Two-stage strategy update rule based on learning cost in weak prisoner's dilemma

When players are dissatisfied with their actual payoffs, they will change the actuality by learning strategy of neighbors. The more effort players put in, the more likely they are to succeed in learning. Inspired by this, this paper proposes a two-stage strategy update rule based on learning cost. The players first decide whether to learn strategy according to the updating willingness. If the players imitate the strategy of neighbors, they need to pay the learning cost. Results show that for the well-mixed population, if the updating willingness is homogeneous and remains unchanged, reducing the updating willingness or increasing the learning cost can extend the life cycle of cooperators. If the updating willingness is heterogeneous and dynamically adjusted based on the difference between the actual payoff and the expected payoff, increasing aspiration value and learning cost promotes cooperation. For the structured population, if the updating willingness is homogeneous and remains unchanged, the moderate learning cost is beneficial for cooperators to resist the temptation of defection, and reducing updating willingness makes the system maintain cooperation within a larger parameter range. If the updating willingness is heterogeneous and dynamically adjusted, the larger learning cost and the appropriate aspiration value promote cooperation. This study highlights the complex dynamics of cooperation in paid strategy learning, contributing to the theory of cooperation in the evolutionary game.