form or another, e.g., using vignettes or eliciting willingness-to-pay. There exist other methods less directly connected to our preference-based framework. For instance, asking directly subjects about their preferred weights (Al-Ajlani et al., 2020) or asking to poor individuals which dimensions matter most to poverty (Kanbur and Lustig, 2001; Bennett, 2021). These latter methods may guide the selection of dimensions<sup>62</sup> and maybe provide a source of legitimacy for weights, although the relationship between such weights and preference u is unclear. In practice, the information on preference u could be collected in a few contexts and then used to select a fixed set of weights to be applied to all contexts.<sup>63</sup> Clearly, these different sources of information on u have important limitations. Conceptually, none of them provides unbiased information on well-being rankings. In practice, each also has its issues.<sup>64</sup> In spite of their important limitations, these sources of data nevertheless offer a promising way of improving on the ad-hoc “equal-weights”. Again, we are looking for better weights, not perfect weights. Observe also that selecting the value of weights from information on preferences does not necessarily imply changing the weights across time and space. Indeed, changing the weights may compromise comparability. The advantage of selecting the values of weights from preferences is that these values are less likely to be implausible than “equal-weights”.

In this section, we present one method to derive a set of weights from (partial information on) preference u. Before that, we argue that seemingly natural methods do not appropriately account for the meaning of the weights used in multidimensional poverty indices. As mentioned in Alkire and Foster (2011b), AF weights do not correspond to marginal rates of substitution. The intuition is illustrated for the two dimensions case, where i is identified as poor when w1si1 + w2si2 ≥ k (Eq. (2)). Assume without loss of generality that w1 > w2 = 1 − w1. Figure 7 illustrates the resulting identification contour (in red) for two different cases. This contour corresponds to the union approach when w2 ≥ k while it corresponds to the intersection approach when w1 < k. Crucially, there is no achievement vector on the identification contour at which the slope of this

> 62The selection of dimension could also be guided by SWB data, as argued by Kingdon and Knight (2006).

> 63This is less of an issue when assuming that preference u is the same for all individuals. In any case, weights should not take equal values if preference data reliably shows that some variables should receive a larger / smaller weight.

> 64 First, subjective well-being questions do not provide unbiased information on well-being. For instance, people make trade-offs between their income and their level of happiness, which is the component of happiness they tend to prefer (Adler et al., 2017). Also, choices need not align with anticipated subjective well-being (Benjamin et al., 2014a). The link between different aspects of SWB questions and different aspects of preferences is still imperfectly understood (Benjamin et al., 2017, 2023). This matters because inter-personal comparisons may change when we change the aspects of SWB from which preference u is elicited (Defloor et al., 2017). See also Ravallion (2014) for additional concerns with the use of SWB data in poverty measurement. However, subjective well-being data are easy to collect. Second, preference elicitation surveys may work relatively well for two or three variables at a time, but multidimensional poverty indicators often rely on more than 10 variables. Also, the ranking that subjects provide on achievement vectors that they did not experience themselves can be criticized as being not sufficiently well-informed (Karimi et al., 2017).

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