Probabilistic punishment proportional to the payoff difference solves the problem of antisocial punishment

This study modifies the model in the previous studies and considers three types of inter-individual relationships: regular, random, and scale-free ring lattices. Furthermore, we introduce defectors, who do not contribute to the public goods; cooperators, who contribute to the public goods; and loners, who do not participate in the public goods framework. We assume that each of these three types of individuals punishes other individuals with a probability proportional to the difference between their own payoff and their opponent's average payoff including them. Using this modified pool punishment model, this study shows the following. Firstly, the damage to the average payoff due to excessive punishment is kept significantly low. Secondly, antisocial punishment is not evolutionarily advantageous, and cooperators always become advantageous. Finally, the final average payoff is always higher than that of pool punishment in existing studies and roughly comparable to that of peer punishment in existing studies. The results of this study provide new insights that the claim of the existing study is not always correct; that is, even if antisocial punishment is possible, it does not have an evolutionary advantage, and cooperators always become advantageous, which in turn solves the problem of antisocial punishment. This study is being conducted as part of efforts to improve specialized education at Kanagawa Institute of Technology.