Unbeatable imitation of a friend

Imitation sometimes achieves success in multi-agent situations even though it is very simple. In repeated games, success of imitation has been characterized by unbeatability against other agents. Previous studies specified conditions under which imitation is unbeatable against co-players in repeated symmetric games, and clarified that the existence of unbeatable imitation is strongly related to the existence of payoff-controlling strategies, called zero-determinant strategies. However, these results were on the situations in which one repeated game is played by one group of players. It was pointed out that imitation of other players in the same group and imitation of other players in the same role in other groups generally result in different outcomes. Here, we investigate the existence condition of unbeatable imitation in the latter situations, described by two concurrent repeated games played by two different groups of players. We find that the existence condition of unbeatable imitation is stronger than the existence condition of unbeatable zero-determinant strategies, whereas both are very limited. Furthermore, we find that adding exploration can make an imitation strategy unbeatable even in games where imitation without exploration can be beaten. Our findings suggest a strong relation between unbeatable imitation and payoff control even in the two concurrent repeated game situations.