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In the popular class of multidimensional poverty measures introduced by Alkire and Foster (2011), a threshold switching function is used to identify who is multidimensionally poor. This paper shows that the weights and cut-off employed in this procedure are generally not unique and that such functions implicitly assume all groups of deprivation indicators of some fixed size are perfect substitutes. To address these limitations, I show how the identification procedure can be extended to incorporate any type of positive switching function, represented by the set of minimal deprivation bundles that define a unit as poor. Furthermore, the Banzhaf power index, uniquely defined from the same set of minimal bundles, constitutes a natural and robust metric of the relative importance of each indicator, from which the adjusted headcount can be estimated. I demonstrate the merit of this approach using data from Mozambique, including a decomposition of the adjusted headcount using a ‘one from each dimension’ non-threshold function.
