ALGORITHMIC FAIRNESS AND DISCRIMINATIONAN APPLICATION IN INSURANCE
Arthur Charpentier, François Hu, Agathe Fernandes-Machado & Philipp Ratz
CRI2 GS, June 2024

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Bio (short)
Arthur Charpentier Professor at Université du Québec à Montréal,  Denuit and Charpentier (2004, 2005) Mathématiques de l’Assurance Non-Vie, Charpentier (2014) Computational Actuarial Science with R, Bénéplanc et al. (2022) Manuel d’Assurance, Charpentier (2024) Insurance: Biases, Discrimination and Fairness.

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Bio (short)
François Hu Postdoctoral fellow, Université de MontréalPhilipp Ratz PhD Student, Université du Québec à Montréal
Agathe Fernandes-Machado PhD Student, Université du Québec à Montréal
Olivier Côté PhD Student, Université Laval, Québec

Also interns Ana Patrón Piñerez and Suzie Grondin,  @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

3 / 49Insurance (and “Actuarial Fairness”)
 Insurance is a risk transfer (from a policyholder to an insurance company)

premium

policyholder

insurer

indemnity

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

4 / 49Insurance (and “Actuarial Fairness”)
 “Insurance is the contribution of the many to the misfortune of the few”

premium

policyholder

insurer

indemnity

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

5 / 49Insurance (and “Actuarial Fairness”)
 “Insurance is the contribution of the many to the misfortune of the few”

premium

policyholder

insurer

indemnity

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

6 / 49Motivation (1. Propublica, Actuarial Justice) Concept of “actuarial justice”
as coined in Feeley and Simon (1994)
 Correctional Offender Management
Profiling for Alternative Sanctions(COMPAS), Perry (2013)§ https://github.com/propublica/compas-analysis

 Angwin et al. (2016) Machine Bias
Dressel and Farid (2018)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

7 / 49Motivation (1. Propublica, Actuarial Justice) From Feller et al. (2016),▶ for White people, among those who
did not re-offend, 22% were wrongly
classified,▶ for Black people, among those who
did not re-offend, 42% were wrongly
classified,▶ problem, since 42% ≫ 22%

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

7 / 49Motivation (1. Propublica, Actuarial Justice) From Dieterich et al. (2016),▶ for White people, among those who
where classified as high risk, 40% did
not re-offend,▶ for Black people, among those who
where classified as high risk, 35% did
not re-offend,▶ no problem, since 40% ≈ 35%

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

7 / 49Motivation (2. Legal Aspects)
 EU Directive (2004/113/EC), 2004 version– Article 5 (Actuarial factors) –1. Member States shall ensure that in all new contracts concluded
after 21 December 2007 at the latest, the use of sex as a factor in thecalculation of premiums and benefits for the purposes of insurance andrelated financial services shall not result in differences in individuals’
premiums and benefits.
2. Notwithstanding paragraph 1, Member States may decide before 21
December 2007 to permit proportionate differences in individuals’ premiums
and benefits where the use of sex is a determining factor in the assessment ofrisk based on relevant and accurate actuarial and statistical data. The
Member States concerned shall inform the Commission and ensure that
accurate data relevant to the use of sex as a determining actuarial factor arecompiled, published and regularly updated.
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

8 / 49Motivation (2. Legal Aspects) Au Québec, Charte des droits et libertés de la personne (C-12)– Article 20.1 –In an insurance or pension contract, a social benefits plan, a retirement, pension or insurance plan, or a public pension or public insurance plan, a distinction, exclusion or preference based on age, sex
or civil status is deemed non-discriminatory where the use thereof is
warranted and the basis therefor is a risk determination factor basedon actuarial data

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

9 / 49Motivation (2. Legal Aspects)
 September 27, 2023, the Colorado Division of Insurance exposed a new proposed regulation entitled Concerning Quantitative Testing of External Consumer Data and Information Sources, Algorithms, and Predictive Models Usedfor Life Insurance Underwriting for Unfairly Discriminatory Outcomes– Section 5 (Estimating Race and Ethnicity) –Insurers shall estimate the race or ethnicity of all proposed insureds
that have applied for coverage on or after the insurer’s initial adoptionof the use of ECDIS, or algorithms and predictive models that use
ECDIS, including a third party acting on behalf of the insurer thatused ECDIS, or algorithms and predictive models that used ECDIS,
in the underwriting decision-making process, by utilizing: BIFSG and
the insureds’ or proposed insureds’ name and geolocation (...)
 Bayesian Improved First Name Surname Geocoding, or “BIFSG”
 External Consumer Data and Information Source, or “ECDIS”
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Motivation (2. Legal Aspects)
 EU Directive (2010/41/EU), 2010 version (on the application of the principleof equal treatment between men and women)– Article 3 (Definition) –(a) ‘direct discrimination’: where one person is treated less favourably
on grounds of sex than another is, has been or would be, treated in acomparable situation;(b) ‘indirect discrimination’: where an apparently neutral provision,
criterion or practice would put persons of one sex at a particular disadvantage compared with persons of the other sex, unless that provision,
criterion or practice is objectively justified by a legitimate aim, and themeans of achieving that aim are appropriate and necessary;

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Motivation (2. Legal Aspects)
 In France, Loi n◦ 2008-496 du 27 mai 2008
– Article 1 –Constitue une discrimination indirecte une disposition, un critère ou
une pratique neutre en apparence, mais susceptible d’entraı̂ner, pourl’un des motifs mentionnés au premier alinéa, un désavantage particulier pour des personnes par rapport à d’autres personnes, à moinsque cette disposition, ce critère ou cette pratique ne soit objectivement
justifié par un but légitime et que les moyens pour réaliser ce but nesoient nécessaires et appropriés.
Extension of ”Loi n◦ 72-546 du 1 juillet 1972”, which removed the requirement
for specific intent.

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Motivation (3. Redlining)(Fictitious maps, inspired by a Home Owners’ Loan Corporation map from 1937)▶ Federal Home Loan Bank Board (FHLBB) ”residential security maps” (forreal-estate investments), Crossney (2016) and Rhynhart (2020)▶ Unsanitary index and proportion of Black inhabitants▶ Discrimination as an “ill-posed problem”?
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Motivation (4. Competition)▶ Québec Province, life table can be gender based▶ Ontario Province, life table must be unisex
Age
x
0
10
20
30
40
50
60
70
80
90
100

men
Lx
10 px
100,000
1%
99,373
0%99,147
1%
98,5891%
97,7272%
96,194
4%
92,482 10%
83,189 25%
62,635 59%
25,452 94%
1,473

 @freakonometrics § freakonometrics

unisexLx
10 px
100,000
1%
99,4510%
99,2690%98,8691%
98,203
1%
97,242
2%
95,115 10%85,891 21%
67,649 53%
31,908 90%
3,142

women
Lx
10 px
100,000
0%
99,5280%99,391
0%
99,1480%
98,6790%98,2901%
97,7489%88,593 18%
72,662 47%
38,364 87%
4,811

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Motivation (5. Proxies)
 On a French motor dataset, average claim
frequencies are 8.94% (men) 8.20% (women).
 Consider some logistic regression to estimate
annual claim frequency, on k explanatory variables excluding gender.
k=0
k=2
k=8
k = 15
empirical

men
8.68%
8.85%
8.87%
8.94%
8.94%

women
8.68%
8.37%
8.33%
8.20%
8.20%

 Models simply tend to reproduce what wasobserved in the data (see “is-ought” problem,in Hume (1739)).
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

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Discrimination and Insurance“Machine learning won’t give you anything like gender neutrality ‘for free’ thatyou didn’t explicitly ask for,” Kearns and Roth (2019)”What is unique about insurance is that even statistical discrimination which
by definition is absent of any malicious intentions, poses significant moral andlegal challenges. Why? Because on the one hand, policy makers would likeinsurers to treat their insureds equally, without discriminating based on race,
gender, age, or other characteristics, even if it makes statistical sense todiscriminate (...) On the other hand, at the core of insurance business lies
discrimination between risky and non-risky insureds. But riskiness often
statistically correlates with the same characteristics policy makers would like toprohibit insurers from taking into account. ” Avraham (2017)“Technology is neither good nor bad; nor is it neutral,” Kranzberg (1986)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

16 / 49

Classifiers (or why actuarial science ̸= computer science)
Classifiers on pictures,→  (cats) –(dogs)→ ! (healthy) – " (sick)
model
!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!







!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!

 @freakonometrics § freakonometrics

!
!
"
!
!
!
"
!

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

17 / 49

Classifiers (or why actuarial science ̸= computer science)
Classifiers on pictures,→  (cats) –(dogs)→ ! (healthy) – " (sick)
model
!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!




!!!

?





!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!

!
!
"
!
!
!
"
!

!!! ?

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

17 / 49

Classifiers (or why actuarial science ̸= computer science)
Classifiers on pictures,→  (cats) –(dogs)→ ! (healthy) – " (sick)
model
!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!


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!!!

?



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!!!
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!
!
"
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!
!
"
!

!!! ?

 @freakonometrics § freakonometrics

Classifiers, we need some “probabilities”→ ☼ (sunny) – Ê (rainy)→ ♀ (woman) – ♂ (man)→ ñ (no claim) – \ (accident)
!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!

☼
☼
Ê
☼
Ê
Ê
☼
Ê

!!! ?

!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!

♂
♀
♀
♂
♀
♂
♂
♀

!!! ?

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!

ñ
ñ
\
ñ
ñ
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\
ñ

!!! ?
17 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)score ∈ [0, 1] ⊂ R @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)score ∈ [0, 1] ⊂ R @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)score ∈ [0, 1] ⊂ R @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)

Following Barocas et al. (2017), standard definitions arescore ∈ [0, 1] ⊂ R

A model m satisfies the independence property if m(X, S) ⊥⊥ S, with respect to the
distribution P of the triplet (X, S, Y )← demographic parity

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)

Following Barocas et al. (2017), standard definitions arescore ∈ [0, 1] ⊂ R

A model m satisfies the independence property if m(X, S) ⊥⊥ S, with respect to the
distribution P of the triplet (X, S, Y )← demographic parity
A model satisfies the separation property if m(X, S) ⊥⊥ S | Y , with respect to the
distribution P of the triplet (X, S, Y )← equalized odds

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
class ∈ {0, 1}
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1( m(x, s) > t)

Following Barocas et al. (2017), standard definitions arescore ∈ [0, 1] ⊂ R

A model m satisfies the independence property if m(X, S) ⊥⊥ S, with respect to the
distribution P of the triplet (X, S, Y )← demographic parity
A model satisfies the separation property if m(X, S) ⊥⊥ S | Y , with respect to the
distribution P of the triplet (X, S, Y )← equalized odds
A model satisfies the sufficiency property if Y ⊥⊥ S | m(X, S), with respect to the
distribution P of the triplet (X, S, Y )← calibration
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers

classical definition of “demographic parity” for a classifier
sensitive
sensitive?

b = 1 | S = A ] = P[ Yb =1|S =B]
P[ Yclass prediction?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ] @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers

classical definition of “demographic parity” for a classifier
sensitive
sensitive?

b = 1 | S = A ] = P[ Yb =1|S =B]
P[ Yclass prediction?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ] @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers

classical definition of “demographic parity” for a classifier
sensitive
sensitive?

b = 1 | S = A ] = P[ Yb =1|S =B]
P[ Yclass prediction?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ] @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers

classical definition of “demographic parity” for a classifier
sensitive
sensitive
b
E[ Y

?

b
| S = A ] = E[ Yclass prediction|S =B]?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ] @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers(weak) definition of “demographic parity” for a classifier
sensitive
sensitive?

E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]
score

?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ] @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers(weak) definition of “demographic parity” for a classifier
sensitive
sensitive?

E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]score(strong) definition of “demographic parity” for a classifier?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ]∀I ⊂ [0, 1],

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers(weak) definition of “demographic parity” for a classifier
sensitive
sensitive?

E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]score(strong) definition of “demographic parity” for a classifier?

P[ m(X, S) ∈ I | S = A ] = P[ m(X, S) ∈ I | S = B ]∀I ⊂ [0, 1], e.g. [40%; 60%].

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

18 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

let FA denote the cumulative distribution functionof scores in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
score

sensitive

if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
sensitive
levels of the score
if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score

score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
quantile
if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
quantile
probabilities associated to the score
if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
then, in group B

quantile

blah blah

if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Fairness for Classifiers using Optimal Transport
score

sensitive









FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
then, in group B

quantile

if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

sensitive
19 / 49

Fairness for Classifiers using Optimal Transport
score








FA (u) = P m(X, S) ≤ u S = A
FB (u) = P m(X, S) ≤ u S = B

and FB denote the cumulative distribution functionof scores in group B
Consider individuals in group A such thatm(X, S) ∈ [0.4; 0.6] | S = A then ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = A
then, in group B

optimalquantile
transport mapping T ⋆if ranks(m(X, S)) ∈ [66.3%; 91.3%] | S = B then m(X, S) ∈ [0.743; 0.861] | S = B
score
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

19 / 49

Formalizing Optimal Transport
Consider the following [0, 1] → [0, 1] mapping
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x )probability p associated with x in group A
⋆

T =

Z 1

argminT (x ) − x2

T :[0,1]→[0,1] 0

dFA (x )Y with Y ∼ FB

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

20 / 49

Formalizing Optimal Transport
Consider the following [0, 1] → [0, 1] mapping
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x )probability p associated with score x in group A
⋆

T =

Z 1

argminT (x ) − x2

T :[0,1]→[0,1] 0

dFA (x )Y with Y ∼ FB

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

20 / 49

Formalizing Optimal Transport
Consider the following [0, 1] → [0, 1] mapping
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x )probability p associated with x in group A
⋆

T =

Z 1

argminT (x ) − x2

T :[0,1]→[0,1] 0

dFA (x )Y with Y ∼ FB

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

20 / 49

Fairness for ClassifiersConsider the following [0, 1] → [0, 1] mapping
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x )probability p associated with x in group A
⋆

T =

Z 1

argminT (x ) − x2

T :[0,1]→[0,1] 0

dFA (x )Y with Y ∼ FB

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

20 / 49

Formalizing Optimal Transport
Consider the following [0, 1] → [0, 1] mapping
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x )probability p associated with x in group A
Z 1

⋆

T =

argminT (x ) − x2

T :[0,1]→[0,1] 0

i.e.



argmin E

T (X ) − X

2 dFA (x )where X ∼ FA ,T :[0,1]→[0,1]Y with Y ∼ FB
corresponding to Monge (1781) problem,
revisited by Kantorovich (1942).
(the minimum value is called Wasserstein distance)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

20 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

optimal transport mapping

quantile of level p in group B

T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

21 / 49

Optimal Transport with a Finite Sample (another interpretation)
m1B ≤ m2B ≤ · · · ≤ mnB
m1A ≤ m2A ≤ · · · ≤ mnA


Consider two samples, m(x i , si = A ) and m(x i , si = B )m1A ≤ m2A ≤· · ·≤ mnA and m1B ≤ m2B ≤· · ·≤ mnB

m is not fair with respect to s if T ⋆ (x ) ̸= x , or miA ̸= miB
optimal transport mapping
quantile of level p in group B
T ⋆ (x ) = FB

−1

◦ FA (x ) ̸= x

probability p associated with u in group A @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

22 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

pile of sandhole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
Monge (1781), “Mémoire sur la théorie des déblais et des remblais ”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

“Optimal Transport” (a side note / a cultural interlude)

hole / excavation site
T⋆
A
This “monotone” (increasing) mapping is optimal xi
yiB déblais remblais

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

23 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
For individual i, why not11mi⋆ = miA + miB
22m⋆ (x , A) =P[S = A]

1 A
1 ⋆ A
m (x ) +T (m (x ))22P[S = B]

 @freakonometrics § freakonometrics

associated score in group B

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
For individual i, why not11mi⋆ = miA + miB
2
2
corresponding tom⋆ (x , A) =P[S = A]

1 A
1 ⋆ A
m (x ) +T (m (x ))22P[S = B]

 @freakonometrics § freakonometrics

associated score in group B

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
For individual i, why not11mi⋆ = miA + miB
2
2
corresponding tom⋆ (x , A) =P[S = A]

1 A
1 ⋆ A
m (x ) +T (m (x ))22P[S = B]

 @freakonometrics § freakonometrics

associated score in group B

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
other “averages” could be considered @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
other “averages” could be consideredthat one (“Wasserstein barycenter”)
is actually optimal in terms of(empirical) risk

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycenters
Mitigation is about finding some m⋆“in-between” (Demographic Parity)
other “averages” could be consideredthat one (“Wasserstein barycenter”)
is actually optimal in terms of(empirical) risk
Given a model m (regression, boosting,random forest, neural nets, etc) we caneasily derive a “fair model”

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

24 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A
P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A) 4
weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A

p = FA m(x, s = A)

P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A)weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A

p = FA m(x, s = A)

P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A)weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A

p = FA m(x, s = A)

P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A)weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A

p = FA m(x, s = A)

P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A)weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

score in group A

p = FA m(x, s = A)

P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A)weightsquantile score in group B associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigating Discrimination with Wasserstein Barycentersm⋆ (x, s = A) = P[S = A] · m(x, s = A)






+ P[S = B] · FB−1 ◦ FA m(x, s = A)m⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B)





+ P[S = B] · m(x, s = B).

p = FB m(x, s = B)score in group B

P[S = A] · FA−1 ◦ FB m(x, s = A)

+ P[S = B] · m(x, s = B)

quantile score in group A associated with probability p

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

25 / 49

Mitigation with Wasserstein Barycenter

We have defined the risk of a model m ∈ M as R(m) = E ℓ(Y , m(X)) .
Define the classes of fair models,(

MDP = m ∈ M s.t. m(X) ⊥⊥ S

MEO = m ∈ M s.t. m(X) ⊥⊥ S | Y

Fairness is achieved by projection onto a fair subspaceb n (m)b fair ∈ argmin Rmm∈Mfair

Given a risk R, a class M and the fair-subclass Mfair , the price of fairness
Efair (M) = @freakonometrics § freakonometrics

minm∈Mfair

R(m) − min R(m) .
m∈Mfreakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

26 / 49

Mitigation with Wasserstein Barycenter
Recall that Bayes estimator is the best model, for the ℓ2 loss,(µ(x) = E Y |X = x and setµA (x) = E Y |X = x, S = AµB (x) = E Y |X = x, S = B

From the definition of Wasserstein distance,Z

W2 (p, q) =infπ∈Π(p,q)!1/2
2|x − y | dπ(x , y )Thus,
E |m(X, S) − µS (X)|2 S = s ≥ W2 (Pm , Ps )2

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

27 / 49

Mitigation with Wasserstein Barycenter
Price of fairness and Wasserstein Barycenter
Efair (M) =minm∈Mfair R(m) − min R(m) ≥ min E W2 (PS , PS,g )2





m∈M



g∈M

where PS is the condition distribution of µ(X, S), given S, and PS,g is the condition
distribution of g(X, S), given S. Moreover, if Mfair = MDP , and if Ps is absolutely
continuous (w.r.t. Lebesgue measure), 

EDP (M) = min E W2 (PS , PS,g )2
g∈M



= min

g∈M

nX

P[S = s] · W2 (Ps , Ps,g )2os

See Gouic et al. (2020) for a complete proof.
We recognize on the right the barycenter, with weights P[S = s] and distance W2 .
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

28 / 49

Back to the COMPAS ExampleS : race (binary), black & whiteY : re-offense (binary), no & yesYb : classifier (risk category), low & high(standard) demographic parity would be translated assensitive

sensitive?

b = high | S = black ] = 58% = P[ Yb = high | S = white ] = 33%,
P[ Y

predictions

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

29 / 49

Back to the COMPAS Example, from Discrimination to CalibrationS : race (binary), black & whiteY : re-offense (binary), no & yesYb : classifier (risk category), low & highsensitive

sensitive?

b = high|Y = no , S = black ] = 42% = P[ Yb = high|Y = no , S = white ] = 22%,P[ Yfalse positive rate?

b = high , S = black ] = 35% = P[ Y = no|Yb = high , S = white ] = 40%.
P[ Y = no|Y

false discovery rate
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

29 / 49

From Discrimination to Calibration
sensitive

sensitive?

demographic parity → E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]scoresensitive

sensitive?

equalized odds → E[ m(X, S) | Y = y , S = A ] = E[ m(X, S) | Y = y , S = B ], ∀yscore

sensitive

sensitive?

calibration → E[ Y | m(X, S) = u , S = A ] = E[ Y | m(X, S) = u , S = B ], ∀uscore @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

30 / 49

From Discrimination to Calibration
sensitive

sensitive?

demographic parity → E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]scoresensitive

sensitive?

equalized odds → E[ m(X, S) | Y = y , S = A ] = E[ m(X, S) | Y = y , S = B ], ∀yscore

sensitive

sensitive?

calibration → E[ Y | m(X, S) = u , S = A ] = E[ Y | m(X, S) = u , S = B ], ∀uscore @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

31 / 49

From Discrimination to Calibration
sensitive

sensitive?

demographic parity → E[ m(X, S) | S = A ] = E[ m(X, S) | S = B ]scoresensitive

sensitive?

equalized odds → E[ m(X, S) | Y = y , S = A ] = E[ m(X, S) | Y = y , S = B ], ∀yscore

sensitive

sensitive?

calibration → E[ Y | m(X, S) = u , S = A ] = E[ Y | m(X, S) = u , S = B ], ∀uscore @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

32 / 49

From Discrimination to Calibration (an Epistemological Detour)

Property E[ Y | m(X, S) = u ] = u, ∀u ∈ [0, 1] corresponds to “calibration”.
“Out of all the times you said there was a 40 percent chance of rain, how often
did rain actually occur? If, over the long run, it really did rain about 40 percentof the time, that means your forecasts were well calibrated,” Silver (2012)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

33 / 49

From Discrimination to Calibration (an Epistemological Detour)
As explained in Van Calster et al. (2019), ”among patientswith an estimated risk of 20%, we expect 20 in 100 tohave or to develop the event,”▶ If 40 out of 100 in this group are found to have thedisease, the risk is underestimated▶ If we observe that in this group, 10 out of 100 have thedisease, we have overestimated the risk.
Most machine learning models can be poorly calibrated, Denuit et al. (2021), Machado et al. (2024).
(picture source: Van Calster et al. (2019))

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

34 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration
Designed to Deceive: Do These People Look Real to You?
by Kashmir Hill and Jeremy White Nov. 21, 2020,

Use of GAN (Generative adversarial network) to generate fakepictures (StyleGAN2 package, implemented in TensorFlow)!!!!!!!!!!

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration

female (0.984)
male (0.016)

female (0.009)
male (0.991)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

From Discrimination to Calibration

S = female
male (0.016)

female (0.009)
male (0.991)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

S = male

35 / 49

From Discrimination to Calibration

female (0.984)
male (0.016)

female (0.983)
male (0.017)

female (0.982)
male (0.018)

female (0.960)
male (0.040)

female (0.009)
male (0.991)

female (0.013)
male (0.987)

female (0.014)
male (0.986)

female (0.015)
male (0.985)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

35 / 49

Individual Fairness
We have counterfactual fairness if “had the protected attributes (e.g., race) of theindividual been different, other things being equal, the decision would haveremained the same,” Kusner et al. (2017)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

36 / 49

Individual Fairness
We have counterfactual fairness if “had the protected attributes (e.g., race) of theindividual been different, other things being equal, the decision would haveremained the same,” Kusner et al. (2017)“Ladder of causation” from Pearlet al. (2009), Pearl and Mackenzie(2018)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

36 / 49

Individual Fairness
We have counterfactual fairness if “had the protected attributes (e.g., race) of theindividual been different, other things being equal, the decision would haveremained the same,” Kusner et al. (2017)“Ladder of causation” from Pearlet al. (2009), Pearl and Mackenzie(2018)

 1. Association(Seeing, “what if I see...”)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

36 / 49

Individual Fairness
We have counterfactual fairness if “had the protected attributes (e.g., race) of theindividual been different, other things being equal, the decision would haveremained the same,” Kusner et al. (2017)“Ladder of causation” from Pearlet al. (2009), Pearl and Mackenzie(2018)

 2. Intervention(Doing, “what if I do...”)
 1. Association(Seeing, “what if I see...”)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

36 / 49

Individual Fairness
We have counterfactual fairness if “had the protected attributes (e.g., race) of theindividual been different, other things being equal, the decision would haveremained the same,” Kusner et al. (2017)“Ladder of causation” from Pearlet al. (2009), Pearl and Mackenzie(2018)
 3. Counterfactuals(Imagining, “what if I had done...”)
 2. Intervention(Doing, “what if I do...”)
 1. Association(Seeing, “what if I see...”)
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

36 / 49

What About Interpretation ?
“Humans think in stories rather than facts, numbers or equations - and thesimpler the story, the better,” Harari (2018)
For Glenn (2000), insurer’s risk selection process has two sides: the one presented to regulators and policyholders (numbers, statistics andobjectivity),
 the other presented to underwriters (stories, character and subjective judgment).
The rhetoric of insurance exclusion – numbers, objectivity and statistics – forms what
Brian Glenn calls “the myth of the actuary,” “a powerful rhetorical situation inwhich decisions appear to be based on objectively determined criteria when they
are also largely based on subjective ones” or “the subjective nature of aseemingly objective process”.

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

37 / 49

What About Interpretation ?
“The fact that the selection of risk factors is subjective and contingent uponnarratives of risk and responsibility has in the past played a far larger role
than whether or not someone with a wood stove is charged higher premiums.”
Going further, “virtually every aspect of the insurance industry is predicated onstories first and then numbers,” Glenn (2003)

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

38 / 49

The Case of Multiple Attributes▶ Consider a machine Learning model m, score predictions and two sensitive
attributes, ethnic origin A1 (White/Black) and gender A2 (Male/Female).
▶ Consider densities of νm|A1 =0 , νm|A1 =1 (left) and νm|A2 =0 , νm|A2 =1 (right)▶ Plot densities of barycenters, νmB and νmB
12

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

39 / 49

The Case of Multiple Attributes▶ Intersectional Fairness, MSA → Single sensitive attribute (SSA), by intersection,
ethnic origin A1
gender A2
a ∈ A = A1 × A2 = {white, black} × {male, female}
Here A corresponds to 4 = 2 × 2 states,n

A = ( white , male ), ( white , female ), ( black , male ), ( black , female )▶ Sequential Fairness, MSA, in Hu et al. (2024) @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

40 / 49oThe Case of Multiple Attributes▶ Given νmB , consider
1▶ the barycenter νmB conditional on A1 (no impact, already fair)1▶ the barycenter νmB conditional on A2
2▶ On the right, distribution of νmB ◦mB2

 @freakonometrics § freakonometrics

1

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

41 / 49

The Case of Multiple Attributes▶ Given νmB , consider2▶ the barycenter νmB conditional on A1
1▶ the barycenter νmB conditional on A2 (no impact, already fair)2▶ On the left, distribution of νmB ◦mB
1

 @freakonometrics § freakonometrics

2

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

42 / 49

The Case of Multiple Attributes▶ The order of this sequential approach leads different interpretations,▶ left hand part, A2 then A1▶ right hand part, A1 then A2

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

43 / 49

Mitigating Discrimination ? (brief conclusion)
If it is mandatory to mitigate, there are robust techniques that can guarantee fairness
Supreme Court Justice Harry Blackmun stated, in 1978,“In order to get beyond racism, we must first take account of race. There is noother way. And in order to treat some persons equally, we must treat themdifferently,” Knowlton (1978), cited in Lippert-Rasmussen (2020)
In 2007, John G. Roberts of the U.S. Supreme Court submits“The way to stop discrimination on the basis of race is to stop
discriminating on the basis of race,” Sabbagh (2007) and Turner(2015)
To go further,
Charpentier (2024) Insurance: Biases, Discrimination and Fairness. [ @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

44 / 49

References
Angwin, J., Larson, J., Mattu, S., and Kirchner, L. (2016). Machine bias: There’s software used acrossthe country to predict future criminals and it’s biased against blacks. ProPublica, May 23.
Avraham, R. (2017). Discrimination and insurance. In Lippert-Rasmussen, K., editor, Handbook of the
Ethics of Discrimination, pages 335–347. Routledge.
Barocas, S., Hardt, M., and Narayanan, A. (2017). Fairness in machine learning. Nips tutorial, 1:2017.
Bénéplanc, G., Charpentier, A., and Thourot, P. (2022). Manuel d’assurance. Presses Universitaires deFrance.
Charpentier, A. (2014). Computational actuarial science with R. CRC press.
Charpentier, A. (2024). Insurance: biases, discrimination and fairness. Springer Verlag.
Crossney, K. B. (2016). Redlining. https://philadelphiaencyclopedia.org/essays/redlining/.
Denuit, M. and Charpentier, A. (2004). Mathématiques de l’assurance non-vie: Tome I Principes
fondamentaux de théorie du risque. Economica.
Denuit, M. and Charpentier, A. (2005). Mathématiques de l’assurance non-vie: Tome II Tarification etprovisionnement. Economica.
Denuit, M., Charpentier, A., and Trufin, J. (2021). Autocalibration and tweedie-dominance for
insurance pricing with machine learning. Insurance: Mathematics & Economics.
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

45 / 49

References
Dieterich, W., Mendoza, C., and Brennan, T. (2016). Compas risk scales: Demonstrating accuracy
equity and predictive parity. Northpointe Inc, 7(7.4):1.
Dressel, J. and Farid, H. (2018). The accuracy, fairness, and limits of predicting recidivism. Scienceadvances, 4(1):eaao5580.
Feeley, M. and Simon, J. (1994). Actuarial justice: The emerging new criminal law. The futures ofcriminology, 173:174.
Feller, A., Pierson, E., Corbett-Davies, S., and Goel, S. (2016). A computer program used for bail andsentencing decisions was labeled biased against blacks. it’s actually not that clear. The Washington
Post, October 17.
Glenn, B. J. (2000). The shifting rhetoric of insurance denial. Law and Society Review, pages 779–808.
Glenn, B. J. (2003). Postmodernism: the basis of insurance. Risk Management and Insurance Review,
6(2):131–143.
Gouic, T. L., Loubes, J.-M., and Rigollet, P. (2020). Projection to fairness in statistical learning.
arXiv, 2005.11720.
Harari, Y. N. (2018). 21 Lessons for the 21st Century. Random House.

 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, CRI2 GS

46 / 49

References
Hu, F., Ratz, P., and Charpentier, A. (2023). Fairness in multi-task learning via wassersteinbarycenters. Joint European Conference on Machine Learning and Knowledge Discovery in
Databases – ECML PKDD.
Hu, F., Ratz, P., and Charpentier, A. (2024). A sequentially fair mechanism for multiple sensitive
attributes. Annual AAAI Conference on Artificial Intelligence.
Hume, D. (1739). A Treatise of Human Nature. Cambridge University Press Archive.
Kantorovich, L. V. (1942). On the translocation of masses. In Doklady Akademii Nauk USSR,
volume 37, pages 199–201.
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