Fairness and Ethics in Insurance Pricing
Arthur Charpentier, François Hu, Agathe Fernandes-Machado & Philipp Ratz
Chaire Generali Data Talk– February 2024

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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.

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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 ECML PKDD 2023 & BIAS 2023, Milano
 AAAI Conference on Artificial Intelligence, Vancouver
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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)
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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%

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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%

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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.
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Motivation (2. Legal Aspects) Au Québec, Charte des droits et libertés de la personne (C-12)– Article 20.1 –Dans un contrat d’assurance ou de rente, un régime d’avantages sociaux, de retraite, de rentes ou d’assurance ou un régime universel
de rentes ou d’assurance, une distinction, exclusion ou préférence
fondée sur l’âge, le sexe ou l’état civil est réputée non discriminatoire
lorsque son utilisation est légitime et que le motif qui la fonde constitue un facteur de détermination de risque, basé sur des données
actuarielles.

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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”
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Motivation (2. Legal Aspects)
 En 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.
Extention de la ”Loi n◦ 72-546 du 1 juillet 1972”, qui supprima l’exigence del’intention spécifique.

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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
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Motivation (4. 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%

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women
8.68%
8.37%
8.33%
8.20%
8.20%

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Discrimination and Insurance”What is unique about insurance is that even statistical discrimination which bydefinition is absent of any malicious intentions, poses significant moral and legal
challenges. Why? Because on the one hand, policy makers would like insurers to treat
their insureds equally, without discriminating based on race, gender, age, or othercharacteristics, even if it makes statistical sense to discriminate (...) 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 to prohibit insurers from taking into account. ” Avraham (2017)”Technology is neither good nor bad; nor is it neutral ” , Kranzberg (1986)”Machine learning won’t give you anything like gender neutrality ‘for free’ that you
didn’t explicitely ask for ”, Kearns and Roth (2019)

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Discrimination and Protected Attributes ?

via The Zebra (2022)
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Fairness for Classifiersx ∈ X ⊂ Rd : ‘explanatory’ variabless ∈ {A, B} : ”sensitive variable”
y ∈ {0, 1} : classification problem
yb ∈ {0, 1} : prediction, classically yb = 1(m(x, s) > t)

Following Barocas et al. (2017), standard definitions are
A model m satisfies the independence property if m(X, S) ⊥⊥ S, with respect to the
distribution P of the triplet (X, S, Y ).
A model satisfies the separation property if m(X, S) ⊥⊥ S | Y , with respect to the
distribution P of the triplet (X, S, Y ).
A model satisfies the sufficiency property if Y ⊥⊥ S | m(X, S), with respect to the
distribution P of the triplet (X, S, Y ).
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Fairness (Demographic Parity) for Classifiers
 Defining ”Demographic Parity”, Corbett-Davies et al. (2017) or Agarwal (2021)
Weak Demographic Parity,b ⊥Decision function yb satisfies weak demographic parity if Y
⊥ S, i.e.
b |S = A] = E[Yb |S = B],
E[YorE[1(m(X, S) > t)|S = A] = E[1(m(X, S) > t)|S = B].
One can easily obtain weak Demographic Parity using different thresholds
E[1(m(X, S) > tA )|S = A] = E[1(m(X, S) > tB )|S = B].

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Fairness (Demographic Parity) for Scores

Strong Demographic Parity,
E[1(m(X, S) ∈ E )|S = A] = E[1(m(X, S) ∈ E )|S = B],
for any E ⊂ [0, 1], or PA [E ] = PB [E ],(PA [E ] = P[m(X, S) ∈ E |S = A]
PB [E ] = P[m(X, S) ∈ E |S = B]

 @freakonometrics § freakonometrics

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Fairness (Demographic Parity) for Scores
 Use some ”distance” between PA and PB(TV, KL, or Wasserstein)infπ∈Π(PA ,PB )E[ℓ(X , Y )], (X , Y ) ∼ π

or
Z

inf

π∈Π(PA ,PB )ℓ(x , y )π(dx , dy ) .

or using a transport mapping TZ

inf

T :T# PA =PB

 @freakonometrics § freakonometricsℓ(x , T (x ))dPA (x ) .

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Fairness and Optimal Transport

Monge (1781), Mémoire sur la théorie des déblais et des remblais
We want to transport optimally sand from a hole (with shape −dP
Z A ) to a pile
Z (withshape dPB ). ”Rien ne se perd, rien ne se créé, tout se transporte”:
 @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, Generali Data Talk

dPA =

dPB .

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Fairness and Optimal Transport

Z

inf

T :T# PA =PB

 @freakonometrics § freakonometricsk

(x − T (x )) dPA (x ) . =

Z 1
0

k
F0−1 (u) − F1−1 (u) du

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, Generali Data Talk

1/k

,

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Fairness and Optimal Transport

Z

inf

T :T# PA =PB
 @freakonometrics § freakonometricsk

(x − T (x )) dPA (x ) . =

Z 1
0

k
FA−1 (u) − FB−1 (u) du

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, Generali Data Talk

1/k

,

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Fairness and Optimal Transport

Optimal transport plan is here T ⋆ : x 7→ y = FB−1 ◦ FA (x ) (increasing function)

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Counterfactual Fairness (and Optimal Transport)

Used to quantify unfairness,m satisfies Strong Demographic Parity
if W2 = 0,
W2 =Z 1 
0

2
FA−1 (u) − FB−1 (u) du

1/2

.

(optimal transport for R-valued measures) @freakonometrics § freakonometrics

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Counterfactual Fairness (and Optimal Transport)“Ladder of causation” from Pearl et al. (2009) 3. Counterfactuals(Imagining, “what if I had done...”)
 2. Intervention(Doing, “what if I do...”)
 1. Association(Seeing, “what if I see...”)
Picture source: Pearl and Mackenzie (2018)
What would be the impact of a treatment T
on a variable of interest Y ?

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Counterfactual Fairness (and Optimal Transport)
 Define individual or counterfactural fairness, Castelnovo et al. (2022)”Individual fairness is embodied in the following principle: similar individuals should begiven similar decisions. This principle deals with the comparison of single individuals
rather than focusing on groups of people sharing some characteristics. ”Following Kusner et al. (2017)

A decision is counterfactually fair if the prediction in the real world is the same as
the prediction in the counterfactual world⋆⋆
E[YS←A|X = x] = E[YS←B|X = x], ∀x,⋆⋆where YS←A
and YS←B
denote ”potential outcomes”.

 since we use the same x it is a ceteris paribus counterfactual.
(is the counterfactual of a man with height 190 cm a woman with height 190 cm ?) @freakonometrics § freakonometrics

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Counterfactual Fairness (and Optimal Transport)sxy(a) propensity score



x
s

y

(b) our approach

Charpentier et al. (2023a) defined mutatis mutandis counterfactural fairness,⋆⋆|X = x] = E[YS←B|X = T ⋆ (x)], ∀x.
E[YS←A(probability to get surgery when delivering a baby for Black / non-Black mother)

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Back to Actuarial Justice

See Berk et al. (2021)

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Mitigation with Wasserstein Barycenter

If W2 ̸= 0 can we mitigate discrimination ?

Use of Wasserstein Barycenter
see Charpentier et al. (2023b)

In Euclidean spaces⋆z = argmin
z



( n
X

)
2

ωi d(z, z i ),i=1

For probability measures⋆

P = argmin
Q

( n
X

)

ωi d(Q, Pi )

2

,

i=1





We have defined the risk of a model m ∈ M as R(m) = E ℓ(Y , m(X)) .
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Mitigation with Wasserstein Barycenter
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, Generali Data Talk

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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|x − y |2 dπ(x , y )Thus,
E |m(X, S) − µS (X)|2 S = s ≥ W2 (Pm , Ps )2

 @freakonometrics § freakonometrics

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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






g∈M

m∈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 .

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Mitigation with Wasserstein Barycenter

Given scores m(x, s = A) and m(x, s = B),
the “fair barycenter score” ism⋆ (x, s = A)
= P[S = A] · m(x, s = A)
+ P[S = B] · FB−1 ◦ FA m(x, s = A) @freakonometrics § freakonometrics

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Mitigation with Wasserstein Barycenter

Given scores m(x, s = A) and m(x, s = B),
the “fair barycenter score” ism⋆ (x, s = B)
= P[S = A] · FA−1 ◦ FB m(x, s = B)+ P[S = B] · m(x, s = B)

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Mitigation with Wasserstein Barycenter

If the two models are balanced, m⋆ is also balanced.
Annual claim occurrence (motor insurance, Charpentier et al. (2023b))
Three models (plain GLM, GBM, Random Forest)

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Mitigation with Wasserstein Barycenter Predictions are different for men (= A) and women (S = B) since W2 ̸= 0 consider post processing mitigation

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Mitigation with Wasserstein Barycenter Given scores m(x, s = A) and m(x, s = B), the “fair barycenter score” ism⋆ (x, s = A) = P[S = A] · m(x, s = A) + P[S = B] · FB−1 ◦ FA m(x, s = A) @freakonometrics § freakonometrics

freakonometrics.hypotheses.org – Arthur Charpentier, 2024, Generali Data Talk

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Mitigation with Wasserstein Barycenter Given scores m(x, s = A) and m(x, s = B), the “fair barycenter score” ism⋆ (x, s = B) = P[S = A] · FA−1 ◦ FB m(x, s = B) + P[S = B] · m(x, s = B)

 @freakonometrics § freakonometrics

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Mitigation with Wasserstein Barycenter We can plot (m(x i , A), m⋆ (x i , A) and (m(x i , B), m⋆ (x i , B)

 @freakonometrics § freakonometrics

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Mitigation with Wasserstein Barycenter Numerical values, for initial occurrence probability of 5%, 10% and 20%, we havem(x) = 5%m(x) = 10%m(x) = 20%

×0.94
4.73%
9.46%
18.91%A (men)
GLM
GBM
4.94%
4.80%
9.83%
9.66%
19.50% 18.68%

RF
4.42%
8.92%
18.26%

×1.11
5.56%
11.12%
22.25%

B (women)
GLM
GBM
5.16%
5.25%
10.38% 10.49%
20.77% 21.63%

RF
6.15%
12.80%
21.12%

 Recent work on the use of Wasserstein Barycenter, in Charpentier et al. (2023b) and
Hu et al. (2023a,b,c), and optimal transport for counterfactual fairness in Charpentier
et al. (2023a).

 @freakonometrics § freakonometrics

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Motivation (5. Source(s) of discrimination - biases everywhere...)claims database underwriters biases• commercial discounts• inferred data
• multiple decisions

id

policyunderwriting database
policysensitive claims biases• fraud detection• sexist mechanic• ageist manager

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Mitigation ? (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 no otherway. And in order to treat some persons equally, we must treat them differently.”
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.

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Mitigation ? (brief conclusion)

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References
Agarwal, S. (2021). Trade-offs between fairness and interpretability in machine learning. In IJCAI 2021
Workshop on AI for Social Good.
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.
Berk, R. A., Kuchibhotla, A. K., and Tchetgen, E. T. (2021). Improving fairness in criminal justicealgorithmic risk assessments using optimal transport and conformal prediction sets. arXiv,
2111.09211.
Bénéplanc, G., Charpentier, A., and Thourot, P. (2022). Manuel d’assurance. Presses Universitaires deFrance.
Castelnovo, A., Crupi, R., Greco, G., Regoli, D., Penco, I. G., and Cosentini, A. C. (2022). Aclarification of the nuances in the fairness metrics landscape. Scientific Reports, 12(1):1–21.
Charpentier, A. (2014). Computational actuarial science with R. CRC press.
Charpentier, A. (2024). Insurance: biases, discrimination and fairness. Springer Verlag.
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References
Charpentier, A., Flachaire, E., and Gallic, E. (2023a). Causal inference with optimal transport. In
Thach, N. N., Kreinovich, V., Ha, D. T., and Trung, N. D., editors, Optimal Transport Statistics for
Economics and Related Topics. Springer Verlag.
Charpentier, A., Hu, F., and Ratz, P. (2023b). Mitigating discrimination in insurance with wassersteinbarycenters. BIAS, 3rd Workshop on Bias and Fairness in AI, International Workshop of ECML
PKDD.
Corbett-Davies, S., Pierson, E., Feller, A., Goel, S., and Huq, A. (2017). Algorithmic decision making
and the cost of fairness. arXiv, 1701.08230.
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.
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.
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References
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.
Gouic, T. L., Loubes, J.-M., and Rigollet, P. (2020). Projection to fairness in statistical learning.
arXiv, 2005.11720.
Hu, F., Ratz, P., and Charpentier, A. (2023a). Fairness in multi-task learning via wassersteinbarycenters. Joint European Conference on Machine Learning and Knowledge Discovery in
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