We propose a strategy with which conversational android robots can handle dialogue breakdowns. For smooth human-robot conversations, we must not only improve a robot's dialogue capability but also elicit cooperative intentions from users for avoiding and recovering from dialogue breakdowns. A cooperative intention can be encouraged if users recognize their own responsibility for breakdowns. If the robot always blames users, however, they will quickly become less cooperative and lose their motivation to continue a discussion. This paper hypothesizes that for smooth dialogues, the robot and the users must share the responsibility based on psychological reciprocity. In other words, the robot should alternately attribute the responsibility to itself and to the users. We proposed a dialogue strategy for recovering from dialogue breakdowns based on the hypothesis and experimentally verified it with an android. The experimental result shows that the proposed method made the participants aware of their share of the responsibility of the dialogue breakdowns without reducing their motivation, even though the number of dialogue breakdowns was not statistically reduced compared with a control condition. This suggests that the proposed method effectively elicited cooperative intentions from users during dialogues.