The evolution of fairness in dyadic relationships has been studied using ultimatum games. However, human fairness is not limited to two-person situations and universal egalitarianism among group members is widely observed. In this study, we investigated the evolution of favoritism and group fairness in a three-person ultimatum game (TUG) under a co-evolutionary framework with both strategy updating and partner switching dynamics. In the TUG, one proposer makes an offer to two responders and the proposal is accepted at the group level if at least one individual responder accepts the offer. Investigating fairness beyond dyadic relationships allows the possibility of favoritism because the proposer can secure acceptance at the group level by discriminating in favor of one responder. Our simulation showed that the proposer favors one responder with a similar type when the frequency of partner switching is low. In contrast, group fairness is observed when the frequency of partner switching is high. The correlation between strategy and neighborhood size suggested that partner switching influences the strategy through the proposer's offer rather than through the responder's acceptance threshold. In addition, the average degree negatively impacts the emergence of fairness unless the frequency of partner switching is high. Furthermore, a higher frequency of partner switching can support the evolution of fairness when the maximum number of games in one time step is restricted to smaller values.Among the mechanisms proposed for the evolution of fairness, the effect of the network (spatial) structure is among the most intensively studied. A seminal study has shown that introducing lattice structure can facilitate the emergence of fairness [12]. This positive effect of the network structure was also confirmed using more complex network structures [13]. The role of the network structure has also been studied in combination with other mechanisms, including empathy [14,15], the fineness of the stratp-1

PACS 87.23.Ge -Dynamics of social systems PACS 87.23.Kg -Dynamics of evolution PACS 02.50.Le -Decision theory and game theory Abstract -Punishment and partner switching are two well-studied mechanisms that support the evolution of cooperation. Observation of human behaviour suggests that the extent to which punishment is adopted depends on the usage of alternative mechanisms, including partner switching. In this study, we investigate the combined effect of punishment and partner switching in evolutionary prisoner's dilemma games conducted on a network. In the model, agents are located on the network and participate in the prisoner's dilemma games with punishment. In addition, they can opportunistically switch interaction partners to improve their payoff. Our Monte Carlo simulation showed that a large frequency of punishers is required to suppress defectors when the frequency of partner switching is low. In contrast, cooperation is the most abundant strategy when the frequency of partner switching is high regardless of the strength of punishment. Interestingly, cooperators become abundant not because they avoid the cost of inflicting punishment and earn a larger average payoff per game but rather because they have more numerous opportunities to be referred as a role agent by defectors. Our results imply that the fluidity of social relationships has a profound effect on the adopted strategy in maintaining cooperation. p-1

The emergence and maintenance of cooperation has attracted intensive scholarly interest and has been analysed within the framework of evolutionary game theory. The role of innovation, which introduces novel strategies into the population, is a relatively understudied aspect of evolutionary game theory. Here, we investigate the effects of two sources of innovation-mutation and heterogeneous updating rules. These mechanisms allow agents to adopt strategies that do not rely on the imitation of other individuals. The model introduces-in addition to canonical imitation-based strategy updating-aspiration-based updating, whereby agents switch their strategy by referring solely to the performance of their own strategy; mutation also introduces novel strategies into the population. Our simulation results show that the introduction of aspirationbased rules into a population of imitators leads to the deterioration of cooperation. In addition, mutation, in combination with heterogeneous updating rules, also diminishes cooperators. This phenomenon is prominent when a large proportion of the population consists of imitators rather than adopters of aspiration-based updating. Nevertheless, a high mutation rate, in combination with a low aspiration level, has positive nonlinear effects, and a heterogeneous population achieves a higher level of cooperation than the weighted average of homogeneous populations. Our results demonstrate the profound role of innovation in the evolution of cooperation.

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