Abstract
Background
Phenomena of instability are widely observed in many dissimilar systems, with punctuated equilibrium in biological evolution and economic crises being noticeable examples. Recent studies suggested that such instabilities, quantified by the abrupt changes of the composition of individuals, could result within the framework of a collection of individuals interacting through the prisoner's dilemma and incorporating three mechanisms: (i) imitation and mutation, (ii) preferred selection on successful individuals, and (iii) networking effects.
Methodology/Principal Findings
We study the importance of each mechanism using simplified models. The models are studied numerically and analytically via rate equations and mean-field approximation. It is shown that imitation and mutation alone can lead to the instability on the number of cooperators, and preferred selection modifies the instability in an asymmetric way. The co-evolution of network topology and game dynamics is not necessary to the occurrence of instability and the network topology is found to have almost no impact on instability if new links are added in a global manner. The results are valid in both the contexts of the snowdrift game and prisoner's dilemma.
Conclusions/Significance
The imitation and mutation mechanism, which gives a heterogeneous rate of change in the system's composition, is the dominating reason of the instability on the number of cooperators. The effects of payoffs and network topology are relatively insignificant. Our work refines the understanding on the driving forces of system instability.
Introduction
Instabilities are widely observed in diversified fields such as sociology, psychology, economics, and biology [1]–[19]. For example, biological evolutions exhibit themselves as intermittent bursts of activities separating relatively long periods of quiescence, with extinctions happening at all scales [5], [6]. This dynamical instability, referred to as punctuated equilibrium, may result from strong interactions among different species [7], [8]. Economic crises, an instability phenomenon in economic systems, are caused not only by the economic and financial policies of individual country, but also the interdependent relations among countries, known as the world trade network and other economic and financial networks [14]–[17].
These systems typically consist of many interacting individuals, each reacting to the environment and other individuals' actions to enhance its own benefit. The relationship among individuals can be described by a network, with nodes and links representing the individuals and their relations, respectively [20], [21]. The reacting strategies are usually modeled by competing games, which were introduced for biological problems [22], [23] and subsequently applied to many other disciplines [24]–[28]. Therefore, a combination of evolutionary games and networks provides an effective approach of research on these systems [29]. To get closer to reality, the mechanisms of “network evolution” [30], [31] and “inheritance and variation” [32], which can also be called imitation or copying mechanisms, were incorporated into subsequent research.
Understanding the underlying mechanisms for system instability has been the focus of recent research. Kim et al. [33] pointed out that an opinion leader could affect a considerable fraction of population yet ordinary people can rarely influence the leader, and this kind of asymmetric influence could result in dynamic instability in prisoners' dilemma game. Schweitzer et al. [15] showed that a single tiny disturbance may lead to the system-level instability through the cascading process on economic networks. Rendell et al. argued that the copying and learning mechanisms would result in instability [34]. Cavaliere et al. [35] proposed a game-theoretic model of dynamic network formation for studying prosperity and instability in which newcomers are more likely to select prosperous individuals as role-models and imitate their strategies and connections. Their model incorporates three mechanisms: (i) imitation and mutation, (ii) preferred selection on successful individuals, and (iii) networking effects, and can exhibit instabilities on both the composition of individuals and the interacting patterns of individuals.
While these mechanisms combined could lead to the system instability, the effects of each individual mechanism are not fully understood. In particular, is there a dominating mechanism for the instability on the composition of individuals? Here, we propose and study simplified models to distinguish the contributions of each mechanism. It is found that imitation and mutation alone can lead to the instability on the composition of individuals in a symmetric way, and the preferred selection mechanism modifies the instability and makes the system exhibit asymmetry. Surprisingly, the co-evolution of network topology and game dynamics is not necessary to the occurrence of instability, in particular, if the new links are added in a global manner, the network topology exerts almost no impact on such instability. The results are further supported by analyzes based on mean-field approximation. This work, therefore, enhances our understanding on the driving forces of system instability.
Results
Our models are constructed under the framework of the snowdrift game, yet qualitatively the same phenomena result also in corresponding models using the prisoner's dilemma (see the Supporting Information for results on prisoner's dilemma). The snowdrift game [36]–[39] is best illustrated by a situation where two drivers are caught in a blizzard and blocked by a snowdrift. Each driver has two choices: either removing the snowdrift by shoveling or staying in the car. If the road is cleared, both drivers get a benefit 
of getting home. There is a cost 
for the labor of shoveling, with 
. If the drivers cooperate in clearing the block, they share the labor and each gets a net benefit of 
. If both choose to stay in the car, they both get zero benefit. If one of the drivers shovels, then both can go home, but the non-cooperative driver (defector) avoids the labor and gains a benefit 
, whereas the cooperator's benefit is 
. Writing 
, the model can be described by the payoff matrix [40]:
 |
(1) |
where 
and 
denote the strategies, say cooperate or defect, of the drivers. In a networked environment, the payoff to an individual in a time step is the sum of payoffs from pair-wise interactions with all her neighbors.
Imitation and Mutation
Considering a system of 
individuals, each of which takes on one of two strategies: cooperate or defect. In every time step, a new individual enters the system, chooses a role-model randomly, and imitates the role-model's strategy with a probability 
or adopts the opposite strategy with probability 
. The parameter 
is thus called the mutation rate and 
is needed to avoid the system from being frozen into a state with all the individuals using the same strategy. To keep 
constant, a randomly chosen individual is removed from the system at the same time. This process is very similar to the well-known Moran process
[41] and thus can be considered as a variant of the Moran process (the Moran process does not take into account the mutation rate, corresponding to the case of 
). In the supplementary information of [35], Cavaliere et al. provided detailed analysis about the differences between birth-death updating and death-birth updating rules. Comparing with the model of Cavaliere et al.
[35], we isolated the effects of imitation and mutation, as the details of the game and thus the payoff and performance of individuals as well as networking effects are all irrelevant. As the statistics are independent of initial configurations, we set the initial condition to be 50% cooperators and 50% defectors.
Figure 1 shows how the number 
of cooperators varies in time. It is observed that for a majority of time steps, most individuals in the system take on the same strategy, but instability sets in to swing the system to the opposite strategy. A state between the two extremes does not stay long. Excluding the time in uniform states where all individuals take on the same strategy, figure 2 gives the probability density function 
of having 
cooperators in the system. The distribution is symmetrical around 
, and the system spends much more time when there are many cooperators or defectors than when there are comparable numbers of them. Analytically, a rate equation approach (see
Analysis
for details) gives
 |
(2) |
for 
, and it gives good agreement with simulation results. This result does not depend on the mutation rate 
as long as 
. The value of 
does determine the relative abundance of the two extreme uniform states, with a smaller value giving a larger 
, as depicted by the simulation and analytic results in figure 3 (see
Analysis
for analytic treatment). The results in figure 1 to figure 3 show that the imitation and mutation mechanism alone would lead to instabilities on the composition of individuals (quantified by the number of cooperators 
). As the selection is made randomly, there is no preference on cooperators or defectors, resulting in a symmetric 
around 
.
Figure 1. Transitions between extreme states consisting of all cooperators and all defectors.
The system size is 
and the mutation rate is 
. The simulation was carried out for 
time steps. Each data point is an average over 
time steps.
Figure 2. Simulation and analytic results of the distribution 
of the number of cooperators.
Results are obtained in a simulation of 
time steps (violet open circles). Considering time steps with 
, the value of 
are found as the fraction of steps with exactly 
cooperators. The parameters are the same as those in Figure 11. The red solid line represents the the analytic results as given by Eq.(2).
Figure 3. Relative abundance of the system being in states of all cooperators and all defectors as a function of the mutation rate 
.
The system size is 
and each data point is obtained by a simulation of 
time steps. The purple dash line represents the analytic solution given in Eq.(10).
Selection Mechanism
To study the effects of preferential selections, we incorporate the snowdrift game into the model. At every time step, each individual plays the snowdrift game with all her connected neighbors and gets a total payoff according to the payoff matrix (1). Individuals will get different payoffs depending on their strategies and their competing neighborhoods. A newcomer then enters the system and selects an individual 
as the role-model with a probability proportional to the total payoff of individual 
. To allow individuals with vanishing payoffs to have a chance to be chosen, we add a small amount 
to every individual's total payoff (see
Analysis
and Supporting Information about the effects of 
on analytical treatment and numerical results, basically speaking, it has almost no impact if 
). The newcomer will follow the role-model's strategy with probability 
or adopts the opposite strategy with probability 
, where 
is the mutation rate. After deciding on the strategy, the newcomer establishes 
links randomly with existing individuals. The time step ends with the removal of one individual randomly from the 
old individuals. Here, we focus on a typical case of 
(
), which favors cooperation. Other values of 
will lead to similar results if 
.
Figure 4 shows the simulation results of 
as a function of time. The preferred selection of more successful individuals leads to a dominance of cooperators. However, the system does not stay in a state full of cooperators all the time. There are instabilities resulting in the sudden occurrence of many defectors that last only for a short duration. As a result, the total payoff of all individuals over time is still high. The situation is similar to the coexistence of prosperity and instability in the model of Cavaliere et al.
[35] (later we will show quantitatively the changes of system profile and strength of instability versus 
, which provide nice evidence on their similarity). Despite the similarity, we stress that the coexistence does not rely on network evolution in the present model, as the 
links are established randomly and the network grows independently of the game dynamics, which is different from that in Ref. [35].
Figure 4. Number of cooperators as a function of time with preferential selection.
The parameters are 
, 
, 
and 
. The simulation was carried out for 
time steps. Each data point is an average over 
time steps. The system spends most of the time in a state of all cooperators, interrupted by instabilities that last for a short duration when defectors suddenly appear. This is analogous to the coexistence of prosperity and instability as observed in the model of Cavaliere et al.
[35].
Network Evolution Mechanism
The independence on network evolution is further illustrated by considering the distributions 
for different values of 
. Figure 5 shows that the distributions 
for 
corresponding to a fully connected network, 
corresponding to a network fragmented into small pieces and 
as an intermediate case are almost the same. Analytic result of 
also shows that 
is irrelevant (see
Analysis
). The preferential selection mechanism makes 
asymmetric and shifts it to the side of larger 
, when compared with figure 2. In contrast, the model of Cavaliere et al.
[35] gives a network that undergoes continual fragmentation and coalescence, which in turn affect the fraction of cooperators in the system. The insensitivity to network topology is further illustrated in figure 6, in which we show time variations of the average degree and the number of disjoint components in the system at short times. These quantities vary in a random fashion, with no observable correlation with 
. Figure 7 reports how the average number of cooperators 
and the average system payoff (i.e., the total payoff of all individuals) 
change with parameters 
and 
. Again, 
has almost no impact on either 
or 
and we display two examples 
and 
in figure 7.
Figure 5. The distributions 
for different 
.
With 
and 
, subgraphs (a) and (b) respectively show the time distributions for 
and 
. The simulations last for 
time steps. Distributions for different 
overlap each other, implying that the number of links in the network has no influence on the prosperity of cooperation and the system instability.
Figure 6. Number of cooperators shown together with (a) the instantaneous average degree of the network and (b) the number of disjoint components in the network.
The parameters are 
, 
, 
and 
. Results are shown for the early stage. Each data point represents an average over 100 time steps.
Figure 7. The changes of (a) the average number of cooperators 
and (b) the average system payoff 
versus parameters 
and 
.
Other parameters are fixed as 
and 
. The simulation lasts for 
time steps. The black squares and red circles represent the cases of 
and 
, respectively.
To further demonstrate that the network evolution mechanism is not a necessary factor leading to the features in figure 5, we study some other mechanisms like good-get-richer
[42]–[44], where the newcomer has a probability 
(
) to connect to the role-model, in addition to a higher probability of selecting the individuals with higher payoffs as role-model, and a probability 
(
) to connect to other individuals. Under such good-get-richer mechanism, the network evolution is related to the game dynamics, but it still gives almost the same distribution 
. Typical simulation results are presented in Fig. S1. Notice that, in the present model, the links are always added in a global manner, while in the model of Cavaliere et al.
[35], the links are added in a localized manner. Therefore, one could infer that the different ways of network construction indeed matter, but the constructing rule is not necessary to be one of the origins of instability.
Effects of Payoff Matrix
In accordance with the mechanism of the snowdrift game, as the increasing of 
, defectors are encouraged and the number of cooperators decreases, leading to the decrease of the system payoff 
. This monotonous changes are illustrated in figure 7.
As shown in figure 8, with preferential selections, insensitive to different values of 
, the system is dominated by cooperators in most time with short-duration instabilities. The number of defectors in the instabilities increases with 
, resulting in the quantitatively different 
as shown in figure 7(c). Analytically, 
can be obtained using the mean-field approximation. Results are also shown in figure 7(c). The result indicates that (i) 
does not depend on the network topology and (ii) the distribution 
depends on 
, in agreement with simulation results. More detailed simulation results on the effects of 
are given in Fig. S2. Analytic treatment is presented in
Analysis
.
Figure 8. Effects of payoff parameter 
on 
.
Number of cooperators as a function of time for (a) 
and (b) 
. The data are obtained from simulations and the lines come from numerical solutions. The parameters are 
, 
and 
. Each data point represents an average over 1000 time steps. (c) Distributions 
for different values of 
. The data points are simulation results and the lines are analytic results.
For the present dynamical process, we quantify the strength of instability on the composition of individuals by counting the total number 
of transitions between all-cooperator state and all-defector state, sharp drops from all-cooperator state and raises from all-defector state (see
Methods
for precise definition). As shown in figure 9, 
again has almost no impact on the strength of instability, while when 
exceeds about 0.3, the strength of instability decreases as the increasing of 
for a wide range of the threshold 
. Recalling figure 7, when 
exceeds about 0.3, the average system payoff 
starts to decrease. Though this paper concentrates on the analysis of the instability about the composition of individuals, the observation about how 
and 
change with 
is to some extent similar to the coexistence of prosperity and instability reported in [35].
Figure 9. How the strength of instability 
changes with the threshold 
for different 
and 
.
The parameter 
has almost no impact on the strength of instability and the plot (a) compares two examples 
and 
, meanwhile 
, 
and 
are fixed. Inset of the plot (a) displays the same curves in log-linear scale. The plot (b) shows the considerable effects of 
on the strength of instability. In fact, 
is a borderline: when 
the tails of 
curves will decay quickly for large 
, namely the change of the composition of individuals is less drastic, while if 
, the strength of instability decreases as the increase of 
, which is of the similar varying tendency to the system payoff. Other parameters are 
, 
and 
. All simulations lasts for 
time steps.
Furthermore, dynamic instability on the number of cooperators is also observed for the prisoner's dilemma, which is associated with different payoff matrix and different selection mechanism (an individual's payoff can be negative and thus we cannot simply apply the linear selecting probability). See simulation results in figure S3. In accordance with the well-known conclusion, in the well-mixed case (i.e., large 
), the defectors get dominant.
Analysis
For the simplest model involving only imitation and mutation, let 
be the probability of having 
cooperators at the time step 
. The averaged probability 
of having 
cooperators can be obtained by averaging 
over a sufficiently long time window 
after the transient, i.e.,
 |
(3) |
where 
is some time after the transient behavior, which is dependent on the initial condition, ends. A newcomer has a probability 
of choosing a cooperator and a probability 
of choosing a defector as the role-model. Therefore, the newcomer has a probability 
of taking on the cooperative strategy. In removing an individual, the probability of eliminating a cooperator is 
, and that for a defector is 
. In one time step, the rates 
and 
at which a system with exactly 
cooperators would evolve into one with 
and 
cooperators are given, respectively, by
 |
(4) |
In the steady state, the rates at which 
increases are balanced by those at which 
decreases. This results in the following set of equations:
 |
(5) |
with the first two terms accounting for an increase in 
and the last term accounting a decrease in 
. Since 
, this set of equations can be solved with the supplementary (boundary) conditions 
. Applying Eq.(5) repeatedly to different values of 
, we have in general 
. Therefore, Eq.(5) can be solved exactly to yield
 |
(6) |
The normalization condition 
serves to fix 
as
 |
(7) |
It follows from Eq.(4)
that
 |
(8) |
This exact solution exhibits several interesting features. In general, 
(
). Thus 
is symmetric around 
, in agreement with that observed in figure 2. For values of 
with 
, 
and 
corresponding to the all-
and all-
states are more probable. For small values of 
, i.e., 
, Eq.(8) gives approximately
 |
(9) |
Equation (9) gives the function form of 
(
) in figure 2. Equations (8) also gives 
as a function of 
, as studied in figure 3. In particular, for 
, Eq.(9) gives
 |
(10) |
which gives the correct behavior as shown in figure 3. In addition, Eq.(8) indicates that for the particular value of 
, 
for all 
. For 
, a bump starts to appear around 
and behaves asymptotically as a gaussian distribution.
Next, we consider the model in which a newcomer selects a role-model preferentially and establishes 
connections randomly. Within a mean-field approximation, we assume that all the cooperators have the same competing environment and all the defectors have the same competing environment. For a cooperator in the system with 
cooperators, there are on average
 |
(11) |
neighbors who are cooperators and 
neighbors who are defectors. Similarly, for a defector, there are on average
 |
(12) |
neighbors who are cooperators and 
neighbors who are defectors. Therefore, the payoff to a cooperator is 
and that to a defector is 
. The probability of choosing a cooperator as the role-model is 
and the probability of choosing a defector is 
, where 
is the total payoff in the system. Here, 
is a small parameter so that every individual would have a finite probability of being chosen as a role-model (as shown in figure S4, the parameter 
has almost no impact on 
if it is close to zero). Including the effect of the mutation rate 
into the case of preferential selection, the rates 
and 
at which a system has exactly 
cooperators would evolve into one with 
and 
cooperators are given by
 |
(13) |
Note that 
and 
depend on the payoff parameter 
. In the steady state, we have in general 
, where 
is the averaged probability of having 
cooperators in the system. Applying the relation recursively to different values of 
, we arrive at
 |
(14) |
with
 |
(15) |
Equations (14) and (15) have exactly the same form as Eqs.(6) and (7), only with 
and 
replaced by 
and 
. Substituting Eq.(13) into Eqs.(14) and (15) gives 
for the case of preferential selection, as that given figure 8(c).
Discussion
Many complex systems display instabilities during their evolutions, yet the driven force of instabilities may not be as complex as being indicated in the literature. By studying on a series of simplified models, we show that imitation and mutation alone can lead to system instability, while the selection strategy and network structure are relatively insignificant. In particular, the co-evolution of network topology and game dynamics is not necessary for the occurrence of instabilities.
In the extremal situation with most of individuals being in the same state, thanks to the imitation mechanism, the new comer and the removed one are very probably of the same state and the composition hardly changes. Therefore, the system tends to stay long in the extremal situation. Given a game where the cooperators are preponderant in profits and individuals prefer to choose successful ones as their role-models, the imitation mechanism makes the system stay long with cooperators in the dominant position. This is known as prosperity in the literature [35]. At the same time, the mutation rate causes to the instability.
Notice that, the prosperity and instability coexists only when 
is very small – this is also reasonable otherwise 
can not be named as mutation rate. In fact, when 
gets larger, the extremal situation will not be preponderant. According to Eq. (8), when 
, 
will become fully uniform, say 
for all 
. For even larger 
, a peak appears at 
and the distribution behaves like a gaussian function. The results for large 
are shown in Figure 10. For an infinite population (i.e., in the thermodynamic limit), 
will show a gaussian form for any finite value of 
. In a word, the instabilities can only be observed for finite-size systems and the critical value 
indeed separate two different behavior.
Figure 10. Simulation and analytic results of the distribution 
of the number of cooperators for large values of 
.
The system size is 
and the mutation rates are 
and 
, respectively. Simulation lasts for 
time steps while the analytical solution is presented in Eq. (8).
In the present model, no matter the network evolution is independent on (rules presented in the main body) or related to (the good-get-richer mechanism in Fig. S1), links associated with the newcomer are added in a global manner. Therefore, even for far different values of 
, the networks can be all considered as random samplings with different densities from well-mixed population, which is also indicated in the domination of defectors in the prisoner's dilemma game in Fig. S3. In comparison, the new links of the model in [35] are added in a localized way. So one could infer that the way of the addition of links (e.g., globally vs. locally) indeed matters. At least, we arrive to a clear conclusion that the co-evolution of network topology and game dynamics is not a necessary condition to the occurrence of dynamic instability. There are still unsolved issues about the precise understanding of networking effects waiting for further study.
Methods
For the present models, we quantify the dynamic instability of the composition of individuals via counting the number of transitions, sharp drops and sharp raises. These three cases are respectively defined as follows: (i) Transition.—A transition is a period where the system goes from all-defector state to all-cooperator state but never return to all-defector state during this period or a period where the system goes from all-cooperator state to all-defector state but never return to all-cooperator state during this period. (ii) Drop.—A drop is a period where the system goes from all-cooperator state to a state with less than 
cooperators and then return to the all-cooperator state, during which, it does not reach the all-defector state. Here 
is a threshold. (iii) Raise.—A raise is a period where the system goes from all-defector state to a state with more than 
cooperators and then return to the all-defector state, during which, it does not reach the all-cooperator state. Figure 11 illustrates a simple example where 
and 
. In this example, one could find 5 transition, 3 drops and 5 raises. We use the total number of transitions, drops and raises, 
, to quantify the strength of dynamic instability. The readers are warned that this definition is suitable for the current case but cannot be directly applied in characterizing instability of a generally dynamical process.
Figure 11. Illustration of how to quantify the strength of instability.
We set 
and 
. In this specific example, 
, contributed by 5 transitions, 3 drops and 5 raises. Transitions, drops and raises are labelled by 
, 
and 
in the plot.
Supporting Information
The distributions
for different parameters
with good-get-richer mechanism.
(PDF)
Effects of
on the distribution of the number of cooperators
.
(PDF)
Dynamic instability in prisoner's dilemma game.
(PDF)
The distributions
for different
.
(PDF)
Acknowledgments
We acknowledge the valuable discussions with Zhi-Hai Rong, Beom Jun Kim, and Chenping Zhu.
Funding Statement
PMH acknowledges the support from the Research Grants Council of the Hong Kong SAR Government under Grant No. CUHK-401109. TZ and ZMY acknowledge the Fundamental Research Funds for the Central Universities. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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Associated Data
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Supplementary Materials
The distributions
for different parameters
with good-get-richer mechanism.
(PDF)
Effects of
on the distribution of the number of cooperators
.
(PDF)
Dynamic instability in prisoner's dilemma game.
(PDF)
The distributions
for different
.
(PDF)