Risk-driven migration and the collective-risk social dilemma

A collective-risk social dilemma implies that personal endowments will be lost if contributions to the common pool within a group are too small. Failure to reach the collective target thus has dire consequences for all group members, independently of their strategies. Wanting to move away from unfavorable locations is therefore all but surprising. Inspired by these observations, we here propose and study a collective-risk social dilemma where players are allowed to move if the collective failure becomes too probable. More precisely, this so-called risk-driven migration is launched depending on the difference between the actual contributions and the declared target. Mobility therefore becomes an inherent property that is utilized in an entirely self-organizing manner. We show that under these assumptions cooperation is promoted much more effectively than under the action of manually determined migration rates. For the latter, we in fact identify parameter regions where the evolution of cooperation is incredibly inhibited. Moreover, we find unexpected spatial patterns where cooperators that do not form compact clusters outperform those that do, and where defectors are able to utilize strikingly different ways of invasion. The presented results support the recently revealed importance of percolation for the successful evolution of public cooperation, while at the same time revealing surprisingly simple ways of self-organization towards socially desirable states.

💡 Research Summary

The paper extends the classic public‑goods framework by incorporating a collective‑risk social dilemma, where a group fails to meet a predefined contribution target and consequently all members lose their endowments. The authors introduce “risk‑driven migration”: each player evaluates the shortfall between the actual total contribution C of its group and the declared target T (a fraction α of the maximal possible contribution). If the absolute difference |C‑T| exceeds a risk threshold β, the player becomes “mobile” and randomly jumps to an adjacent empty site on a two‑dimensional lattice. This mobility is not imposed externally; it emerges automatically from the perceived risk of collective failure.

The model is implemented on a square lattice (typically 100 × 100) with periodic boundaries. Each site is occupied by either a cooperator (who contributes 1 to the common pool) or a defector (who contributes 0). Groups are formed by the focal player and its nearest neighbours (Moore neighbourhood, size 5). After contributions are summed, the group’s payoff is calculated: if C ≥ T the group receives a benefit B that is equally shared; otherwise every member suffers a loss L. Following the payoff calculation, the risk‑driven migration step occurs, and finally strategies are updated by a pairwise imitation rule proportional to payoff differences (the usual replicator dynamics).

Systematic simulations explore the parameter space (α ∈