Mixed update rules point to the dynamics of cooperation in simulated populations

The question of how collective cooperation emerges and perseveres in a population of self-interested individuals has vexed theologians and behavioral ecologists alike. Dynamic simulations have pointed to the use of update rules, in which individuals update their own behavioral rules as important for the evolution of cooperation. It remains unclear, however, how update rules influence the evolution of cooperation when they incorporate mixed preferences from payoff-driven and conformity-driven factors.

Researchers have demonstrated that a modification to the evolutionary prisoner’s dilemma game can help predict which portion of a population will cooperate with one another. Under a mixed update rule, individuals in the model chose a conformity update rule at various probabilities. Lin et al. then used the pair approximation approach and mean-field approach to derive differential equations that allowed them to calculate the fraction of the population that opted for cooperation.

“In this work, we investigate how social learning influences the evolution of collective cooperation in population games, which is a topic that is closely related to collective intelligence and multi-agent learning,” said author Xiaojie Chen.

Their modeling demonstrated that one stable, interior equilibrium point exists in systems under weak selection, and that cooperators can survive alongside non-cooperators in such a population.

The group found that even if interacting individuals were given very low probabilities of adopting more conformity-driven rules, a stable number of cooperators exist in the population.

As the probability of conformity-driven rules increases, so does the fraction of cooperators, regardless of the perceived benefit value of cooperating.

The group next plans to study how the ways of social learning influence collective performance of decision-making in any population structure.

Source: “Evolutionary dynamics of cooperation driven by a mixed update rule in structured prisoner’s dilemma games,” by Longhao Lin, Chengrui Li, and Xiaojie Chen, Chaos (2025). The article can be accessed at https://doi.org/10.1063/5.0245574 .

This paper is part of the Intelligent Game on Networked Systems: Optimization, Evolution and Control Collection, learn more here .