This study reconsiders the Ohdaira and Terano's previous studies regarding the prisoner's dilemma game with the sequential strategy and the second-best decision by introducing the notion of the bounded rationality, and also presents a new knowledge. The studies regarding cooperation without kin relationships are based on either interactions of two types, i.e. pairwise interactions like the prisoner's dilemma game or group interactions like the public goods game. On the other hand, as Ohdaira and Terano also argue, these previous studies discuss the very limited case that each player knows whole payoff matrix of the game, i.e. all players know how their payoffs should be in relation to each pair of strategy between the player and his/her opponent player. Therefore, this study newly introduces the notion of the bounded rationality that each player does not know whole payoff matrix, and also shows the property of cooperation when each player makes his/her bounded rational second-best decision. This paper is organized as follows. Firstly, the author sophisticates the Ohdaira and Terano's previous studies by introducing the notion of the bounded rationality and its formulation. Secondly, this paper presents the new knowledge utilizing the extension of the Ohdaira and Terano's model and discusses the scientific significance of the results.