Proximity inheritance explains the evolution of cooperation under natural selection and mutation

Abstract

In this paper, a mechanism called proximity inheritance is introduced in the birth–death process of a networked population involving the Prisoner's Dilemma game. Different from the traditional birth–death process, in the proposed model, players are distributed in a spatial space and offspring is distributed in the neighbourhood of its parents. That is, offspring inherits not only the strategy but also the proximity of its parents. In this coevolutionary game model, a cooperative neighbourhood gives more neighbouring cooperative offspring and a defective neighbourhood gives more neighbouring defective offspring, leading to positive feedback among cooperative interactions. It is shown that with the help of proximity inheritance, natural selection will favour cooperation over defection under various conditions, even in the presence of mutation. Furthermore, the coevolutionary dynamics could lead to self-organized substantial network clustering, which promotes an assortment of cooperative interactions. This study provides a new insight into the evolutionary mechanism of cooperation in the absence of social attributions such as reputation and punishment.

1. Introduction

The puzzle of cooperative behaviours lies in the essence of many problems where cooperative behaviour benefits others at a cost, yet free-riding behaviour enjoys cooperative benefits without any punishment [1–3]. Though selfish free-riding behaviour is the better choice by rational reasoning, cooperative behaviour has been widespread in nature and human society [4–7]. Therefore, understanding the evolution of cooperation becomes a major challenge. Over the last decades, the evolutionary game theory, together with the paradigm of the Prisoner's Dilemma game, has been a major framework to study the emergence of cooperation in populations. Within this framework, various cooperative mechanisms, such as direct reciprocity [8], indirect reciprocity [9] and group selection [10], have been put forward as possible ways towards cooperation. Among these cooperative mechanisms, network reciprocity has recently been attracting increasing attention with the development of network sciences [11–14].

The interaction networks exist ubiquitously in biological and social populations yet have been neglected for a long time owing to their topological complexity [15–17]. The network reciprocity mechanism demonstrates that the network structure could provide a moderate environment for interaction and cooperation among individuals. It has been shown that cooperation can be promoted in various network structures such as spatial lattices, random graphs and scale-free networks [18–20]. Indeed, survival of cooperation in networked populations is mainly attributed to possible clustering of neighbouring cooperators, which thus avoid being exploited by defectors.

The finding that static interaction networks can benefit from the evolution of cooperation has motivated investigations on the effect of dynamic networks on cooperation. In fact, the interaction structure among individuals may be inherently time-varying or evolves under evolutionary selection, hence resulting in the coevolution of strategy and structure in the evolutionary game process. In this aspect, a variety of coevolutionary rules have been proposed to describe the coupling updating of strategies and network topology and to reveal the effect of dynamic networks on the evolution of cooperation [21]. Among them, one major class of coevolutionary rules assume that individuals have the ability to control their interactions according to the strategy, payoff, reputations or other tags of other individuals. Hence, besides strategy updating, individuals can form new links and break existing links during the evolutionary game process. Theoretical and empirical studies have shown that such kinds of coevolutionary game dynamics give rise to more notable and stable cooperation patterns [22–27].

In this article, we consider a coevolutionary rule based on the mechanism of proximity inheritance. In this model, individuals are distributed in a spatial space and interact only with those individuals within a threshold distance. Under the selection pressure, some individuals will be eliminated and new individuals will be reproduced according to the fitness landscape of the population. The eliminated individual together with all its edges will be removed from the network, and the newborn individual will inherit the strategy of its parents with some probability of mutation and be placed in the neighbourhood of its parents as a model of proximity inheritance. Note that the above updating of strategy and network structure is based on the inheritance mechanism. The requirement of interaction control, which is a kind of social behaviour, is abandoned. Instead, based on the observation that offspring often inherits social, moral, religious or political beliefs or just lives by proximity to its parents [28–30], the mechanism of proximity inheritance is used as the driving force of network dynamics.

Based on the above proposed coevolutionary dynamics of strategy and network structure, we address the problem of the evolution of cooperation in populations with the mechanism of proximity inheritance. Firstly, it is shown that self-organized cooperator clustering could emerge during the evolutionary process and that defectors lose the battle with cooperators and are eliminated eventually. In fact, the fraction of cooperators in the population remains high for a large range of temptation of defection and selection strength, even when mutation happens with some probability. Hence, the emerged cooperation pattern is stable and robust in this coevolutionary game process. Next, a set of comparative experiments are designed to justify the effect of proximity inheritance. It is shown that the cooperation level will be reduced to a certain extent if offspring is distributed within or beyond the neighbourhood of its parents. Furthermore, if the interaction network is static or newborn individuals are randomly distributed in the spatial space, then cooperation is never favoured in the evolutionary process. Finally, the structural change of the interaction network is investigated in terms of the clustering coefficient. A greater degree of clustering is observed in the dynamic networks with game interactions than that in dynamic networks without game interactions or random geometric graphs [31].

2. Material and methods

(a). Model description

In this article, the applied evolutionary model is a repeated procedure of game interaction and strategy updating. The game-interaction procedure depicts the interplay between cooperators and defectors and results in a fitness landscape of the population, and the strategy-updating procedure gives the rule of how a new generation of population generates based on the above resulting fitness landscape.

In the game-interaction procedure, individuals are assumed to be involved in a Prisoner's Dilemma game. In detail, an individual can be a cooperator or a defector. A cooperator can get a payoff R = 1 from a cooperator for mutual cooperation and nothing S = 0 from a defector as a sucker's payoff. By contrast, a defector can get a payoff T = 1 + c from a cooperator and P = c for a defector for the punishment of mutual defection, where the parameter 0 < c < 1 is a key variant to measure the temptation of defection.

Individuals are also assumed to be organized as a network. Each individual plays the above Prisoner's Dilemma game only with its neighbours. The accumulated payoff of an individual is the sum of payoffs acquired from interacting with each of its neighbours. Furthermore, to diminish the accumulation effect of payoff in large-degree nodes, the payoff of each individual is normalized by the number of its neighbours. To take it formally, let $p_i^t$ be the payoff of individual i at step t, then the fitness of individual i is calculated as $f_i^t=1-\omega +{\omega p}_i^t$. Here, the parameter ω ≥ 0 is called selection intensity, which adjusts the effect of game interaction on population evolution.

We used the Moran process [32] to model the strategy-updating procedure of populations to encompass the effect of competition, selection and heredity of natural evolution on the emergence of cooperation and social structures. In detail, at each time step of strategy updating, an individual is randomly chosen to reproduce a new individual and another individual is randomly chosen for death. Hence, during the evolutionary process, the population size remains constant. To characterize the effect of selection, the reproducing individual is chosen with a probability proportional to the fitness of individuals. Thus, the individual with higher fitness is more likely to be chosen to produce offspring. To characterize the effect of heredity and mutation, the newborn individual will take the same strategy as its parents if mutation does not happen. Let μ be the mutation probability. If mutation happens, then the newborn individual will choose a random strategy from cooperator and defector.

While the strategy is updated, the interaction structure of individuals is also updated in our model. We consider a geometric network model, where individuals are distributed in a two-dimensional plane and any pair of individuals are connected given that their Euclidean distance is less than a given threshold r₁ > 0. During the strategy-updating procedure, when an individual is chosen for death, then the corresponding individual and all its edges will be deleted from the interaction network of individuals. Additionally, when a newborn individual is produced, then the newborn individual will be randomly placed in the circle with a radius of r₂ > 0 and centred on its parents. See figure 1 for an illustration of the geometric network of the population and updating of its structure. Here, the boundary of the population is assumed to be open in our model. That is, newborn individuals may be distributed outside the initial boundary given that its pervading circle comes across the initial boundary.

Figure 1.

Illustration of the geometric network and updating of its structure. (a) Each pair of individuals are connected if their Euclidean distance is within some threshold, (b) if an individual is chosen for death, then the corresponding node and all its edges will be removed, (c) if a newborn individual is produced, then a new node will be added and randomly placed at a location in a circle centred on its parents.

Because individuals are more likely to interact when they live near to each other or possess similar social, moral, political or religious beliefs, the geometric network provides a proper model to characterize the interactions among the population. Furthermore, newborn individuals will often inherit social, moral, political or religious beliefs just by proximity to their parents, hence newborn individuals are placed in a circle centred on their parents. For simplicity, the above structure evolving mechanism is called proximity inheritance.

To understand the dynamical behaviour of the above evolutionary model, a variety of simulations will be carried out in the following with different conditions. Yet, some general set-up will be used otherwise specified. In detail, initially a number of N = 100 individuals are randomly placed in the [0, 1) × [0, 1) plane, and the connection threshold r₁ is set to be $\sqrt{(\ln N/\pi N)}\approx 0.12$ to promise the connectivity of the geometric graph. Each individual chooses to be a cooperator or defector with equal probability. The pervading radius r₂ is selected to be the same as the connection threshold r₁. The selection intensity ω and the mutation rate μ are generally set to be 0.2 and 0.01, respectively. The temptation of defection c is set to be 0.2. For convenience, the set of parameters used in the above model and their default values are summarized in table 1.

Table 1.

The set of parameters and its default value.

notation	meaning	default value
N	population size	100
r1	connection threshold	0.12
r2	pervading radius	0.12
ω	selection intensity	0.2
μ	mutation probability	0.01
c	temptation of defection	0.2

Furthermore, for each group of fixed parameters, 80 independent simulations will be carried out so as to ease up the randomness effect of the evolutionary dynamics. In detail, the statistics data are obtained by firstly averaging over from the 10 × 10³-th generation to 30 × 10³-th of a single simulation (here the first 10 × 10³ generations are required to reach dynamical equilibrium) and then averaging over 80 independent realizations. Further details about experiment design are given in the main Results section for the convenience of illustrations.

3. Results

(a). Emergence of cooperation

To show the effect of proximity inheritance on the evolutionary process of a population of cooperators and defectors in the Moran process, some typical snapshots of the evolutionary process are first given to depict the distributions of cooperators and defectors on the plane space, as shown in figure 2 (see also the electronic supplementary material, figure S1 for the case with mutation). In the simulation, strategy mutation is not considered and all the other parameters are set to the default value (see Material and methods). In this simulation, 49 cooperators and 51 defectors are randomly distributed in a space initially. In this stage, the defectors can easily take advantages of the cooperators through the defector–cooperator (CD) links, which take up a proportion of 50.5% of the total links. By contrast, the cluster effect of cooperators is low owing to the limited proportion (20.7%) of cooperator–cooperator (CC) links. As the evolutionary process goes on, defectors are more likely to distribute around defectors while cooperators are more likely around cooperators as a result of the mechanism of proximity inheritance. In this case, the chance of free riding for defectors will be greatly reduced, while the cluster effect of cooperators will be strengthened. Indeed, the proportion of CD links reduces to 23.7% at the 2 × 10³-th generation and to zero at the 4 × 10³-th generation. As a comparison, the proportion of CC links increases to 47.3% at the 2 × 10³-th generation and to 98.3% at the 4 × 10³-th generation. As a result, in this simulation, cooperators win the competition with defectors and the fraction of cooperators in the population eventually reaches 100%.

Figure 2.

Typical snapshots in a simulation of the evolutionary model. The mutation probability is set to be μ = 0 and other parameters are as default. Red and blue nodes denote cooperators and defectors, respectively. The figure illustrates the evolution of cooperation in a spatially distributed population under the coevolutionary rule. (a) The initial generation, cooperation frequency = 0.49, the proportion of CC, CD and defector-defector (DD) links are 20.7%, 50.5% and 28.8%, respectively. (b) The 2 × 10³-th generation, cooperation frequency = 0.6, the proportion of CC, CD and DD links are 47.3%, 23.7% and 29%, respectively. (c) The 4 × 10³-th generation, cooperation frequency = 0.95, the proportion of CC, CD and DD links are 98.3%, 0% and 16.7%, respectively. (d) The 10 × 10³-th generation, cooperation frequency = 1, the proportion of CC, CD and DD links are 100%, 0% and 0%, respectively.

In order to get a clear understanding of the conditions for the emergence of cooperation, we obtained the statistics data of cooperation frequency in evolutionary processes with a variety of different parameters. The results are shown in figures 3 and 4. Firstly, we fix the selection intensity and the mutation probability so as to see how the cooperation frequency depends on the temptation of defection c. Obviously, the cooperation frequency decreases as the temptation of defection increases. There exists a critical value: when the temptation of defection c is within the critical value, the final cooperation frequency is over 50% and cooperation is favoured, yet when the temptation of defection is beyond the critical value, defection takes more proportion of the population. Generally, in our model, the critical value is within the [0.3, 0.6] interval, depending on the value of selection intensity and the mutation probability. For example, given ω = 0.2 and μ = 0.01, the final cooperation frequency is over 56% even if the temptation of defection is c = 0.6, yet for ω = 0.8 and μ = 0.05, the cooperation frequency is less than 40% when the temptation of defection is c = 0.4.

Figure 3.

The fraction of cooperators as a function of temptation of defection c under different values of selection intensity. (a) The mutation probability μ = 0.01, (b) the mutation probability μ = 0.05. In this figure, the fraction of cooperators is over its initial value 50% and increases with the selection intensity for a large scale of temptation of defection, indicating that selection favours cooperation over defection. (Online version in colour.)

Figure 4.

The fraction of cooperators as a function of temptation of defection c under different values of mutation probability. (a) The selection intensity ω = 0.2, (b) the selection intensity ω = 0.8. The figure shows that cooperation can be favoured over defection even when mutation happens with some probability. (Online version in colour.)

Now we fix the temptation of defection and the mutation probability to compare the value of cooperation frequency for different selection intensity ω. As shown in figure 3, the relationship between selection intensity and cooperation frequency is not simply linear. For small temptation of defection (less than 0.2), cooperation is favoured over defection, and thus an increase in the selection intensity promotes the fraction of cooperators. For large temptation of defection (larger than 0.4), defection is favoured over cooperation, and thus an increase in the selection intensity leads to a decrease in the cooperation frequency. Yet, when the temptation of defection is neither the above two cases, from the continuity in the evolution proofs, there exists an optimal selection intensity (neither too small or too large) resulting in the largest cooperation frequency. All the above phenomena are observed for both mutation probabilities μ = 0.01 in figure 3a and μ = 0.05 in figure 3b (see also the electronic supplementary material, figure S2a for the case with a larger population size).

An important question of cooperation evolution is whether the emerged cooperator clusters can be maintained under the invasion of defectors. Indeed, if mutation is considered, defectors can be produced in the cooperator clusters and then exploit cooperators, leading to decomposition of cooperative clusters (electronic supplementary material, figure S3). To know whether cooperation can resist the invasion of defectors with the mechanism of proximity inheritance, the fraction of cooperators is obtained under different values of mutation probability, as shown in figure 4. Though the fraction of cooperators decreases as the mutation probability increases, it can be observed that the fraction of cooperators can remain dominant in the population even when mutation happens with a probability from 0.01 to 0.1, given that the temptation of defection is within a critical value. In fact, a defector in a cooperator cluster can get a higher fitness temporarily. Yet owing to the mechanism of proximity inheritance, more defectors will be reproduced around in the cooperator cluster and cooperators around this defector will be eventually eliminated owing to selection. As a result, the fitness of defectors becomes low again and these mutated defectors are eventually eliminated again (electronic supplementary material, figure S4).

(b). The effect of proximity inheritance

To get a clear understanding of the effect of proximity inheritance on the evolution of cooperation, a set of comparative experiments are designed in the following. Firstly, the fraction of cooperators as a function of the pervading radius r₂ is obtained to reveal the role of pervading radius in the coevolutionary model. The results are shown in figure 5a (see also the electronic supplementary material, figure S2b for the case with a larger population size). As expected, with the increase in the pervading radius r₂, newborn individuals are more likely to spread out of the proximity of their parents. As a result, the clustering effect of cooperators will be reduced and defectors have more chance to exploit cooperators. That is why the fraction of cooperators decreases when the pervading radius r₂ increases from r₁ = 0.12, 0.18, 0.24 to 0.9. Interestingly, the fraction of cooperators increases with r₂ ∈ [0, r₁] and reaches its maximal when the pervading radius r₂ equals the connection threshold r₁. The above phenomena can be observed for different r₁, as shown in figure 5a. A possible explanation is that it becomes a bit harder for cooperators to get rid of those mutated defectors when the pervading radius r₂ is smaller than the connection threshold r₁.

Figure 5.

The fraction of cooperators as a function of (a) the pervading radius r₂ under different values of connection threshold, and (b) the connection threshold r₁ under different values of pervading radius. In both cases, the fraction of cooperators reaches its maximum given that the pervading radius of offspring equals the connection threshold of its parents. (Online version in colour.)

The fraction of cooperators as a function of the connection threshold r₁ shows similar performance with that of the pervading radius. As shown in figure 5b, the fraction of cooperators increases as the connection threshold r₁ increases from 0.06 to r₂ = 0.12, 0.18, 0.24, and decreases as the connection threshold increases from r₂ to 0.9. The maximal fraction of cooperators is obtained when the connection threshold r₁ equals the pervading radius r₂. Indeed, when r₁ < r₂, the probability for newborn individuals to spread out of the proximity of their parents will be reduced with the increase in the connection threshold r₁, hence the clustering effect of cooperators will be strengthened and results in a high level of cooperation. Yet when r₁ > r₂, newborn individuals will stay in the proximity of their parents, and all the individuals in the population are more likely to interact with each other with an increase in r₁. Hence, it becomes harder for cooperators to get rid of those defectors. In this case, the fraction of cooperators decreases with an increase in r₁.

We now consider two different evolutionary models to compare and to reveal the effect of proximity inheritance. In the first evolutionary model, newborn individuals are randomly placed in the [0, 1) × [0, 1) plane. In the second evolutionary model, a newborn individual is placed at the location of the dead individual. For convenience, the first and second comparative models are called the ‘random-location' model and ‘replacing-the-death’ model, respectively. Obviously, in both the random-location model and the replacing-the-death model, newborn individuals will be not placed in the proximity of their parents. Moreover, the network structure of the population is static in the replacing-the-death model. Figure 6 shows the fraction of cooperators as a function of the temptation of defection c in the three models. Obviously, without the mechanism of proximity inheritance, cooperation is not favoured over defection even for a very small temptation of defection in the above two evolutionary processes.

Figure 6.

The fraction of cooperators as a function of temptation of defection c in a different evolutionary model. Other parameters are as default. It can be observed that cooperation is never favoured if offspring is placed in a random location (square data) or the location of the eliminated individual (diamond data) instead of the neighbourhood of its parents (circle data). (Online version in colour.)

(c). Emergence of clustering

Besides promotion of cooperation, various topological characteristics of the interacting networks of individuals emerge in the coevolutionary model (electronic supplementary material, figures S5–S7). Of particular interest is that a greater degree of clustering is also observed in the dynamic networks of the coevolutionary model than in random networks. To reveal this, the average clustering coefficient is used to capture the degree of ‘friends of friends' in networks. Firstly, we check whether the average clustering coefficient achieved in the dynamic networks with the inheritance of proximity differs from its initial random geometric networks. To determine this, we generate 100 random geometric networks (the node number and connection threshold are as default) and record the average clustering coefficient of the initial network and the final 20 × 10³ rounds of dynamic networks. The simulation results show that the mean of the average clustering coefficient is improved from 0.555 in the random geometric graphs to 0.583 in the coevolutionary networks with selection strength ω = 0.4.

To show the effect of game interactions on clustering, we then compare the clustering coefficient in the dynamic networks with game interactions (ω ≠ 0) and without game interactions (ω = 0). Note that all individuals have the same fitness in the case of the selection strength ω = 0, hence the game interaction has nothing to do with the evolution of the network. It can be observed that the clustering coefficient increases from 0.563 for ω = 0 to 0.583 for ω = 0.4 and 0.602 for ω = 0.8 (electronic supplementary material, figure S8). Indeed, the role of game interactions is enhanced as there is an increase in the selection strength. The above results, together with the results of evolution of cooperation, clearly reveal the mutual promotion of cooperation and clustering in populations under the mechanism of proximity inheritance.

4. Discussion

Our simulations show that clustering cooperators can emerge and preserve under the selection pressure and possible mutation with the help of proximity inheritance. In detail, the fraction of cooperators in our model remains dominant of the evolving population for a large range of temptation of defection, selection intensity and specifics of mutation rate. The cooperation level reaches maximal given that the distribution range of newborn individuals coincides with the neighbourhood of its parents. Besides the evolution of cooperation, the interplay between strategy and interactions in our model could also promote clustering among individuals in the population. Indeed, it is shown that the clustering coefficient of networks increases in the coevolution model with game interactions as compared with the model without game interactions.

The mechanism for the resulting emergence of cooperation can be illustrated by an extreme case of our coevolutionary model. Consider a population of spatially distributed cooperators with a single defector. Under the coevolutionary rule, at first, the defector acquires higher fitness and produces more defectors in its neighbourhood. At the same time, the cooperators in the neighbourhood of the defector get lower fitness and will be eventually eliminated. Yet, as the production of defectors and elimination of cooperators in the neighbourhood of defector decrease, the fitness of defectors decreases rapidly. Thus, the defectors will lose the competition with the cooperators out of the defective neighbourhood and thus are eventually eliminated. From an overall viewpoint, under the mechanism of natural selection and proximity inheritance, the cooperators can spontaneously cut off those neighbouring cooperators of defectors to isolate and then eliminate the defectors. Our results provide a different potential explanation for the cooperation and clustering in biological and social populations. The mechanism of proximity inheritance, which supposes that offspring is more likely to be distributed around its parents, could not only lead to a strong assortment of cooperative interactions but also improve the possibility of direct reciprocity, indirect reciprocity and kin assortment among individuals [33–35].

The underlying setting of our model is quite simple and solid from a practical perspective. First, the assumption that individuals are distributed in a spatial space and interact only with those individuals within a threshold distance is a common manner of interaction in biological or social populations [36]. Note that the spatial space can mean not only geographical spaces but also lingual, moral and other social spaces. Then, the birth–death updating rule is one of the fundamental models of population evolution under natural selection and mutation. Finally, the mechanism of proximity inheritance, which induces the coevolution of strategy and interactions, is based on the ubiquitous observations that offspring often inherit social, moral, religious or political beliefs or just live by proximity to its parents [37,38].

Data accessibility

No data were collected in order to conduct the research presented in this manuscript.

Competing interests

The authors declare no competing financial interests.

Funding

This work was supported by the National Natural Science Foundation of China under grant 61873088, 61503130 and the National Natural Science Foundation of Hunan Province under grant no. 2016JJ3044.