Evolutionary games with conformists

We consider evolutionary social dilemmas on either the scale-free or the Erdős-Rényi random network (see Methods for details), or on interaction networks with a uniform degree distributions, such as the random regular graph and the square lattice, where each player x is initially designated either as cooperator (C) or defector (D) with equal probability. Each instance of the game involves a pairwise interaction where mutual cooperation yields the reward R, mutual defection leads to punishment P and the mixed choice gives the cooperator the sucker’s payoff S and the defector the temptation T. We predominantly consider the weak prisoner’s dilemma, such that T > 1, R = 1 and P = S = 0, but we also consider the true prisoner’s dilemma where S < 0.

The standard Monte Carlo simulation procedure comprises the following elementary steps. First, according to a random sequential update protocol, a randomly selected player x acquires its payoff Π x by playing the game with all its neighbors. Next, player x randomly chooses one neighbor y, who then also acquires its payoff Π y in the same way as previously player x. To avoid payoff-related effects that are due to heterogeneous interaction topologies29, we normalize the payoff with the degree of the corresponding player . After both players acquire their payoffs, player x adopts the strategy s y from player y with a probability determined by the Fermi function

where K = 0.1 quantifies the uncertainty related to the strategy adoption process22,69. In agreement with previous works, the selected value ensures that strategies of better-performing players are readily adopted by their neighbors, although adopting the strategy of a player that performs worse is also possible70,71. This accounts for imperfect information, errors in the evaluation of the opponent and similar unpredictable factors.

To introduce conformity, we designate a fraction ρ of the population as being conformity-driven, which influences the strategy adoption rule. Instead of Eq. 1, if player x is a conformist, we use

where is the number of players adopting strategy s x within the interaction range of player x, while k h is one half of the degree of player x. By using Eq. 2, player x is most likely to adopt the strategy that is, at the time, the most common in its neighborhood. As in Eq. 1, here too K = 0.1 introduces some uncertainty to the process, such that it is not completely impossible for a conformist to adopt a strategy that is in the minority among its neighbors. If, however, the number of cooperators and defectors in the neighborhood is equal, the conformity-driven player will change its strategy with probability 1/2. We also emphasize that, in this study, a conformist uses only local information. In particular, a conformity-driven player simply tends to follow the majority in its local neighborhood. An alternative approach might entail a conformist having available global information about the whole population and act accordingly. In such a case, however, the global information may likely suppress the vital importance of local information stemming from the direct neighbors. Due to several open questions and options on implementation, we leave it to future studies to determine the merits of global conformity.

A key consideration in this paper is which players make up the fraction ρ of the population that act as conformists. We consider and compare several different options. The simplest option is to assign conformists uniformly at random, which we use as the benchmark case. Secondly, on heterogeneous interaction networks, such as scale-free and Erdős-Rényi random networks, we use the degree k x of each player, whereby the probability to be designated as a conformist can be either k x /k max or 1 − k x /k max for degree-related or inversely-degree-related assignment, respectively. In the former case, high-degree players, i.e., the hubs or leaders, most certainly end up as conformists, while in the latter case low-degree players, i.e., the masses or the periphery, are the most likely conformists. Thirdly, again of relevance on heterogeneous networks, we use the concept of collective influence of players to distinguish them67. In particular, we calculate the collective influence CI x at depth 1 for every player x (see67 for details), where the maximal value is CI max . The incentive for player x to act as a conformist is then simply based on the ratio CI x /CI max . Accordingly, players with the highest collective influence at depth 1 in the network will most likely end up using Eq. 2 rather than Eq. 1 for the strategy adoption. In principle, this approach is the same as the degree-related assignment of conformity, in that the leaders are the likeliest to conform. Lastly, we check our observations on homogeneous networks, where all players have an identical number of neighbors. Instead of considering topological differences, we thus assign a different strategy pass capacity w x to each player as a pre-factor in Eq. 1 68, such that the distribution is , where w x ∈ [0.01, 1] interval. This practically means that the majority of the players has a very low w x value, while only a few have w x ≈ 1 (the distribution is shown in the inset of Fig. 4). As with the degree, we consider w-related (the probability is simply equal to w x because w max = 1) or inversely-w-related (1 − w x ) assignment of conformity, such that the leaders or the masses are preferentially considered as conformists, respectively.

Figure 4 Fraction of cooperators f C in dependence on the fraction ρ of the population that act as conformists, as obtained on the square lattice for two different selection rules indicated in the legend. As in Figs 1 and 2, here assigning the conformist status to players with the highest strategy pass capacities (high) leads to lower levels of cooperation than are obtained if the same status is assigned to players with the lowest strategy pass capacities (low). Thus, even on lattices and regular networks (results for random regular graphs are practically identical), where the rank of a player is determined by a property not related to the topology of the network, it is clear that for socially optimal outcomes the conformists should be the masses, not the leaders. A square lattice with N = 800 × 800 nodes and the weak prisoner’s dilemma (T, S) = (1.1, 0) parametrization have been used. Presented results are averages over 10 independent realizations. The inset shows the applied distribution of strategy pass capacities in the population . Full size image

Evolutionary dynamics

We begin by presenting the main results on scale-free networks in Fig. 1, where we show how the fraction of cooperators (f C ) in the stationary state depends on the fraction ρ of the whole population that is made up of conformist players. As stated in the Introduction, these players do not aspire to maximal payoffs as is traditionally assumed in evolutionary games, but rather, they prefer to adopt the most common strategy within their interaction range. As we have reported in16, when conformity is assigned randomly and to a sufficiently high fraction of the population, the spatial selection for cooperation is enhanced and the level of cooperation in the stationary state is higher than in the absence of conformist players. Results presented in Fig. 1 show, however, that this effect can be either enhanced or destroyed based on the preference of who will be made to conform. Specifically, if conformists are players with a high degree or high collective influence in the network, the evolution of cooperation is significantly impaired. The constructive interplay between heterogeneity and coordination is almost completely lost. Naturally, we can still observe a gradual increase of f C as the fraction of conformists increases, but this is just a simple consequence of the fact that we reach a strategy neutral state at ρ = 1.

Figure 1 Fraction of cooperators f C in dependence on the fraction ρ of the population that act as conformists, as obtained on the scale-free network for different selection rules indicated in the legend. Assigning the conformist status to players with the highest degree (degree) or collective influence (collective) in the scale-free network strongly impairs the constructive interplay between heterogeneity and coordination, thus leading to lower levels of cooperation than are obtained if the same status is assigned randomly (uniform) or to low-ranking players (inverse). Scale-free networks with N = 104 nodes and the weak prisoner’s dilemma (T, S) = (1.1, 0) parametrization have been used. Presented results are averages over 5000 independent realizations. Full size image

On the other hand, if conformists are preferentially selected from low-degree players, i.e., from the masses, then an even higher level of cooperation in the stationary state is attainable, indicating an optimization of the aforementioned interplay. This result is intuitive and understandable, because high-degree players are naturally in a position to influence a large neighborhood and the effectiveness of this influence is facilitated further if the neighborhood is made up predominantly of conformist players. This interplay of heterogeneity (the ability of a small number of select players, i.e., the leaders, having a wide influence) and coordination (the pressure of conformity to select the most common strategy, even if it is not payoff-maximizing) ultimately leads to large homogenous groups or clusters competing in the population, which in the long-run reveals the benefits of cooperation by virtue of network reciprocity. Naturally, this argument fails if leaders are made to conform because they then become unable to capitalize on their central position within the network. In the latter case, the coordination becomes less efficient, because players having the capacity to search for a more successful strategy are unable to lead others, which ultimately hinders the successful evolution of cooperation.

These details of the described macroscopic dynamics are generally valid and apply regardless of the properties of the interaction network and game parametrization. Indeed, as indicated by the results presented in Fig. 2, which were obtained on Erdős-Rényi random networks, it is better if conformists are selected specifically from those players who have lower degree or collective influence. On the contrary, if highly influential players are targeted as conformists and made to follow the majority, then the level of cooperation in the stationary state drops significantly. Similarly, as evidenced by the results presented in Fig. 3 for both the scale-free (left panel) and Erdős-Rényi random (right panel) networks, even if the weak prisoner’s dilemma game parametrization is replaced with the more stringent variant of the same social dilemma, the targeted selection of conformists amongst the masses still ensures high levels of cooperation in the stationary state. We note that, although the level of cooperation might appear to be modest at first, due to the application of degree-normalized payoffs, there would actually be almost no cooperators able to survive without the help of conformity in the considered T − S range.

Figure 2 Fraction of cooperators f C in dependence on the fraction ρ of the population that act as conformists, as obtained on the Erdős-Rényi random network for different selection rules indicated in the legend. As for the scale-free network in Fig. 1, here assigning the conformist status to players with the highest degree (degree) or collective influence (collective) also leads to lower levels of cooperation than are obtained if the same status is assigned randomly (uniform) or to low-ranking players (inverse). Erdős-Rényi random networks with N = 104 nodes and the weak prisoner’s dilemma (T, S) = (1.1, 0) parametrization have been used. Presented results are averages over 1000 independent realizations. Full size image

Figure 3 Color-coded fraction of cooperators f C on the T − S parameter plane, as obtained when conformists are selected from low-degree players. Left panel shows the results obtained on scale-free networks, while the right panel shows the results obtained on Erdős-Rényi random networks. In both cases the fraction of the population that act as conformists was ρ = 0.7. We note that, in the absence of conformists and since we use degree-normalized payoffs, cooperation is practically absent in the depicted T − S range. Networks with N = 104 nodes have been used and the presented results are averages over 1000 independent realizations. Full size image

Previous research has also shown that, when we normalize payoffs on heterogeneous interaction network, we destroy the heterogeneity among players, which drastically weakens the positive impact of enhanced network reciprocity72,73,74. While this argument is of course valid, it is nevertheless important to keep in mind that players are still different because of the remaining differences in degree and the resulting diverse opportunities to spread their strategies. This difference can be revealed if we choose conformists selectively, by considering the individual differences of players. As we have shown, it is detrimental to force leaders into following their own neighborhood. We note that this observation remains valid even if we apply less realistic absolute payoff values to determine strategy imitation probabilities12. On the contrary, expecting the same conformist attitude from the masses could be very useful for the whole population. This argument can be supported indirectly by considering also collective influence, as recently proposed in67, instead of the degree. As can be observed in both Figs 1 and 2 (see label collective), exploiting the concept of collective influence to target the most influential players as conformists results in the same fall of cooperation as we have observed when high-degree players were made to conform.