Santos et al., 2008. Social Diversity in Public Goods Game

Ref: Santos, F. C., Santos, M. D., & Pacheco, J. M. (2008). Social diversity promotes the emergence of cooperation in public goods games. Nature, 454, 213-216

Individuals are connected in a scale-free random graph in heterogeneous networks. Each individual plays several n-person prisoners dilemma games (aka public goods game, PGG) with its neighbors. There is a game centered on each individual, and consists of the focal individual and all of its neighbors.

Methods
Each individual and its k neighbors define a group. Individuals were either Cooperators or defectors.
Evolution: At each time step, each individual will adopt the strategy of a random neighbor (if more fit) with a probability proportional to the fitness difference.
They used populations of 1,000 individuals starting with 50% cooperators.
Equilibrium fraction of cooperators was average of 2,000 generations after a transient period of 10,000 generations. Repeated 100 times for 10 different realizations of each class of graph. In all cases, they used graphs with average connectivity z = 4 (average of 4 connections per node).

In each game, individuals either cooperated or defected. The income to a defector and cooperator in one game were given by

 Where the c is the contribution of a cooperator, r > 1 is the production function of the PGG, n_C is the number of cooperators in the game, k_x is the number of neighbors for focal individual x. As a global measurement, they used a normalized enhancement factor u = r/(z + 1).


Two mechanisms of cooperating: 
C1: Each cooperator contributes c to each game. This may be unrealistic, since it imposes no limitations on how much of individual's resources he can contribute.
C2: Each cooperator contributes c/(k + 1) to each game. Thus, each C has the same amount he can contribute, and it may be spread over many games.

C2 does not increase the emergence of cooperation in a regular graph, but did so in a scale-free graph (z= 4 in both cases). See the figure to the right. The reason there is no change in the regular graph is because the connections are homogeneous, so each node has the same number of connections, so the relative payoffs are the same. However, in the scale-free graph, there is now heterogeneity in the returns from each game. Thus here, the relative fitness of a cooperator increases with its connectivity.

Defectors are victims of their own success, with the immediate switching of strategies, they take over a local hub, and then are out-competed by a neighboring hub of cooperators.

Two-fold role of cooperators:
1. Effectively disseminate C strategy across social networks.
2. Get a stronghold on hubs by minimizing potential loss from exploitation by Ds.

NOTE: I think this is a feature of the instant updating - the fact that strategy switches are not only synchronous, but the fact that games are not repeated and node switches are absolute (i.e., change is immediate and not slow). The fact that strategy change should be slow has been noted for cultural evolution (Bednar & Page, 2006) as well as for more cognitive and neurological research.

Economics Perspective: Considered populations of 100% cooperators and looked at their "wealth" (fitness) distribution. While a homogeneous graph yields an egalitarian distribution, the heterogeneous graph yields a power law distribution with many poor and few rich. This part was not covered that much in the main body of the paper, and is less interesting to me right now.

NOTE: The authors state that selection here is strong (e.g., p. 214). However, they are using essentially the same replicator dynamics as the Nowak crew, who claim (more correctly, in my opinion), that selection is weak. This is because the number of individuals in the population is not changing, but only the strategies they play. The whole strong/weak selection classification scheme is seriously in need of some revision...