In the homogeneous case of one-dimensional objects, we show that any preference relation that is positive and homothetic can be represented by a quantitative utility function and unique bias. This bias may favor or disfavor the preference for an object. In the first case, preferences are complete but not transitive and an object may be preferred even when its utility is lower. In the second case, preferences are asymmetric and transitive but not negatively transitive and it may not be sufficient for an object to have a greater utility for be preferred. In this manner, the bias reflects the extent to which preferences depart from the maximization of a utility function.Intransitive preferences, incomplete preferences, irrational behavior, bias, procedural concerns, process of choice

Bertrand Lemaire

Marc Le Menestrel

English

Research Papers in Economics

Biased Representation of Homothetic
Preferences on Homogeneous Sets
Marc Le Menestrel∗ and Bertrand Lemaire†
November 2004
Abstract
In the homogeneous case of one-dimensional objects, we show that
any preference relation that is positive and homothetic can be repre-
sented by a quantitative utility function and a unique bias. This bias
may favor or disfavor the preference for an object. In the ﬁrst case,
preferences are complete but not transitive and an object may be pre-
ferred even when its utility is lower. In the second case, preferences
are asymmetric and transitive but not negatively transitive and it may
not be suﬃcient for an object to have a greater utility to be preferred.
In this manner, the bias reﬂects the extent to which preferences depart
from the maximization of a utility function.
Keywords: Intransitive preferences, Incomplete preferences, Ir-
rational Behavior, Bias, Procedural concerns, Process of Choice.
∗Universitat Pompeu Fabra, Departament d’Economia i Empresa, Ramon Trias Fargas
25-27, 08005 Barcelona, España. E-mail: Marc.Lemenestrel@upf.edu.
†UMR 8628 du CNRS, Université de Paris-Sud, Mathématiques (bât. 425), 91405
Orsay cedex, France. E-mail: Bertrand.Lemaire@math.u-psud.fr.
11 Introduction
Biased representation of preferences consists in representing preferences with
a utility function and a multiplicative bias outside this utility function. The
utility function provides for a measurement of the objects of preferences while
the bias explains why preferences may be inconsistent with the maximization
of the utility function. In this manner, the bias allows for a larger class of
preferences to be represented and captures the extent to which preferences
may depart from the maximization of a utility function.
In the homogeneous case of one-dimensional objects, we prove in this pa-
per that there exist a positively-valued function u unique up to multiplication
by a positive constant and a unique bias 0 ≤ α ≤ 1 such that
x Â y ⇔ αu(x) > (1 − α)u(y). (i)
If α = 1
2, preferences are represented by the maximization of u.I fα>1
2
then the bias “favors” the preference over x,a n dx may be preferred even if
its utility is lower. If α<1
2 then the factor “disfavors” the preference over x,
and it is not suﬃcient for x to have a greater utility to be preferred. If α =0
then preferences are empty and if α =1 , they hold for any pair of objects.
Note that these values of 0 and 1 for α are representing two limit cases where
t h eb i a sd e t e r m i n e sp r e f e r e n c e si n d e p e ndently of the utility of objects. Also,
when α 6=0 , we may like to write the inequality as
u(x)
u(y) >
(1−α)
α and make
more explicit the proportional threshold of preferences (for threshold models,
see Suppes et Al. 1989).
In the homogeneous case presented here, two main axioms are required
to obtain this result. Firstly, preferences are assumed to be positive:i fa n
object x i sp r e f e r r e dt oa no b j e c ty, then a relative increase of the quantity of
x does not change its preference. Secondly, preferences verify a form of scale
invariance called homotheticity: a preference for x over y remains invariant
if x and y are replicated an identical number of times.
As no speciﬁcp r o p e r t yf o rt h eb i n a r yr e l a t i o nÂ alone is required, this
model covers a broad class of preferences over homogeneous sets. If α>1
2,
the relation Â is complete but not transitive. If α<1
2,t h er e l a t i o nÂ
is asymmetric and transitive but its negation is not transitive, leading to
intransitivity of indiﬀerence. Of course, Â is asymmetric and negatively
transitive, i.e. it is a weak order, if and only if α = 1
2.
We obtain the representation (i) assuming that the primitive relation,
noted Â, veriﬁes an Archimedean axiom. When the primitive relation is
non-Archimedean, we note it % and we obtain a representation of the form
2x % y ⇔ αu(x) ≥ (1 − α)u(y). (i0)
In that case, we deﬁne two relations Â and ∼ such that Â is represented
by (i) and ∼ is represented by
x ∼ y ⇔ αu(x)=( 1− α)u(y). (i00)
In this manner, we naturally have x % y ⇔ x Â y or x ∼ y and the rela-
tion ∼ deﬁnes a particular type of “indiﬀerence”. In terms of interpretation,
x is indiﬀerent to y if and only if x is preferred to y but any “increase” in
the quantity of y negates this preference (note that in general, such “indiﬀer-
ence” is neither symmetric nor transitive). This later representation should
be useful to experimentally elicit the value of the bias.
While this model requires weak assumptions on preferences to prove the
existence of the function u, the biased representation obtained has strong
uniqueness properties. Indeed, the function u is unique up to multiplication
by a positive constant, hence allowing the measurement of both diﬀerences
and ratios of utility, which corresponds to a full quantitative measurement
(also called a ratio-scale, see Krantz et Al. 1971). As to the factor α, it
is uniquely determined by the preferences, thus allowing a precise measure-
ment of the departure from utility maximization. In this manner, relaxing
the maximization hypothesis does not impede the measurement of utility
while oﬀering a way to model preferences that would have been considered
“irrational”.
The result presented in this paper is an extension of the biased represen-
tations introduced in Le Menestrel and Lemaire (2004). There, the relation
of preferences Â was assumed to be asymmetric and transitive, and corre-
sponds, in the present paper, to the case when the bias disfavors the preferred
object, i.e. when 0 <α6 1
2. An Archimedean condition was also assumed
and the interpretation was developed around the idea of a proportional lack
of discrimination (also known as Weber’s law). In the present paper, the bias
can also favor preferences and may be used to interpret situations in which
individuals forgo part of their utility because of considerations that cannot
be easily included in the utility function. Typical examples are considera-
tions proper to the process of choice (i.e. procedural concerns in Sen, 1997)
or ethical considerations. Future work shall develop such interpretations.
Finally, we have generalized Le Menestrel and Lemaire (2004) to the non-
homogeneous case in Lemaire and Le Menestrel (2004). Then, the biasing
factor is not necessarily constant. Similarly, it is probably possible to gener-
alize the result presented here to non-homogeneous sets.
32 Extending the homogeneous case of biased
representation
Let A be a nonempty set of elements x,y,z... ∈ A.D e n o t e N∗ the set
of positive integers, and assume A t ob ee n d o w e dw i t ham a pN∗ × A →
A,(m,x) 7−→ mx such that (mm0)x = m(m0x) and 1x = x. Such a A is
called a N∗ − set. In this manner, nx can be naturally interpreted as the
quantity n of an object x. Denoting R>0 the set of positive real numbers,
note that a R>0 −set is a special case of a N∗ −set. Hence, the main results
presented here (Theorems 1 and 2) would remain valid if one wants to let n
be a positive real number instead of a positive natural number. Here, we use
natural numbers because we do not need quantities to be non-denumerable
to obtain our result.
We say that A is homogeneous if, for all x,y ∈ A, there exists (m,n) ∈
N∗ × N∗ such that mx = ny. A homogeneous set can hence be readily inter-
preted as consisting of quantities of one-dimensional objects. We note Q>0
the set of positive rational numbers.
Let Â be a binary relation on A and consider the three following axioms
(x,y ∈ A;m,n ∈ N∗):
A1 (Positivity): ∀(x,y,m,n) such that m>n ,we have x Â y ⇒ mx Â
ny.
A2 (Homotheticity): ∀(x,y,m) we have x Â y ⇔ mx Â my;
A3 (Archimedean): ∀(x,y) such that x Â y, ∃(m,n) such that m<n
and mx Â ny.
Theorem 1 Let A be a N∗ − set endowed with a binary relation Â that
veriﬁes A1, A2 and A3. Suppose A is homogeneous. Then there exist a
function u : A → R>0 and a number 0 ≤ α ≤ 1 such that, for all x,y ∈ A
and m ∈ N∗, we have
x Â y ⇐⇒ αu(x) > (1 − α)u(y), (i)
u(mx)=mu(x). (ii)
Moreover, the pair (u,α) o f ( i )i su n i q u eu pt or e p l a c i n gu by λu for
λ>0.
Proof. We can always choose a function u : A → R>0 such that u(mx)=
mu(x) for all (x,m) ∈ A×N∗. Since A is homogeneous, such a function exists
4a n di su n i q u eu pt om u l t i p l i c a t i o nb yap o s i t i v es c a l a r ;i no t h e rw o r d s ,g i v e n
an element a ∈ A, u is uniquely determined by its value at a.I ft h er e l a t i o n
Â (respectively 6Â)i se m p t y ,w et a k eα =0(resp. α =1 ). So in both cases,
we have
x Â y ⇐⇒ αu(x) > (1 − α)u(y).
From now on, suppose that both the relation Â and the relation 6Â are
nonempty. For x,y ∈ A,w ed e ﬁne the subset of Q>0
Px,y = {
m
n
: mx Â ny,∃(m,n) ∈ N
∗ × N
∗}.
Let x,y ∈ A. By A1 and A2, if q ∈ Px,y then Q≥q ⊂ Px,y. And since Â is
nonempty and A is homogeneous, we have Px,y 6= ∅. Put tx,y =i n f R≥0 Px,y.
By A1 and A2, we have Q>t ⊂ Px,y with t = tx,y.
Now, if q ∈ Px,y, then by A2 and A3, there exists q0 ∈ Q<q ∩ Px,y, which
implies q>t x,y. Hence, we have Px,y ⊂ Q>t and then Px,y = Q>t.
We thus have
x Â y ⇐⇒ 1 ∈ Px,y ⇐⇒ tx,y < 1.
Since Pmx,ny = n
mPx,y, we have tmx,ny = n
mtx,y. Using the homogeneity of
A, we obtain tx,y > 0 (recall the relation 6Â is supposed to be nonempty).
Now, choose a ∈ A. Since we can always replace u by λu with λ = u(a)−1ta,a,
we can suppose u(a)=ta,a. Since ta,nx = nta,x, we thus have u(x)=ta,x ∈
R>0. Also put
σ(x,y)=t
−1
a,xt
−1
x,yta,y ∈ R>0.
Since Pmx,ny = n
mPx,y, we have tmx,ny = n
mtx,y. Hence u(mx)=mu(x)
and
σ(mx,ny)=( mta,y)
−1(
n
m
tx,y)
−1nta,y = σ(x,y).
Since A is homogeneous, σ is constant on A × A; let α =
σ(A×A)
σ(A×A)+1. We
have 0 <α<1 (both Â and 6Â being nonempty). Also
x Â y ⇔ t−1
x,y > 1
⇔ (t−1
a,xt−1
x,yta,y)ta,x >t a,y
⇔ αu(x) > (1 − α)u(y).
As for the uniqueness property (in the general case, i.e. without hypothesis
on the emptiness of Â and 6Â), let (v,β) be a pair such that v : A →
R>0,v(mx)=mv(x),0 ≤ β ≤ 1,a n dxÂ y ⇔ βv(x) > (1 − β)v(y). Since
A is homogeneous, we necessarily have v = λu for some λ ∈ R>0. It is then
easy to deduce that β = α.
5We can summarize the properties of Â in the following corollary:
Corollary 1 Let Â be a binary relation on a homogeneous N∗ − set A
that veriﬁes A1, A2 and A3, and let (u,α) be a pair that veriﬁes conditions
(i) and (ii) of.Theorem 1. The relation Â is
• nonempty if and only if α>0,
• asymmetric and transitive if and only if α ≤ 1
2,
• complete if and only if α>1
2.
Note also (with the notation of Corollary 1) that the relation 6Â is given
by
x 6Â y ⇐⇒ αu(x) ≤ (1 − α)u(y).
In particular, 6Â is nonempty if and only if α<1.
In Theorem 1, we assume that the primitive relation Â is Archimedean.
Now, starting with a binary relation % on A that veriﬁes A1 and A2, we
deﬁne two binary relations Â and ∼ on A as follows:
- x Â y ⇔ (x % y and ∃(m,n) such that m<nand mx % ny);
- x ∼ y ⇔ (x % y and x 6Â y).
As suggested by the notation, we have x % y ⇔ (x Â yo rx∼ y). Since %
is homothetic and positive, then Â and ∼ are homothetic, and Â is positive
and Archimedean. Note that ∼ may not be symmetric (i.e. it may not verify
x ∼ y ⇒ y ∼ x); and Â may not be asymmetric (i.e. it may not verify
x Â y ⇒ y 6Â x). Note also that if x 6Â y ⇔ y % x for all x,y ∈ A, then the
relation ∼ is given by
x ∼ y ⇔ (x % y and y % x) ⇔ (x 6Â y and y 6Â x);
in which case it is clearly symmetric.
The relation ∼ is empty if and only if the relation % veriﬁes A3, in which
case we have Â=% . Hence assuming that ∼ is not empty amounts to assume
that % veriﬁes the following axiom (x,y ∈ A;m,n ∈ N∗):
A3’ (Non-Archimedean): ∃(x,y) such that x % y and, ∀(m,n) such that
m<n ,we have mx 6% ny.
This leads us to a slightly diﬀerent formulation of Theorem 1:
6Theorem 2 Let A be a N∗ − set endowed with a binary relation % that
veriﬁes A1, A2 and A3’. Suppose A is homogeneous. Then there exist a
function u : A → R>0 a n dan u m b e r0 <α≤ 1 such that, for all x,y ∈ A
and m ∈ N∗, we have
x % y ⇐⇒ αu(x) ≥ (1 − α)u(y), (i0)
u(mx)=mu(x). (ii)
Moreover, the pair (u,α) o f ( i )i su n i q u eu pt or e p l a c i n gu by λu for
λ>0.
Proof. Choose a function u : A → R>0 such that u(mx)=mu(x) for all
(x,m) ∈ A×N∗ (Cf. the proof of Theorem 1). If 6% is empty, we take α =1 ;
hence the pair (α,u) veriﬁes (i0).
We now suppose 6% is not empty. Since % veriﬁes A3’, it is not empty.
Also, ∼ is not empty. Because A is homogeneous, for all (x,y) ∈ A×A, there
exists (m0,n 0) ∈ N∗ ×N∗ such that m0x ∼ n0y. We deﬁne Px,y and tx,y as in
the proof of Theorem 1. We also deﬁne the subset of Q>0
Qx,y = {
m
n
: mx % ny,∃(m,n) ∈ N
∗ × N
∗}.
So we have the inclusion Px,y ⊂ Qx,y. If q ∈ Qx,y, then by A1 and A2, we
have Q≥q ⊂ Qx,y; and by the deﬁnition of Â,w eh a v eQ>q ⊂ Px,y = Q>tx,y,
and q ≥ tx,y. Since Â6=%, the inclusion Px,y ⊂ Qx,y is strict. Hence we have
Qx,y = Q≥tx,y, and
x % y ⇐⇒ 1 ∈ Qx,y ⇐⇒ tx,y ≤ 1.
Since 6% is supposed to be nonempty, we also have tx,y > 0.F r o mT h e o r e m
1, there exists a 0 ≤ α ≤ 1 such that x Â y ⇐⇒ αu(x) > (1 − α)u(y). We
then have
x % y ⇐⇒ αu(x) ≥ (1 − α)u(y)
and
x ∼ y ⇐⇒ αu(x)=( 1− α)u(y).
Since the relation % is nonempty, the relation Â is also nonempty, and
we have α>0.
Finally, if (v,β) is another pair verifying conditions (i0) and (ii), then
it veriﬁes condition (i) of Theorem 1, which implies (v,β)=( λu,α) for a
positive scalar λ.
7We can summarize the properties of % in the following corollary:
Corollary 2 Let % be a binary relation on a homogeneous N∗ − set A
that veriﬁes A1, A2 and A3’, and let (u,α) be a pair that veriﬁes conditions
(i0) and (ii) of.Theorem 2. The relation % is
• asymmetric if and only if α<1
2,
• transitive if and only if α ≤ 1
2,
• complete if and only if α ≥ 1
2.
Note also (with the notation of Corollary 2) that the relation 6% is given
by
x 6% y ⇐⇒ αu(x) < (1 − α)u(y).
In particular, 6% is nonempty if and only if α<1.
The following two corollaries help to further understand the link between
Theorem 1 and Theorem 2. A key condition is whether the biasing factor is
a rational number.
Corollary 3 Let % be a binary relation on a homogeneous N∗ − set A
that veriﬁes A1, A2 and A3’, and let (u,α) be a pair that veriﬁes conditions
(i0) and (ii) of.Theorem 2. Then α ∈ Q>0.
Proof. For all (x,y) ∈ A × A, there exists (m0,n 0) ∈ N∗ × N∗ such that
m0x ∼ n0y and we have tx,y = m0
n0 (Cf. the proof of Theorem 2). From the
deﬁnition of α (Cf. the proof of Theorem 1), we conclude that α ∈ Q>0.
Corollary 4 Let Â be a nonempty binary relation on a homogeneous
N∗ −set A that veriﬁe sA 1 ,A 2a n dA 3 ,a n dl e t(u,α) be a pair that veriﬁes
conditions (i) and (ii) of.Theorem 1. Let % be the binary relation deﬁned by
x % y ⇐⇒ αu(x) ≥ (1 − α)u(y). Then % veriﬁes A1 and A2, and it veriﬁes
A 3 ’i fa n do n l yi fα ∈ Q>0.
Proof. Clearly, the relation % veriﬁes A1 and A2. And we have
x Â y ⇔ (x % y and ∃(m,n) with m<nsuch that mx % ny.
Hence if the relation % veriﬁes A3’, from Corollary 3, we have α ∈ Q>0.
Now suppose α ∈ Q>0. Since α ∈ Q>0, there exists (m,n) ∈ N∗ × N∗ such
that αm =( 1− α)n. Hence for all x ∈ A, we have mx % nx but mx 6Â nx.
So Â6=%, and % veriﬁes A3’.
83 References
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dations of measurement (Vol. I). San Diego: Academic Press.
2. Lemaire, B. and Le Menestrel, M. (2004), Homothetic interval orders,
submitted.
3. Le Menestrel, M. and Lemaire, B. (2004), Biased extensive measure-
ment: the homogeneous case, Journal of Mathematical Psychology 48:
9-14.
4. Sen, A. (1997),Maximization and the Act of Choice, Econometrica 65:
745—779.
5. Suppes, P. , Krantz, D. H., Luce, R. D. and Tversky, A. (1989). Foun-
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Biased representation of homothetic preferences on homogeneous sets

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