The seminal contribution of Sen (1976) led to a new way to conceptualize and measure absolute poverty, by arguing for the need to ‘take note of the inequality among the poor’ (Sen 1976: 227). Since then, the ‘Inequality’ of poverty has become the third ‘I’ of poverty, which together with the ‘Incidence’ and the ‘Intensity’ of it constitute the dimensions deemed relevant for poverty evaluation. In this paper, we first argue that the interest in the third ‘I’ of poverty actually (...) originates from aprioritarian(Parfit 1995) rather than an egalitarian attitude. Further, we illustrate the inability of the three ‘I's to fully comprise the criteria for the assessment of poverty which are de facto adopted by existing poverty indices. Some of them resolve distributional conflicts by following leximin, hence assigning a pivotal role to the worst off. We question the desirability of leximin, and conclude that giving absolute priority to the worst off is plausible only in cases where the latter has been identified by an exogenous threshold demarcating a significant difference in human suffering. Finally, we explore to what extent prioritarianism and the sufficiency argument of Frankfurt (1987), Crisp (2003) and Casal (2007) can help conceptualize giving absolute priority to individuals or groups indentified by exogenous (poverty and ultra-poverty) thresholds. (shrink)