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

CIRM - MLISTRAL - September 2022

@freakonometrics

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AgendaI 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
��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)@freakonometrics

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Motivation
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))
woman

woman
man

man

queen
homemaker

kingcomputerprogrammer

nurse
doctor@freakonometrics

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Actuarial Pricing I
I “at the core of insurance business lies discrimination between risky and non-risky
insureds”, Avraham (2017)x ∈ X ⊂ Rd : ‘explanatory’ variables

(



x c ∈ Xc ⊂ Rdc : car-vehicule ‘explanatory’ variables



0


(xc = mc (x c ; x) ∈ R : car-vehicle scoringx g ∈ Xg ⊂ Rdg : geographic ‘explanatory’ variables
xg0 = mg (x g ; x) ∈ R : geographic scoring





y ∈ {0, 1}, N or R+ : variable of interestyb = m(x, xc0 , xg0 ) : prediction (or score)Classically, a two-stage approach is considered to create a geographic score
I Fit a model mb to predict y based on x, compute residuals εb
I Fit a model mb g (·; x) to predict εb based on x g
I define a score xg0 = m
b g (x g ; x) (that is function of x, too)@freakonometrics

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Protected Attributes ? Motor insurance in North America
CA

HI

GA

NC

NY

MA PA

FL

TX

AL

ON NB

NL

QC

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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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Protected Attributes ? Motor insurance in North America
Spatial information and racial bias (redlining)@freakonometrics

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Defining Group Fairness when y is binary Ix ∈ X ⊂ Rd : ‘explanatory’ variables





p

( ∈ {0, 1} : protected variable or sensitive attributex c ∈ Xc ⊂ Rdc : car-vehicule ‘explanatory’ variables
 x 0 = mc (x c ; x, p) ∈ R : car-vehicle scoring( cx g ∈ Xg ⊂ Rdg : geographic ‘explanatory’ variables
xg0 = mg (x g ; x, p) ∈ R : geographic scoring






y ∈ {0, 1} : variable of interest (binary)s = m(x, p, xc0 , xg0 ) : score
yb = 1(s > threshold) : prediction
Fairness Through Unawareness, Kusner et al. (2017)
Protected attribute p is not explicitly used in decision function yb.
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Defining Group Fairness when y is binary II
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[YWe can compare s(X) conditional on Y , but also on P

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Defining Group Fairness when y is binary III
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 when y is binary IV
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

10 / 36Dependence measures I
Group fairness is characterized be independence or conditional independence properties.
Given two random variables U and V ,corr[U, V ] Pearson’s correlation
C (U, V ) = corr[FU (U), FV (V )] Spearman’s rank correlationτ [U, V ] Kendall’s tauthat can be extended to conditional measures, as Lawrance (1976), since(E[U] = E[V ] = 0
corr[U, V ] = E UV whenE[U 2 ] = E[V 2 ] = 1(b ⊥⊥ P
Demographic Parity : Yb ⊥⊥ P|Y
Equalized Odds : Y@freakonometrics

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b , P) = 0=⇒ C (Yb , P|Y = y ) = 0, ∀y=⇒ C (Y11 / 36Dependence measures II
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}.
This measure can be used to characterize independence,
U ⊥⊥ V ⇐⇒ HGR(U, V ) = 0,
and if (U, V ) is a Gaussian vector, HGR(U, V ) = |corr(U, V )|.
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12 / 36Dependence measures III

Thus, this measure can be used to characterize fairness,(b ⊥⊥ P ⇐⇒ HGR(Yb , P) = 0
Demographic Parity : Yb ⊥⊥ P|Yb , P|Y = y ) = 0, ∀yEqualized Odds : Y⇐⇒ HGR(YHGR can be difficult to estimate, but one can use some Neural Networks

HGRNN (U, V ) = max E[fωf (U)gωg (V )]ω f ,ω g

See also Breiman and Friedman (1985) on the estimation of this maximal correlation,in the context of regression@freakonometrics

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13 / 36Dependence measures IV
More generally, V can be a vector on V ⊂ Rk , then
HGR(U, V ) = max HGR[U, h(V )] = max

f ∈SU ,g∈SV

h:V→R


E[f (U)g(V )]where SV = {g : V → R : E[g(V )] = 0 and E[g(V )2 ] = 1}. A conditional version
exists, and one can estimate that measure using a neural network,

HGRNN (U, V ) = max E[fωf (U)gωg (V )]ω f ,ω g(b ⊥⊥ P
Demographic Parity : Yb ⊥⊥ P|Y
Equalized Odds : Y@freakonometrics

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b , P) = 0⇐⇒ HGR(Yb , P|Y = y ) = 0, ∀y⇐⇒ HGR(Y14 / 36

Adversarial Approach I

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)θ

i=1either related to some loss, or some log-likelihood,
To avoid over-fit: penalize complexity (penalty P)argmin L hθ (x), y + λP(hθ )θ@freakonometrics

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Adversarial Approach II
Inspired by Goodfellow et al. (2018) (but also Bechavod and Ligett (2017) or Cho
et al. (2020)), to avoid un-fairness: penalize according to HGR(by , p) (for demographic
parity),argmin L hθ (x), y + λHGRωf ,ωg (hθ (x), 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))

oor, more generally when y ∈ ΩY (e.g. {0, 1, 2, 3+}), if k = #Ωy@freakonometrics

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Adversarial Approach IIInargminθmax

ω f 0 ,ω g0 ,··· ,ω fk ,ω gk.ℒ(0%& 1 ,3)
.4&

𝑋

+𝜆

o
X

L(hθ (X), Y ) +λy E(X,P)∼Dy (bfωfy (hθ (X))bgωgy (P))
y ∈Ωy.5(6%%' (0%& 1 )7*%( 8 ).4&Predictor ℎWeights : 𝑤!

−

𝜕𝐸(𝑓%!! (ℎ!" 𝑋 )𝑔*!# 𝑃 )𝜕𝑤"

Adversary 𝑓
Weights : 𝑤"

𝑌"

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

𝑃−

@freakonometrics

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𝜕𝐸(𝑓%!! (ℎ!" 𝑋 )𝑔*!# 𝑃 )𝜕𝑤#

17 / 36

Dealing with high dimension I

I geographic / spatial information, X g
I car type / make / model, X c
I other classical ratemaking variables, X (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 𝑃 ), 𝑌 + 𝜆 HGR 𝑌,
0),0'

𝑑𝑤(

X4

Encoder 𝑐
Weight: 𝑤+

C1 𝑃)) !(ℒ $# (%%,'#& %' ,(#( (%()),* +, -./ (%$,'))
𝑑ℒ ℎ0$(𝑋1, 𝛾0& 𝑋2 , 𝑐0((𝑋()), 𝑌 + 𝜆 HGR (𝑌,
$
# ,#'
0),0'

)

!0$

𝑑𝑤'

X!

Encoder 𝛾
Weight: 𝑤"

G

Predictor ℎWeight: 𝑤!

−

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

𝑌"

Adversary 𝑓*
Weight: 𝑤,

P

Adversary 𝑔2
Weight: 𝑤/

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

X!

@freakonometrics

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−

0),0'

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

21 / 36

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

22 / 36

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), and p is the (binary) gender
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

24 / 36

Application on real data (pricing game 2015) IIy ∈ {0, 1, 2+} (claim frequency), and p is the (binary) gender
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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Application on real data (pricing game 2015) IIIy ∈ {0, 1, 2+} (claim frequency), and p is spatial information (redlining)@freakonometrics

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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 .

@freakonometrics

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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, conditional average treatment effect (CATE) ?
?
τ (x) = CATE(x) = E Yi,T
←1 − Yi,T ←0 X = x@freakonometrics

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