Impact of discrepant accumulations strategy on collective cooperation in multilayer networks

Abstract

Understanding large-scale cooperation among related individuals has been one of the largest challenges. Since humans are in multiple social networks, the theoretical framework of multilayer networks is perfectly suited for studying this fascinating aspect of our biology. To that effect, we here study the cooperation in evolutionary game on interdependent networks. Importantly, a part of players are set to adopt Discrepant Accumulations Strategy. Players employing this mechanism not only use their payoffs in the multilayer network as the basis for the updating process as in previous experiments, but also take into account the similarities and differences in strategies across different layers. Monte Carlo simulations demonstrate that considering the similarities and differences in strategies across layers when calculating fitness can significantly enhance the cooperation level in the system. By examining the behavior of different pairing modes within cooperators and defectors, the equilibrium state can be attributed to the evolution of correlated pairing modes between interdependent networks. Our results provide a theoretical analysis of the group cooperation induced by the Discrepant Accumulations Strategy. And we also discuss potential implications of these findings for future human experiments concerning the cooperation on multilayer networks.

Introduction

Cooperation exists widely in the real world ranging from natural to social systems^1,2. The emergence and maintenance of cooperation among selfish individuals, even though inconsistent with the theory of natural selection proposed by Darwin³, has received lots of attention. Researchers have traditionally adopted evolutionary game theory⁴ as common formal framework for studying the dynamics of strategy change and games such as the prisoner’s dilemma and the snowdrift game as metaphors for the tension between group welfare and individual selfishness. Several mechanisms have been invoked in order to explain why altruism can actually emerge, such as kin selection, group selection, direct reciprocity, indirect reciprocity and network reciprocity⁵. After Nowak and May⁶ first combined evolutionary games and spatial chaos to explain the existence of cooperators in the population, numerous studies spring up to show that the spatial structures, referred to as network reciprocity, enhance the cooperative behaviors among the individuals^7,8.

From the end of the last century to the present, significant progress has been achieved in the field of network science, as evidenced by notable works^9,10. The extensive exploration of the impact of interaction topology on cooperative behavior has been a focal point during this period^11–15. Utilizing complex network models, numerous studies have constructed models to simulate real systems¹⁶, providing robust spatial structures for the analysis of human cooperation. As research delves deeper, researchers have come to the realization that single-layer networks are inadequate for accurately describing reality. Research in many realm has highlighted that seemingly irrelevant changes in one network can have very much unexpected consequences in another networks^17,18. Indeed, the multilayer network models are able to account for the variety of different social contexts an individual may be involved in, and are thus crucial for an in-depth understanding of human cooperation than the single-layer network models^19,20. Huang conducted a study on the impact of external forcing on evolutionary games within complex networks²¹. The findings revealed that external forcing serves as a potent catalyst for cooperation. Kleineberg²² delved into the concept of topological enslavement in evolutionary games on correlated multiplex networks. The study concluded that in scenarios where the multiplex comprises numerous layers and exhibits strong degree correlations, the system’s topology dictates the dynamics. Consequently, the final outcome, specifically the cooperation density, becomes independent of the payoff parameters. In another study, Chen explored the evolution of cooperation driven by collective interdependence on multilayer networks²³. Theses researches highlighted the significant impact of collective interdependence on interdependent network reciprocity.

The aforementioned conclusion, along with various studies on interdependent network reciprocity, serves to extend the exploration of multilayer networks. Mechanisms such as coupled evolutionary fitness²⁴, interconnection^25,26, biased imitation^27,28, information sharing^29,30, and biased resource allocation^31,32 have collectively demonstrated that the multi-layer network structure plays a pivotal role in fostering the evolution of cooperation. These findings contribute to the existing literature on interdependent network reciprocity within the framework of evolutionary game theory.

In the field of game theory, fitness is one of the key factors influencing strategy updates. In a single-layer network, various strategies often exert their influence by affecting fitness. In multilayer networks, fitness not only depends on strategies but also requires consideration of interdependencies between networks. Without considering additional mechanisms such as conformity, there are two common methods for computing fitness in multi-layer networks. The first method relies solely on single-layer payoffs as the source of fitness, where the payoffs from external players or the players themselves directly influence the current strategy^33,34. Some studies employ specialized multi-layer network models to endow players with particular properties, such as community status^35,36 or additional learning partners³¹. In the second method, fitness stems from the cumulative payoffs across layers. Mechanisms that directly accumulate cross-layer outcomes have been shown to be effective strategies for rapidly promoting strategy persistence³⁷. Building upon this, Santos introduces bias mechanism, finding that that the stronger the bias in the utility function, the higher the level of public cooperation^27,28. However, both of these methods have their drawbacks. The first method overlooks the impact of interdependencies between networks on payoffs. The specialized network models used often only apply to specific scenarios, lacking generality. The second cumulative strategy method tends to unconditionally accumulate payoffs across layers when computing fitness, disregarding individual characteristics and their performance differences across different networks. Both approaches fail to fully reflect reality, neglecting nuances, which results in biased model design and experimental results compared to actual scenarios, hindering further research progress. In practical situations involving diverse social relationships, individuals not only consider multiple factors simultaneously but also tend not to blindly attribute all payoffs to a single strategy. Instead, people are inclined to judiciously categorize and distinguish the outcomes of different strategies based on specific environments.

Hence, this experiment emphasizes a unique feature that we term the “Discrepant Accumulations Strategy mechanism (DAS)”. In essence, individuals existing in multiple networks, when updating their strategy on a particular network, differentiate and calculate the benefits from different strategies at other levels based on the current network strategy and obtain a more accurate measure of fitness. In detail, we model populations on a multilayer network, considering social structures as compositions of multilayer lattice networks with varying connection ratios between layers. The learning process is governed by the Discrepant Accumulations Strategy mechanism, employing the weak prisoner’s dilemma to characterize interactions among agents. To the best of our knowledge, there is limited literature discussing the impact of the Discrepant Accumulations Strategy mechanism on evolutionary dynamics. Therefore, this study holds significance in addressing challenges related to solving social dilemmas.

In the subsequent sections of this paper, we commence by providing a comprehensive illustration of the game model, elucidating the Discrepant Accumulations Strategy mechanism shared across two interdependent networks in detail (“Model” section). Following that, in “Results” section, we present extensive numerical simulation results along with detailed explanations. Finally, we offer concluding remarks summarizing the key characteristics of cooperation and potential applications within the current model.

Model

The evolutionary games are staged on two square lattice with periodic boundary conditions, each of size $L\times L$, where initially each player x is designated as a cooperator($s_x$ = C) or defector ($s_x$ = D) with equal probability. Likewise randomly, a fraction $c$ of players on each lattice is selected to form an external link with a corresponding player in the other lattice, the larger the value of $c$, the more players have external link. Games and strategy propagation occur only within layers, while payoffs can spread across layers.

The weak prisoner’s dilemma game is characterized by the temptation to defect T = b, reward for mutual cooperation R = 1; and both the punishment for mutual defection P as well as the suckers payoff S equaling 0, where 1 < b $\le$ 2. To better describe the intensity of the dilemma when introducing specific reciprocity mechanisms, Wang et al. introduced a new set of scaling parameters, defining the strength of gamble-intending dilemma (GID) $D_g^{\prime }$ and the strength of the risk-averting (RAD) $D_r^{\prime }$ respectively^38,39:

$$\begin{array}{c}\hfill D_g^{\prime }=\frac{T-R}{R-P}=\frac{D_g}{R-P},\end{array}$$

and

$$\begin{array}{c}\hfill D_r^{\prime }=\frac{P-S}{R-P}=\frac{D_r}{R-P}.\end{array}$$

As only $D_g^{\prime }$ is positive, the weak prisoner’s dilemma used in this experiment can also be classified as the so-called Chicken game. In this case a player receives its payoff by playing the game with all its neighbors. And the payoff matrix can be expressed as:

$$\begin{array}{c}\hfill M=\left(\begin{array}{cc}1& 0\\ b& 0\end{array}\right).\end{array}$$

One round of the games consists of each individual playing game with each of her neighbors in each layer in the multiplex. Furthermore, we consider the evolution of the system to be governed by imitation dynamics⁸, reflecting that individuals tend to adopt the strategy of more successful neighbors. And it is worth noting that in multilayer networks, due to the absence of physical links, strategy imitation does not occur across layers. We simulate the evolutionary process on both networks in accordance with the standard Monte Carlo simulation procedure comprising the following elementary steps. First, a player x is selected randomly and its payoff ${\mathrm{\Pi }}_x$ is determined on the governing evolutionary game. Next, a neighbor y from the neighborhood is chosen randomly and acquires its payoff ${\mathrm{\Pi }}_y$ in the same way. Lastly, player x adopts the strategy of player y with the probability $P(s_x\leftarrow s_y)$, specified by the Fermi-Dirac distribution. This rule is widely recognized in evolutionary games because it balances both rational and irrational behavior when players update their strategies⁴⁰.

$$\begin{array}{c}\hfill P\left(s_x,\leftarrow ,s_y\right)=\frac{1}{1+exp\left[\left(U_x,-,U_y\right),/,K\right]}.\end{array}$$ 1

This imitation dynamic reflects that individuals tend to adopt the strategy of more successful neighbors. Parameter K acts as a temperature and quantifies the irrationality of the players. Without lacking the generality, we set K = 0.1, while $U_x$ and $U_y$ are the utilities of player x and y, respectively. All the players that have an external link have the utility

$$\begin{array}{c}\hfill U_x={\mathrm{\Pi }}_x+a{\mathrm{\Pi }}_{x^{\prime }},\end{array}$$ 2

where the ${\mathrm{\Pi }}_x$ and ${\mathrm{\Pi }}_{x^{\prime }}$ are used to represent, respectively, the aggregated payoff of x and x′. While those players do not have an external link retain $U_x={\mathrm{\Pi }}_x$. Besides, the player with an external link has a probability of adopting the Discrepant Accumulations Strategy, which we refer as the DAS strength parameter $\beta$ $(0\le \beta \le 1)$. The larger the value of $\beta$, the more likely players are to adopt the DAS mechanism. In this mechanism, if x and x’ adopt the same strategy, the utility remains unchanged. However, if their strategies are different, the utility can be expressed as:

$$\begin{array}{c}\hfill U_x={\mathrm{\Pi }}_x-a{\mathrm{\Pi }}_{x^{\prime }}.\end{array}$$ 3

The parameter $(0\le a\le 1)$ determines the strength of external links, i.e., the larger its value the higher the potential increase of utility of two players that are connected by the external link. Based on the preceding study²⁴, where Wang had studied the general impact of the value of a and the related optimal interdependence between two networks, we here use a fixed value a = 0.5 without loosing generality.

Monte Carlo simulations are performed on sufficiently large lattice networks ranging in size from $50\times 50$ and $200\times 200$ near transition points to avoid accidental extinction of the two competing strategies. The stationary fraction of cooperators ($f_c$ ) is recorded after the system reaches dynamical equilibrium, i.e., when the average cooperation level becomes time independent. More specifically, we perform ${10}^4$ Monte Carlo steps (MCS) to reach the stationary state, and subsequently ${10}^6$ more steps to record. Moreover, we average the final outcome over up to 100 independent initial conditions to further improve accuracy. By comparing the results of different network sizes, we found that network size only affects the evolution time but does not change the nature of the evolutionary outcomes. For convenience in presentation, the subsequent experimental results are all derived from a $100\times 100$ network.

Results

To begin with, we investigate the cooperation frequency as a function of Discrepant Accumulations Strategy (DAS) strength parameter $\beta$ for fixed values of the connect ratio c on the interdependent networks (Fig. 1). In scenarios where the interdependence between layers is relatively weak, characterized by the value of c ($c<0.3$), both network reciprocity and the Discrepant Accumulations strategy mechanism encounter challenges in achieving effective operation, irrespective of the value of b. Cooperation emerges exclusively in exceptionally favorable environments, specifically $\mathit{b}\le 1.02$. Under these conditions, the cooperation rate can be maintained at approximately 60% (Fig. 1a). When the interdependence between layers strengthens, ($c\ge 0.3$), both network reciprocity and the Discrepant Accumulations Strategy mechanism begin to manifest their effects. In the Fig. 1b,c plots, multiple curves appear above the x-axis, indicating that, at equilibrium, the majority of players in the system are cooperators. Additionally, we observe that as the value of $\beta$ increases, the cooperation rate at equilibrium in the system gradually rises. When the environment becomes more stringent (e.g., $\mathit{b}\ge 1.15$ in Fig. 1b and $\mathit{b}\ge 1.20$ in Fig. 1c), network reciprocity and the DAS mechanism fail, leading the cooperation rate to drop to zero. Furthermore, by comparing these three graphs, we observe that with the increase in the value of $c$, the number of curves appearing above the x-axis also increases. The cooperation rate at equilibrium shows an upward trend. This indicates that, with the assistance of network reciprocity and the DAS mechanism, the living space for cooperators expands. In summary, we note a robust correlation between the DAS mechanism and inter-layer dependency. The fortification of this dependency amplifies the efficacy of the DAS mechanism. Not only does the DAS mechanism play a pivotal role in enhancing cooperation within the system, but it also broadens the survival space for cooperators.

Figure 1.

Cooperation frequency as a function of the Discrepant Accumulations Strategy strength parameter $\beta$ ($0.0\le \beta \le 1.0$) on networks. Panels (a–c) are the results under $\mathit{c}=0.1$, $\mathit{c}=0.5$, $\mathit{c}=1.0$, respectively.

To delve into further details, we depict the dependence of cooperation frequency on $b$ and $\beta$ in Fig. 2. The contour plots visually represent the cooperation levels within the system, revealing three distinct regions: pure cooperators, pure defectors, and the coexistence of both states. When $\beta$ is 0.0 and b is in the range (1.00, 1.06), the cooperation rate of the system is maintained at around 0.8. However, when $\beta$ is equal to 1.0, to keep the cooperation rate in the same range, the range of b narrows down to (1.15, 1.17). In addition to confirming that the mechanism enhances cooperation, these results highlight a noteworthy discovery under the DAS mechanism, the region where cooperators and defectors coexist is considerably smaller than under other mechanisms, exhibiting a stark black-and-white effect. Furthermore, as the strength of the DAS mechanism increases, this narrowing effect becomes more pronounced: as $\beta$ increases(depicted in Fig. 2c), the coexistence area decreases.

Figure 2.

Contour plots of cooperation frequency with the Discrepant Accumulations Strategy parameter $\beta$ ($0.0\le \beta \le 1.0$) under different values of c. Panels (a–c) are the results under $\mathit{c}=0.1$, $\mathit{c}=0.5$, $\mathit{c}=0.9$, respectively.

To elucidate the origins of distinct cooperation behaviors facilitated by the Discrepant Accumulations Strategy, we investigate the time series of cooperation levels on networks with varying values of b when $c$ = 1.0 (Fig. 3) through the enduring (END) and expanding (EXP) analysis suggested in various previous studies⁴¹. In the enduring (END) period, cooperators are plundered by defectors, and only a few cooperators survive by the formation of compact cooperator-clusters. In the expanding (EXP) period, cooperator-clusters expand and may dominate the system.

Figure 3.

Panels depict the time series of cooperation level for different b and different values of $\beta$ of the multiplex.For the purpose of illustration, the value of c is set to 1 in the above figures. Panels (a–d) are the results under $\mathit{b}=1.05$, $\mathit{b}=1.10$, $\mathit{b}=1.15$, $\mathit{b}=1.18$, respectively.

When $b=1.05$ Fig. 3a, regardless of the value of $\beta$, the cooperation rate on the network exhibits a typical “down first, then up” trend. In the END period, due to the random distribution of individual strategies on the networks and the characteristics of the weak Prisoner’s Dilemma, cooperators are easily exploited by defectors. As a result, the cooperation rate on the network declines rapidly. The larger the value of $\beta$, the slower the rate of decline in cooperation, and the higher the cooperation rate at the end of the END period. When $\beta =1.0$, the cooperation rate reaches 40%, but when $\beta =0.1$, the cooperation rate is only around 20%. In the subsequent EXP period, the remaining cooperators from the previous phase begin to expand. An increase in the value of $\beta$ will simultaneously enhance the rate at which the system reaches a steady state and the cooperation rate of the system at equilibrium. However, as the value of $b$ increases, the DAS mechanism begins to fail. In some systems, the cooperation rate no longer rises after the END phase but instead approaches the x-axis at a slower rate compared to before. When the value of $b$ exceeds 1.18 Fig. 3d, no system’s cooperation rate can surpass 10%.

To probe into the distributions of cooperators and defectors in the physical space, we investigate the snapshots of state transition. As players on different layers do not differ in terms of status and attributes across various networks, we plot the players from different layers on the same image. Figure 4 shows the time evolution of the spatial strategy distributions for different values of $\beta$ and $c$ when $b$ = 1.11. In the early stages of evolution, from $T=1$ to $T=10$, cooperators rapidly change their strategies due to exploitation. Both defectors and cooperators cluster together on the network, as seen in the second column of Fig. 4 where blue and orange spots coexist, with the blue modules being more prominent. This indicates that defection is predominant at this stage. The remaining clusters of cooperators are crucial for the subsequent transmission of cooperative strategies. Subsequently, regardless of the increase in $c$ value or $\beta$ value, which represent the enhancement of inter-layer connections and the intensity of the DAS mechanism, the cooperative clusters that persist during the END period will expand. This expansion preserves the seeds for the propagation of cooperative strategies in the later stages. If the probability of inter-layer connections is less than 50% ($c<0.5$) or the cumulative strategy is below 0.3, defection becomes the dominant strategy in the system. This is well demonstrated by the Fig. 4s,t images in the last column of the figure. If only the inter-layer connection rate or the cumulative strategy intensity is increased individually, it does not change the dominance of defection as the primary strategy (as shown in Fig. 4u,w). The cooperative clusters just disappear at a slower rate. Only when both factors act together ($c>0.5$ and $\beta >0.5$) can the cooperative clusters expand in the subsequent EXP phase, making cooperation the dominant strategy when the system reaches a steady state (as shown in Fig. 4v,x). In summary, the combined cooperative cumulative strategy of inter-layer connections can help the group retain cooperative clusters during the END phase and propagate cooperative strategies during the EXP phase. This is considered key to enhancing the group’s cooperation rate, as suggested in the literature.

Figure 4.

Series of distribution snapshots of cooperate (orange) and defect (blue) for different Discrepant Accumulations Strategy strength parameter c. Columns from left to right: c = 0.1, c = 0.5, c = 1.0. In each column, the left sub-column $\beta$ = 0.1, while the right sub-column $\beta$ = 1.0. And rows from top to bottom: T = 1, T = 10, T = 100, T = 10,000. The simulations were obtained for b = 1.10 in all panels.

The cross-layer pairing between two-layer network is the key to the Discrepant Accumulations Strategy. In this model, there are three pairing modes, which can be divided into two major categories: same-strategy matching, including C–C (cooperate–cooperate) pairing mode and D–D (defect–defect) pairing mode, and different-strategy C–D (cooperate–defect) pairing mode. The result is shown in Fig. 5. To further explain the equilibrium state of cooperation, we first study the changes in the overall ratio of different pairing modes in various scenarios. When the values of $c$ and $b$ are mismatched (i.e., $c<0.1$ and $b>1.04$), the DAS mechanism will fail. In this scenario, regardless of the value of $\beta$, only one strategy will prevail in the system, predominantly the D–D pairing Fig. 5c,i. Based on this, we will primarily analyze the situations where multiple pairing modes coexist.

Figure 5.

Panels illustrates the proportions of various pairing modes in the system upon reaching a steady state under different conditions. Detailed conditions are provided in the upper left corner of each subplot.

When the interdependence between networks is weak and the environment is favorable Fig. 5a, multiple pairing modes exist in the system. However, as the value of $\beta$ increases, the proportion of C–C pairings mode will rise. However, this dominance of C–C pairings is fragile. Even a slight increase in the value of $b$ (Fig. 5b) can lead to a shift in the dominant pairing strategy from C–C to D–D. As the interdependence between network layers deepens, this situation is alleviated. By comparing the images in the first and second rows, it can be observed that C–C pairings gradually establish a stable dominant position. Additionally, although the number of C–D pairings decreases with an increase in $\beta$, they still remain at a level greater than zero. As the value of $c$ increases, the effect of the DAS mechanism becomes more significant, C–C pairings mode becomes the dominant pairing mode in more environments. When the environment becomes harsher, the impact of the DAS mechanism becomes even more pronounced, especially beyond a certain intensity threshold (such as when $\beta >0.5$, Fig. 5f), where the DAS mechanism can rapidly change the dominant pairing strategy in the system. It is noteworthy that when $c>0.5$, different pairings almost no longer appear. In summary, we can conclude that the DAS mechanism influences the pairing modes to alter the cooperation level in the system.

Next, we will continue to explore the temporal evolution process of different pairing modes under varying values of $\beta$. Figure 6 illustrates the evolution of the proportion of the three pairing modes within the group. In this module, to better demonstrate and analyze the effects of the DAS mechanism, we have set the interdependence coefficient $c$ between networks to 1. In the phase where $b<1.05$, if the DAS mechanism is not implemented, all three pairing modes will coexist within the group. Otherwise, due to the significantly lower payoffs for individuals in different-strategy pairings compared to same-strategy pairings, these individuals will gradually shift to same-strategy pairings. Ultimately, the DAS mechanism will reduce the proportion of different-strategy pairings in the spatial structure. With the assistance of external layer income, the number of remaining cooperators in the END phase increases. The sustained returns from cooperator clusters cause individuals in different-strategy pairings to preferentially shift to C–C pairings, thereby making cooperation the dominant strategy.

Figure 6.

Panels depict the time series of ratio of different pairing modes for different b and different values of $\beta$. For the purpose of illustration, the value of c is set to 1 in the above figures. Panels (a–c) are the results under $\mathit{b}=1.05$, $\mathit{b}=1.10$, $\mathit{b}=1.20$, respectively.

When $1.05<b<1.20$, the situation changes. The randomness introduced by update strategies and the DAS mechanism causes the proportion of individuals using different-strategy pairings in the system to fluctuate for a period rather than steadily decline. This indirectly ensures the retention of cooperative strategies, thereby promoting the subsequent spread of cooperation. By comparing the curves in Fig. 6d with $\beta$ values of 0 and 0.5, it can be observed that the DAS mechanism can significantly alter the evolutionary outcomes during this phase. When $b=1.20$ (Fig. 6e,f), the curves split into two categories and almost overlap. In this situation, the assistance provided by the DAS mechanism from the external network is much less than the benefits brought by the defection strategy. Although the DAS mechanism can delay the process of individuals choosing the defection strategy, it cannot change the final outcome. At this point, the DAS mechanism fails, and defection becomes the dominant strategy.

In summary, the DAS mechanism works synergistically with inter-layer connections and is most effective when the interdependence among multilayer networks is strong. The DAS mechanism helps spread cooperative strategies within the system by converting different-strategy pairings into same-strategy pairings and retaining cooperative strategies. It is particularly effective when $b<1.2$, as it can both increase the cooperation rate and accelerate the evolution process. However, as the environment continues to deteriorate, the DAS mechanism eventually fails.

Conclusion

We propose an evolutionary game model on a two-layer interdependent network to explore the evolution of cooperation, wherein the interdependency is implemented through strategy state sharing, and cooperation is promoted through the Discrepant Accumulations Strategy. With an appropriate value of b, we found that the cooperation frequency exhibits a non-monotonous phenomenon with the increment of $\beta$. To delve into the microscopic mechanism of cooperation development, we examined the evolutionary process of the system. Furthermore, we explored the different pairing modes of players and found that the aforementioned phenomenon occurs because a specific pairing mode can dominate the system at certain time points. It is shown that the increase in the parameter $\beta$ restricts the living space of players adopting the C–D pairing mode, and in most cases, this space is occupied by players adopting the C–C pairing mode. Moreover, before the system reaches equilibrium, network reciprocity and the mechanism we proposed assist cooperators in avoiding exploitation by defectors and hasten the disappearance of antisocial pairing strategies. This aid contributes to the integration of cross-layer strategies and the formation of cooperative clusters.

In summary, compared with previous methods for calculating fitness, the Discrepant Accumulations Strategy (DAS) aligns more closely with reality. This mechanism relies on inter-network dependencies and accelerates the system towards higher cooperation rates. In this experiment, we employed a single multilayer network model and game model. It is potentially valuable to refine this model by incorporating more complex network structures and improved update strategies to draw conclusions related to multilayer networks.

Author contributions

Xin Ge and Xi He conceived the experiments, Yixiang Zhao and Xi He conducted the experiments, Jian Yang and Yue Liu analysed the results. All authors reviewed the manuscript.

Data availability

The data that support the findings of this study are available from the corresponding author, Xin Ge, upon reasonable request.

Competing interests

The authors declare no competing interests.