Difference of reciprocity effect in two coevolutionary models of presumed two-player and multiplayer games

Unlike other natural network systems, assortativity can be observed in most human social networks; however, it has been reported that a social dilemma situation represented by a 2×2 prisoner's dilemma game favors dissortativity to enhance cooperation. Our simulations successfully reveal that a public goods game with coevolution for both agents’ strategy and network topology encourages assortativity, although it only slightly enhances cooperation as compared to a 2×2 donor and recipient game with a strong dilemma to be solved. This outcome occurs because the network dynamics in a multiplayer game discourages emerging cooperation unlike its beneficial result in a 2×2 prisoner's dilemma game.

Figure 1

Relations of dilemma weakness $r$ with (a) cooperation fraction, (b) maximum degree, (c) assortative coefficient, (d) maximum degree of cooperators, and (e) maximum degree of defectors. Sigma means standard deviation.

Figure 2

Comparison of cooperation fractions by the proposed coevolution model for PGG [red closed circle, same as Fig. 1a]; the monoevolution model for PGG (blue open square); and coevolution (gray bold line) and monoevolution (black bold line) models for a 2×2 D&R game.

Figure 3

Ensemble time evolution episode only when agents can attain cooperative coexisting equilibrium when $r$ $=$ 5.0: (a) cooperation fraction, assortative coefficient, and cooperation fractions of neighbors to cooperators and defectors; and (b) maximum degree among all agents and average degrees of cooperators and defectors.

Figure 4

Degree distribution at time step $=$ 100 of one of the episodes, which attain cooperative coexisting phase (showing ensemble average time evolution). Cooperation fraction $P_c$, assortative coefficient $\mathit{Ac}$, average cluster coefficient ⟨$C$⟩, average path length ⟨$L$⟩, maximum degree $k_{\max }$, and average degree of cooperators and defectors ⟨$k_C$⟩, ⟨$k_D$⟩ at time step $=$ 100 are shown. The broken line represents the initial distribution (Poisson distribution), the bold line represents the distribution of all agents, and the red and blue plots represent those of defectors and cooperators, respectively.

Figure 5

Schematic of what happens in respective coevolutionary processes for a 2×2 D&R and PGG to explain why coevolution destroys cooperation in a stronger dilemma situation. In a 2×2 D&R (a), network adaptation may improve the focal agent's payoff by rewiring a cooperator. In a PGG (b), the situation seems more complicated. The maximum cooperator among the Agent $i$'s neighbor and Agent $i$ herself is Agent $k$. Hence, $\frac{24}{5}r-5<\frac{14}{5}r\iff r<\frac{5}{2}$ is valid, Agent $j$ becomes “maximum”. Under a strong dilemma, the focal $i$ copies D from the defective neighbor $j$ who gains maximum payoff among $i$'s neighbors and $i$ when focal $i$ updates her strategy. Thus, when focal $i$ updates her links with neighbors, the link with $j$ is severed even if the cooperative fraction among the participants of the game $j$ hosts is not as low as the global cooperative fraction. In addition, focal $i$ encounters difficulty in establishing a new link with an appropriate agent who has relatively denser cooperative neighbors than does agent $j$. Consequently, focal $i$ builds a new link with the inappropriate agent who herself is a cooperator but has a high percentage of defective neighbors.