Wisdom of groups promotes cooperation in evolutionary social dilemmas

Abstract

Whether or not to change strategy depends not only on the personal success of each individual, but also on the success of others. Using this as motivation, we study the evolution of cooperation in games that describe social dilemmas, where the propensity to adopt a different strategy depends both on individual fitness as well as on the strategies of neighbors. Regardless of whether the evolutionary process is governed by pairwise or group interactions, we show that plugging into the "wisdom of groups" strongly promotes cooperative behavior. The more the wider knowledge is taken into account the more the evolution of defectors is impaired. We explain this by revealing a dynamically decelerated invasion process, by means of which interfaces separating different domains remain smooth and defectors therefore become unable to efficiently invade cooperators. This in turn invigorates spatial reciprocity and establishes decentralized decision making as very beneficial for resolving social dilemmas.

Figure 1. The “wisdom of groups” promotes the evolution of cooperation in the prisoner's dilemma game on the square lattice.

Depicted is the fraction of cooperators ρ_C in dependence on the temptation to defect b, as obtained for different values of α (see figure legend). It can be observed that the larger the value of α the higher the value of b where cooperators are able to dominate and survive the evolutionary competition with defectors.

Figure 2. The “wisdom of groups” promotes the evolution of cooperation irrespective of the properties of the interaction network.

In this figure the prisoner's dilemma game was staged on the honeycomb lattice. Depicted is the fraction of cooperators ρ_C in dependence on the temptation to defect b, as obtained for different values of α (see figure legend). As by the results obtained on the square lattice depicted in Fig. 1, it can be observed that the larger the value of α the higher the value of b where cooperators outperform defectors.

Figure 3. The “wisdom of groups” promotes the evolution of cooperation irrespective of the type of the game.

In this figure the public goods game was staged on the square lattice. Depicted is the fraction of cooperators ρ_C in dependence on the multiplication factor r, as obtained for different values of α (see figure legend). As by the results obtained for the prisoner's dilemma game in Figs. 1 and 2, it can be observed that the larger the value of α the lower the multiplication factor r that is needed for cooperators to survive and dominate the population.

Figure 4. Interfaces that separate domains of cooperators and defectors remain smooth if the “wisdom of groups” is taken into account.

Depicted is the evolution of a prepared initially rough interface, as obtained for α = 0 [top panels from (a) to (e)], i.e., ignoring the “wisdom of groups”, and for α = 4 [bottom panels from (f) to (j)]. It can be observed that by taking into account the wider knowledge of the group (bottom row), the roughening of the interfaces is prevented. Cooperators therefore rise to dominance (the final pure C phase is not shown). In the upper row the defectors are able to invade the cooperative domains aggressively, which in turn further roughens the interfaces and eventually leads to a pure D phase (not shown). Note that to distinguish different learning activities w of players, we have used different shades of blue (for cooperators) and red (for defectors), as indicated in the figure legend. Lighter colors correspond to higher learning activity while darker colors denote players with lower learning activity. For α = 0 all players constantly have w = 1, and are accordingly depicted by the brightest shades of blue and red. The snapshots were taken at MCS = 0, 10, 30, 70 and 100 for the top row, and at MCS = 0, 200, 1400, 6000 and 19000 for the bottom row. In both cases the temptation to defect was set equal to b = 1.07 and the system size was L = 80 (small solely to ensure a proper resolution of the relevant spatial patterns).

Figure 5. Evolution of cooperation from a random initial state under the influence of the “wisdom of groups”.

Depicted are characteristic spatial patterns, as obtained for α = 4 and b = 1.08. The color code is the same as used in Fig. 4. Defectors with the highest learning activity, depicted light red, can be observed at the interfaces that separate ordered domains. Cooperators are able to overtake these defectors, which results in smoother interfaces and ultimately in a pure C phase (not shown). Also note that only the cooperative domains with “straight” borders and of sufficiently large size are able to prevail against the invading defectors. Smaller circularly shaped cooperative domains are unable to grow and surrender to the evolutionary pressure rather fast. The snapshots were taken at MCS = 0 (a), 10000 (b), 30000 (c) and 41000 (d). The system size was L = 80.

Figure 6. Schematic presentation of the leading invasion process and the corresponding difference of invasion probabilities with and without taking into account the “wisdom of groups”.

Arrows in the top two panels depict the leading invasion process for α = 0 (left) and α = 4 (right). The invasion that is marked by the gray arrow in the right panel becomes practically irrelevant due to large α (see main text for details). The color code is the same as used in Figs. 4 and 5 to distinguish players with different learning activities. Lower panel depicts the normalized difference of invasion probabilities Φ of C → D and D → C strategy changes in dependence on the temptation to defect b at K = 0.5. It can be observed that for α = 4 the invasion front changes sign at a higher value of b than for α = 0, thus corroborating the reported promotion of cooperation due to the “wisdom of groups” at the microscopic level of the evolutionary process.

Figure 7. The “wisdom of groups” dynamically modifies the learning activity of players.

Depicted is the learning activity w in dependence on the fraction of neighboring players that have a different strategy than the focal player, as obtained for different values of α (see figure legend). Traditional versions of both the prisoner's dilemma and the public goods game are recovered if α = 0, as then the “wisdom of groups” is ignored and does not influence the learning activity w. On the other hand, for larger values of α the impact of the neighbors becomes increasingly stronger, virtually prohibiting strategy changes that would introduce a strategy different from their own.