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Biased imitation in coupled evolutionary games in interdependent networks

We explore the evolutionary dynamics of two games—the Prisoner's Dilemma and the Snowdrift Game—played within distinct networks (layers) of interdependent networks. In these networks imitation and interaction between individuals of opposite layers is established through interlinks. We explore an update rule in which revision of strategies is a biased imitation process: individuals imitate neighbors from the same layer with probability p, and neighbors from the second layer with complementary probability 1 − p. We demonstrate that a small decrease of p from p = 1 (which corresponds to forbidding strategy transfer between layers) is sufficient to promote cooperation in the Prisoner's Dilemma subpopulation. This, on the other hand, is detrimental for cooperation in the Snowdrift Game subpopulation. We provide results of extensive computer simulations for the case in which layers are modelled as regular random networks, and support this study with analytical results for coupled well-mixed populations.

represent distinct environments, characterized by their game rules, which can interact or be coupled. Consider the example of two companies and the joint network of contacts between their employees. The code of conduct in each company may be different, depending on its internal organization and goals. Nevertheless, employees from one company may interact with employees from the other company, because they know each other personally or for instance when establishing a business transaction. By interacting with an acquaintance from the other company, each employee can acquire new working strategies, which can afterwards be imitated by their contacts inside each company. Another natural example is interaction between subpopulations of different cultural backgrounds.
Here we investigate how imitation and interaction between individuals playing different games influence the final cooperation levels. For that, we consider a population organized in two layers of equal size, in which one distinct game is played in each of the layers, as schematized in Fig. 1 (see also Methods for details). Nodes establish intralinks with neighbors of the same layer, and interlinks with neighbors of the opposite layer. In the following, we start by studying by computer simulations the case in which both layers are regular random graphs. Afterwards, we also consider the limit case of a well-mixed population of infinite size, also organized in two layers. We introduce bias in imitation: individuals imitate a neighbor from the same layer with probability p, or a neighbor of the opposite layer with probability (1 − p). We investigate the impact of varying p on the final level of cooperation reached in each of the layers (see Methods).

A. Regular random layers
For reference, we present in Fig. 2 the final fraction of cooperators for the PD and the SG played in isolated populations. Panel A) shows the results of computer simulations in regular random networks, while panel B) shows the stationary solutions of the replicator equatioṅ for an infinite well-mixed population, that is a population in which every individual has the same probability of interacting with anyone else [41]. In the replicator equation, x represents the fraction of cooperators in the population, f C (x) and f D (x) stand for the mean fitness of cooperators and defectors respectively (with f C (x) = xR + (1 − x)S and f D = xT + (1 − x)P ). The PD is the harshest social dilemma for cooperation, and therefore only a small region characterized by S close to 0 and T close to 1 is able to escape full defection, in regular random networks. The SG, on the other hand, is a coexistence game, in which the final fraction of cooperators x in an infinite well-mixed population is given by x = (P − S)/(R − S − T + P ).
To evaluate the impact of an interdependence between the PD and the SG on the evolution of cooperation, we start by considering a population in which each of the layers is modeled by a regular random graph with nodes of degree k = 4. This graph has no finite loops in the infinite size limit. We assume that each node establishes one interlink with a uniformly randomly chosen node from the opposite layer. That is, individuals engage in four interactions with neighbors of the same layer and one interaction with a neighbor from the other layer.
The contour plots in Fig. 3 show the result of our simulations, namely, the final fraction of cooperators in each of the layers as a function of the game parameters T and S, for several , where f C (x) and f D (x) stand for the mean fitness of cooperators and defectors respectively [41]. Red corresponds to full defection, blue corresponds to full cooperation. The parameter β is set to 1.0 for panel A). When p < 1 strategy transfer between layers becomes possible. Comparing the results for the PD layer for p = 0.9, p = 0.95 and p = 1, we reach an interesting conclusion: although the fraction of cooperators is still low, even a small probability of imitating a neighbor playing a distinct social dilemma (in this case, the SG) is sufficient to prevent full defection in the majority of the PD quadrant.
As p further decreases, for a given pair of game parameters T, S the fraction of cooperators in the stationary state becomes approximately equal in both layers (note that S SG = S, S PD = −S with S ∈ [0, 1]). However, a careful inspection of the contour plots presented in Fig. 3 for 0.1 ≤ p ≤ 0.5 shows that the fraction of cooperators in both layers is not exactly identical (i.e. the contours are not exactly symmetrical): for a fixed pair of T and S values, the PD layer tends to have a slightly lower fraction of cooperators.   for decreasing p. Excluding the results for p = 0 (which are a direct consequence of allowing exactly one interlink for each node) we observe that the fraction of cooperators monotonously decreases for decreasing p in the SG layer, while in the PD layer it monotonously increases.
As we will see in the following, this behavior will not be maintained when this population structure is replaced by a well-mixed population.

B. Well-mixed population
We now proceed to analyze the general case of a population modelled by a fully connected network organized in two layers. Each layer has N/2 nodes and 1 2 (N/2)(N/2 − 1) intralinks. Contrary to the previous case of regular random layers, in which each node established one interlink, we now assume that each node establishes N/2 interlinks to the N/2 nodes of the opposite layer, in accordance with the structure of a fully connected network. Therefore, each node has N − 1 neighbors: N/2 − 1 in the same layer as itself, and N/2 in the opposite layer. n PD and n SG stand for the number of cooperators in the PD (SG) layer, respectively.
In this case, mean-field theory is exact and the evolution of population can be expressed completely by n PD (t) and n SG (t).
We obtain the solutions for the stationary state of this model in the infinite population limit, N → ∞. For that, we adopted the variables x PD = n PD /(N/2) and x SG = n SG /(N/2) (x PD ∈ [0, 1] and x SG ∈ [0, 1]). The mean fitness of individuals in each of the layers is expressed in terms of the game parameters and the fraction of cooperators and defectors. In the PD layer, the fitnesses of cooperators (f C,PD ) and defectors (f D,PD ) are expressed as: Similarly, in the SG layer the fitnesses of cooperators (f C,SG ) and defectors (f D,SG ) are We obtain the following rate equations for the evolution of the fraction of cooperators in the PD and SG layers: For instance, in Eq. (S1) the factor (1 − x PD ) stands for the probability of selecting a defector from the PD layer. The second factor, in the square brackets, accounts for the two possible  At p = 0 the difference between the coupled regular random networks and well-mixed population is dramatic. While for the coupled well-mixed populations, the final state is full defection for the whole T − S plane (Fig. 6), for regular random networks we observed coexistence of defection and cooperation for the whole T − S plane. The reason for this controversy is the fundamental difference between the structures of the absorbing states in these two systems. Clearly, in the mean-field system (well-mixed) there are two absorbing states: full cooperation and full defection. For p = 0, the full defection scenario is realized.
In contrast, in the coupled regular random populations, at p = 0 the system has an infinite number of absorbing states. We already explained that this state is organized as an arbitrary It is interesting to relate our findings with observations in other works [39,40] which studied the role of biased fitness (utility) on the level of cooperation in interdependent structured populations. Although these works explored a bias and network architectures very different from our study, they reported the existence of an optimal control parameter value for cooperation, similarly to what we notice for a well-mixed population.
We can also relate our model to the multi-network model [42,43], where links between nodes are of two sorts: the interaction links through which individuals play and the imitation links through which individuals revise their strategies. All links in our model are used both for interaction and imitation. The principal difference is that interaction occurs with probability 1 for every link, while imitation takes place with probability p along intralinks and the complementary probability 1 − p along interlinks.

II. DISCUSSION
We explored the evolution of cooperation in two games-the Prisoner's Dilemma and the Snowdrift Game-played within an interdependent network. We demonstrated that as the probability to imitate neighbors from the opposite layer increases, the final level of cooperation in each of the layers behaves differently. A small probability 1 − p is always detrimental for the level of cooperation in the SG layer, regardless of the population structure (well-mixed or when layers are modelled as regular random networks). In the case of a wellmixed population we observed an intermediate optimal value of p for which the fraction of cooperators in the steady state is maximal in the PD quadrant. On the other hand, when layers are regular random graphs, the level of cooperation in the PD layer registers a significant increase from full defection as p decreases from 1, as is shown in Fig. 3. We have also shown that different arrangements of interconnections between layers result in fundamentally different structures of absorbing states for p = 0, that is the limit case in which individuals can only imitate neighbors through the interlinks.
The constraint between the game parameters of the PD and the SG implemented in this work is one of possible options. Based on our mean-field approach we have verified that our conclusions remain qualitatively valid for other relationships between game parameters [44].
In this work we studied a representative case of β = 1. In the Supplementary Information we consider other values of β.
The Prisoner's Dilemma and the Snowdrift Game are two of the possible two-person games. Future research directions include establishing an interdependence between other two-and N -person games, as well as extending the biased imitation process here reported to the case in which layers are strongly heterogeneous, such as scale-free networks [45].

III. METHODS
Evolutionary games. We implement standard two-person games. Each individual can act as an unconditional cooperator (C) or as an unconditional defector (D). The possible payoffs resulting from the interaction between two players are resumed in the payoff matrix in which the entries represent the payoff earned by the row player. When both individuals opt for cooperation (defection), they both receive a Reward, R (Punishment, P ). When a cooperator meets a defector, the cooperative player received the Sucker's payoff (S), while the defective player receives the Temptation (T ) for defecting.
The relative ordering of the game parameters R, S, T and P defines four different twoperson games [46]. In this study, we consider the Prisoner's Dilemma (PD) (characterized by T > R > P > S) [23] and the Snowdrift Game (SG) (characterized by T > R > S > P ) [24]. We fix R = 1, P = 0 as is conventional [26].
Population structure. We consider a population organized in two layers (schematized in Fig. 1), in which each node represents a different individual. In each layer all individuals play the same game, the PD in one layer and the SG in the other. We refer to links between nodes of opposite layers as interlinks, and links between nodes of the same layer as intralinks.
In the absence of interlinks, each layer stands as an isolated population in which one single game is played. When layers are modelled as regular random graphs, we assume that each node establishes one interlink to a randomly chosen node from the opposite layer. On the other hand, when studying a two-layer fully connected network, we assume that each node establishes interlinks to each of the nodes in the opposite layer.
Relation between game parameters. We assume that individuals in the PD layer adopt the game parameters R PD , S PD , T PD and P PD (with T PD > R PD > P PD > S PD ). Similarly, individuals in the SG layer consider R SG , S SG , T SG and P SG (with T SG > R SG > S SG > P SG ).
The game parameters for the PD and the SG can be related in several ways. We opt for while the payoff matrix for the PD layer is given by The difference between matrix (8) for the SG layer and matrix (9) for the PD layer is in the payoff obtained by a cooperator when interacting with a defector, the sucker's payoff.
When interacting with a defector, a cooperator located in the SG layer earns S, while a cooperator located in the PD layer earns −S. We stress that this is only one of possible relations between the parameters of the two games.  [47,48], (10) f A (f B ) stands for the fitness of A (B), and β represents the intensity of selection, which regulates the accuracy of the imitation process. In the limit β → 0 evolution proceeds by random drift, while β → ∞ corresponds to the deterministic limit of pure copying dynamics: A imitates B iff f B > f A , with probability 1. We consider the respresentative case of β = 1.0.
Simulations. The results of Figs. 3 and 5 were obtained for a population of total size N = 2000, in which both layers had an equal size, N/2 = 1000. Each point in the contour plots corresponds to an average over 10 3 trials. In each trial, we start by randomly selecting one over 10 2 realizations of a regular random network to model each layer, and establish N/2 interlinks between the layers. Each interlink is established by uniformly randomly choosing two end nodes, one from each layer, allowing each node to establish exactly one interlink. Each trial starts by randomly distributing 50% of cooperators and defectors in each of the layers. We allow the population to evolve for 10 4 generations (where 1 generation corresponds to N/2 strategy revisions), after which we average the fraction of cooperators over the next 10 3 generations.

SUPPLEMENTARY INFORMATION
In the main text we focused on the case of β = 1. In particular, for this value of β, we considered a fully connected network organized in two layers. In this Supplementary Information we provide additional results for other β values for this easily treatable network, which demonstrate that the value β = 1 is representative. We present the rate equations in the specific case p = 1, for which the results in the steady state prove to be independent of β.
Furthermore, we show detailed numerical results for p < 1, which depend on the particular β value chosen.
For p < 1, on the other hand, Eqs. (5) and (6)  Comparing the results shown for the various β values, we observe that the final level of cooperation shows the same qualitative behavior discussed in the main text regardless of the β value. That is, the fraction of cooperators monotonously decreases with decreasing p in the SG layer, and there is an optimal p value for the level of cooperation in the PD layer.
As β increases from β = 0.1, there is a shift in that optimal p value: for β = 0.1 it is close to p = 0.9, while on the other hand for β = 10 it is close to p = 0.25. That is, the optimal p value for the level of cooperation in the PD layer decreases for increasing β.
The results are also independent of the β value adopted for p = 0: the population always reaches full defection in both layers. and SG (S > 0) layers as a function of T and S, for β = 0.1 (first row), β = 1 (second row) and β = 10 (third row). The results for β = 1 are reproduced from the main text for easier comparison.
The dashed line separates the results for the SG layer (upper quadrants) from the results for the PD layer (lower quadrants). Red corresponds to full defection, blue corresponds to full cooperation.