Group-size effects on the evolution of cooperation in the spatial public goods game

Attila Szolnoki and Matjaž Perc

Phys. Rev. E 84, 047102 – Published 28 October 2011

Abstract

We study the evolution of cooperation in public goods games on the square lattice, focusing on the effects that are brought about by different sizes of groups where individuals collect their payoffs and search for potential strategy donors. We find that increasing the group size does not necessarily lead to mean-field behavior, as is traditionally observed for games governed by pairwise interactions, but rather that public cooperation may be additionally promoted by means of enhanced spatial reciprocity that sets in for very large groups. Our results highlight that the promotion of cooperation due to spatial interactions is not rooted solely in having restricted connections among players, but also in individuals having the opportunity to collect payoffs separately from their direct opponents. Moreover, in large groups the presence of a small number of defectors is bearable, which makes the mixed-phase region expand with increasing group size. Having a chance of exploiting distant players, however, offers defectors a different way to break the phalanx of cooperators and even to resurrect from small numbers to eventually completely invade the population.

Received 11 July 2011

DOI:https://doi.org/10.1103/PhysRevE.84.047102

Authors & Affiliations

Attila Szolnoki¹ and Matjaž Perc²

¹Research Institute for Technical Physics and Materials Science, P.O. Box 49, H-1525 Budapest, Hungary
²Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, SI-2000 Maribor, Slovenia

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Issue

Vol. 84, Iss. 4 — October 2011

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Images

Figure 1
Left: Schematic presentation of different group sizes $G$ on the square lattice. Depicted are groups containing $G = 9$ (open squares), 45 (gray squares), 97 (pluses), 185 (solid squares), and 301 (crosses) players, respectively. The focal player is marked by a star. Right: Schematic presentation of possible sources of strategy invasion. Three different cases are considered, namely, the adoption can only be attempted from the nearest neighbors (open squares), from the focal group (white patch), or from all the groups (gray areas) where the focal player is a member.Reuse & Permissions
Figure 2
Critical multiplication factor $r_{c}$ in dependence on $G$ . Strategy donors were selected only from the four nearest neighbors (solid squares), from within the group of players where player $x$ is focal (open circles), or among all the groups where player $x$ is a member (solid circles). The uncertainty by strategy adoptions was $K / G = 0.1$ (the normalization of $K$ with $G$ takes into account the number of groups participating in the accumulation of payoffs).Reuse & Permissions
Figure 3
Critical multiplication factor $r_{c}$ in dependence on $G$ , as obtained when the payoffs are acquired from a single group where the corresponding players are focal. The symbols correspond to those used in Fig. 2, determining the set of potential strategy donors. The uncertainty by strategy adoptions was $K = 0.1$ (note that the normalization with $G$ is unnecessary since all the payoffs originate from a single group).Reuse & Permissions
Figure 4
Characteristic snapshots of the spatial grid for $G = 5$ (top row) and $G = 301$ (bottom row), using the focal group imitation range as obtained for $K / G = 0.1$ and a system size of $L = 200$ . Cooperators are marked by light gray (green) and defectors by dark gray (red). In both cases the final outcome is a full $D$ phase where normalized synergy factors are almost equal far from the transition points ( $r / G = 0.74$ and $0.24$ , respectively). Prepared initial states were used to highlight the two significantly different strategy invasion processes. Snapshots in the top row were taken at 0, 50, 100, 200, and 1500 MCSs, while in the bottom row they were taken at 0, 30, 100, 300, and 400 MCSs.Reuse & Permissions
Figure 5
Time courses of the density of cooperators $ρ_{C}$ for $G = 5$ (dashed blue line) and $G = 445$ (solid red line), starting from the mentioned prepared initial state (see Fig. 4) and using all-group imitation range, as obtained for $K / G = 0.1$ and a system size of $L = 800$ . Synergy factors were $r / G = 0.7$ and $0.6$ , respectively.Reuse & Permissions

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