equality of opportunity versus equality of opportunity sets

Soc Choice Welfare (2007) 28: 209–230DOI 10.1007/s00355-006-0165-4

ORIGINAL PAPER

Erwin Ooghe · Erik SchokkaertDirk Van de gaer

Equality of opportunity versus equalityof opportunity sets

Received: 31 March 2003 / Accepted: 7 March 2005 / Published online: 31 May 2006© Springer-Verlag 2006

Abstract We characterize two different approaches to the idea of equality of oppor-tunity. Roemer’s social ordering is motivated by a concern to compensate for theeffects of certain (non-responsibility) factors on outcomes. Van de gaer’s socialordering is concerned with the equalization of the opportunity sets to which peoplehave access. We show how different invariance axioms open the possibility to gobeyond the simple additive specification implied by both rules. This offers scopefor a broader interpretation of responsibility-sensitive egalitarianism.

1 Introduction

The traditional idea of equality of opportunity has been interpreted in two differentways in the recent literature. A first approach starts from the premise that individualoutcomes are determined by two types of factors: compensation factors (such asinnate abilities, handicaps, called “type” in the sequel) and responsibility factors(such as ambition, preferences, called “effort” in the sequel).1 The basic idea ofequality of opportunity then is to equalize individual outcomes to the extent thatthey reflect differences in “type”, while allowing for different outcomes wheneverthey are due to “effort”. This focus on compensation of outcomes is prominent

E. Ooghe · E. SchokkaertCenter for Economic Studies, KULeuven, Naamsestraat, 69, 3000 Leuven, BelgiumE-mail: [email protected]: [email protected]

D. Van de gaer (B)SHERPPA, Vakgroep Sociale Economie, UGent, Hoveniersberg 24, 9000 Gent, BelgiumE-mail: [email protected]

E. Schokkaert · D. Van de gaerCORE, Université Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium

1 Roemer (2002) calls these “arbitrary” and “responsible” factors.

210 E. Ooghe et al.

in the larger part of the axiomatic literature – see, e.g., Bossert (1995), Fleurbaey(1994, 1995a,b), Bossert and Fleurbaey (1996), Iturbe-Ormaetxe (1997), Maniquet(1998, 2004). It is also the basic inspiration of Roemer’s rule (Roemer 1993, 1998,2002).

A second approach to equality of opportunity focuses on the opportunity setto which people have access, and tries to make these sets as equal as possible –see, e.g., Kranich (1996), and Ok and Kranich (1998). Compensation is definedin terms of opportunity sets. The concern for responsibility implies that only theset matters, while individuals remain responsible for their choice. A rule whichstarts from this inspiration has been proposed by Van de gaer (1993) and is furtherexplored in Bossert et al. (1999).

Of course the two approaches are related, since the opportunity set to which aperson with a given type has access gives rise to the vector of possible outcomes thathe can obtain for different values of effort. Yet their basic inspiration is different.

Roemer’s approach has been very popular in the recent past and it has resultedin a series of thought provoking empirical applications (Llavador and Roemer2001; Roemer et al. 2003; Roemer 2002). As a matter of fact, in his recent “pro-gress report”, Roemer (2002) explicitly criticizes the alternative “opportunity set”approach, claiming that it has not resulted in any empirical work because it remainstoo abstract and fails to take sufficient cues from popular and long-standing con-ceptions of equality of opportunity. This criticism is certainly not valid for the Vande gaer rule. In many cases, both the Roemer rule and the Van de gaer rule lead toexactly the same policy prescriptions. This happens to be true in all the empiricalapplications referred to earlier. In cases where the rules differ, the Van de gaer ruleis even computationally simpler than the Roemer rule, at least in a continuous set-ting, because the latter has to integrate the lower countour set of the different types.Empirical applicability therefore cannot be an argument to discard the former.

In this paper, we want to explore in more detail the differences between theRoemer rule and the Van de gaer rule. We briefly review both rules in Sect. 2.We illustrate how Roemer’s rule is an example of the CO (compensating out-comes) approach, while Van de gaer’s rule is an example of the CS (compensatingsets) approach. Moreover, both rules assume an additive aggregation procedure.In Sect. 3, we introduce notation and present both basic rules, together with someextensions. Section 4 discusses the axioms and Sect. 5 characterizes and discussesthe basic versions of the Roemer rule and the Van de gaer rule. The axioms usedin the characterization reveal the strong and weak points of both. They also yielda better insight into the implications of the additive aggregation rule underlyingboth social orderings. In Sect. 6 we show how relaxing this additivity assumptionpaves the way for a characterization of some alternative rules (also proposed inKolm 2001). Section 7 concludes.

2 Two alternative rules

To set the scene, let us first briefly summarize both approaches in a finite setting.Let X be the set of social options, e.g. the different policy prescriptions. Individ-uals who are homogeneous with respect to compensation factors are of the same“type”; we gather the different types i = 1, . . . , m in a set M . Individuals who arehomogeneous with respect to responsibility characteristics have exerted the same

Equality of opportunity versus equality of opportunity sets 211

“effort”; the different effort levels j = 1, . . . , n are collected in a set N . We assumethat effort can be observed, either directly (for example, labour hours) or indirectly(for example, via Roemer’s percentile approach: on the basis of the rank order ofan individual within the distribution of outcomes of his type).2 To simplify ourexposition, we assume that each type-effort couple (i, j) ∈ M × N is representedby exactly one individual; we abbreviate i j ∈ M N . Later on, we mention how toadapt the framework to relax this assumption. Outcomes are defined by the socialoption, the type, and the effort of the individual. Let U = (

U 11 , U 2

1 , . . . , U nm

)be

a profile of real-valued outcome functions U ji , one for each type-effort couple i j ,

defined as

U ji : X → R(or R++) : x �→ U j

i (x) .

To illustrate both approaches, we consider an economy with two types and fiveeffort levels. Figure 1 presents the outcomes of both types under social option x forthe different effort levels, i.e. the vectors Ui (x) = (U 1

i (x) , . . . , U 5i (x)

), i = 1, 2.

Either one tries to equalize the outcomes of different types at the same effortlevel: this is the CO (compensating outcomes) approach to which Roemer’s rulebelongs. Define the social option x j which maximizes the minimal outcome of thedifferent types at effort level j , i.e.

x j = arg maxx∈X

mini∈M

U ji (x) . (1)

It is rather unlikely that x j will be the same for all effort levels. Therefore, it isnecessary to look for an adequate aggregation rule over the different effort levels.Roemer proposes – without much justification – a simple additive specification forthis purpose. Roemer’s social objective becomes

maxx∈X

∑

j∈N

mini∈M

U ji (x) . (2)

Figure 2 presents the minimal outcomes for each effort level in black; the Roemerrule wants to maximize the sum of these outcomes.

Or one tries to equalize the opportunity sets of the different types; this is in linewith the CS (compensating sets) approach to which Van de gaer’s rule belongs.The Van de gaer rule follows Bossert (1997), who values an opportunity set solelyon the basis of the corresponding outcomes. Here again one has to define howto aggregate these outcomes. Again without too much justification, Van de gaerproposes a simple additive specification: the value of the opportunity set of type iunder option x becomes

∑

j∈N

U ji (x) .

Equalizing opportunity sets boils down to equalizing their valuation; the maxi-min-rule readily suggests itself. Van de gaer’s rule maximizes the value of theopportunity set of the most disadvantaged type:

2 Roemer proposes to identify effort on the basis of the quantiles of the outcome distributionfor each type. For a critical assessment of this quantile hypothesis, see Fleurbaey (1998) andKolm (2001). For a strong defence, see Roemer (2002).

212 E. Ooghe et al.

Fig. 1 Relation between effort and outcome for two types

Fig. 2 Roemer’s sum-of-mins rule

maxx∈X

mini∈M

∑

j∈N

U ji (x). (3)

Figure 3 shows the outcomes of the most disadvantaged (lowest sum of outcomes)– here type 1 – in black; the Van de gaer rule wants to maximize the sum of theseoutcomes.

Comparing (2) and (3) shows that the only formal difference is the interchangeof the min and the sum operator.3 If the same type is the least advantaged ateach effort level, both criteria lead to the same policy. However, although theyare formally similar, the basic intuition behind both rules – CO versus CS – is

3 In fact, Roemer (2002) presents the Van de gaer-rule as an alternative compromise procedurein case the options x j , defined in Eq. (1), are different for some effort levels. Roemer has nostrong preference for any of these alternatives over the other. In this spirit, our paper can be readas a comparison of different compromise procedures in the Roemer Approach. However, we feelthat the differences are deeper than suggested by this interpretation.


Fig. 3 Van de gaer’s min-of-sums rule

very different.4 We provide an example where the ranking of two social options isdifferent between both rules.

Figure 4 presents the opportunity sets of two types (bullets and diamonds) overfive effort levels in black; except for the lowest effort level, type 1 is everywherebetter off. Now, suppose we can redistribute outcomes for the lowest effort levelonly, such that outcome levels after the transfer (in white) coincide. According tothe Roemer rule, this policy is an improvement, although type 1 dominates type 2after the change; thus, the CO-approach neglects opportunity set considerations.The Van de gaer rule, would prefer the inverse policy; thus, the CS-approach mayincrease the outcome differences for individuals who exert the same effort.

This example suggests that there is really a basic difference between the Ro-emer rule and the Van de gaer rule – a difference which goes deeper than a simplenormalization. At the same time, however, both rules have analogous weaknesses.Consider a situation with only one type. Both the Roemer rule and the Van de gaerrule will then evaluate different policies in a utilitarian fashion, i.e. as the sum ofthe outcomes at the different effort levels. It has been argued (see, e.g., Fleurbaey1998; Fleurbaey and Maniquet 2004) that this goes against the intuition of the ideaof equal opportunity because the government should not intervene and should sim-ply accept the laissez-faire outcomes, if the differences between outcomes resultonly from differences in effort.

It is true that this is the usual “natural reward” interpretation of equality ofopportunity. However, a different idea of responsibility–sensitivity, called “utili-tarian reward” by Fleurbaey and Maniquet (2004) does not necessarily imply anabsolute respect for laissez-faire. It is possible to take the position that the socialevaluation should focus on the conditional distribution of outcomes rather thanthe way these outcomes are obtained while at the same time postulating a defi-nite idea about the optimal conditional distribution. In the latter interpretation itseems useful to think about ways to formulate a more flexible approach than thesum-specification of both the Roemer rule and the Van de gaer rule.

4 Kolm (2001) suggests to interpret Roemer’s criterion via the normative concept “desert” andVan de gaer’s in terms of its dual, “merit”.

214 E. Ooghe et al.

Fig. 4 Roemer versus Van de gaer

The main aim of this paper is to further explore the basic differences betweenthe standard Roemer rule and the Van de gaer rule (Sect. 5), and to look for moreflexible extensions (Sect. 6). Sections 3 and 4 introduce the necessary notation.

3 Social orderings

Notation We assume strong neutrality as defined by Roberts (1995) for his “exten-sive” social choice framework. As a consequence, we can focus on a social order-ing5 R over outcome vectors u = (

u11, u2

1, . . . , unm

), rather than on an extensive

social welfare functional defined over profiles U = (U 1

1 , U 21 , . . . , U n

m

). For con-

venience, we will consider outcome matrices:

u =[u j

i

]=

⎡

⎢⎢⎢⎢⎣

u11 . . . un

1. . . . . .

u1i . . . u j

i . . . uni

. . . . . .

u1m . . . un

m

⎤

⎥⎥⎥⎥⎦

Row ui = (u1

i , . . . , uni

)represents the outcome vector for type i (the opportuni-

ties), while column u j = (u j

1, . . . , u jm)

represents the outcome vector for effortlevel j . Our framework with one individual in each cell is more general than itappears at first sight. Suppose that the number of individuals is finite and that typesand effort are independently distributed (a natural assumption since it is hard tohold people responsible for their effort level if it is correlated with their type –see also Fleurbaey 1998, p. 221). If certain combinations of effort and types occurmore than once, we can always construct a matrix u with one individual in eachcell, but where some columns have the same effort level and some rows are of thesame type.

We introduce three rank-operators. The first one ranks outcomes within eachcolumn of a matrix u in an increasing way; the resulting matrix equals u, with

5 An ordering is a complete and transitive binary relation. P and I denote the correspondingasymmetric and symmetric relation.


u j1 representing the lowest outcome for effort level j in the original matrix u and

so on. The second one reranks the rows in an increasing way, on the basis of the

sum of their outcomes; the resulting matrix equals+u, with

+u1=

(+u

1

1, . . . ,+u

n

1

)the

opportunities of the type with the lowest sum of outcomes in the original matrixu and so on. The final one also reranks the rows of a strictly positive matrix in

an increasing way, but on the basis of the product of its outcomes;xu is the matrix

obtained from u in this way.

The Roemer rule and the Van de gaer rule: We present Roemer and Van degaer’s proposal, slightly modified to satisfy the strong Pareto principle; we there-fore replace maximin by leximin. The Roemer rule can be rephrased as lexico-graphically maximizing the sum of the minimal outcomes at each effort level; u Rvholds, if and only if

{either,

∑j∈N u j

i =∑ j∈N vji , i = 1, . . . , m,

or, ∃s, 1 ≤ s ≤ m :∑ j∈N u ji =∑ j∈N v

ji , i < s and

∑j∈N u j

s >∑

j∈N vjs .

In the same vein, the Van de gaer rule can be reformulated as lexicographicallymaximizing the minimal sum of outcomes for the different types; u Rv holds, ifand only if

⎧⎨

⎩either,

∑j∈N

+u

j

i =∑ j∈N+v

j

i , i = 1, . . . , m,

or, ∃s, 1 ≤ s ≤ m :∑ j∈N+u

j

i =∑ j∈N+v

j

i , i < s and∑

j∈N+u

j

s >∑

j∈N+v

j

s .

The main focus of this paper will be on the characterization of these two basicsocial orderings. However, as mentioned already in the previous section, the sum-component in both rules has hardly been justified in the literature until now. As aby-product of our characterization we will better understand the consequences ofthis additive specification. In Sect. 6, we will characterize some alternative rules.

The multiplicative Roemer rule and the Van de gaer rule: Instead of adding,we may also multiply outcomes over different effort levels. Restricting the domainto strictly positive outcome matrices, the product-Roemer rule ranks u Rv, if andonly if:

{either,

∏j∈N u j

i =∏ j∈N vji , i = 1, . . . , m,

or, ∃s, 1 ≤ s ≤ m :∏ j∈N u ji =∏ j∈N v

ji , i < s and

∏j∈N u j

s >∏

j∈N vjs .

The product-Van de gaer rule ranks u Rv, if and only if

⎧⎨

⎩either,

∏j∈N

xu

j

i =∏ j∈Nxv

j

i , i = 1, . . . , m,

or, ∃s, 1 ≤ s ≤ m :∏ j∈Nxu

j

i =∏ j∈Nxv

j

i , i < s and∏

j∈Nxu

j

s >∏

j∈Nxv

j

s .

216 E. Ooghe et al.

Flexible Roemer rule and Van de gaer rule Finally, we introduce a Kolm–Atkin-son–Sen (KAS) and Kolm–Pollak (KP) specification, as suggested by Kolm (2001).Let g : R

m++ → R++ and h : Rn++ → R++ be strictly increasing, symmetric sep-

arable and homogeneous functions, and define exp(u j) = (

exp u j1, . . . , exp u j

m)

and exp (ui ) = (exp u1i , . . . , exp un

i

). The KAS-Roemer family and the KP-Roemer

family contain social orderings preferring u to v, if and only if respectively6

⎛

⎝∑

j∈N

(g(

u j))ρ

⎞

⎠

1ρ

≥⎛

⎝∑

j∈N

(g(v j))ρ

⎞

⎠

1ρ

and

⎛

⎝∑

j∈N

(g(

exp(

u j)))ρ

⎞

⎠

1ρ

≥⎛

⎝∑

j∈N

(g(

exp(v j)))ρ

⎞

⎠

1ρ

.

The KAS-Van de gaer family and KP-Van de gaer family invert the aggregationsteps. They contain social orderings preferring u to v, if and only if respectively:

(∑

i∈M

(h (ui ))ρ

) 1ρ

≥(∑

i∈M

(h (vi ))ρ

) 1ρ

and

(∑

i∈M

(h (exp (ui )))ρ

) 1ρ

≥(∑

i∈M

(h (exp (vi )))ρ

) 1ρ

.

4 Properties

We focus on a social ordering defined over matrices in a domain D , with D equalto R

m×n or Rm×n++ :

• UD (unrestricted domain). R is defined on D = Rm×n .

• PD (positive domain). R is defined on D = Rm×n++ .

Continuity is defined as:• CON (continuity).

For each u ∈ D; the better-than and worse-than sets {v ∈ D |vRu } and{v ∈ D |u Rv } are closed.Efficiency boils down to the strong Pareto principle, defined as:

• SP (strong Pareto).For all u, v ∈ D; if u j

i ≥ vji for all i j ∈ M N , then u Rv. If, in addition, there

exists a kl ∈ M N such that ulk > vl

k , then u Pv.

The main differences between the CO-approach and the CS-approach are linkedto separability, impartiality and compensation. The columns in Table 1 presentthe different axioms within these three classes, while the rows indicate the twomain approaches. The symbol + (resp. –) indicates whether an axiom is typical

6 As usual, when ρ = 0 the KAS-specification is equal to a log-utilitarian rule.


Table 1 Compensating outcomes versus compensating sets

Separability (SE) Impartiality (SI) Compensation (C)

Effort (N ) type (M) effort (N ) type (M) (for type only)

SE∗N SEN SE∗

M SEM SI∗N SIN SI∗M SIM C DC

CO + + − + − • • + + •CS − + + + • + − • − •

(resp. atypical) for one of both approaches; the symbol • refers to axioms that willbe used in the next section to characterize the Roemer rule and the Van de gaerrule.

Separability: For reasons that will become clear later on, separability assump-tions do not play a role in the characterization of the basic Roemer rule and theVan de gaer rule (see Table 1). They are more important to characterize the flexibleextensions in Sect. 6. In a CS-approach it seems straightforward to impose a sep-arability requirement between the different rows (opportunities of a type). Strongseparability between types eliminates indifferent types, i.e. types with identicalopportunities under two social options do not play a role.

• SE∗M (strong separability between types).

For all u, v, u′, v

′ ∈ D and for all subsets G ⊆ M ; if u ji = v

ji and u

′ ji = v

′ ji for

all i j ∈ G N , while u ji = u

′ ji and v

ji = v

′ ji for all i j ∈ M N\G N , then u Rv if

and only if u′Rv

′.

In the same way we can define strong separability between effort (columns). Thisaxiom eliminates indifferent effort levels, i.e. effort levels with identical outcomevectors. This axiom is very intuitive for the CO approach.

• SE∗N (strong separability between effort).

For all u, v, u′, v

′ ∈ D and for all subsets G ⊆ N ; if u ji = v

ji and u

′ ji = v

′ ji for

all i j ∈ MG, while u ji = u

′ ji and v

ji = v

′ ji for all i j ∈ M N\MG, then u Rv if

and only if u′Rv

′.

For both separability requirements, we can also introduce a weaker version, whichrequires separability in case of specific subdomains. DM contains matrices whereoutcomes in each row are constant: effort has no effect on the outcomes of eachtype. DN consists of matrices where outcomes in each column are constant: typehas no effect on the outcomes for each effort level.

• SEM (separability between types).SE∗

M holds for all u, v, u′, v

′ ∈ DM .• SEN (separability between effort).

SE∗N holds for all u, v, u

′, v

′ ∈ DN .

218 E. Ooghe et al.

Impartiality: Much more important are the impartiality requirements. Suppesindifference implies that the “names” of the types (resp. effort levels) do not matter.We will define it in two different strengths: we can permute rows (resp. columns),but stronger, we can also permute outcomes within each column (resp. row) sepa-rately. Let σ denote a permutation of a set. When reasoning within a CS-approach,the following combination of axioms seems reasonable:

• SIM (Suppes indifference for types).

For all u, v ∈ D and for each permutation σ on M ; if v =[u j

σ(i)

], then uIv.

• SI∗N (strong Suppes indifference for effort).For all u, v ∈ D and for each m-tuple (σ1, . . . , σm) of permutations on N ; if

v =[uσi ( j)

i

], then uIv.

The first axiom SIM is a natural requirement from a CS perspective. It imposes thatthe different types should be treated in a symmetric way: their opportunity vectorscan be permuted without influencing the social evaluation. The second one, SI∗N ,is also a plausible requirement if we look at the rows of the matrices as embodyingthe values of the opportunity sets to which different individuals have access. Itimposes that only the whole vector of opportunities matters, not the specific linkbetween each effort level and the resulting outcome.

On the other hand, in the CO-approach we can reasonably assume:

• SI∗M (strong Suppes indifference for types).For all u, v ∈ D and for each n-tuple

(σ 1, . . . , σ n

)of permutations on M ; if

v =[u j

σ j (i)

], then uIv.

• SIN (Suppes indifference for effort).

For all u, v ∈ D and for each permutation σ on N ; if v =[uσ( j)

i

], then uIv.

In the CO-setting, SI∗M is a very natural requirement. When comparing the out-comes for different types at the same effort level, the identity of the types shouldnot matter. SIN permutes entire columns, or the names of the effort levels do notmatter. We could also consider a version of Suppes indifference in between SINand SI∗N . Recall the rank operator ∼; we define

D = {u ∈ D |∃v ∈ D such that u = v } .

Notice that the Roemer rule and the Van de gaer rule coincide on D. Therefore,both rules satisfy the following version of Suppes indifference:

• SIN (ordered Suppes indifference for effort).

SI∗N holds for all u, v ∈ D.

Compensation: In the CO approach, compensation requires equalization of theoutcomes of the different types who exerted the same effort. In the spirit of Ham-mond (1976), we define:

• C (compensation).For all u, v ∈ D and for all i j, k j ∈ M N ; if v

jk > u j

k ≥ u ji > v

ji , while uh

g = vhg

for all other gh �= i j, k j , then u Rv.


In the CS-approach however, compensation focuses on the equalization of oppor-tunity vectors. Valuing opportunity vectors is a difficult problem and we do notwant to impose strong assumptions about it at this stage. We therefore start fromthe simple idea that an opportunity set is “better” than another if it gives rise tohigher outcomes for all effort levels. Given two types i and k, we write ui ≥ uk todenote u j

i ≥ u jk , for all j in N ; in this case, we say that type i dominates type k.

Adding such a dominance requirement to the previous compensation axiom givesus dominance compensation:7

• DC (dominance compensation).For all u, v ∈ D and for all i j, k j ∈ M N ; if vi ≥ vk and v

ji > u j

i ≥ u jk > v

jk ,

while uhg = vh

g for all other gh �= i j, k j , then u Rv.

The Van de gaer rule implies much more than only a dominance condition withrespect to the evaluation of opportunity sets. It incorporates the following utilitariancompensation principle:8

• UC (utilitarian compensation).For all u, v ∈ D and for all types i, k ∈ M ; if

∑j∈N v

jk >

∑j∈N u j

k ≥∑

j∈N u ji >

∑j∈N v

ji , while uh

g = vhg for all gh /∈ i N ∪ k N , then u Rv.

We do not use this stronger axiom for the characterization of the Van de gaer rule,but we show that it is implied by the combination of other axioms. For later ref-erence, we finish by introducing the idea of utilitarian neutrality. If the sum ofopportunities remains the same for all types, then we should rank the matrices asindifferent:

• UN (utilitarian neutrality).For all u, v ∈ D; if

∑j∈N u j

i =∑ j∈N vji for all i in M , then uIv.

5 Results

The Roemer rule and the Van de gaer rule: Roemer’s CO-approach focuses onthe differences between the outcomes for different types at the same effort level,i.e. on the inequality within the columns of the outcome matrix. It then seemsnatural to require that the social evaluation should be invariant if all outcomeswithin the columns are changed by the same absolute number or are changed in thesame proportion since such changes do not affect absolute or relative inequality,respectively. We will see that the Roemer rule focuses on absolute inequality withina column. Roemer remains agnostic, however, about the differences between thedifferent columns. Therefore, the social evaluation does not depend on the differ-ences between outcomes at different effort levels.

7 This differs from the approach taken by Ok and Kranich (1998) who advocate to defineequalization of opportunity sets for a given “cardinality-based” ordering of all possible opportu-nity sets. Our “ordinality-based” notion of dominance compensation is weaker than “opportunitydominance for a policy” as introduced by Hild and Voorhoeve (2004).

8 The cartesian product {i} × N is abbreviated as i N .

220 E. Ooghe et al.

To formalize this idea, recall DN , the domain of matrices with all elements equalin a column. We define the following axiom:9

• Itntf (invariance axiom).

For all u, v ∈ D and for all α ∈ DN ; u Rv if and only if (u + α) R (v + α).

Notice that Itntf implies separability between effort levels SEN . The axiom Itn

tf meansthat the social judgment does not change provided that the absolute differences inoutcomes at each effort level, i.e. within each column, are preserved. At the sametime, since the elements of matrix α may differ over different effort levels, differ-ences in outcomes between the columns are irrelevant for the social evaluation.Within the CO-approach, the Itn

tf axiom forces us to focus on the inequality ofoutcomes, irrespective of the level at which these inequalities are evaluated.

The Itntf axiom is equally relevant in the CS-approach. Indeed, if the difference

in outcomes for each effort level remains the same, then the difference in the val-uation of the opportunity sets also remains the same. If one wants to focus on theinequality of the sets, the α-transformation (defined in Itn

tf ) should therefore notlead to a change in the relative evaluation of different opportunity sets.

We are now ready to formulate our two basic characterization results:10

Proposition 1 For each row in the table, there is only one social ordering R whichsatisfies the axioms indicated by •; it is the rule in the first column. In addition, +(resp. –) means that the rule satisfies (violates) the corresponding axiom:

UD Itntf SP DC SI∗M SIM SI ∗

N SIN

Roemer • • • • • + − •Van de gaer • • • • − • • +

Discussion: In the proof of Proposition 1 (see Appendix), we show that all axi-oms are independent. The separability axioms play no role in this characterization.Their potential role is taken over by the invariance axiom Itn

tf and the equity axiomDC.11 More surprising is the fact that we use in both cases the same dominancecompensation axiom DC. The only substantial differences are to be found in theimpartiality axioms SI∗M and SI∗N . As we show next, they have strong implications.

The proof of Proposition 1 for the Van de gaer rule makes use of the followingLemma 1, showing that the combination of the invariance axiom Itn

tf with strongSuppes indifference between effort levels SI∗N implies utilitarian neutrality UN:

Lemma 1 If a social ordering R defined on D satisfies UD, Itntf and SI∗N , then it

also satisfies UN.

9 From a formal point of view, the invariance axiom Itntf is identical to a two-dimensional

invariance assumption, introduced in Ooghe and Lauwers (2005). It amounts to translation-scalemeasurability (T ) combined with full comparability (F) between the different types (for a giveneffort level) and translation-scale measurability (T ) and no comparability (N ) between the differ-ent effort levels (for a given type).

10 All the proofs are put together in the Appendix.11 Ooghe and Lauwers (2005) provide an alternative characterization of the Roemer-rule, via

(additional) separability axioms, combined with a minimal equity requirement.


It is clear that UN makes sense only in the CS approach, where sets are measured bythe sum of the outcomes of those that have the same compensation characteristic.Lemma 1 shows that, in combination with UD and Itn

tf , the axiom SI∗N brings usalready a long way in the direction of the Van de gaer rule. This becomes evenmore transparent when we impose also the dominance compensation axiom DC,because one can show

Lemma 2 If a social ordering R defined on D satisfies UD, Itntf , SI∗N and DC, then

it also satisfies UC.

The other impartiality requirement SI∗M plays an equally important role in thecharacterization of the Roemer rule. Imposing it indeed implies a considerablestrengthening of the rather weak idea of dominance compensation. This is summa-rized in the following lemma:

Lemma 3 If a social ordering R defined on D satisfies UD, SI∗M and DC, then italso satisfies C.

Note that we described in the previous section how compensation C captures themain intuition of the CO-approach. The basic incompatibility between the Roemerrule and the Van de gaer rule can be summarized in the following lemma:

Lemma 4 Suppose m, n ≥ 2. A social ordering R defined on D cannot satisfyUD, SP, C and UN simultaneously.

Combining these various lemmas, we obtain the following message. If we want tokeep UD, Itn

tf , SP, and DC, we will have to choose either SI∗M , or SI∗N . ChoosingSI∗M implies choosing C and choosing SI∗N implies choosing UN, and even UC.The choice between both impartiality axioms can therefore be seen as the choicebetween compensating outcomes via C, or compensating opportunity sets via UC,where the value of the opportunity sets is measured as the sum of the outcomesachieved by individuals with the same type. This dichotomy is also the drivingforce behind the example given in Sect. 2 (Fig. 4).

6 Extensions

The multiplicative extensions: In the previous section, we have shown that boththe Roemer rule and the Van de gaer rule satisfy the invariance axiom Itn

tf . It isnot obvious that this is an attractive idea. First, it implies an absolute approachto the evaluation of inequality of opportunity. Such an absolute approach is notthe most popular in the literature on inequality measurement. Second, it impliesa complete agnosticism about the ethical evaluation of differences between theoutcomes related to different effort levels.

The most obvious alternative to Itntf is to follow the dominant relative approach

to inequality measurement: within a column we look at relative, rather than abso-lute differences in outcomes. This means that we only allow for ratios of outcomesbetween types with the same effort to be relevant. Consider � the domain of diag-onal matrices β ∈ R

n×n with strictly positive diagonal entries β ii > 0. We then

propose:

222 E. Ooghe et al.

• Irnrf (invariance axiom)

For all u, v ∈ D and for all β ∈ �; u Rv if and only if (uβ) R (vβ).

Similar to Itntf , this invariance axiom Irn

rf implies separability between effort levelsSEN . Furthermore, Irn

rf preserves the relative inequality between outcomes for agiven effort level. Since this holds for all effort levels, it can be maintained thatthis does not change the inequality of opportunity sets either, if one takes a relativeperspective on inequality. With the necessary change in domain, we get:

Proposition 2 For each row in the table, there is only one social ordering R whichsatisfies the axioms indicated by •; it is the rule in the first column. In addition, +(resp. –) means that the rule satisfies (violates) the corresponding axiom:

PD Irnrf SP DC SI∗M SIM SI ∗N SIN

Product-Roemer • • • • • + − •Product-Van de gaer • • • • − • • +

The flexible extensions: We can go further and drop the assumption of non-comparability between columns. Again, this is easily done by exploiting the anal-ogy with the classical invariance requirements in the social choice literature. Define1n

m ∈ Rm×n , a matrix consisting of ones:

• Itftf (invariance axiom).

For all u, v ∈ D and for all a ∈ R; u Rv if and only if(u + a1n

m

)R(v + a1n

m

).

• Irfrf (invariance axiom).

For all u, v ∈ D and for all b ∈ R++; u Rv if and only if (bu) R (bv).

Itftf (resp. Irf

rf ) does not imply separability between effort levels; separability axi-oms will be imposed directly. Furthermore, Itf

tf (resp. Irfrf ) only compares situations

that keep the absolute (resp. relative) differences between outcomes at all effortlevels constant. These weaker axioms allow us to express a wide spectrum of opin-ions about inequalities that are due to type and effort. The following propositioncharacterizes the KAS and KP specification for the Roemer rule and the Van degaer rule:

Proposition 3 For each row in the table, there is only one family of social order-ings R which satisfies the axioms indicated by •; it is the family in the first column.In addition, + (resp. –) means that each rule in the family satisfies (violates) thecorresponding axiom:

PD Irfrf SP CON SE∗

M SEM SE∗N SEN SI∗M SIM SI∗N SIN

KAS-R • • • • − • • + • + − •KAS-VDG • • • • • + − • − • • +

UD Itftf

KP-R • • • • − • • + • + − •KP-VDG • • • • • + − • − • • +R stands for Roemer and VDG for Van de gaer

It remains true that the main differences between the (broad) Roemer rule and(broad) Van de gaer rule (and therefore also between the CO and the CS approaches


to equality of opportunity) are revealed by the impartiality axioms, but they arehere reinforced by specific separability axioms.

We can conclude that the invariance axiom Itntf is crucial in explaining the sum-

assumption which is imposed both by Roemer and Van de gaer. Alternatives tothese rules are readily available for those who do not like the (strong) implicationsof Itn

tf : absolute inequality measurement within a column, i.e. for a given effort level,and complete neglect of differences in outcome levels between columns, i.e. fordifferent effort levels. Of course, none of these alternatives counters the criticismthat with one type the laissez-faire outcomes should be respected. However, if oneaccepts the broader interpretation of responsibility-sensitivity in which the socialevaluation also depends on the form of the opportunity sets, the increased flexibilitygained by relaxing Itn

tf seems a definite improvement. Note that by imposing evenweaker invariance requirements than Itf

tf and Irfrf , many additional functional forms

become possible.

7 Conclusion

Although the Roemer rule and the Van de gaer rule are formally similar, their basicintuition is very different. This difference goes deeper than a mere difference in nor-malization. The former exploits the idea of compensating for different outcomes,if these differences follow from differences in non-responsibility or compensationfactors. The latter focuses on the evaluation of opportunity sets and wants to maketheir “value” as equal as possible. As we have shown, both approaches are incom-patible in general. While they lead to similar policy prescriptions in many cases(and to identical policy prescriptions if there is one type whose opportunity set isdominated by all the other types), this should not detract us from the differencesin ethical inspiration. It would be interesting to further explore these differences inconcrete applications.12 More empirical work could be very fruitful in this regard.

Both rules satisfy an invariance axiom which we have labeled Itntf . It is reflected

in their additive specification. The axiom Itntf imposes that the social evaluation does

not change if the same constant is added to the outcomes of all types at a giveneffort level. This essentially implies an absolute approach to inequality measure-ment. Moreover, these constants may differ at different effort levels, implying thatdifferences between the outcomes related to different effort levels do not matter.All these assumptions are debatable. However, we have shown that introducingother invariance axioms leads to rules of the Roemer family and the Van de gaerfamily with a more flexible specification.

The importance of this increased flexibility should not be underestimated. Ashas been noted by Fleurbaey (1998), neither the Roemer rule nor the Van de gaerrule satisfy the traditional notion of equality of opportunity implying that the lais-sez-faire outcome should be respected in cases where there is only one type. Thisis one specific notion of responsibility-sensitive egalitarianism, however. It canalso be argued that responsibility matters, i.e. that effort should be rewarded, whileat the same time putting forward definite ideas about how it should be rewarded

12 Van de gaer et al. (2001) explore the justification of the rules in the context of the measure-ment of intergenerational mobility. Schokkaert et al. (2004) calculate the optimal linear incometax for these rules.

224 E. Ooghe et al.

and, more specifically, without accepting that the laissez-faire rewards to effortare necessarily ideal from an ethical point of view. This requires that the form ofopportunity sets is evaluated, even if there is only one type. The additive (or utili-tarian) specification underlying the original Roemer rule and the Van de gaer rulemay not be the most attractive candidate for such an evaluation exercise. The moreflexible specifications introducing a parameter of inequality aversion can then beseen as a first step into the direction of a broader notion of responsibility-sensitiveegalitarianism.

8 Appendix

8.1 Proof of lemmas

Proof of Lemma 1 If a social ordering R defined on D satisfies UD, Itntf and SI∗N ,

then it also satisfies UN. Consider a matrix u ∈ D and two couples i j, ik in M N .Construct a matrix v by permuting elements i j and ik in matrix u. Using SI∗N wehave uIv. Add an arbitrary amount a ∈ R to the outcomes in the j-th columnof both matrices u and v to get u′ and v′; using Itn

tf , we have u′ Iv′. Permute theelements i j and ik in matrix u′ to get u′′ and by SI∗N and transitivity of R, we haveu′′ Iv′. Subtract the same amount a from the elements in the j-th column of bothmatrices u′′ and v′ to get u′′′ and v′′ = v; from Itn

tf , we have u′′′ Iv. By construction,

the matrix u′′′ is everywhere equal to v, except for outcomes u′′′ ji = v

ji − a and

u′′′ki = vk

i + a. Hence, we may remove outcome mass within a row of a matrix v,without changing social welfare, more precisely:

• Transfer principle.For all v ∈ D, for all i j, ik ∈ M N and for all a ∈ R; uIv, with u j

i = vji + a,

uki = vk

i − a and uhg = vh

g for all gh ∈ M N\ {i j, ik}.On the basis of this “transfer principle” (used repeatedly, if necessary), UN musthold, as required. ��Proof of Lemma 2 If a social ordering R defined on D satisfies UD, Itn

tf , SI∗N andDC, then it also satisfies UC. From Lemma 1, we know that UD, Itn

tf and SI∗Nimplies UN. We prove that adding DC also implies UC. Suppose the antecedentsof UC are true for two matrices u, v ∈ D and two types i, k in M , i.e.

∑j∈N v

jk >

∑j∈N u j

k ≥ ∑j∈N u j

i >∑

j∈N vji , while uh

g = vhg for all gh /∈ G = i N ∪ k N .

Construct two matrices u′ and v′ as follows:

(i) v′1k =

∑

j∈N

vjk (ii) u

′1k =

∑

j∈N

u jk (iii) u

′1i =

∑

j∈N

u ji (iv) v

′1i =

∑

j∈N

vji

(v) u′hg = v

′hg = 0,∀gh �= i1, k1 & gh ∈ G (vi) u

′hg = uh

g & v′hg = vh

g ,∀gh /∈ G.

From UN, we have uI u′ and v′ Iv. But by construction, DC applies and we haveu′ Rv′. From transitivity also u Rv holds. ��


Proof of Lemma 3 If a social ordering R defined on D satisfies UD, SI∗M and DC,then it also satisfies C. Suppose the antecedents of C are true for u, v ∈ D and fori j, k j in M N , i.e. v

jk > u j

k ≥ u ji > v

ji , while uh

g = vhg for all other gh �= i j, k j .

Construct two matrices u′ and v′ by reranking all columns different from j , eitherincreasingly or decreasingly depending on whether i < k or i > k. From SI∗M , wehave uI u′ and v Iv′. By construction, we can apply DC to get u′ Rv′ and thus alsou Rv must hold, as required. ��

Proof of Lemma 4 Suppose m, n ≥ 2. A social ordering R defined on D can-not satisfy UD, SP, C and UN simultaneously. First, consider the following fourmatrices

u =[

3 83 6

], v =

[2 84 6

], u

′ =[

2 104 5

], v

′ =[

3 93 6

]

Using C, we have u Rv and vRu′. Using UN, we have u′ Iv′. Everything together,we obtain by transitivity that u Rv′ must hold, which contradicts SP. We can alwaysembed these four matrices in larger ones belonging to R

m×n , which completes theproof.13 ��

8.2 Proof of Proposition 1

8.2.1 A. Characterization of the Roemer rule:

It is easy to verify existence: the Roemer rule satisfies all axioms UD, Itntf , SP, DC,

SI∗M and SIN . To prove uniqueness, we need the following lemma, which allowsus to use a transfer principle, as in Lemma 1, but for ordered matrices:

Lemma 5 If a social ordering R defined on D satisfies UD, Itntf , and SIN , then it

also satisfies the following “ordered” transfer principle: for all v ∈ D, for alli j, ik ∈ M N such that v

ji−1 = vk

i−1 (if i �= 1) and for all a with (if i �= 1)

vji−1 − v

ji ≤ a ≤ vk

i − vki−1; u Iv holds, with (i) u j

l = vjl + a and uk

l = vkl − a,

for all l ∈ G = {i, i + 1, . . . , m} (ii) uhg = vh

g for all gh ∈ M N\ (G j ∪ Gk).14

Proof Consider a matrix v ∈ D, i j, ik ∈ M N such that vji−1 = vk

i−1 (if i �= 1) and

a with vji−1 − v

ji ≤ a ≤ vk

i − vki−1. Suppose j < k; the other case is analogous.

Consider the following sequence of transformations on v (in vector notation); anexplanation follows:

13 This result also holds (i) on the positive domain, and/or (ii) without completeness of thesocial relation.

14 G j = G × { j}.

226 E. Ooghe et al.

v =⎛

⎜⎝v1; . . . ; v

j1 , . . . , v

ji , . . . , v

jm︸︷︷︸

v j

; . . . ; vk1, . . . , vk

i , . . . , vkm︸︷︷︸

vk

; . . . ; vn

⎞

⎟⎠ ,

v′ =(v1; . . . ; v

j1 +a, . . . , v

ji +a, . . . , v

jm +a; . . . ; vk

1, . . . , vki , . . . , vk

m; . . . ; vn),

v′′ =

(v1; . . . ; v

j1 + a, . . . , v

ji−1 + a, vk

i , . . . , vkm; . . . ; vk

1, . . . , vki−1, v

ji

+a, . . . , vjm + a; . . . ; vn

),

v′′′ =

(v1; . . . ; v

j1 , . . . , v

ji−1, v

ki − a, . . . , vk

m − a; . . . ; vk1, . . . , vk

i−1, vji

+a, . . . , vjm + a; . . . ; vn

),

v′′′′ =

(v1; . . . ; v

j1 , . . . , v

ji−1, v

ji + a, . . . , v

jm + a; . . . ; vk

1, . . . , vki−1, v

ki

−a, . . . , vkm − a; . . . ; vn

).

v′ is derived from v by adding a to column j , v′′ is derived from v′, by permut-ing elements i jand ik for all i in G (notice that the constraint for a ensures thatv′′ ∈ D), v′′′ is derived from v′′ by subtracting a from column j , and v′′′′ is derivedfrom v′′′ by permuting again elements i jand ik for all i in G (and again, v′′′′ ∈ D).We have v′′ Rv′ ⇔ v′′′ Rv and v′ Rv′′ ⇔ vRv′′′ (via Itn

tf ), and v′′ Iv′ and v′′′′ Iv′′′(via SIN ). Combining both, we have v′ Rv′ ⇔ v′′′′ Rv and v′ Rv′ ⇔ vRv′′′′, anddue to reflexivity, we must have v′′′′ Iv. It suffices to see that u = v′′′′, and thusuIv, as required. ��Proof of Proposition 1A. Consider an ordering R on D which satisfies UD, Itn

tf , SP,DC, SI∗M and SIN .First, we show, for all a, b ∈ D, that

a Rb ⇔⎡

⎢⎣

1n

∑j∈N a j

1 . . . 1n

∑j∈N a j

1. . . . . .1n

∑j∈N a j

m . . . 1n

∑j∈N a j

m

⎤

⎥⎦ R

⎡

⎢⎣

1n

∑j∈N b j

1 . . . 1n

∑j∈N b j

1. . . . . .1n

∑j∈N b j

m . . . 1n

∑j∈N b j

m

⎤

⎥⎦ .

Consider a, b ∈ D such that a Rb and consider the following steps:

• Recall the rerank operator ∼. Define c, d ∈ D with c = a and d = b; using SI∗Mwe have cRd .

• Use the ordered transfer principle described in Lemma 5 (repeatedly, if neces-sary) to obtain matrices e, f from c, d with

e1 =⎛

⎝1

n

∑

j∈N

c j1 , . . . ,

1

n

∑

j∈N

c j1

⎞

⎠ ; f1 =⎛

⎝1

n

∑

j∈N

d j1 , . . . ,

1

n

∑

j∈N

d j1

⎞

⎠ .

From Lemma 5 and transitivity, we have eR f .


• Use the ordered transfer principle described in Lemma 5 (repeatedly, if neces-sary) – in a way which leaves the first row unchanged – to obtain matrices g, hfrom e, f with

g1 = e1 and g2 =⎛

⎝1

n

∑

j∈N

e j2 , . . . ,

1

n

∑

j∈N

e j2

⎞

⎠ ; h1 = f1 and

h2 =⎛

⎝1

n

∑

j∈N

f j2 , . . . ,

1

n

∑

j∈N

f j2

⎞

⎠ .

Via Lemma 5 and transitivity, we have gRh.• . . .• Repeating these steps for all rows (in an upward way), we obtain the desired

result.

Second, due to the previous result, we might restrict attention to matrices in thedomain DM (see axiom SEM ). Define an ordering R◦ on R

m such that u Rv ⇔u1 R◦v1, for all u, v ∈ DM ; this ordering is well-defined. We show that R◦ must bethe leximin rule; then, R is the Roemer social ordering. The induced ordering R◦inherits (from the properties SP, SI∗M and DC for R) the strong Pareto principle,Suppes indifference (anonymity) and (a slightly stronger version of) HammondEquity, suitably redefined on the new domain R

m . But then, R◦ has to be theleximin rule (see, for instance, Bossert and Weymark (2004) Theorem 15). ��

8.2.2 B. Characterization of the Van de gaer rule:

It is easy to verify existence: the Van de gaer rule satisfies all axioms UD, Itntf , SP,

DC, SI∗N and SIM . We prove uniqueness. From Lemma 1, R has to satisfy UN.Using UN and transitivity (repeatedly if necessary), we have, for all a, b ∈ D, that

a Rb ⇔⎡

⎢⎣

1n

∑j∈N a j

1 . . . 1n

∑j∈N a j

1. . . . . .1n

∑j∈N a j

m . . . 1n

∑j∈N a j

m

⎤

⎥⎦ R

⎡

⎢⎣

1n

∑j∈N b j

1 . . . 1n

∑j∈N b j

1. . . . . .1n

∑j∈N b j

m . . . 1n

∑j∈N b j

m

⎤

⎥⎦ .

From here, the proof is the same as in the second part of the proof of Proposition1A. ��

8.2.3 C. Independence of the axioms:

We show that, for both rules, dropping the axioms one by one results in at leastone new rule, which satisfies all other axioms.

C1. Roemer rule

• Drop Itntf : consider the leximin rule applied to vectors

(u1

1, u21, . . . , un

1; u12, . . . , un

2; . . . ; u1m, . . . , un

m

) ∈ Rmn .

228 E. Ooghe et al.

• Drop SP: consider the original “maxmin”-version of the Roemer rule, whichmaximizes the sum of the mins.

• Drop DC: consider the utilitarian rule applied to vectors(u1

1, u21, . . . , un

1; u12, . . . , un

2; . . . ; u1m, . . . , un

m

) ∈ Rmn .

• Drop SI∗M : consider the Van de gaer rule• Drop SIN : adapt the Roemer rule using (strictly positive) weights when summing

(the mins).

C2. Van de gaer rule

• Drop Itntf : consider the “extensive” leximin rule, defined in C1.

• Drop SP: consider a “maxmin”-version of the Van de gaer rule, which maximizesthe minimal sum.

• Drop DC: consider the “extensive” utilitarian rule, defined in C1.• Drop SI∗N : consider the Roemer rule.• Drop SIM : adapt the Van de gaer rule using weights (∈ N0) when leximinning

(the sums).15


First, consider a social ordering R satisfying PD and Irnrf . Define a social ordering

R◦ such that for all u, v ∈ Rm×n++ : ln u R◦ ln v ⇔ u Rv, with ln x equal to the

matrix[ln x j

i

]for x = u, v. It is easy to see that R◦ has to satisfy UD and Itn

tf

Second, if we impose one of the axioms SP, DC, SI∗M , SIM , SI∗N or SIN on R, thenthe same axiom must hold for R◦. Finally, we can use Proposition 1 to characterizeR◦: depending on whether we additionally impose the set SP, DC, SI∗M , SIN or theset SP, DC, SIM , SI∗N , we obtain that R◦ must be the Roemer rule (resp. the Vande gaer rule). By definition of both rules and by definition of R◦, we get that Rmust be the Roemer rule (resp. the Van de gaer rule), but applied to logarithmicallytransformed matrices; this is equal to the product-Roemer rule (resp. product-Vande gaer rule) applied to the original matrices u, v ∈ R

m×n++ . ��


We prove the characterization for the KAS–Roemer family; the other cases areanalogous. All orderings in the family satisfy the axioms PD, Irf

rf , SP, CON, SEM ,SE∗

N , SI∗M , and SIN . The other way around, consider a social ordering R satisfyingthese axioms. Using CON and SE∗

N , this social ordering can be represented bycontinuous functions f and g, such that for all u, v ∈ D:

u Rv ⇔ f(g(u1) , . . . , g

(un)) ≥ f

(g(v1) , . . . , g

(vn)) .

15 As it is rather unusual, weighting the leximin rule with positive integers has to be under-stood as applying the leximin rule to numbers, which are replicated by its weight. For examplecomparing (1, 2, 3) with (2, 1, 3) and using (1, 2, 1) as weighting vector, amounts to comparing(1, 2, 2, 3) and (2, 1, 1, 3) via the leximin rule.


The axioms SP, SE∗N , SI∗M and SP, SEM and SIN require f and g to be strictly

increasing, separable and symmetric. The axiom Irfrf imposes homogeneity for g

and homotheticity for f (because any strictly increasing transformation of f repre-sents the same rule). As a consequence, f is a KAS social welfare function definedover vectors

(g(u1), . . . , g (un)

), with g strictly increasing, separable, symmetric

and homogeneous, as required. ��

Acknowledgements We thank Marc Fleurbaey and two anonymous referees for their commentson an earlier version. We also wish to acknowledge financial support from the InteruniversityAttraction Poles Programme - Belgian Science Policy [Contract No. P5/21].

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