and illustrates how to use them in the context of inequality of opportunity estimations. An empirical illustration based on the EU Survey of Income and Living Conditions is contained in section 4, in which we will also highlight the particular advantages of using tree-based estimation methods. Lastly, section 5 concludes.

# **2 Empirical Approaches to Equality of Opportunity**

Consider a population of size _N_ indexed by _i ∈{_ 1 _, ..., N }_ and an associated vector of incomes _Y_ = _{y_ 1 _, ..., yi, ..., yN }_ . Individual _i_ ’s outcome of interest _yi_ is the result of two sets of factors. First, a set of _circumstances_ beyond her control of size _P_ : Ω _i_ = _{Ci_<sup>1</sup><sup>_, ..., C_</sup> _i_<sup>_p, ..., C_</sup> _i_<sup>_P}_.Second,a</sup> set of _efforts_ of size _Q_ : Θ _i_ = _{Ei_<sup>1</sup><sup>_, ..., E_</sup> _i_<sup>_q, ..., E_</sup> _i_<sup>_Q}_.Ingeneral,theoutcomegeneratingfunction</sup> _g_ : Ω _×_ Θ _→_ R+ can therefore be written as Each circumstance _C_<sup>_p_</sup> _∈_ Ωis characterized by a total of _X_<sup>_p_</sup> possible realizations, where each realization is denoted as _x_<sup>_p_</sup> . Based on the realizations _x_<sup>_p_</sup> we can partition the population into a set of non-overlapping _types T_ = _{t_ 1 _, ..., tm, ..., tM }_ . A type is a subgroup of the original population uniform in terms of circumstances, i.e. individuals _i_ and _j_ belong to the same type _tm ∈ T_ if _x_<sup>_p_</sup> _i_<sup>=</sup><sup>_xp_</sup> _j_<sup>_∀Cp∈_Ω.Theybelongtodifferenttypes</sup><sup>_tm∈T_if</sup><sup>_∃Cp∈_Ω:</sup><sup>_xp_</sup> _i_<sup>=</sup><sup>_xp_</sup> _j_<sup>.</sup> The number of types in the population is given by _M_ =<sup>�</sup><sup>_P_</sup> _p_ =1<sup>_Xp_.FollowingRoemer(1998)we</sup> assume that the joint realizations of the effort variables _E_<sup>_q_</sup> _∈_ Θ can be summarized by a scalar _π ∈_ [0 _,_ 1]. Individuals sharing the same expression of effort are called a _tranche_ . Hence, types and tranches define two particular ways of partitioning the population into subgroups, where group membership either indicates uniformity in circumstances (types) or effort (tranches).

In the literature we can distinguish two broad classes of equality of opportunity definitions.<sup>3</sup> First, the _ex-ante_ view focuses on between-type differences in the value of opportunity sets without paying attention to the specific effort realizations of individual type members. According to this perspective, equality of opportunity is satisfied if the aggregate value of opportunity sets is equalized across types. One example in case is the _ex-ante utilitarian_ perspective according to which the value of opportunity sets is indicated by the average outcome within the specific type. Thus, equality of opportunity would be realized if the mean outcome of each type was equal to the population mean. Second, the _ex-post_ view focuses on individual outcomes conditional on effort exertion. According to this perspective, equality of opportunity would be satisfied if individual outcomes were equalized within each tranche, i.e. individuals with equal levels of effort exertion realize the same outcomes. A comprehensive discussion of the ex-ante and expost principles of equality of opportunity can be found in Fleurbaey and Peragine (2013). In the context of this paper we will restrict ourselves to the ex-ante utilitarian approach only.

**Tests and Measures** The extant literature has witnessed the development of empirical _tests_ and _measures_ for ex-ante utilitarian inequality of opportunity. A prominent example for the former category is provided by Lefranc et al. (2009), who show that rejecting the null hypothesis of no first-order stochastic dominance in type-specific outcome distributions is sufficient to reject the existence of equal opportunities in the population from an ex-ante utilitarian perspective. Furthermore, in a recent contribution Kanbur and Snell (2017) develop likelihood ratio tests that can serve to test for ex-ante utilitarian equality of opportunity. A widely adopted example of the latter category, is the measure developed by Van de gaer (1993) and Chec-

> 3Measures different from the ones illustrated here, have been proposed in the literature. The interested reader is referred to Van de gaer and Ramos (2016) for a comprehensive overview.

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