The academic construction of ambidexterity articulated around notions such as exploration, exploitation (J. March 1991) has been flourishing over the years with a strong background in organisational theory to explain levels of performance and innovation. However, they have also made a call for in-depth studies to understand managerial capabilities such as decision-making (Birkinshaw & Gupta 2013; O’Reilly & Tushman 2013; Benner & Tushman 2015) supporting the tension of competing objectives. In this paper, we show that organisational ambidexterity can kill innovation as the underlying decision theories are not fully supporting the nature of decision required in regimes such as contextual ambidexterity (Gibson & Birkinshaw 2004). Two case studies from the aircraft cabin equipment industry are presented and analysed at the project management level with descriptors from organisational ambidexterity and decision-making. We propose to consider unconventional decision theories, taking into account non-expected utilities such as potential regret of imagined prospects, as a means to support management tools enabling ambidexterity at the decisional and contextual levels. First, we show that common decision models based on expected utility encoded in management tools mobilised for contextual ambidexterity can fail to support innovation. Second, we propose that a non-expected utility, such as potential regret of imagined prospects, serves the management of competing exploration/exploitation objectives. Third, the case studies help contouring a management tool extending observed attempts to sustain or extend contextual ambidexterity through unconventional decision-making.