Evolutionary Games defined at the Network Mesoscale: The Public Goods game

Evolutionary game dynamics on graphs has become a hot topic of research during the last years. The attention has been mainly focused on 2-players games, such as the Prisoner’s Dilemma game, since the pairwise interactions can be easily implemented on top of networked substrates. However, for m-players game, such as the Public Goods game, the microscopic description about the pairwise interactions contained in the network is not enough, since m-players game are intrinsically defined at the mesoscopic network level. This mesoscopic level describes how individuals engage into groups where the Public Goods games are played. However, the actual group structure of networks has not been considered in the literature, being automatically substituted by a fictitious one. In this work, we study the emergence of cooperation in collaboration networks, by incorporating the real group structure to the evolutionary dynamics of the Public Goods game. Our results are compared with those obtained when the mesoscopic structure is ignored. We show that cooperation is actually enhanced when the group structure is taken into account, thus providing a novel structural mechanism, relying on the mesoscale level of large social systems, that promotes cooperation. Moreover, we further show that the particular characteristics of the group structure strongly influence the survival of cooperation.

Evolutionary game theory on graphs is attracting lately a lot of interest among the community of physicists working on complex systems [1,2]. This is a very appealing research topic because it combines two important ideas. First, interactions take place on a (possibly complex) network [3,4], generalizing the lattice perspective; and, second, that the dynamics taking place on that substrate needs not be the traditional one, but rather it can arise from an evolutionary approach [5]. On the other hand, from the applications viewpoint, studying evolutionary games on graphs is one of several avenues proposed to understand the emergence of cooperation in different contexts [6]. This is a most relevant issue that arises, for instance, in understanding the origin of multicellular organisms [7], of altruistic behavior in humans and primates [8], or the way advanced animal societies work [9,10], to name a few.

Research on evolutionary game theory on graphs focused on the problem of the emergence of cooperation has considered mainly the Prisoner’s Dilemma game [11,12]. The Prisoner’s Dilemma game (PDG) describes a situation in which cooperation is hampered by the players’ temptation to defect (defecting yields more payoff than cooperating when facing a cooperator) and by the risk arising from cooperation (cooperating with a defector yields the lowest payoff) [13]. This leads to a social dilemma in so far as when players cooperate both the total benefit and the individual benefit are higher than when mutual defection occurs. While evolutionary dynamics leads all the individuals to defection when interactions take place in a well-mixed population (every player interacts with every other one), the existence of a network structuring the population can sometimes promote the emergence of cooperation [14], but this depends strongly on the details of the network and the dynamics [2,15].

Much less attention has been paid to the m-player generalization of the PDG, also called Public Goods Game (PGG) [16]: Cooperators contribute an amount c (“cost”) to the public good; defectors do not contribute. The total contribution is multiplied by an enhancement factor r < m and the result is equally distributed between all m members of the group. Hence, defectors get the same benefit of cooperators at no cost, i.e., they free-ride on the cooperators’ effort. This is an alternative view of the social dilemma posed by the so-called tragedy of the commons [17]. As with the PDG, the evolutionary outcome of the PGG differs if played on a well-mixed population (where once again defection is selected) or on a network structure. Thus, Brandt et al. [18] showed that local interactions can promote cooperation in the sense that full cooperation is obtained for values of r well below the critical value r = m. This result, arising from simulation in an hexag