Two-step method of equalization of opportunity

Ranking the circumstances: Inequality of opportunity

As a first step to test for opportunity equalization we need to get unambiguous ranking of all the concerned circumstance types, separately for each exogenous social states. This in effect is the test for the existence of IOP under each given social state, which is executed upon exploiting the notion of stochastic dominance. Although the use of stochastic dominance is not new to economics and finance, Lefranc, Pistolesi & Trannoy (2008, 2009) applies this concept for the first time in the literature ofIOP⁶. The privileged type is identified as the one, the distribution of which dominates that of the other types at certain order of stochastic dominance.

The basic theoretical underpinning of stochastic dominance is provided in the expected utility theory. For any non-decreasing utility function, one distribution, F(·), yields unambiguously better return than another, G(·), if the former first order stochastically dominates the latter. This implies, F(·) ≤ G(·), with F and G, being the cumulative distribution functions⁷. The same concept is applied in the set up of IOP, where F(·) and G(·), corresponds to the different cumulative distributions conditional on different types.

Consider any two types, c and c⁰, such that c 6= c⁰. Therefore under an exogenous social state, π, and for a given level of effort, their respective type-specific distributions can be written by the cumulative distribution functions, F_π(y|c, e) and F_π(y|c⁰, e), or equivalently, by the quantile functions F_π⁻¹(p|c, e) and F_π⁻¹(p|c⁰, e), for all values of the cumulative population percentile, p, within the range of [0,1]. Then type, c, will be identified as the privileged type as compared to type, c⁰, if the outcome distribution corresponding to the former dominates that of the latter at order one (c₁ c⁰), that is if

6For other applications of stochastic dominance inIOP, see for example,Peragine & Serlenga(2008), Trannoy, Tubeuf, Jusot & Devaux (2010). See Harris & Mapp (1986), Broske & Levy (1989) for its applications in other areas of economics and finance.

7See (Mas-Colell, Whinston & Green 1995, Chapter 6).

equation (2.1) holds in the form of following inequality Therefore, the first order stochastic dominance of type c, over c⁰, indicates that the cumulative distribution corresponding to the former type should lie to the right of that of the latter. Equivalently, the above condition can also be concluded from the inverse stochastic dominance of type c over c⁰, at order one, if the quantile distribution of the former type lies above than that of the latter⁸. Borrowing fromLefranc et al. (2009), we will say that there exist strong IOP in the society, under the social state,π, if the above condition is satisfied and further, type c, is enjoying an unethical privilege over type, c⁰. Ranking the social states: Equalization of opportunity

Once we have the unambiguous ranking of types within each of the exogenous social states we can proceed to rank the social states themselves, by applying the same concept of stochastic dominance, but in adifference-in-difference set up. The first difference mea-sures the gap between the type-specific distributions for each given social state, whereas the second difference measures the gap between the social states in terms of the respective gaps in their type-specific distributions.

Figure2.1 illustrates the basic concept of opportunity equalization across the different social states, for a pair of types (c, c⁰). The left and the right panel of the figure corre-sponds to the type-specific cumulative distributions under two different exogenous social states, π_m and π_n, respectively. Notice that irrespective of the social states, type c, has always turned out to be the privileged one, as the cumulative distribution corresponding to this type always lies to the right of that of the other type, for either of the social states.

But clearly, as compared to social state π_m, the privilege enjoyed by the advantageous type is less under social state π_n. Since the gap in the distributions between the advan-tageous and the disadvanadvan-tageous type is lesser under social stateπ_n, we can say that the economic opportunity equalizes between those types, if we move from social state π_m to

8The first and second order stochastic dominance is equivalent to the inverse stochastic dominance of the same order (Shorrocks 1983). However, the equivalence does not hold beyond the second order.

π_n.

y F(y)

F_πm(y|c⁰, e)

F_πm(y|c, e)

(a) Social state πm

y F(y)

F_πn(y|c⁰, e)

F_πn(y|c, e)

(b) Social stateπn

Figure 2.1: A simple illustration of equalization of opportunity

However, it is possible that the privileged dominant type in social stateπ_m, turned out to be the disadvantageous one under social state π_n. Nevertheless, the principle of IOP holds for either social states and a fall in the gap between the type-specific distributions under social state π_n, is still indicative of opportunity equalization. Thus, from the perspective of responsibility sensitive egalitarian justice, the direction of dominance does not matter per se, what matters instead is the absolute gap between the types. Therefore as far as equalization of opportunity is concerned, we can claim for one if we see a fall in the absolute gap between the type-specific distributions for different social states.

For notational simplicity, letF_π⁻¹ and F_π⁰⁻¹, denote the distributions ofF_π⁻¹(p|c, e) and F_π⁻¹(p|c⁰, e), respectively. So for a pair of different exogenous social states, π_m and π_n, the absolute gap between the type-specific distributions for a pair of types, (c, c⁰), can be expressed as

-Γ(F_π⁻¹_m, F_π⁰⁻¹_m , p) = |F_π⁻¹_m −F_π⁰⁻¹_m | for social state: π_m (2.3a) Γ(F_π⁻¹_n , F_π⁰⁻¹_n , p) = |F_π⁻¹_n −F_π⁰⁻¹_n | for social state: π_n (2.3b) Given the unequivocal first order dominance between the types c and c⁰ for each of the exogenous social states (by equation2.2), the right hand sides of equation (2.3) are always positive.

Further, let ∆^(π_(c,c^m0^,π) ⁿ⁾ denote the difference in difference between the type-specific dis-tributions of two social states, π_m and π_n, for the same pair of types, (c, c⁰). Hence as compared to social stateπ_m the economic opportunity will equalize under social state π_n, if the extent of IOP between the same pair of types is lesser for the latter social state.

Therefore the criteria for opportunity equalization while moving from social state π_m to π_n, requires that for all p∈[0,1]

-∆^(π_(c,c^m0^,π)ⁿ⁾ =|F_π⁻¹_m −F_π⁰⁻¹_m | − |F_π⁻¹_n −F_π⁰⁻¹_n | ≥0

⇒Γ(F_π⁻¹_m, F_π⁰⁻¹_m , p)≥Γ(F_π⁻¹_n, F_π⁰⁻¹_n , p) (2.4) Provided the expression of equation (2.4) we can invoke the criteria of first order dom-inance as presented in equation (2.2), but in a difference-in-difference set up. Therefore we can say that equation (2.4) essentially means that the distribution of Γ(F_π⁻¹_m, F_π⁰⁻¹_m, p) dominates that of the Γ(F_π⁻¹_n, F_π⁰⁻¹_n , p) at first order inverse stochastic dominance. Since Γ(·) is nothing but the gap between the type-specific distributions, the above criteria of opportunity equalization between a pair of social states is also referred as the criteria of gap curve dominance in Andreoli et al. (2019). Provided unambiguous ranking among the pair of types, (c, c⁰), at order one, equation (2.4) provides a necessary and sufficient condition for opportunity equalization across a pair of social states.

However if in the first step a pair of types can not be ranked at dominance of order one for any of the social state, the above condition of equalization of opportunity is no longer sufficient. In that case,Andreoli et al.(2019) showed that upon further restricting the class of preferences, it is always possible to rank a pair of types by higher order inverse stochastic dominance so that a necessary and sufficient condition for opportunity equalization can always be formulated for that subset of preferences. In particular, if for all the concerned social states,πi ∈ {πm, πn}, the distribution Fπi dominates that ofF_π⁰_i, by orderk of inverse stochastic dominance, then for a subset of preferences,R^k⊂ R, the general criteria of equalization of opportunity requires that, for all p∈[0,1]

-Γ^k(F_π⁻¹_m, F_π⁰⁻¹_m , p)≥Γ^k(F_π⁻¹_n, F_π⁰⁻¹_n , p) (2.5)

In the above expression, Γ^k(·) is the integrated cumulative distribution gap, defined for the social state π as, Γ^k(F_π⁻¹, F_π⁰⁻¹, p) = Λ^k_π(p)−Λ^0k_π(p), where Λ^k_π(p) and Λ^0k_π(p), are respectively, the distributions of F_π⁻¹ and F_π⁰⁻¹, integrated at orderk−1⁹.

Therefore if for any of the social states we fail to rank a pair of types (c, c⁰) by dominance of order one, we can not conclude on equalization of opportunity among the social states by condition (2.4), for the class of all rank-dependent preferences, R. However, if for all the social states, types can still be ranked at order two, for example, we can proceed to test for opportunity equalization by condition (2.5), for the sub-class of risk-averse preferences, R² ⊂ R, and so on.