Evolutionary dynamics in finite populations with zealots
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Abstract
We investigate evolutionary dynamics of twostrategy matrix games with zealots in finite populations. Zealots are assumed to take either strategy regardless of the fitness. When the strategy selected by the zealots is the same, the fixation of the strategy selected by the zealots is a trivial outcome. We study fixation time in this scenario. We show that the fixation time is divided into three main regimes, in one of which the fixation time is short, and in the other two the fixation time is exponentially long in terms of the population size. Different from the case without zealots, there is a threshold selection intensity below which the fixation is fast for an arbitrary payoff matrix. We illustrate our results with examples of various social dilemma games.
Mathematics Subject Classification
91A22 60J701 Introduction
A standard assumption underlying evolutionary game dynamics, regardless of whether a player is social agent or gene, is that players tend to imitate successful others. In actual social evolutionary dynamics, however, there may be zealous players that stick to one option according to their idiosyncratic preferences regardless of the payoff that they or their peers earn. Collective social dynamics in the presence of zealots started to be examined for nongame situations such as the voter model representing competition between two equally strong opinions (i.e., neutral invasions) (Mobilia 2003; Galam and Jacobs 2007; Mobilia et al. 2007; Xie et al. 2011; Singh et al. 2012). Zealots seem to be also relevant in evolutionary game dynamics. For example, voluntary immunization behavior of individuals when epidemic spreading possibly occurs in a population can be examined by a publicgoods dilemma game (Fu et al. 2011). In this situation, some individuals may behave as zealot such that they try to immunize themselves regardless of the cost of immunization (Liu et al. 2012).
In our previous work, we examined evolutionary dynamics of the prisoner’s dilemma and snowdrift games in infinite populations with zealots (Masuda 2012). Specifically, we assumed zealous cooperators and asked the degree to which the zealous cooperators facilitate cooperation in the entire population. We showed that cooperation prevails if the temptation of unilateral defection is weak or the selection strength is weak. For the prisoner’s dilemma, we analytically obtained the condition of cooperation.
In the present paper, we conduct a finite population analysis of evolutionary dynamics of a general twoperson game with zealots. Evolutionary games in finite populations have been recognized as a powerful analytical tool for understanding properties of evolutionary games such as conditions of cooperation in social dilemma games. In addition, the outcome for finite populations is often different from that for infinite populations (Nowak et al. 2004; Taylor et al. 2004; Nowak 2006). We take advantage of this method to understand evolutionary dynamics of games with zealots for general matrix games.
It should be noted that the fixation probability, i.e., the probability that a given strategy eventually dominates the population as a result of stochastic evolutionary dynamics, is a primary quantity to be pursued in evolutionary dynamics in finite populations. Nevertheless, fixation trivially occurs in the presence of zealots if all zealots are assumed to take the same strategy; the zealots’ strategy always fixates. For example, if there is a single zealous cooperator in the population, cooperation always fixates even in the conventional prisoner’s dilemma game. However, in this adverse case, fixation of cooperation is expected to take long time; the relevant question here is the fixation time (Antal and Scheuring 2006; Traulsen et al. 2007; Altrock and Traulsen 2009; Altrock et al. 2010; Assaf and Mobilia 2010; Ewens 2010; Wu et al. 2010; Altrock et al. 2011; Assaf and Mobilia 2012; Kreindler and Young 2013). Here we examine the mean fixation time of the strategy selected by the zealots. This quantity serves as a probe to understand the extent to which zealots influence nonzealous players in the population. The fixation time would be affected by the payoff matrix, population size, number of zealous players, and strength of selection. We derive the asymptotic dependence of the mean fixation time on the population size when the fraction of zealots in the population is fixed. Mathematically, we extend the approach taken in Antal and Scheuring (2006) to the case with zealots.
2 Model
We assume that \(N\) players may flip the strategy according to the Moran process (Moran 1958; Ewens 2010). We call these players the ordinary players. The other \(M\) players are zealots that never change the strategy irrespectively of their fitness. Because our primary interest is in the possibility of cooperation in social dilemma games induced by zealous cooperators, we assume that all zealots take strategy \(A\); \(A\) is identified with cooperation in the case of a social dilemma game. We also assume that \(a, b, c, d \ge 0\) for the Moran process to be welldefined.
Because we have assumed a wellmixed population, the state of the evolutionary process is specified by the number of ordinary players selecting \(A\), which we denote by \(i\). In each time step, we select an ordinary player with the equal probability \(1/N\). The strategy of the selected player is updated. Then, we select a player, called the parent, whose strategy replaces that of the previously selected player. The parent is selected with the probability proportional to the fitness among the \(N+M\) players including the zealots and the player whose strategy is to be replaced. The population size \(N\) is constant over time. It should be noted that a player is updated once on average in time \(N\).
Because the zealots always select \(A\), the Moran process ends up with the unanimous population of \(A\) players (we impose \(a>0\) for this to be true). In other words, fixation of \(A\) always occurs such that the issue of fixation probability is irrelevant to our model.
3 Results
We calculate the mean fixation time and its approximation in the case of a large population size by extending the framework developed in Antal and Scheuring (2006) (also see Van Kampen 2007; Redner 2001; Krapivsky et al. 2010; Ewens 2010).
3.1 Mean fixation time: exact solution
Consider the state of the population in which \(i\) (\(0\le i\le N\)) ordinary players select strategy \(A\). A total of \(i+M\) and \(Ni\) players, including the zealots, select strategies \(A\) and \(B\), respectively. The Moran process is equivalent to a random walk on the \(i\) space in which \(i=0\) is a reflecting boundary, and \(i=N\) is the unique absorbing boundary.
3.2 Deterministic approximation of the random walk
In this section we classify the deterministic dynamics driven by the expected bias of the random walk (i.e., \(T_i^+  T_i^\)) into three cases, as is done in the analysis of populations without zealots (Taylor et al. 2004; Antal and Scheuring 2006). The obtained classification determines the dependence of the mean fixation time on \(N\), as we will show in Sect. 3.3.
Classification of the three cases of the mean fixation time when \(N\) is large
\(abc+d\le 0\)  \(abc+d>0\)  

\(c<(m+1)a\)  Case (i)  Case (i) or (iii) 
\(c>(m+1)a\)  Case (ii)  Case (ii) 
 Case (i): \(f(y)> 0\) holds true for all \(y\) (\(0\le y\le 1\)) such that the dynamics starting from any initial condition tends to \(y=1\) (Fig. 1a). In an infinite population, \(A\) dominates \(B\). In a finite population, we expect that the fixation time is short. This case occurs when \(c < (m+1)a\) and one of the following conditions is satisfied:

\(abc+d \le 0\).

\(abc+d > 0\) and \(y_1^*\le 0\) (i.e., \(2ma+(m+1)bmcd\ge 0\)).

\(abc+d > 0\), \(0<y_1^*<1\) (i.e., \(2ma+(m+1)bmcd<0\) and \((2m+2)a+(m+1)b+(m+2)cd<0\)), and \(D \le 0\).

\(abc+d > 0\) and \(y_1^*\ge 1\) (i.e., \((2m+2)a+(m+1)b+(m+2)cd\ge 0\)).

 Case(ii)
: \(f(y)=0\) has a unique solution \(y^*_1\) (\(0< y^*_1<1\)) such that the dynamics starting from any initial condition converges to \(y^*_1\) (Fig. 1b). In an infinite population, \(A\) and \(B\) coexist. In a finite population, we expect that the fixation time is long. This case occurs when \(c > (m+1)a\).
 Case (iii): \(f(y)=0\) has two solutions \(0<y^*_1 < y^*_2<1\). Dynamics starting from \(0\le y<y^*_2\) converges to \(y^*_1\), and that starting from \(y^*_2<y<1\) converges to \(y=1\) (Fig. 1c). In an infinite population, a mixture of \(A\) and \(B\) and the pure \(A\) configuration are bistable. In a finite population, we expect that the fixation time is long if the dynamics starts with \(0\le y<y^*_2\) and short if it starts with \(y^*_2<y<1\). This case occurs when$$\begin{aligned}&c < (m+1)a, \end{aligned}$$(14)$$\begin{aligned}&abc+d > 0,\end{aligned}$$(15)and$$\begin{aligned}&0<\tilde{y}<1, \end{aligned}$$(16)are satisfied.$$\begin{aligned} D > 0 \end{aligned}$$(17)
The condition given by Eq. (14) is related to the socalled cooperation facilitator assumed in a previous model (Mobilia 2012) as follows. Consider a hypothetical infinite population in which almost all players select \(A\), i.e., \(y\approx 1\). Then, the payoff that a player with strategy \(A\) gains by being matched with the other ordinary players and zealous players is equal to \((m+1)a\). The payoff that a player with strategy \(B\) gains by being matched with the other ordinary players, but not zealous players, is equal to \(c\). Therefore, Eq. (14) represents the condition for the stability of the homogeneous population of strategy \(A\) against invasion by \(B\) when zealous players somehow contribute to the payoff of ordinary \(A\) players and not to that of ordinary \(B\) players. Such a zealous player is equivalent to the cooperation facilitator assumed in Mobilia (2012).
In the corresponding model without zealots, there are four scenarios: \(A\) dominates \(B\) (Fig. 1d), \(B\) dominates \(A\) (Fig. 1e), a mixture of \(A\) and \(B\) is stable (Fig. 1f), and \(A\) and \(B\) are bistable (Fig. 1g) (Antal and Scheuring 2006). The cases shown in Fig. 1d, f, and g are analogous to cases (i), (ii), and (iii), respectively, for the game with zealots. The case shown in Fig. 1e never occurs in the game with zealots because \(y\) tends to increase in the absence of \(A\) owing to the fact that unanimity of \(B\) among the ordinary players is a reflecting boundary of our model. In fact, this case corresponds to case (ii) for the presence of zealots (Fig. 1b). If we set \(m\rightarrow 0\), we obtain case (i) when \(ac>0\) and \(bd>0\), case (ii) when \(ac<0\), and case (iii) when \(ac>0\) and \(bd<0\). As is consistent with Antal and Scheuring (2006), the classification depends only on the \(ac\) and \(bd\) values. However, the scenario in which \(B\) dominates \(A\) (Fig. 1e) does not happen even with the vanishing density of zealots (i.e., \(m\rightarrow 0\)) because the unanimity of \(B\) remains to be a reflecting boundary as long as there is at least one zealot.
3.3 Mean fixation time: large \(N\) limit
3.3.1 Case (i)
3.3.2 Case (ii)
3.3.3 Case (iii)
In this case, \(q_i\) takes a local minimum at \(i=Ny^*_1\) and a local maximum at \(i=Ny^*_2\). Therefore, behavior of the random walk in the range \(0\le i <Ny^*_2\) is qualitatively the same as that for case (ii), and that in the range \(Ny^*_2<i<N\) is qualitatively the same as that for case (i). Because the former part makes the dominant contribution to the fixation time, the scaling of the mean fixation time is given by Eq. (28).
Case (iii) occurs when strategy \(A\) is disadvantageous when it is rare and advantageous when it is frequent. The coordination game provides such an example (Sect. 5.4).
3.3.4 Summary and the borderline case
In summary, the mean fixation time in the limit of large \(N\) is given by \(t_0 \propto N\ln N\) in case (i) and \(t_0 \propto \sqrt{N}\exp (\gamma N)\) (\(\gamma >0\)) in cases (ii) and (iii). For the parameter values on the boundary between the two scaling regimes, the same arguments as those for the model without zealots (Antal and Scheuring 2006) lead to \(t_0 \propto N^{3/2}\).
4 Dependence of the mean fixation time on the selection strength
5 Examples
We compare the mean fixation time for some games with that for the neutral game, i.e., \(a = b = c = d >0\). In the absence of zealots, the neural game yields \(T_i^+=T_i^\) (\(1\le i\le N1\)). The random walk is unbiased, and the socalled mean conditional fixation time is equal to \(N(N1)\) (Antal and Scheuring 2006). The mean conditional fixation time is defined as the mean fixation time starting from state \(i=1\) under the condition that the absorbing state at \(i=N\), not \(i=0\), is reached.
5.1 Constant selection
5.2 Prisoner’s dilemma game
Consider the prisoner’s dilemma game with a standard payoff matrix given by \(a=1\), \(b=0\), \(c=T\), and \(d=0\), where \(T>1\). Strategies \(A\) and \(B\) represent cooperation and defection, respectively. It should be noted that \(abc+d < 0\). With a general selection strength, the conditions derived in Sect. 3.2 imply that \(t_0 \propto N\ln N\), i.e., case (i), if \(T < 1+m/w\), and \(t_0 \propto \sqrt{N}\exp (\gamma N)\) with case (ii) if \(T > 1+m/w\). This condition coincides with that for the dominance of cooperators in the case of the infinite population (Masuda 2012).
Next, to examine the effect of the selection strength, we set \(T=1.2\) and \(m = 0.1\). The mean fixation time as a function of \(N\) and \(w\) is shown in Fig. 3b. Equation (34) implies that \(t_0 \propto N\ln N\) when \(w < w_\mathrm{c}= 0.5\). Consistent with this result, \(t_0\) grows fast as a function of \(N\) when \(w\) is large (i.e., \(w=0.7\) and 1). In particular, for \(w=1\), \(400 \sqrt{N}\exp (\gamma N)\) normalized by the \(t_0\) value for the neutral game (dashed line in Fig. 3b) agrees well with the exact results (thin solid line). For small \(w\) (i.e., \(w=0.4\)), \(t_0\) seems to scale with \(N\ln N\) (thick solid line).
Figure 3c shows the dependence of \(t_0\) on \(N\) for different densities of zealots (i.e., \(m\)). It should be noted that the baseline \(t_0\) value derived from the neutral game depends on the value of \(m\). Because we set \(T=1.2\) and \(w=1\) in Fig. 3c, the threshold value of \(m\) is equal to 0.2. In fact, the normalized \(t_0\) diverges according to \(\propto \sqrt{N}\exp (\gamma N)\) when \(m=0.1\) (dashed line and thick solid line), whereas it seems to converge to a constant value when \(m=0.3\) (thin solid line).
Figure 3 indicates that \(t_0\) for the prisoner’s dilemma game is always larger than that for the neutral game (i.e., the normalized \(t_0\) is larger than unity). This is consistent with the intuition that cooperation is difficult to attain in the prisoner’s dilemma game as compared to the neutral game.
5.3 Snowdrift game
In this section, we examine the snowdrift game (Maynard Smith 1982; Sugden 1986; Hauert and Doebeli 2004) defined by \(a=\beta 0.5\), \(b=\beta 1\), \(c=\beta \), and \(d=0\), where \(\beta >1\). Strategies \(A\) an \(B\) are identified as cooperation and defection, respectively. Each player is tempted to defect if the other player cooperates, as in the prisoner’s dilemma game. However, different from the prisoner’s dilemma game, a player is better off by cooperating if the partner defects; mutual defection is the worst outcome. In the infinite wellmixed population without zealots, the game has the unique mixed Nash equilibrium in which the fraction of cooperation is equal to \((2\beta 2)/(2\beta 1)\).
5.4 Coordination game
The firstpassage time increases slowly as \(i\) increases when \(i\) is small. It rapidly increases with \(i\) for intermediate values of \(i\), Once the random walker passes the critical \(i\) value, it feels a positive bias such that the firstpassage time only gradually increases with \(i\) for large \(i\). The values of \(i\) that separate the three regimes are roughly consistent with the analytical estimates \(y^*_1 = 0.1\) and \(y^*_2 = 0.2\) [Eqs. (12), (13)]. It should be noted that the firstpassage time shows representative behavior of case (iii) although \(w\) is only slightly larger than \(w_\mathrm{c}\).
6 Discussion
We extended the results for the fixation time under the Moran process (Antal and Scheuring 2006) to the case of a population with zealous players. Similar to the case without zealots (Antal and Scheuring 2006), we identified three regimes in terms of the payoff matrix, number of zealots, and selection strengths. In one regime, the fixation time is small (i.e., \(\propto N\ln N\)). In the other two regimes, it is large (i.e., \(\propto \sqrt{N}\exp (\gamma N)\) with \(\gamma >0\)). We illustrated our results with representative games including the prisoner’s dilemma game, snowdrift game, and coordination game.
Zealots have several impacts on evolutionary dynamics in finite populations. First, fixation of one strategy \(A\) always occurs with zealots because we assumed that all zealots permanently take \(A\). Second, there is a case in which fixation is fast if the fraction of \(A\) players is sufficiently large, whereas fixation is slow if the fraction of \(A\) is small. This scenario occurs for the coordination game. In the absence of zealots, the same game shows bistability such that the fixation to the unanimity of \(A\) or that of \(B\) occurs fast (Antal and Scheuring 2006). Third, for a selection strength smaller than a threshold value, the fixation is fast for any payoff matrix. In the absence of zealots, the dependence of the mean fixation time on \(N\) for large \(N\) values is completely determined by the signs of \(ac\) and \(bd\) (Antal and Scheuring 2006). Therefore, the scaling of the mean fixation time on \(N\) is independent of the selection strength because manipulating the selection strength does not change the sign of the effective \(ac\) or \(bd\) value. If the payoff matrix is given in the slow fixation regime, the fixation is exponentially slow even for a small selection strength. In contrast, in the presence of zealots, slow fixation can be accelerated if we lessen the selection strength.
Mobilia examined the prisoner’s dilemma game with cooperation facilitators (Mobilia 2012). A cooperation facilitator was assumed to cooperate with cooperators and not to play with defectors. The cooperation facilitator and zealous cooperator in the present study are common in that they never change the strategy. However, they are different. First, zealous cooperators are embedded in a wellmixed population such that they myopically cooperate with defectors as well as cooperators. Second, the ordinary players may imitate the zealous cooperator’s strategy (i.e., cooperation). In contrast, players do not imitate the cooperation facilitator’s strategy (i.e., cooperation) in Mobilia’s model. As a consequence, cooperation does not always fixate in his model.
Examination of the case of imperfect zealots, in which zealots change the strategy with a small probability (Masuda 2012), warrants future work.
Notes
Acknowledgments
We thank Bin Wu for carefully reading the manuscript. NM acknowledges the support provided through GrantsinAid for Scientific Research (No. 23681033) from MEXT, Japan, the Nakajima Foundation, CREST JST, and the Aihara Innovative Mathematical Modelling Project, the Japan Society for the Promotion of Science (JSPS) through the “Funding Program for WorldLeading Innovative R&D on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy (CSTP).
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