eωðπ X  π Y Þ Þ. In the Mixed process, the symmetric 2  2 game can be described by the payoff matrix:

C D




C a

D  b

c

d

ð1Þ

A co-operator interacting with another co-operator obtains a while a defector interacting another defector receives d. If a co-operator interacts with defector, it obtains b, whereas defector receives c in this situation. Let ðn; mÞ denote the state of the system, where the number of co-operators with the update mechanism of the Moran process is n, and the number of co-operators with the update mechanism of Fermi process is m. Only ðn; mÞ ¼ ð0; 0Þ and ðn; mÞ ¼ ðN; M Þ are absorbing states. In state ð0; MÞ, the individuals with the same update mechanism as that of the Moran process are defectors, while the individuals with the same update mechanism as that of the Fermi process are co-operators. These two kinds of individuals can influence each other: Co-operators can be transformed into defectors with a certain probability, while defectors can be transformed into co-operators with a certain probability. The transformations give the system a chance to change the state. Therefore we ensure that state ð0; MÞ is not absorbing. Similar results can be obtained to state ðN; 0Þ. Thus, the average payoffs are:

πC ¼

aðn þ m 1Þ þbðN þM  n mÞ NþM 1

ð2Þ

πD ¼

cðn þ mÞ þ dðN þ M  n m  1Þ NþM 1

ð3Þ

for a co-operator and a defector, respectively. We denote fitness for co-operators and defectors by f C and f D , respectively. Several choices are possible to define fitness function, and a common point that fitness is a monotonically increasing function of the payoff has been established. Following some research (Nowak, 2006; Sigmund, 2010; Wu et al., 2010), we choose: f C ¼ eωπ C

ð4Þ

f D ¼ eωπ D

ð5Þ

where ω denotes the intensity of selection. The population size N þ M is constant in time, in each time step either the number of co-operators with the update mechanism of the Moran process or the number of co-operators with the update mechanism of the Fermi process can at most change by one. P þ ;0 ðn; mÞ represents the transition probability moving from ðn; mÞ

244

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to ðn þ 1; mÞ in a given time step. P  ;0 ðn; mÞ represents the transition probability moving from ðn; mÞ to ðn  1; mÞ in a given time step. P 0; þ ðn; mÞ represents the transition probability moving from ðn; mÞ to ðn; m þ 1Þ in a given time step. P 0;  ðn; mÞ represents the transition probability moving from ðn; mÞ to ðn; m  1Þ in a given time step. P 0;0 ðn; mÞ denotes the probability that the population remains in state ðn; mÞ. These transition probabilities are given by: Nn ðn þ mÞf C P þ ;0 ðn; mÞ ¼ N þ M ðn þ mÞf C þ ðN þ M  n  mÞf D

ð6Þ

P  ;0 ðn; mÞ ¼

n ðN þ M  n  mÞf D N þ M ðn þ mÞf C þ ðN þ M  n  mÞf D

ð7Þ

P 0; þ ðn; mÞ ¼

M m n þ m 1 N þ M N þ M 1 þ eωðπ D  π C Þ

ð8Þ

P 0;  ðn; mÞ ¼

m N þM  n  m 1 NþM NþM 1 þ eωðπ C  π D Þ

ð9Þ

P 0;0 ðn; mÞ ¼ 1 P þ ;0 ðn; mÞ  P  ;0 ðn; mÞ  P 0; þ ðn; mÞ  P 0;  ðn; mÞ ð10Þ We choose a ¼ 0:5; b ¼  0:5; c ¼ 1; d ¼ 0; N ¼ M ¼ 100; ω ¼ 0:01 to observe the transition probabilities as seen in Fig. 1.

Then we focus on the following backward master equation: qt þ 1 ½ðn; mÞjðn0 ; m0 Þ ¼ qt ½ðn; m  1Þjðn0 ; m0 ÞP 0; þ ðn; m  1Þ þ qt ½ðn; m þ 1Þjðn0 ; m0 ÞP 0;  ðn; m þ 1Þ þ qt ½ðn  1; mÞjðn0 ; m0 Þ P þ ;0 ðn  1; mÞ þqt ½ðn þ 1; mÞjðn0 ; m0 ÞP  ;0 ðn þ 1; mÞ  þ qt ½ðn; mÞjðn0 ; m0 Þ 1  P 0; þ ðn; mÞ  P 0;  ðn; mÞ  P þ ;0 ðn; mÞ   P  ;0 ðn; mÞ 0

ð12Þ

0

where qt þ 1 ½ðn; mÞjðn ; m Þ denotes the probabilities that the system reaches state ðn; mÞ for the first time exactly t time steps after having been started in state ðn0 ; m0 Þ. Summing over all t, 0 0 Φðn0 ;m0 Þ↦ðn;mÞ ¼ ∑1 t ¼ 0 qt ½ðn; mÞjðn ; m Þ is the probability that the system reaches state ðn; mÞ at any later time after starting in state ðn0 ; m0 Þ. This yields Φðn; mÞ ¼ Φðn0 ;m0 Þ↦ðn;mÞ . The fixation probability in state ð0; 0Þ and ðN; M Þ is given by:

Φð0; 0Þ ¼ 0

ð13Þ

ΦðN; MÞ ¼ 1

ð14Þ

For 1 r n r N  1; 1 r m r M  1, there is a balance equation for the fixation probabilities

Φðn; mÞ ¼ P þ ;0 ðn; mÞΦðn þ 1; mÞ þ P  ;0 ðn; mÞΦðn  1; mÞ þ P 0; þ ðn; mÞΦðn; m þ 1Þ þ P 0;  ðn; mÞΦðn; m  1Þ

  þ 1  P þ ;0 ðn; mÞ  P  ;0 ðn; mÞ  P 0; þ ðn; mÞ  P 0;  ðn; mÞ Φðn; mÞ

ð15Þ 3. Fixation probabilities In finite populations, the common concern is the fixation probability of co-operators, i.e., the probability to end up in the state ðN; M Þ with the initial state ðn; mÞ. We here use Φðn; mÞ to describe it. Because this system has only two absorbing states, the probability to end up in the state ð0; 0Þ with the initial state ðn; mÞ is 1  Φðn; mÞ. In order to obtain the fixation probability, let us consider the probabilities qt ðn; mÞ describing the event that the system is found in state ðn; mÞ at time t. The master equation of qt ðn; mÞ is qt þ 1 ðn; mÞ ¼ qt ðn; m  1ÞP 0; þ ðn; m 1Þ þqt ðn; m þ 1ÞP 0;  ðn; m þ1Þ þ qt ðn  1; mÞP þ ;0 ðn  1; mÞ þ qt ðn þ 1; mÞP  ;0 ðn þ1; mÞ   þ qt ðn; mÞ 1  P 0; þ ðn; mÞ  P 0;  ðn; mÞ  P þ ;0 ðn; mÞ P  ;0 ðn; mÞ ð11Þ

The boundary conditions are For n ¼ 0; 1 r m rM  1, the equation is:

Φð0; mÞ ¼ P þ ;0 ð0; mÞΦð1; mÞ þP 0; þ ð0; mÞΦð0; m þ 1Þ  þ P 0;  ð0; mÞΦð0; m  1Þ þ 1  P þ ;0 ð0; mÞ P 0; þ ð0; mÞ   P 0;  ð0; mÞ Φð0; mÞ

ð16Þ

For 1 r n r N 1; m ¼ 0, the equation is

Φðn; 0Þ ¼ P þ ;0 ðn; 0ÞΦðn þ 1; 0Þ þ P  ;0 ðn; 0ÞΦðn  1; 0Þ  þ P 0; þ ðn; 0ÞΦðn; 1Þ þ 1  P þ ;0 ðn; 0Þ   P  ;0 ðn; 0Þ  P 0; þ ðn; 0Þ Φðn; 0Þ

ð17Þ

For n ¼ N; 1 r m rM  1, the equation

ΦðN; mÞ ¼ P  ;0 ðN; mÞΦðN  1; mÞ þ P 0; þ ðN; mÞΦðN; m þ 1Þ  þ P 0;  ðN; mÞΦðN; m 1Þ þ 1  P  ;0 ðN; mÞ   P 0; þ ðN; mÞ  P 0;  ðN; mÞ ΦðN; mÞ

ð18Þ

For 1 r n r N 1; m ¼ M, the equation is

Φðn; MÞ ¼ P þ ;0 ðn; MÞΦðn þ 1; MÞ þ P  ;0 ðn; MÞΦðn  1; M Þ  þ P 0;  ðn; M ÞΦðn; M  1Þ þ 1  P þ ;0 ðn; M Þ P  ;0 ðn; M Þ   P 0;  ðn; M Þ Φðn; M Þ ð19Þ For n ¼ 0; m ¼ M, the equation is:

Φð0; MÞ ¼ P þ ;0 ð0; MÞΦð1; MÞ þP 0;  ð0; MÞΦð0; M  1Þ   þ 1  P þ ;0 ð0; M Þ  P 0;  ð0; M Þ Φð0; M Þ

ð20Þ

For n ¼ N; m ¼ 0, the equation is:

ΦðN; 0Þ ¼ P  ;0 ðN; 0ÞΦðN  1; 0Þ þ P 0; þ ðN; 0ÞΦðN; 1Þ   þ 1  P  ;0 ðN; 0Þ  P 0; þ ðN; 0Þ ΦðN; 0Þ

ð21Þ

4. Relationship between the Moran and Fermi processes Fig. 1. Results of transition probabilities with N ¼ M ¼ 100; ω ¼ 0:01, where a ¼ 0:5; b ¼  0:5; c ¼ 1; d ¼ 0.

Let us now focus on the relationship between the Moran and Fermi processes. Here, we consider an evolutionary game between

X. Liu et al. / Journal of Theoretical Biology 364 (2015) 242–248

5. Results and discussion

two types A and B with the following payoff matrix:

A B




A B  a b c

5.1. Donation game under weak selection ð22Þ

d

Let idenote the number of A individuals in a population of ~ and it is the state of the system. The average constant size N, payoffs for individuals of either type are given by i1

N~  i

aþ b π A ðiÞ ¼ ~ N 1 N~  1 i

ð23Þ

N~  i  1

d π B ðiÞ ¼ ~ cþ N~ 1 N 1

ð24Þ

For the Fermi process, the transition probabilities are given by:  i N~  i 1 T i7 Fermi ¼ ~ N N~ 1 þe 8 βðπ A ðiÞ  π B ðiÞÞ

ð25Þ

where β denotes the intensity of selection. For the Moran process, the transition probabilities are given by:  if A N~  i T iþ Moran ¼ ~ N if A þ ðN~ iÞf B

ð26Þ

 i ðN~ iÞf B T i Moran ¼ ~ N if A þ ðN~  iÞf B

ð27Þ

where f A denotes the fitness of type A, and f B denotes the fitness   T T ¼ T iþ  ¼ of type B. For f A ¼ eβπ A ; f B ¼ eβπ B , we have T iþ  i

Moran

i

1  β þ βπ B ðiÞ  1  βðπ A ðiÞ  π B ðiÞÞ ¼ 1  β þ βπ A ðiÞ

ð28Þ

We also have a Taylor expansion with the Fermi process under weak selection  T i  ¼ e  βðπ A ðiÞ  π B ðiÞÞ  1  β ðπ A ðiÞ  π B ðiÞÞ þ T i Fermi

In Section 5.1, we focus on the fixation of a single co-operator in a population of defectors. We denote that ρC is the fixation probability of a single co-operator in Moran process while ρ0C is the fixation probability of a single co-operator in the Fermi process. In addition, we denote that Φð0; 1Þ is the fixation probability of a single co-operator with the update mechanism of the Fermi process in the Mixed process, and Φð1; 0Þ is the fixation probability of a single co-operator with the update mechanism of the Moran process in Mixed process. We now consider the Donation game with the payoff matrix is ! 0 b c0  c0 ð30Þ 0 b 0 0

here we assume b 4 c0 4 0 to receive the prisoner's dilemma. For the Moran process, we can calculate the fixation probability of a single co-operator to ρC  N1 þ ð½ðN þ1Þu þ 3vÞ=ð6NÞω o N1 , where 0 u ¼ 0; v ¼ c0 ðN  1Þ  b =ðN 1Þ o0 (Taylor et al., 2004; Nowak, 2006; Traulsen et al., 2006a). For the Fermi process, we can obtain ρC ¼ ρ0C as seen in Section 4 (Karlin and Taylor, 1975; Karlin and Taylor, 1981; Pacheco et al., 2006; Traulsen et al., 2006a; Altrock and Traulsen, 2009; Wu et al., 2010). Now let us consider the Mixed process with N ¼ M ¼ 1. When the initial state of the system is ð0; 1Þ, the transition probabilities are p þ ;0 ð0; 1Þ ¼

1 1  0 2 1 þeωðb þ c0 Þ

p0;  ð0; 1Þ ¼

1 1 eωðb þ c Þ   0 2 2 1 þeωðb þ c0 Þ

0

Fermi

e  βðπ A ðiÞ  π B ðiÞÞ . In general, the probability that the system will fixate to the pure state all A at any later time after starting with state i is     Φi ¼ ∑ik¼10 ∏kj ¼ ¼ 1 T j =T jþ = ∑Nk ¼ 10 ∏kj ¼ ¼ 1 T j =T jþ . Thus, we   can obtain Φi Moran ¼ Φi Fermi . The Moran process with exponential fitness mapping and the Fermi process are identical in terms of fixation probabilities for any games and any selection intensity (Traulsen and Hauert, 2009; Wu et al., 2010). In comparison with the Moran process, the Fermi process is invariant to adding a constant to all entries of the payoff matrix, as it only depends on payoff differences (Traulsen et al., 2007). For f A ¼ 1  β þ βπ A ; f B ¼ 1  β þ βπ B , under weak selection (i.e. β⪡1), a Taylor expansion up to first order in β yields  T i  T iþ Moran

245

ð29Þ

Thus, under weak selection, the approximations of the fixation probabilities with the Moran and Fermi processes are identical (Altrock and Traulsen, 2009; Traulsen et al., 2007). In addition, for the Moran process there is a restriction that the fitness values have to be positive, while for the Fermi process, the payoff matrix can contain unrestricted positive and negative entries. This is an advantage of the Fermi process over the Moran process (Traulsen et al., 2007).

ð31Þ 0

ð32Þ

Then we can calculate the fixation probability

Φð0; 1Þ ¼

p þ ;0 ð0; 1Þ 1 ¼ 0 p þ ;0 ð0; 1Þ þp0;  ð0; 1Þ 1 þ 12eωðb þ c0 Þ

ð33Þ

When the initial state of the system is ð1; 0Þ, the transition probabilities are p0; þ ð1; 0Þ ¼

1 1 1   0 2 2 1 þeωðb þ c0 Þ

p  ;0 ð1; 0Þ ¼

1 eωðb þ c Þ  0 2 1 þeωðb þ c0 Þ

0

ð34Þ

0

ð35Þ

Then we can calculate the fixation probability

Φð1; 0Þ ¼

p0; þ ð1; 0Þ 1 ¼ 0 p0; þ ð1; 0Þ þp  ;0 ð1; 0Þ 1 þ 2eωðb þ c0 Þ

ð36Þ

Easily, for the corresponding Moran process with N ¼ 2, the fixation probability is

ρC ¼

1 0 1 þ eωðb þ c0 Þ 0

ð37Þ

Because b 4 c0 4 0 and ω is sufficiently small, here we can obtain an inequation Φð0; 1Þ 4 1=2 4 ρC 4 Φð1; 0Þ for two individual's Donation game. In comparison with the results of the Moran process (i.e. ρC ), the fixation probabilities of a single co-operator with the update mechanism of the Fermi process (i.e. Φð0; 1Þ) is higher than 1=2 (i.e. the results of neutral evolution 1=ðN þMÞ). This suggests natural selection can favor C replacing D (Nowak, 2006). 0 0 In Fig. 2, we choose r ¼ c0 =b ; b ¼ 1; N ¼ M ¼ 20; ω ¼ 0:01. Both the fixation probability Φð0; 1Þ and Φð1; 0Þ decrease with the growth of r, and Φð1; 0Þ is always less than the fixation probability

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0

0

Fig. 2. Results of theoretical value with r ¼ c0 =b ; b ¼ 1; N ¼ M ¼ 20; ω ¼ 0:01. The fixation probabilities of a single co-operator with the update mechanism of the Fermi process are represented by a red line, and the fixation probabilities of a single co-operator with the update mechanism of the Moran process are represented by a blue line. To make a comparison, the theoretical result for the Moran process with 40 individuals is shown by black line, and the result for neutral evolution is shown by dotted line (for interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

under neutral evolution (i.e. 1=ðN þ M Þ). Similar results have been obtained in the Moran and Fermi processes under same conditions (Traulsen et al., 2006a; Wu et al., 2010). However, we can see that Φð0; 1Þ is higher than 1=ðN þ MÞ. Specially, it's noteworthy that Φð0; 1Þ is higher than the fixation probabilities of a single cooperator in the Fermi process all the time. On the other hand, Φð1; 0Þ is less than the fixation probabilities of a single co-operator in the Moran process. Here we conclude that the system with the Mixed process can support a single co-operator mutant with the update mechanism of the Fermi process to capture the whole population, but this system goes against the fixation of a single co-operator with the update mechanism of the Moran process. For the Donation game under weak selection, we choose 0 0 r ¼ c0 =b ¼ 0:5; b ¼ 1; N ¼ M ¼ 20; ω ¼ 0:01. Fig. 3(a) demonstrates the change law of the difference value between the fixation probabilities in the Mixed and in Moran processes with N ¼ 40. As shown in Fig. 3(a), when the initial n is fixed, the difference value increases with the growth of the initial m. The fixation probability ρC increases as the number of initial co-operators grows in the Moran process. Accordingly, the growth of the initial m results in adding the fixation probability Φðn; mÞ.Similarly, when the initial m is fixed, the difference value decreases with the growth of the initial n. Next, to further analyze this finding, we assume that the total number of co-operators is fixed with n þm ¼ 20 (see Fig. 3(b)). The fixation probability Φðn; mÞ will reduce when n increases and m decreases. This result indicate that the growth of co-operators with the update mechanism of the Fermi process results in adding the fixation probability, when the total number of co-operators in the system is fixed in the Mixed process. 5.2. Prisoner's Dilemma game and the Stag-hunt game In Section 5.2, we focus on two typical examples in the Mixed process (Traulsen et al., 2006a). For c 4 a 4 d 4 b, the game of this system belongs to the prisoner's dilemma(Sigmund, 2010; Hilbe et al., 2013). Let us combine Fig. 4(a) and (b) to analyze. Here we choose a ¼ 3; b ¼ 1; c ¼ 4; d ¼ 2; N ¼ M ¼ 20. In Fig. 4(a), the growth of the initial co-operators results in adding the fixation probability Φðn; mÞ. In additon, more co-operators with the update mechanism of the Fermi process lead to higher fixation probabilities when cooperators' quantity is the same. In Fig. 4 (b), defectors always

0

Fig. 3. Results of the fixation probabilities, where r ¼ 0:5; b ¼ 1; N ¼ M ¼ 20; ω ¼ 0:01. (a) The lattice represents the difference between the fixation probabilities in Mixed process and in Moran process with 40 individuals. (b) We make n þ m ¼ 20 changeless. Filled circles represent the fixation probabilities in state ðn; 20 nÞ.

dominate in the system. When the intensity of selection is weak ðω ¼ 0:01Þ, the co-operators have reasonable chances to fixate in the finite population with N ¼ M ¼ 20. However, when the intensity of selection is a little bit stronger, the fixation becomes scarcely possible unless the initial number of co-operators is close to 40. Now let us consider the stag-hunt game for a 4 c; b od. We choose a ¼ 4; b ¼ 1; c ¼ 3; d ¼ 2; N ¼ M ¼ 20. As shown in Fig. 4(c), the growth of the initial co-operators results in adding the fixation probability Φðn; mÞ. Besides, similarly to the results of prisoner's dilemma, more co-operators with the update mechanism of the Fermi process lead to higher fixation probabilities when co-operators' quantity is the same. As shown in Fig. 4(d), the fixation probabilities under weak selection is higher than that under a little bit stronger intensity of selection when the number of co-operators is sufficiently small, while the fixation probabilities under weak selection is lower than that under a little bit stronger intensity of selection when the number of co-operators is sufficiently big.

6. Conclusion Evolutionary dynamics, a rapidly growing field of considerable importance, focuses on the evolutionary processes: Moran and Fermi processes, which belong to the “birth-death” and imitation processes respectively (Nowak, 2006; Traulsen et al., 2006a; Altrock and Traulsen, 2009; Wu et al., 2010; Altrock et al., 2012). For the 2  2 game, we

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Fig. 4. Results of fixation probabilities for prisoner's dilemma where a ¼ 3; b ¼ 1; c ¼ 4; d ¼ 2; N ¼ M ¼ 20 and stag-hunt game where a ¼ 4; b ¼ 1; c ¼ 3; d ¼ 2; N ¼ M ¼ 20. In (a) and (c), the simulation results are shown by solid circles, and the theoretical results are shown by lines. There are two different meanings for x-axis. The first is that when x o 20, we set n ¼ 0 and when x Z 20, we set m ¼ 20. The second is that when xo 20, we set m ¼ 0 and when x Z 20, we set n ¼ 20. Red circles and green circles show the fixation probabilities under ω ¼ 0:01 with the first meaning and the second meaning, respectively. Blue circles and yellow circles show the fixation probabilities under ω ¼ 0:5 with the first meaning and the second meaning, respectively. In Fig. 4(b) and (d), the filled surface shows the fixation probabilities for ω ¼ 0:01, and the unfilled surface shows the fixation probabilities for ω ¼ 0:5 (for interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

have introduced a model in a well-mixed population with two strategies: cooperation and defection. This paper has presented a numerical study of the fixation probabilities of the Mixed process. The findings suggest more co-operators with the update mechanism of the Fermi process lead to higher fixation probabilities when co-operators' quantity is the same. In comparison with the results of the Moran process, the fixation probabilities of a single co-operator with the update mechanism of Fermi process is higher than 1=ðN þ M Þ all the time, which suggests natural selection can favor C replacing D (Nowak, 2006). Considering the special condition in Section 1, the results suggest that more co-operators with local information lead to more chance of the formation of cooperation in the system with information asymmetry when co-operators' quantity is the same. Acknowledgments We thank the anonymous Editors and Reviewers' warm work earnestly. References Altrock, P.M., Traulsen, A., 2009. Fixation times in evolutionary games under weak selection. New J. Phys. 111, 013012. Altrock, P.M., Traulsen, A., Galla, T., 2012. The mechanics of stochastic slowdown in evolutionary games. J. Theor. Biol. 311, 94–106. Antal, T., Scheuring, I., 2006. Fixation of strategies for an evolutionary game in finite populations. Bull. Math. Biol. 688, 1923–1944. Doebeli, M., Hauert, C., 2005. Models of co-operation based on the Prisoner's Dilemma and the Snowdrift game. Ecol. Lett. 87, 748–766. Fisher, R.A., 1922. On the dominance ratio. Proc. R. Soc. Edinb. 50, 204–219. Fisher, R.A., 1930. The evolution of dominance in certain polymorphic species. Am. Nat. 64, 385–406.

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