Replicator population dynamics of group interactions: Broken symmetry, thresholds for metastability, and macroscopic behavior

Emmanuel Artiges, Carlos Gracia-Lázaro, Luis Mario Floría, and Yamir Moreno

Phys. Rev. E 100, 052307 – Published 20 November 2019

Abstract

The effect of group structure on cooperative behavior is not well understood. In this paper, we study the dynamics of a public goods game involving $n$ -agent interactions. In the proposed setup, the population is organized into groups. We associate the individual fitness to group performance, while the evolutionary dynamics takes place globally. We derive analytical expressions and show that the model exhibits several fixed points, including the symmetric homogeneous states of total cooperation and total defection, which are unstable and stable, respectively. Interestingly, even if both individual and group levels are organized as well-mixed populations, the dynamics displays intermediate values of cooperation under the replicator dynamics. Namely, as soon as one of the groups, at least, is fully cooperative, intermediary fixed points appear for the rest of the groups. In addition to the analytical approach, we have performed numerical simulations that reproduce the internal fixed points obtained theoretically, showing coexisting intermediate levels of cooperation. Potential implications of these results in terms of group selection and the role of social norms are also discussed.

Received 12 July 2019

DOI:https://doi.org/10.1103/PhysRevE.100.052307

Physics Subject Headings (PhySH)

Complex systems Social systems

Nonlinear DynamicsInterdisciplinary Physics

Authors & Affiliations

Emmanuel Artiges¹, Carlos Gracia-Lázaro ^2,3, Luis Mario Floría^2,4, and Yamir Moreno^2,3,5

¹École Normale Supérieure de Lyon, Université Claude Bernard Lyon I, 69342 Lyon Cedex 07, France
²Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Zaragoza 50009, Spain
³Departamento de Física Teórica. University of Zaragoza, Zaragoza 50009, Spain
⁴Departamento de Física de la Materia Condensada, University of Zaragoza, Zaragoza 50009, Spain
⁵ISI Foundation, Turin, Italy

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Vol. 100, Iss. 5 — November 2019

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Images

Figure 1
Structure of the model. Strategist agents are disposed in groups and can either cooperate or defect. Agents' payoffs are proportional to the number of cooperators in their group, cooperators having to pay an extra cost. Although agents obtain their payoffs from their group, imitation takes place from any group. In the diagram, solid red arrows represent the cooperation invasion flows, and blue dashed arrows the defection flows. See the text for further details.
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Figure 2
Phase portrait for the two-groups case. A red dot indicates an unstable fixed point, a black dot a stable one, and a blue dot indicates a saddle point. The green lines correspond to the $f_{1}^{C} = f_{2}^{D}$ and $f_{1}^{D} = f_{2}^{C}$ singularities. The green dots correspond to the points $(1, p_{2}^{th})$ and $(p_{1}^{th}, 2)$ . Gray (respectively, purple) lines correspond to nullclines $d p_{1} = 0$ ( $d p_{2} = 0$ ), while arrows represent the trajectories. Here, $r = 4$ . See the text for further details.
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Figure 3
Three-groups case. (a) Representation of the unit cube $(p_{1}, p_{2}, p_{3})$ . Colored areas correspond to the planar regions, codimension 1 stable manifolds, where one of the three groups is fully cooperator. (b) Phase portrait for the three-groups case restricted to the plane $p_{3} = 1$ . The white area is the basin of the full defection state, while blue area corresponds to the stable manifold. The green lines correspond to the $f_{1}^{C} = f_{2}^{D}$ and $f_{1}^{D} = f_{2}^{C}$ singularities, blue lines to the nullclines, and arrows to trajectories. Inner black dot corresponds to the fixed point $(p^{*}, p^{*}, 1)$ , and black dots located on the coordinate axes correspond to the fixed points $(\hat{p}, 1, 1), (1, \hat{p}, 1)$ . In this plot, $r = 4$ . See the text for further details.
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Figure 4
Ground and upper fixed points. The graphs show the theoretical values of $\hat{p}$ (solid blue line) and $p^{up}$ (dashed orange line) as a function of $r$ . The number of groups has been fixed to $m = 6$ , and the number of full-C groups to $n_{f} = 1$ . See the text for further details.
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Figure 5
Stochastic evolution of the system. The graphs show the time evolution of the fractions of cooperators for two representative realizations of the six-groups case, each solid line corresponding to a group. Dotted lines correspond to the theoretical predictions for $\hat{p}$ and $p^{up}$ . (a) For small group sizes ( $N = 200$ ) fluctuations allow groups to exchange their levels of cooperation. (b) For larger group sizes ( $N = 1000$ ) one group clearly detaches from the rest to occupy the upper fixed point, and the fluctuations are not large enough to allow exchange. In these plots, $r = 6$ .
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Figure 6
Histogram of cooperation level in groups. The histogram counts the number of groups having a certain cooperation rate after 200 time steps. The counts have been accumulated over 100 independent realizations. Dashed (respectively, dotted) line corresponds to the theoretical predictions for $\hat{p}$ ( $p^{up}$ ). Here, the parameters are assigned the same values as in the upper panel of Fig. 5: $m = 6, r = 6$ , and $N = 200$ . The initial cooperation rates in the non full-C groups are given by a Gaussian determined by $μ = σ^{2} = n / 4 .$
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