# **1 Introduction**

John Roemer’s (1998) seminal contribution, _Equality of Opportunity_ , has incited a flourishing empirical literature on the measurement of unequal opportunities. At the heart of Roemer’s formulation is the idea that factors that determine individual outcomes can be divided into two categories: factors over which individuals have control, which he calls _effort_ , and factors for which individuals cannot be held responsible, which he calls _circumstances_ . Individuals characterized by exactly the same exogenous circumstances are assigned to a circumstance _type_ . Members of a type have the same background conditions to transform resources into outcomes. Therefore, while within-type inequality, as caused by the differential exertion of effort, is morally irrelevant, between-type differences in achievements are inequitable and call for compensation. Thus, opportunity-equalizing policies have the objective of neutralizing the impact of circumstances on the distribution of the desirable outcome.

Following Roemer’s approach, a battery of methods to measure inequality of opportunity have been proposed (see Roemer and Trannoy, 2015; Van de gaer and Ramos, 2016, for recent overviews).<sup>1</sup> Today, well established empirical methods include summary indexes that quantify the extent of unequal opportunities (Alm ̊as et al., 2011; Bourguignon et al., 2007; Checchi and Peragine, 2010) as well as statistical tests that detect the mere existence thereof (Kanbur and Snell, 2017; Lefranc et al., 2009). In either case, empirical results are sensitive to critical choices of model selection which are under complete discretion of the researcher.

First, researchers have to make a decision on which circumstance variables to consider for estimation.<sup>2</sup> Observable circumstances beyond individual control are typically a subset of the real number of exogenous variables affecting individual outcomes. This issue has been largely discussed by the literature, and the prevailing view is that partial observability implies downwardbiased inequality of opportunity estimates (Ferreira and Gignoux, 2011). To counteract this downward bias, one strategy is to resort to high-quality datasets that provide very detailed information with respect to individual circumstances (Hufe et al., 2017). Naturally, the scope of improvement of this approach is limited by sample sizes. Consider for example the increasing availability of genetic datasets with billions of polymorphisms per person (Altshuler et al., 2015). While the genetic make-up of individuals clearly is beyond individual control and must be considered a circumstance, the number of circumstances exceeds the available degrees of freedom which forces the researcher to choose selectively from the available set of circumstances.

Second, the influence of circumstances may be dependent on the expression of other circumstance characteristics. For example, it is a well-established finding that the influence of similar child-care arrangements on various life outcomes varies strongly by biological sex (Garc ́ıa et al., 2017). In contrast to such evidence, however, many empirical applications presume that the effect of circumstances on individual outcome is fixed and additive (Bourguignon et al., 2007; Ferreira and Gignoux, 2011). On the one hand, analogous to partial observability, this functional form assumption forces a downward bias on inequality of opportunity estimates. On the other hand, limitations in the available degrees of freedom may prove the estimation of fully saturated models impractical. Again the researcher is left to her own devices in selecting the best model for estimating inequality of opportunity.

While the downward bias of inequality of opportunity estimates is prominently discussed in the extant literature, the reliability of estimates has been largely disregarded. Holding the

> 1Note that a number of contributions from the social choice literature on fair allocation had previously proposed similar methods (Fleurbaey, 1995, 2008; Van de gaer, 1993).

> 2Roemer does not provide a fixed list of variables that are to be considered as circumstances. Rather he suggests that the set of circumstances should evolve from a political process (Roemer and Trannoy, 2015). In empirical implementations typical circumstances are biological sex, socioeconomic background, race, or the area of birth.

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