The evolution of n-player cooperation—threshold games and ESS bifurcations

Variability in group size and the evolution of collective action

“…5. We note that the situation illustrated in this figure is analogous to the one considered in Bach et al (2006), who also show that a downward shift in a unimodal concave gain function may cause the number of interior rest points of the replicator dynamics to drop from two to zero via a saddle-node bifurcation. The key difference between the scenario considered in Bach et al (2006) and the one we consider here is that in their model the disappearance of the interior rest points is caused by a downward shift in the gain sequence d, whereas in our model the gain sequence is given and it is a shift in the experienced group-size 22 .…”

“…. 25, 2015; the total number of players is required to produce a collective good (Bach et al, 2006;Pacheco et al, 2009;Archetti and Scheuring, 2011), (ii) games of multiplayer reciprocity (Boyd and Richerson, 1988), and (iii) variants of the volunteer's dilemma where the cost of providing the good is shared among cooperators (Weesie and Franzen, 1998), sometimes referred to as multiplayer snowdrift game (Zheng et al, 2007;Souza et al, 2009). No unambiguous, general conclusions concerning the effect of variable group sizes can be obtained in these cases.…”

“…25, 2015; conditions appearing in Proposition 2 -for instance, Example 2 provides conditions on the parameter values of the public goods game with synergy or discounting for which this is the case-these conditions are much more stringent that the ones in Proposition 1. In the cases not covered by Proposition 2, e.g., when the gain sequence is decreasing and convex (as in the volunteer's dilemma from Example 1) or is concave and unimodal (as it is the case in the model of Bach et al (2006) …”

“…As shown in the vast literature on nonlinear public goods games (e.g., Dugatkin, 1990;Motro, 1991;Bach et al, 2006;Hauert et al, 2006;Cuesta et al, 2008;Pacheco et al, 2009;Archetti and Scheuring, 2011) cooperative behavior may arise in the evolutionary solution of such games even when other mechanisms potentially promoting cooperation such as relatedness (Eshel and Motro, 1988;Archetti, 2009;Peña et al, 2015) and reciprocity in repeated interactions (Boyd and Richerson, 1988;Hilbe et al, 2014) are absent.…”

“…This paper has followed Peña (2012) in investigating the evolutionary consequences of variation in group size using the replicator dynamics of two-strategy multiplayer games. While this is a common approach is the literature on collective action problems (Motro, 1991;Bach et al, 2006;Peña et al, 2014), alternative approaches are possible. In particular, the very same question we are interested in has been explored by Brännström et al (2011) in the framework of continuous strategies and adaptive dynamics (Metz et al, 1996).…”