Fairness and discrimination in actuarial pricing
Arthur Charpentier1 , Laurence Barry2 , Vincent Grari3 ,Lamprier Sylvain3 , Detyniecki Marcin4
1 Univeristé du Québec à Montréal

2 Chaire Pari

3 Sorbonne Université

4 AXA

Séminaire de la chaire DIALog (Digital Insurance And Long-term risks) 2022

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AgendaI Barry and Charpentier (2022) The Fairness of Machine Learning in Insurance:
New Rags for an Old Man?, ArXiv:2205.08112 �
I Charpentier (2022) Insurance: Discrimination, Biases and Fairness, Institut Louis
Bachelier �
I Grari et al. (2022) A fair pricing model via adversarial learning, ArXiv:2202.12008
��

Ethics, fairness and discrimination
Protected Attributes ?
Big Data and Proxies
Big/Small Data and Possible Pitfalls@freakonometrics

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Group fairness
From correlation to causality
Counterfactual and optimal transport
Adversarial Approach2 / 75Ethics, Fairness and Discrimination I@freakonometrics

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3 / 75Ethics, Fairness and Discrimination II@freakonometrics

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4 / 75Ethics, Fairness and Discrimination III@freakonometrics

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5 / 75Ethics, Fairness and Discrimination IV
I “at the core of insurance business lies discrimination between risky and non-risky
insureds”, Avraham (2017)
I “Technology is neither good nor bad; nor is it neutral ” , Kranzberg (1986)
I “Machine learning won’t give you anything like gender neutrality ‘for free’ thatyou didn’t explicitely ask for ", Kearns and Roth (2019)
It is a complex problem...
I Accuracy : π(x) = EP [Y |X = x] (P historical probability) (is)I Fairness : π ? (x) = EP? [Y |X = x] (P? targeted probability) (ought, Hume (1739))@freakonometrics

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6 / 75Ethics, Fairness and Discrimination V
Word embedding exhibiting gender bias, Bolukbasi et al. (2016)
woman

woman
man

man

queen
homemaker

kingcomputerprogrammer

nurse@freakonometrics

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7 / 75Ethics, Fairness and Discrimination VI
Spatial information and racial bias (redlining)@freakonometrics

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Protected Attributes ?
CA

HI

GA

NC

NY

MA PA

FL

TX

AL

ON NB

NL

QC

x
x
•
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x∗ •
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x
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x
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CA: Californie, HI: Hawaii, GA: Georgia, NC: Caroline du nord, NY: New York, MA:
Massachusetts, PA: Pennsylvanie, FL: Floride, TX: Texas, AL: Alberta, ON: Ontario,
NB: Nouveau-Brunswick, NL: Terre-Neuve-et-Labrador, QC: Québec
GenderAge
Driving experience
Credit history
Education
Occupation
Employment status
Marital status
Housing situation
Address/ZIP code
Insurance history@freakonometrics

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Big Data and Proxies I
More and more features, possibly (strongly) correlated with protected variables
I location (policyholder home adress)
Jean et al. (2016), Seresinhe et al. (2017), Gebru et al. (2017), Law et al. (2019),
Ilic et al. (2019), Kita and Kidziński (2019), see also redlining@freakonometrics

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Big Data and Proxies II
Bohren et al. (2019) on statistical discrimination, or efficient discrimination, as
in Becker (1957) (inspired by Edgeworth(1922) up to Phelps (1972))
Becker (2005) says “if young Moslem Middle
Eastern males were in fact much more likely
to commit terrorism against U.S. than wereother groups, putting them through tighter
security clearance would reduce current airport terrorism”,“racial profiling” is “effective”, even though “such profiling is ‘unfair’ to the many
young male Moslems who are not terrorists, and to the many minority shoppers whoare honest ...”
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Big Data and Proxies III“... That could be made up in part by compensating groups who are forced to go
through more careful airport screening through putting them in shorter security lines,
or in other ways. Similarly, innocent shoppers who are stopped and searched could becompensated for their embarrassment and time”
See also Boczar et al. (2021) (Insurance Data Science Conference in London)
phrenology (Lombroso (1876) and Bertillon and Chervin (1909)) and ugly laws
(TenBroek (1966) and Burgdorf and Burgdorf Jr (1974))

source https://nvlabs-fi-cdn.nvidia.com/stylegan2-ada-pytorch/, cf Karras et al. (2020)@freakonometrics

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Big/Small Data and Possible Pitfalls I
Missing an important covariate is an important issue>
I yi = β0 + x >1 β 1 + x 2 β 2 + εi : true model
I yi = b0 + x >1 b 1 + ηi : estimated model
Maximum likelihood estimator of b 1 isb1 = (X > X 1 )−1 X > y
b
1
1
−1 >
= (X >
1 X 1 ) X 1 [X 1 β 1 + X 2 β 2 + ε]
−1 >
>
−1 >
>
−1 >
= (X >
1 X 1 ) X 1 X 1 β 1 + (X 1 X 1 ) X 1 X 2 β 2 + (X 1 X 1 ) X 1 ε= β 1 + (X 01 X 1 )−1 X >X β + (X > X )−1 X >1ε|{z 1 2 }2 | 1 1{z
}
β 12

νi

b1 ] = β + β 6= β .
so that E[b1
121@freakonometrics

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Big/Small Data and Possible Pitfalls II
From Bickel et al. (1975) (see also Alipourfard et al. (2018))

Total
Top 6
A
B
C
D
E
F

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Total
5233/12763 ∼ 41%
1745/4526 ∼ 39%
597/933 ∼ 64%
369/585 ∼ 63%
321/918 ∼ 35%
269/792 ∼ 34%
146/584 ∼ 25%
43/714 ∼ 6%

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Men3714/8442 ∼ 44%
1198/2691 ∼ 45%
512/825 ∼ 62%
353/560 ∼ 63%
120/325 ∼ 37%
138/417 ∼ 33%
53/191 ∼ 28%
22/373 ∼ 6%

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Women1512/4321 ∼ 35%
557/1835 ∼ 30%
89/108 ∼ 82%
17/ 25 ∼ 68%
202/593 ∼ 34%
131/375 ∼ 35%
94/393 ∼ 24%
24/341 ∼ 7%

Proportions
66%-34%59%-41%88%-12%96%- 4%
35%-65%53%-47%
33%-67%52%-48%14 / 75

Big/Small Data and Possible Pitfalls III
As mentioned in Cohen (1986),
P[T ≤ 1|X = Costa Rica] < P[T ≤ 1|X = Sweden]
P[T ≤ 1|X = (Costa Rica, x )] > P[T ≤ 1|X = (Sweden, x )], ∀x@freakonometrics

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Defining Group Fairness Iy ∈ {0, 1}





p ∈ {0, 1}
x ∈ Rds ∈ [0, 1]yb ∈ {0, 1}

variable of interest (classically binary)
protected variable (sensitive)‘explanatory’ variables
score, classically s = s(x, p)classifier, classically yb = 1(s > t)

Fairness Through Unawareness, Kusner et al. (2017)
Protected attribute p is not explicitly used in decision function yb.
I what is y ?
I how to define a fair pricing ?

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Defining Group Fairness II

policyholder(x i , pi )@freakonometrics

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Defining Group Fairness III
policyholder(x i , pi )infered data
possibles scores
md (x, p)@freakonometrics

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Defining Group Fairness IV
policyholder

underwriting(x i , pi )mu (x, p)• commercial discount• multiple decisions from policyholder(related to various personal features)

infered data

• multiple decisions from agent / broker

md (x, p)@freakonometrics

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Defining Group Fairness V
policyholder

underwriting

claim a loss

(x i , pi )mu (x, p)`(x, p)(true value of the loss)

infered data
md (x, p)@freakonometrics

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Defining Group Fairness VI
policyholder

underwriting

claim a loss

(x i , pi )mu (x, p)`(x, p)• go to court ?

infered data

claims manager

md (x, p)

mc (x, p)@freakonometrics

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Defining Group Fairness VII
policyholder

underwriting

claim a loss

(x i , pi )mu (x, p)`(x, p)infered data

claims manager

md (x, p)

mc (x, p)fraud detection
mf (x, p, x 0 (p))@freakonometrics

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Defining Group Fairness VIII
policyholder

underwriting

claim a loss

(x i , pi )mu (x, p)`(x, p)infered data

claims manager

md (x, p)

mc (x, p)

indemnityyifraud detection
mf (x, p, x 0 (p))@freakonometrics

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Defining Group Fairness IX
policyholder

underwriting

claim a loss

(x i , pi )mu (x, p)`(x, p)infered data

claims manager

md (x, p)

mc (x, p)

indemnityyidataset(yi , x i , p i )fraud detection
mf (x, p, x 0 (p))@freakonometrics

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Defining Group Fairness X

Not to mention multiple biases, from feedback bias (telematics and gamification) to
selection bias

(see Hand (2020) on dark data)@freakonometrics

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Defining Group Fairness XIy ∈ {0, 1}





p ∈ {0, 1}
x ∈ Rds ∈ [0, 1]yb ∈ {0, 1}

variable of interest (classically binary)
protected variable (sensitive)‘explanatory’ variables
score, classically s = s(x, p)classifier, classically yb = 1(s > t)

Demographic Parity, (Corbett-Davies et al. (2017), Agarwal (2021))b ⊥⊥ P, i.e.
Decision function yb satisfies demographic parity if Y
b = y |P = 0] = P[Yb = y |P = 1], ∀y or E[Yb |P = 0] = E[Yb |P = 1]
P[YSince y is binary, a classical intermediary component is the score s(x)and a classical tool is the ROC curve (obtained by changing threshold t)@freakonometrics

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26 / 75x2

false positivesŷ = 1ŷ = 0ŷ = 1TPR

Defining Group Fairness XII

negatives

positives

y =0
y =1

positives
false negativeŷ = 0x1@freakonometrics

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FPR

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false positivesŷ = 1ŷ = 0ŷ = 1TPR

Defining Group Fairness XIII

negatives

positives

y =0
y =1

positives
false negativeŷ = 0x1@freakonometrics

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FPR

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Defining Group Fairness XIV
x2

negatives

positivesŷ = 1ŷ = 0x1@freakonometrics

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score

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Defining Group Fairness XVx2ŷ = 1

negatives

positivess(x)|y = 1

s(x)|y = 0

ŷ = 0x1@freakonometrics

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score

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Defining Group Fairness XVI
x2ŷ = 1s(x)|y = 1

negatives

positives

true positivess(x)|y = 0
false positivesŷ = 0x1

thresholdŷ = 0@freakonometrics

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Defining Group Fairness XVII
We can compare s(X) conditional on Y , but also on P

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Defining Group Fairness XVIII
Equal Opportunity, Hardt et al. (2016)
True positive parity
b = 1|P = 0, Y = 1] = P[Yb = 1|P = 1, Y = 1]P[Yor false positive parity
b = 1|P = 0, Y = 0] = P[Yb = 1|P = 1, Y = 0]
P[Y@freakonometrics

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Defining Group Fairness XIX
statistical parity
conditional statistical parity

Dwork et al. (2012)
Corbett-Davies et al. (2017)

equalized odds
equalized opportunity
predictive equality
balance (positive)
balance (negative)

Hardt et al. (2016)
Hardt et al. (2016)
Corbett-Davies et al. (2017)
Kleinberg et al. (2017)
Kleinberg et al. (2017)

conditional accuracy equality
predictive parity
calibration
well-calibration

Berk et al. (2017)
Chouldechova (2017)
Chouldechova (2017)
Chouldechova (2017)

accuracy equality

Berk et al. (2017)

treatment equality

Berk et al. (2017)@freakonometrics

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b = 1|P = p] = cst, ∀pP[Yb = 1|P = p, X = x ] = cstx , ∀p, yP[Yb = 1|P = p, Y = y ] = csty , ∀p, yP[Yb = 1|P = p, Y = 1] = cst, ∀pP[Yb = 1|P = p, Y = 0] = cst, ∀p
P[Y
E[S|P = p, Y = 1] = cst, ∀p
E[S|P = p, Y = 0] = cst, ∀pb = y ] = csty , ∀p, y
P[Y = y |P = p, Yb
P[Y = 1|P = p, Y = 1] = cst, ∀p
P[Y = 1|P = p, S = s] = csts , ∀p, s
P[Y = 1|P = p, S = s] = s, ∀p, s
b = Y |P = p] = cst, ∀p
P[YFNp= cstp , ∀p
FPp

independenceb ⊥Y⊥Pseparation
b ⊥
Y
⊥P |Y
S⊥
⊥P |Y
sufficiency
b
Y ⊥
⊥P |Y

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From correlation to causality I
I “classifying projection methods as using demographic/actuarial models or
non-demographic/causal models”
Keilman (2003) and Hudson (2007)
I “Article 5(2) allowed Member States to Permit proportionate differences inindividuals premiums and benefits where the use of sex is a determining factor in
the assessment of risk based on relevant and accurate actuarial and statistical
data.”
Thiery and Van Schoubroeck (2006) and Schmeiser et al. (2014)
I “Two judges on the Supreme Court dissented in the Zurich case. In their view, aninsurer must not only prove a statistical correlation between a particular groupand higher risk, but a causal connection”
Gomery et al. (2011)@freakonometrics

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From correlation to causality II
p

x

y
p

x

x

mediatoryp

I DAGs are important

confoundercollidery

I Looking for a counterfactual@freakonometrics

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From correlation to causality III
Consider some distances D on {0, 1} × {0, 1} or [0, 1] × [0, 1], and d on Rp × Rp ,Lipschitz property, Duivesteijn and Feelders (2008)
D(byi , ybj ) or D(si , sj ) ≤ d(x i , x j ), ∀i, j = 1, · · · , n.

Conterfactual fairness, Kusner et al. (2017) If the prediction in the real world is thesame as the prediction in the counterfactual world where the individual would havebelonged to a different demographic group, we have counterfactual equity, i.e.
 ?

 ?

0
P YP←p= y |X = x = P YP←p
0 = y |X = x , ∀p , x, y .

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From correlation to causality IV
I counterfactuals(what if I had done...? )I intervention
I association(what if I see...? )what would have happened if this person had hadtreatment 1 instead of treatment 0 ?

(picture Pearl & Mackenzie (2018))@freakonometrics

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From correlation to causality V
Causal inference litterature,I t some binary treatment (t ∈ {0, 1})
I x some covariates?
?
I y denote the observed outcome, yi,T
←1 and yi,T ←0 the potential outcomes1
2
3

treatement
ti
1
0
1

yi
121
109
162

outcome
?
?
yi,T
yi,T
←1
←o
121
?
?
109
162
?

age
x1,i
37
28
53

genderx2,iFFMheight
x3,i
160
156
190

weight
x4,i
56
54
87

There will be a significant impact of treatment t on y if yT? ←0 6= yT? ←1 (see Rubin(1974), Hernán and Robins (2010) or Imai (2018)).
?
?
The causal effect for individual i is τi = yi,T
←1 − yi,T ←0@freakonometrics

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From correlation to causality VI
One can define the sample average treatment effect (SATE)
n

1X ?
?
SATE =
yi,T ←1 − yi,T
←0
n
i=1

the average treatment effect (ATE) ?
?
τ = ATE = E Yi,T
←1 − Yi,T ←0and, for possibly heterogeneous effects, sample average treatment effect (CATE) ?
?
τ (x) = CATE(x) = E Yi,T
←1 − Yi,T ←0 X = x@freakonometrics

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From correlation to causality VII
Naive difference-in-means estimators, but Splawa-Neyman et al. (1990)
n

n

n

i=1

i=1

i=1

X
1 X
1 X
τb =
ti yi −
(1 − ti )yi , where nj =tin1
n0
or the regression-based estimators (also called single learner)
τb(x) = µb1 (x) − µb0 (x)
where µbt (x) = E[Y |T = t, X = x]
One can also use the propensity score πi = P[T = 1|X = x i ]nτb =1 X ti yi(1 − ti )yi−nπi
1 − πii=1

to derive a inverse-propensity score weighted estimator.
This can be use to derive a doubly-robust estimator@freakonometrics

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Counterfactual and optimal transport I

One can also use matching technique,
I for treated individual i (ti = 1), match that individual to someone in the
non-treated group (tj = 0), ji? = argmin{d(x i , x j )},j:tj =0E.g. xi is the height of a person, ti is the gender of that person, then
n

1

1 X
τb =
ti · yi − yji?
n1

i=1

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Counterfactual and optimal transport II@freakonometrics

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Counterfactual and optimal transport III

I consider a permutation of all treated individuals (ti = 1),
I for treated individual i (ti = 1), match that individual to someone in the
non-treated group (tj = 0), ji? = argmin{d(x i , x j )},j:tj =0

I then we remove the untreated observation from the database, and iterate (so as
to match all treated individuals with an untreated person@freakonometrics

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Counterfactual and optimal transport IV@freakonometrics

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Counterfactual and optimal transport V@freakonometrics

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Counterfactual and optimal transport VI@freakonometrics

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Counterfactual and optimal transport VII@freakonometrics

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Counterfactual and optimal transport VIII

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Counterfactual and optimal transport IX@freakonometrics

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Counterfactual and optimal transport X
See Monge (1781)’s problem(déblais/remblais - excavation/infill)
We want to go from P0 to P1 (distributions
on Rk ). Given T : Rk → Rk , define the“push-forward” measureP1 (A) = T# P0 (A) = P T −1 (A) , ∀A ⊂ Rk .
An optimal transport T ? (in Brenier’s
sense) from P0 towards P1 will be solution ofZ?
T ∈ arginf
kx − T (x)k2 dP(x).
T :T# P=Q Rk@freakonometrics

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Counterfactual and optimal transport XI@freakonometrics

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Counterfactual and optimal transport XII
One can prove (see) that T ? = ∇ψ where ψ is a convex function. If k = 1, T is anincreasing function, i.e. if F0 (x ) = P0 [X ≤ x ] and F1 (x ) = P1 [X ≤ x ], then
? P = P (since F (x ) = F (T ?−1 (x )) and T ? isT ? (x ) = F1−1 ◦ F0 (x ) satisfies T#01
10
optimal.

@freakonometrics

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Counterfactual and optimal transport XIII
That is an optimal coupling problem, that we can writeZminc(x , y )dF (x , y ), where F (x , y ) = C (F0 (x ), F1 (y ))copula C

R2

As Gordaliza et al. (2019) and Black et al. (2020), a classifier m obtained from score s(m(x, p) = 1(s(x, p) > seuil)) is unfair if m(x, 0) 6= m(T ? (x), 1). Black et al. (2020)
define the FlipSet as
XF (m, T ? ) = x ∈ X : m(x, 0) 6= m(T ? (x), 1) .
(
XF+ (m, T ? ) = x ∈ X : m(x, 0) > m(T ? (x), 1) .

XF− (m, T ? ) = x ∈ X : m(x, 0) < m(T ? (x), 1) .
? P = P , the average causal effect isand if T#
01

ACE = E[YT? ←1 ] − E[YT? ←0 ] = P0 [XF− (m, T ? )] − P0 [XF+ (m, T ? )]@freakonometrics

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Counterfactual and optimal transport XIV@freakonometrics

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Adversarial Approach I
Hirschfeld (1935), Gebelein (1941) and Rényi (1959)
HGR(U, V ) = max corr[f (U), g(V )] = max

f ∈SU ,g∈SVE[f (U)g(V )]where SU = {f : U → R : E[f (U)] = 0 and E[f (U)2 ] = 1} and similarly SV .
One can also consider a conditional version,no
HGR(U, V |Z ) =max
E[f (U)g(V )|Z ]f ∈SU|Z ,g∈SV |Z

where SU|Z = {f : U → R : E[f (U)|Z ] = 0 and E[f (U)2 |Z ] = 1}.
(b ⊥⊥ P
Demographic Parity : Yb ⊥⊥ P|Y
Equalized Odds : Y@freakonometrics

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i.e. HGR(Yb , P|Y ) = 0
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Adversarial Approach II
HGR can be difficult to estimate, but one can use some Neural Net,HGRNN (U, V ) = max E[fωf (U)gωg (V )]ωf ,ωg

In a classical ML or econometric pricing model, solve
n


 X

argmin L yb, y , where L yb, y =` ybi , yi and yb = hθ (x )θi=1

To avoid over-fit: penalize complexity (penalty P)argmin L hθ (x), y + λP(hθ )θ@freakonometrics

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Adversarial Approach III
Inspired by Goodfellow et al. (2018), to avoid un-fairness: penalize according toHGR(by , p) (for demographic parity)
argmin L hθ (x ), y + λHGRωf ,ωg (by , p)
θ,ω f ,ω g

i.e.

n
o

argmin max L(hθ (X), Y ) + λE(X,S)∼D (bfωf (hθ (X))bgωg (P)θω f ,ω g

or HGR(by , p|y ) (for equalized odds), i.e. when y ∈ {0, 1}nargminmax
L(hθ (X), Y ) + λ0 E(X,P)∼D0 (bfωf0 (hθ (X))bgωg0 (P))θω f 0 ,ω g0 ,ω f 1 ,ω g1+ λ1 E(X,P)∼D1 (bfωf1 (hθ (X))bgωg1 (P))@freakonometrics

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Adversarial Approach IVor, more generally when y ∈ ΩY (e.g. {0, 1, 2, 3+}), if k = #ΩynoXargminmax
L(hθ (X), Y ) +λy E(X,P)∼Dy (bfωfy (hθ (X))bgωgy (P))θω f 0 ,ω g0 ,ω fk ,ω gk

%ℒ(!$

%

( ,*)

%# %

𝑋

+𝜆

y ∈Ωy

%,("-$ (!$% ( )$. $' / )
&

−

%# %

Predictor ℎWeights : 𝑤!

𝜕𝐸(𝑓%!! (ℎ!" 𝑋 )𝑔*!# 𝑆 )𝜕𝑤"Adversary 𝑓
Weights : 𝑤"

𝑌"

𝐸(𝑓&#! (ℎ#" 𝑋 )𝑔)## 𝑆 )Adversary 𝑔
Weights : 𝑤$

𝑆−@freakonometrics

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𝜕𝐸(𝑓%!! (ℎ!" 𝑋 )𝑔*!# 𝑆 )𝜕𝑤#

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Dealing with high dimension I

I geographic / spatial information, X g
I car type / make / model, X g
I other classical ratemaking variables, X p (non protected)
Some features can be in high dimension, natural solution would be PCA or
autoencoders (see Shi and Shi (2021) about feature embedding in high dimension)@freakonometrics

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Dealing with high dimension II
Principal Component Analysis (PCA)min kX − P > PXk2F s.t. rank(P) = k
P∈Π

x1

x10
z1

x2

where Π is the set of projection matrices.

x20
z2

x3

X ∈X

X0 ∈ X

Z ∈Z

z3

P>

P

xp

min kX − X 0 k2 = min kX − P > PXk2
= min

n
X

encoder>

P Px i − x i>>P Px i − x ix30"bottleneck"hidden layer

xp0
decoderi=1@freakonometrics

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Dealing with high dimension III
x1

Autoencoder

x10
z1


min kX − ψ ◦ ϕXk2F

x2

ψ

x20
z2

X ∈X

X0 ∈ XZ ∈Z
ϕmin kX − X k = min kX − ψ ◦ ϕ(X)k
n
X

x30
z3

ψ
0 2

min

x3

ψ ◦ ϕ(x i ) − x i>2ψ ◦ ϕ(x i ) − x ixp encoder"bottleneck"hidden layer

xp0
decoderi=1@freakonometrics

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Dealing with high dimension IV𝑑(ℒ ℎ0$ 𝑋1, 𝛾0& 𝑋2 , 𝑐0( 𝑋(

1 𝑆 ), 𝑌 + 𝜆 HGR 𝑌,
0),0'

𝑑𝑤(

X4

Encoder 𝑐
Weight: 𝑤+

C

1 𝑆))
𝑑ℒ ℎ0$(𝑋1, 𝛾0& 𝑋2 , 𝑐0((𝑋()), 𝑌 + 𝜆 HGR (𝑌,$,'))!(ℒ $#$ (%%,'#& %' ,(#( (%( )),* +, -./ (%

𝑑𝑤'

!0$

0),0'

X!

Encoder 𝛾
Weight: 𝑤"

G

#),#'

Predictor ℎWeight: 𝑤!

−

𝜕𝐸(𝑓*-! (ℎ-" (𝑋. , 𝛾0& 𝑋/ , 𝑐0( (𝑋+ ))𝑔̂ # 𝑆 ))𝜕𝑤,

𝑌"

Adversary 𝑓*
Weight: 𝑤,

S

Adversary 𝑔2
Weight: 𝑤/

5 𝑆 = 𝐸(𝑓80 (ℎ0 𝑋1 , 𝐶, 𝐺 )𝑔<0 𝑆 )HGR 𝑌,)$'

X!

@freakonometrics

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−

0),0'

𝜕𝐸(𝑓*-! (ℎ-" (𝑋. , 𝛾-$ 𝑋/ , 𝑐-% (𝑋+ ))𝑔̂ # 𝑆 ))𝜕𝑤/

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Application on synthetic data I
I S: sensitive / protected (gender)

I Age: age of the driver

I Col: color of the car

I Ina: inattention

I Sp: maximum speed of the car

I Agg: aggressivity

I Sal: average salary of thepolicyholders area

I Y: total costSSal
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Agg

Col
Sp

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Y

Ina
age

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Application on synthetic data II
Performance Accuracy vs Fairness

BASE
OURS

0.6
BASE
OURS

0.7

0.850

0.5

0.825
0.800
0.775

HGR(Y,Col)

0.6

HGR(C,S)Classification accuracy %Dependence between Predictions and Color by

Dependence between Car risk and S by

BASE
OURS

0.875

0.5

0.4
0.3

0.4

0.750

0.2

0.725

0.3

0.1

0.700
40

50

60
70
p-rule %

80

90

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

I λ: fairness penalty()
b = 1|P = 1) P(Yb = 1|P = 0)P(YI p-rule: min,b = 1|P = 0) P(Yb = 1|P = 1)
P(Y@freakonometrics

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Application on real data (pricing game 2015) Iy ∈ {0, 1} (claim occurrence)
Performance Accuracy vs Fairness

0.84400.8435

0.5

BASE
OURS

BASE
OURS

0.50
0.45

0.4

0.84300.40

0.84250.84200.84150.3

HGR(Y,G)HGR(Y,C)Classification accuracy %Dependence between Geo. risk and Predictions byDependence between Car risk and the Predictions by

BASE
OURS

0.2

0.30
0.25

0.84100.20

0.1

0.84050.35

0.15

0.840082@freakonometrics

84

86

88 90
p-rule %92

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94

96

0.0

0.1

0.2

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0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

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Application on real data (pricing game 2015) IIy ∈ {0, 1, 2+} (claim frequency)
Dependence between Car risk and Predictions by

GINI vs FairQuant
BASE
OURS

0.5

Dependence between Geo. risk and Predictions by

BASE
OURS

0.46

0.4

0.40

0.3

HGR(Y,G)HGR(Y,C)

GINI

0.44

0.42

BASE
OURS

0.45

0.2

0.35
0.30
0.25

0.40

0.20

0.1

0.15

0.38
0.00250.00200.0015
FairQuant0.00100.0005

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

see Grari et al. (2022) for more examples (including the case where y ∈ R+ )@freakonometrics

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67 / 75References I
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