Robust ``pro-poorest'' poverty reduction with counting measures: the non-anonymous case

When measuring poverty with counting measures, are there conditions ensuring that poverty reduction not only reduces the average poverty score further but also decreases deprivation inequality among the poor more, thereby emphasizing improvements among the poorest of the poor? In the case of a non-anonymous assessment, i.e. when we can track poverty experiences of the same individuals or households using panel datasets, we derive three conditions whose fulfillment allows us to conclude that multidimensional poverty reduction is more egalitarian in one experience vis-à-vis another one, for a broad family of poverty indices which are sensitive to deprivation inequality among the poor, and from an ex-ante conception of inequality of opportunity. We illustrate these methods with an application to multidimensional poverty in Peru before and after the 2008 world financial crisis.