arXiv:2111.11630v1 [econ.TH] 23 Nov 2021
AGGREGATION OF MODELS,
CHOICES, BELIEFS, AND PREFERENCES
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
Abstract.
A natural notion of rationality/consistency for aggregati
ng
models is that, for all (possibly aggregated) models
A
and
B
, if the out-
put of model
A
is
f
p
A
q
and if the output model
B
is
f
p
B
q
, then the out-
put of the model obtained by aggregating
A
and
B
must be a weighted
average of
f
p
A
q
and
f
p
B
q
. Similarly, a natural notion of rationality for
aggregating preferences of ensembles of experts is that, fo
r all (possibly
aggregated) experts
A
and
B
, and all possible choices
x
and
y
, if both
A
and
B
prefer
x
over
y
, then the expert obtained by aggregating
A
and
B
must also prefer
x
over
y
. Rational aggregation is an important element
of uncertainty quantification, and it lies behind many seemi
ngly differ-
ent results in economic theory: spanning social choice, bel
ief formation,
and individual decision making. Three examples of rational
aggregation
rules are as follows. (1) Give each individual model (expert
) a weight
(a score) and use weighted averaging to aggregate individua
l or finite
ensembles of models (experts). (2) Order/rank individual m
odel (ex-
pert) and let the aggregation of a finite ensemble of individu
al models
(experts) be the highest-ranked individual model (expert)
in that ensem-
ble. (3) Give each individual model (expert) a weight, intro
duce a weak
order/ranking over the set models/experts (two models may s
hare the
same rank), aggregate
A
and
B
as the weighted average of the highest-
ranked models (experts) in
A
or
B
. Note that (1) and (2) are particular
cases of (3) (in (1) all models/experts share the same rank, a
nd in (2)
the ranking is strict). In this paper, we show that all ration
al aggrega-
tion rules are of the form (3). This result unifies aggregatio
n procedures
across many different economic environments, showing that t
hey all rely
on the same basic result. Following the main representation
, we show
applications and extensions of our representation in vario
us separated
economics topics such as belief formation, choice theory, a
ggregation of
optimal models, and social welfare economics.
1. Introduction
This paper presents a general framework characterizing rat
io-
nal/consistent aggregation (of models, choices, beliefs,
and preferences,
which we simply refer to as features) with applications to ec
onomic the-
ory. In this framework, individual features have outcomes,
and aggregation
rules identify the outcome of groups of features. We focus on
a recursive
Affiliation: Division of Computing and Mathematical Science
s (CMS), California In-
stitute of Technology. Email:
hhamzeyi@caltech.edu
and
owhadi@caltech.edu
Date
: November 24, 2021.
1
2
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
form of aggregation, which is the case in the Cased-Based dec
ision the-
ory developed by Gilboa and Schmeidler,
2003
; Billot, Gilboa, Samet, and
Schmeidler,
2005
, where the aggregate outcome for larger collections of fea-
tures results from aggregating the outcomes of smaller subs
ets. Specifically,
the aggregate outcome of the union of two disjoint collectio
ns of features is
a weighted average of the outcome of each collection of featu
res separately.
We show that this form of recursive aggregation is a common st
ructure that
lies behind many seemingly unrelated results in economic th
eory.
Our central axiom, the
weighted averaging axiom/property
(we will use
the terms
axiom
and
property
interchangeably), is a simple formalization of
the recursivity. It imposes a structure on how the outcome of
the union of
two disjoint subsets of features relates to the outcome of ea
ch of the subsets
separately. The axiom states that the outcome of a set of feat
ures can be
recursively computed by first partitioning the set of featur
es into two disjoint
subsets. Then, the aggregated outcome is a weighted average
of the outcome
of each of the two smaller subsets.
Our contribution is three-fold: (1) We find all aggregation p
rocedures that
satisfy the weighted averaging axiom, which generalized th
e result of Billot
et al.,
2005
. Moreover, by enhancing the procedure with continuity axio
m,
we connect the axiom to the path independent axiom, which is s
tudied in
the choice literature. (2) With a simple geometrical dualit
y argument, we
connect the weighted averaging to the combination axiom of G
ilboa et al.,
2003
and Extended Pareto of Shapley and Shubik,
1982
. (3) We present
applications and extensions to different domains of economic
s, notably in
the context of Belief Formation, Choice Theory, and Welfare
Economics.
Formally, we define an aggregation rule as a function on the se
t of subsets
of features that maps each subset of features to an outcome. O
ur main
result finds all aggregation rules that satisfy recursivity
in the form of our
weighted average axiom. We show that as long as for any two dis
joint
subsets of features, the outcome of their union is a weighted
average (with
non-negative weights) of the outcome of each subset, then th
e aggregation
rule has a simple form (with a technical richness condition)
:
There exist a strictly positive weight function and a weak ord
er (a tran-
sitive and complete order) over the set of features, with the
outcome of any
subset of features being the weighted average of the outcome
s of each of the
highest-ordered features of the subset separately.
The importance of the result is that the weight of each featur
e is indepen-
dent of the group of features being aggregated. The role of th
e weak order
in the main representation is to partition the set of feature
s into different
equivalence classes and rank them from the highest class to t
he lowest class.
If all features of a subset of features are in the same class, t
hen the outcome
is the weighted average of the outcomes of each member of the s
et. However,
if some features have a higher ranking than others, then the a
ggregation rule
will ignore lower-ordered features.
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
3
Following the main result, we discuss two special cases of ou
r main re-
sult. In the first case, we introduce the
strict weighted averaging
axiom to
represent the case where the outcome of the union of two disjo
int subsets of
features is contained in the “relative” interior of the outc
omes of each subset
separately. We then show that the strict weighted averaging
axiom is the
necessary and sufficient condition for the weak order, in the m
ain represen-
tation, to have only one equivalence class. Hence, the outco
me of a subset
of features is just the weighted average (with strictly posi
tive weights) of
the outcomes of each feature separately.
In the second case, we model the space of features as a subset o
f a vector
space. By considering the distance between vectors, we capt
ure the notion
of similarity or closeness of features. In this context, we c
an consider the
following notion of continuity of outcomes with respect to f
eatures: replacing
a feature in a subset of features with another closely simila
r feature, the
outcome of this new subset stays close to the outcome of the pr
evious one.
Under this property, which we define as the
continuity
property, we show
that all similar enough features attain the same ranking wit
h respect to the
weak order. Moreover, the weight function is a continuous fu
nction over the
set of features. In other words, the weight between two close
(or similar)
features should be close. In a special case, where the space o
f features is
a convex set, we show that all features attain the same rankin
g. In this
case, there is no difference between the weighted averaging an
d the strict
weighted averaging property.
Depending on the application, features and the aggregation
rules may
have different interpretations. A feature may represent a sig
nal or an event
containing some information about the true state of nature.
In this case, the
role of an aggregation rule is to form a belief about the true s
tate of nature.
In the context of choice theory, features may represent choi
ce objects, where
an aggregation rule behaves as a decision-maker that select
s a lottery or a
random choice out of a group of choice objects. Another inter
pretation is
in the context of welfare economics, where each feature repr
esents a prefer-
ence of an individual over some alternatives. In this case, a
n aggregation
rule represents a social welfare function that associates w
ith each preference
profile, a single preference ordering over the set of alterna
tives.
To describe a natural interpretation of our result, conside
r the problem
of modeling an agent who seeks to make a prediction about the t
rue state of
nature, conditional on observing a set of events. In this con
text, a feature
represents an event, and the outcome of the model conditiona
l on observing
a set of events is a belief about the true state of nature. Our m
ain result
provides a necessary and sufficient condition for the belief f
ormation pro-
cess to behave as a
Bayesian Updater
. Under the averaging property of the
belief formation process, there exists a conditional proba
bility system asso-
ciated with the set of events, and the belief formation proce
ss conditional on
observing a set of events behaves like a conditional probabi
lity. The weak
order of the main result is capturing the idea that condition
al on observing
4
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
even a zero probability event, the belief formation still be
haves as a Bayesian
updater.
To motivate the proposed framework, sections
4
,
5
,
6
, and
7
present
applications and extensions of our main representation res
ults. We show
that the weighted averaging axiom is closely related to many
known axioms
in different topics, from the Pareto axiom in Social Choice The
ory to the
path independence axiom in Choice Theory.
2. Model aggregation
Let
X
be a nonempty set
1
. Write
X
̊
for the set of all nonempty finite
subsets of
X
. One may interpret
X
as a (possibly infinite) set of models,
A
P
X
̊
as a finite set of models, and
X
̊
as the set of all nonempty finite
sets of models. Let
H
be a separable Hilbert space with
H
“
R
n
as a
prototypical example.
Definition 1.
An
aggregation rule
on
X
is a function
f
:
X
̊
Ñ
H
, that
associates with every
A
P
X
̊
a vector
f
p
A
q P
H
.
2
For
x
P
X
and
A
P
X
̊
, one may interpret
f
pt
x
uq
as the output of the
model
x
, and
f
p
A
q
as the output of the aggregation of models contained in
A
. The purpose of this section is to characterize aggregation
rules satisfying
the weighted averaging axiom/property defined below.
Definition 2.
We say that an aggregation rule
f
satisfies the
weighted
averaging
axiom/property if for all
A, B
P
X
̊
such that
A
X
B
“ H
, it
holds true that
f
p
A
Y
B
q “
λf
p
A
q ` p
1
́
λ
q
f
p
B
q
(2.1)
for some
λ
P r
0
,
1
s
(which may depend on
A
and
B
). We say that
f
sat-
isfies the
strict weighted averaging
axiom/property if (
2.1
) holds true
for
λ
P p
0
,
1
q
. We say that
f
satisfies the
extreme weighted averaging
axiom/property if (
2.1
) holds true for
λ
P t
0
,
1
u
.
Two simple examples of aggregation rules satisfying the wei
ghted averag-
ing property are as follows.
Example 1.
Write
R
``
for the set of strictly positive real numbers and
let
w
:
X
Ñ
R
``
be a weight function on
X
. For
x
P
X
, let
3
f
p
x
q
be the
output of model
x
. For
A
P
X
̊
, define
f
p
A
q “
ÿ
x
P
A
̈
̊
̋
w
p
x
q
ř
y
P
A
w
p
y
q
f
p
x
q
̨
‹
‚
.
(2.2)
Then
f
satisfies the strict weighted averaging property.
1
We make no assumptions about the cardinality or topology of
X
2
All discussions of this paper continue to hold if
H
is replaced by any general (possibly
infinite dimensional) normed vector space.
3
Abusing notations we write
f
p
x
q
for
f
pt
x
uq
for
x
P
X
.
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
5
Example 2.
Consider a complete strict order
ą
on
X
. Given any feature
x
P
X
, let
f
p
x
q P
H
be the output of the model
x
. For
A
P
X
̊
, write
M
p
A,
ą
q
for the highest order element
4
in
A
. For
A
P
X
̊
, define
f
p
A
q “
f
p
M
p
A,
ą
qq
.
(2.3)
Then
f
satisfies the extreme weighted averaging property.
We will now show that all aggregation rules satisfying the st
rict weighted
averaging property must be as in Example
1
if
X
contains at least three
elements
x, y, z
such
f
p
x
q
, f
p
y
q
and
f
p
z
q
are not collinear.
Definition 3.
An aggregation rule
f
:
X
̊
Ñ
H
is
rich
if the range of
f
is
not a subset of a line.
Theorem 1.
Let the aggregation rule
f
:
X
̊
Ñ
H
be rich. The following
are equivalent:
(1)
The aggregation rule
f
satisfies the strict weighted averaging prop-
erty.
(2)
There exists a weight function
w
:
X
Ñ
R
``
such that for every
A
P
X
̊
:
f
p
A
q “
ř
x
P
A
w
p
x
q
f
p
x
q
ř
x
P
A
w
p
x
q
.
(2.4)
Moreover, the function
w
is unique up to multiplication by a positive number.
We will now show that all aggregation rules satisfying the we
ighted av-
eraging property must be of the form of a combination of Examp
le
1
and
2
if for all
x
P
X
we can find
y, z
P
X
such that
f
pt
x
uq
, f
pt
y
uq
, and
f
pt
z
uq
are not collinear and the pairwise aggregation of
x, y, z
does not satisfy the
extreme aggregation property.
Definition 4.
An aggregation rule
f
:
X
̊
Ñ
H
is
strongly rich
if for any
x
P
X
there exist
y, z
P
X
such that:
(1)
f
pt
x
uq
, f
pt
y
uq
, and
f
pt
z
uq
are not on the same line.
(2)
f
pt
x, y
uq R t
f
p
x
q
, f
p
y
qu
and
f
pt
x, z
uq R t
f
p
x
q
, f
p
z
qu
5
.
Definition 5.
A binary relation
ě
on
X
is a
weak order
on
X
, if it is
reflexive (
x
ě
x
), transitive (
x
ě
y
and
y
ě
z
imply
x
ě
z
), and complete
(for all
x, y
P
X
,
x
ě
y
or
y
ě
x
). We say that
x
is equivalent to
y
, and
write
x
„
y
, if
x
ě
y
and
y
ě
x
.
Theorem 2.
Let the aggregation rule
f
:
X
̊
Ñ
H
be strongly rich. Then
the following are equivalent:
(1)
The aggregation rule
f
satisfies the weighted averaging axiom.
4
M
p
A,
ą
q P
A
and
M
p
A,
ą
q
ą
x
for
x
P
A
zt
M
p
A,
ą
qu
.
5
In the proof of our main result, we show that as long as
f
pt
x
uq
, f
pt
y
uq
, and
f
pt
z
uq
are not on the same line, then
f
pt
x, y
uq R t
f
p
x
q
, f
p
y
qu
and
f
pt
x, z
uq R t
f
p
x
q
, f
p
z
qu ñ
f
pt
y, z
uq R t
f
p
y
q
, f
p
z
qu
.
6
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
(2)
There exist a unique weak order
ě
on
X
and a weight function
w
:
X
Ñ
R
``
such that for every
A
P
X
̊
:
f
p
A
q “
ÿ
x
P
M
p
A,
ě
q
̈
̊
̋
w
p
x
q
ř
y
P
M
p
A,
ě
q
w
p
y
q
̨
‹
‚
f
p
x
q
.
(2.5)
Moreover in this case, the function
w
is unique up to multiplication by a
positive number in each of the equivalence classes of the wea
k order
ě
.
The representation (
2.5
) has two components: one is captured by the
weak order
ě
; the other is the weight function
w
. The weak order partitions
the set of features into equivalence classes and ranks them f
rom top to
bottom. If all models
x
P
A
have the same ranking, then the outcome
f
p
A
q
of
A
P
X
̊
is the weighted average of the outcomes of each element
x
P
A
. However, if some elements have a higher ranking than others
, then
the aggregation rule will ignore the lower-ordered element
s. Hence, the
assessment of the aggregation rule has two steps. First, it o
nly considers
the highest-ordered elements. Then, it uses the weight func
tion and finds
the weighted average among the highest-ordered elements.
The richness condition is necessary for both theorems
1
and
2
. Example
3
shows that without this condition, aggregation rules may sa
tisfy the strict
weighted averaging axiom without having a weighted average
representation.
Example 3.
Let
X
“ t
x, y, z
u
with
f
pt
x
uq “
0
, f
pt
y
uq “
1
{
2
, f
pt
z
uq “
1
, f
pt
x, y
uq “
1
{
4
, f
pt
y, z
uq “
3
{
4
, f
pt
x, z
uq “
3
{
8, and
f
pt
x, y, z
uq “
7
{
16.
Assume that there exists a positive weight function on
X
, and the aggre-
gation rule over any coalition of
X
has a representation as a weighted av-
erage of its elements. Assume that
w
:
X
Ñ
R
``
is the corresponding
weight function. In order to have such a representation, we s
hould have
f
pt
x, y
uq “
w
p
x
q
f
p
x
q`
w
p
y
q
f
p
y
q
w
p
x
q`
w
p
y
q
. By considering the value of
f
pt
x, y
uq
, f
pt
x
uq
,
and
f
pt
y
uq
, we get
w
p
x
q
w
p
y
q
“
1. Similarly, by considering the coalition
t
y, z
u
we get
w
p
y
q
w
p
z
q
“
1. By combining these two observations, we get
w
p
x
q
w
p
z
q
“
1. However, considering the coalition
t
x, z
u
, and the representation
f
pt
x, z
uq “
w
p
x
q
f
p
x
q`
w
p
z
q
f
p
z
q
w
p
x
q`
w
p
z
q
, we get
w
p
x
q
w
p
z
q
“
5
{
3, which is a contradiction.
Hence, the representation does not work in this case.
Assume
X
is a subset of a normed vector space. We will now show that
weight
w
in the representation (
2.5
) must be continuous if
f
is continuous
as defined below.
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
7
Definition 6.
An aggregation rule
f
:
X
̊
Ñ
H
is
(continuous)
if, for
any
A
P
X
̊
Y tHu
, any
x
P
X
z
A
, and any sequence
p
x
n
q
8
n
“
1
P
X
,
x
n
Ñ
x
implies
6
f
p
A
Y t
x
n
uq Ñ
f
p
A
Y t
x
uq
.
Theorem 3.
Let X be a subset of a normed vector space and let
f
:
X
̊
Ñ
H
be a strongly rich continuous aggregation rule satisfying t
he weighted aver-
aging property. Then the representation
(
2.5
)
holds true with a continuous
weight function
w
:
X
Ñ
R
``
. Furthermore, for any
x
P
X
there exists
ǫ
ą
0
such that
y
„
x
for all where
y
P
X
such that
|
y
́
x
| ă
ǫ
.
We will now show that if
X
is a convex subset of a normed vector space,
then any continuous aggregation rule on
X
under the weighted averaging
axiom can only have a single equivalence class, and as a conse
quence, both
the weighted averaging and strict weighted averaging prope
rties lead to the
representation (
2.4
) for
f
.
Theorem 4.
Let
X
be a convex subset of a normed vector space, and
f
:
X
̊
Ñ
H
a rich continuous aggregation rule satisfying the weighted
averaging
property. Then, there exists a continuous weight function
w
:
X
Ñ
R
``
such that the representation
(
2.4
)
holds true.
3. Preference aggregation and duality
In many cases, where the range of the aggregation rule is the s
et of linear
functionals, a simple geometrical interpretation of the we
ighted averaging
axiom results in a related but different consistency axiom. Le
t
H
be a
Hilbert space and write
x ̈
,
̈y
for the associated inner product. Let
S
Ă
H
be a convex subset of
H
. Every
h
P
H
induces a weak order (reflexive,
transitive, and complete binary relation)
Á
h
over the set
S
by:
s
1
Á
h
s
2
ô x
h, s
1
y ě x
h, s
2
y
(3.1)
Let
X
be a non empty set. In this section we define an aggregation rul
e
7
as
a function
f
mapping
X
̊
to
H
. Since we may interpret each
h
P
H
as a
linear ranking of the elements of the set
S
, the goal of an aggregator
f
is to
attach an aggregated linear ranking to every finite subset
A
of
X
.
Example 4.
A simple example of interpretation of
X, S
and
f
is as follows.
Let
X
be a set of experts and
S
a set of alternatives (models, decisions,
choices). An expert
x
P
X
defines a ranking/preference
f
pt
x
uq
over
S
.
An aggregation rule
f
is a voting mechanism enabling the aggregation of
the preferences of a finite set of experts. A rational notion o
f consistency
(employed here and formally introduced below in Definition
7
) is that if
A, B
P
X
̊
are two disjoint sets of experts such that both (
f
p
A
q
and
f
p
B
q
)
6
The convergence in
X
is with respect to the norm on
X
, and the convergence in the
range of the aggregation rule is with respect to norm of
H
.
7
By the Riesz representation theorem,
f
can also be defined as function mapping
X
̊
to the space of continuous linear functionals on
H
, in which case for
A
P
X
̊
,
f
p
A
q
is
identified with the unique element
h
P
H
such that
f
p
A
qp
x
q “ x
h, x
y
for
x
P
H
.
8
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
prefer
s
1
P
S
over
s
2
P
S
, then their aggregate
f
p
A
Y
B
q
must also prefer
s
1
over
s
2
.
Observing that the order (
3.1
) is invariant under scaling of
h
we will
restrict the range of aggregation rules to the set
N
ν
“ t
h
P
H
| x
h, ν
y “
1
u
for
some
ν
P
H
. This restriction also avoids entirely opposite ranking di
rections
by imposing a shared rank on
ν
. The condition of existence of such a
ν
is
what we call a
minimal agreement
condition.
Definition 7.
An aggregation rule
f
:
X
̊
Ñ
N
v
is
weakly consistent
if
for all disjoint sets
A, B
P
X
̊
, and for all
s
1
, s
2
P
S
,
s
1
Á
f
p
A
q
s
2
, s
1
Á
f
p
B
q
s
2
ñ
s
1
Á
f
p
A
Y
B
q
s
2
(3.2)
Moreover, it is
consistent
if it also satisfies the following condition:
s
1
ą
f
p
A
q
s
2
, s
1
Á
f
p
B
q
s
2
ñ
s
1
ą
f
p
A
Y
B
q
s
2
(3.3)
A simple duality argument (Farkas’s lemma) results in the fo
llowing the-
orem.
Theorem 5.
Let
f
:
X
̊
Ñ
N
ν
be an aggregation rule. Then, the followings
are equivalent:
(1)
f
is consistent.
(2)
f
satisfies the strict weighted averaging property.
Moreover, the followings are also equivalent:
(1)
f
is weakly consistent.
(2)
f
satisfies the weighted averaging property.
Using Theorem
1
, we immediately attain the representation of the con-
sistent aggregation rules.
Corollary 1.
Let
f
:
X
̊
Ñ
N
ν
be a consistent rich aggregation rule. Then,
there exists a weight function
w
:
X
Ñ
R
``
such that for every set of
features
A
P
X
̊
,
f
p
A
q “
ÿ
x
P
A
̈
̊
̋
w
p
x
q
ř
y
P
A
w
p
y
q
̨
‹
‚
f
p
x
q
.
(3.4)
Moreover, the weight function is unique up to multiplicatio
n by a positive
number.
Note that we can generalize the result to the case of weakly co
nsistent
rules.
Remark
1
.
The notion of consistency obtained as a dual interpretation
of
the weighted averaging is the same axiom as
Extended Pareto
introduced
by Shapley et al.,
1982
. Similarly, it is the
Combination axiom
in Gilboa
et al.,
2003
; Gilboa and Schmeidler,
2012
.
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
9
4. Belief Formation
In this section, we interpret the set of features as signals.
Each signal
contains some information about the distribution of states
of nature. The
role of an aggregation rule is an agent who makes a prediction
about the
true state of nature-based on observing some signals. In thi
s context, the
range of an aggregation rule is that of probability distribu
tions over the
states of nature. Following Billot et al.,
2005
, an aggregation rule is a
belief
formation process
that associates with each finite set of signals, a
belief
over
the states of nature.
The representation of the belief formation process under th
e weighted
averaging axiom is a straightforward application of the mai
n results. Using
our representation, on the one hand, we propose an extension
, where the
timing of signals may be important. We consider the case wher
e an agent
can receive signals in different time zones in the past. The age
nt tries to
form a prediction at the present time, and it may perceive sig
nals closer to
the time of the prediction as more credible. To capture the re
presentation,
we introduce the
stationarity
axiom, in which a belief induced by a set of
received signals and their timing is the same as the belief in
duced by shifting
the timings of all signals by a constant number to the past.
Under stationarity, any belief formation process satisfyi
ng the strict
weighted averaging axiom has a weight function over the set o
f signals and
an exponential discount factor over each time zone. The beli
ef associated
with a set of received signals is the discounted weighted ave
rage of the beliefs
associated with each signal. In this case, the weight functi
on captures the
time-independent value of each signal.
On the other hand, we interpret the set of signals as the infor
mation
structure of an agent who wants to predict the true state. We i
nterpret each
subset of signals as an event in her information structure. W
e show that as
long as the information structure has a finite cardinality, t
he strict weighted
averaging axiom is the necessary and sufficient condition for
a rich belief
formation process to appear as a Bayesian updater. This resu
lt answers the
question in Shmaya and Yariv,
2007
, regarding finding a necessary and suf-
ficient condition for a belief formation process to act as a
Bayesian updating
rule
.
4.1. Belief Formation Processes.
Let Ω
“ t
1
,
2
, . . . , n
u
be a set of states
of nature and let ∆
p
Ω
q
be the set of all probability distributions over Ω.
We interpret the elements of the set
X
as disjoint signals or events. The
role of an aggregation rule over a finite subset of
X
̊
is to predict the true
state of nature by assigning probabilities to each state of Ω
. Therefore,
following Billot et al.,
2005
, the aggregation rules can be interpreted as a
belief formation process
, which assigns a
belief
to the set of states of nature
after observing a finite subset of signals.
10
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
Definition 8.
A
belief formation process
is a function
f
:
X
̊
Ñ
∆
p
Ω
q
,
that associates with every finite set of signals
A
P
X
̊
, a
belief
f
p
A
q P
∆
p
Ω
q
on the states of nature.
Theorem
2
shows that if the belief induced by the union of two disjoint
finite sets of signals is on the line segment connecting the be
liefs induced
by each set of signals separately, then, under the strong ric
hness condition,
there exists a strictly positive weight function and a weak o
rder over the set
of signals such that the belief over any finite subset of signa
ls is a weighted
average of the beliefs induced by each of the highest-ordere
d signals of that
subset.
By enforcing the belief formation process to use both of the i
nduced be-
liefs,
i.e.
, the belief induced by the union of two disjoint finite sets of
signals
is on the “interior” of the line segment connecting the induc
ed belief of each
set of signals separately, we can use Theorem
1
to find the representation.
Formally, we have:
Corollary 2.
Let
f
:
X
̊
Ñ
∆
p
Ω
q
be a strongly rich belief formation process
satisfying the weighted averaging property. Then, there exi
st a unique weak
order
ě
on
X
and a weight function
w
:
X
Ñ
R
``
such that for every
A
P
X
̊
:
f
p
A
q “
ÿ
x
P
M
p
A,
ě
q
̈
̊
̋
w
p
x
q
ř
x
P
M
p
A,
ě
q
w
p
x
q
̨
‹
‚
f
p
x
q
.
(4.1)
Moreover, if
f
satisfies the strict weighted averaging property, then the w
eak
order
ě
has only one equivalence class and for every
A
P
X
̊
:
f
p
A
q “
ÿ
x
P
A
̈
̋
w
p
x
q
ř
x
P
A
w
p
x
q
̨
‚
f
p
x
q
.
(4.2)
Although representation (
4.2
) is, under the strict weighted averaging
property, similar to the one in Billot et al.,
2005
, their belief formation
process is defined over
sequences
of signals, in which each sequence can have
multiple copies of the same signal. In contrast, we define the
belief forma-
tion process over
sets
of signals, and there can be only one copy of a signal
in each set. Billot et al.’s main axiom, concatenation axiom
, is defined over
any two sequences of signals, and counts the number of each si
gnal in each
sequence. However, our strict weighted averaging property
expressed for
f
p
A
Y
B
q
does not allow
A
and
B
to both contain the same signal.
4.2. Role of Timing.
We now explore the role of the timing of signals
by associating signals with time labels. In that setting, a s
ignal closer to
the time of the prediction may be perceived as more credible (
have more
weight) compared to the same signal if it was received furthe
r in the past.
Formally, let X be the set of signals. The present time denote
d by 0, and
time
t
P
N
represents
t
units of time before the present time. For a given
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
1
1
finite subset of signals
A
P
X
̊
, let a function
T
A
:
A
Ñ
N
, represent the
timing of each signal in the set
A
,
i.e.
, for any signal
x
P
A
,
T
A
p
x
q
is the
time of receiving the signal
x
. Given a
c
P
N
,
T
A
`
c
represents a
time shift
of size
c
over the timing
T
A
of a set of received signals
A
. Finally, the set
X
T
“ tp
A, T
A
q |
A
P
X
̊
, T
A
:
A
Ñ
N
u
represents all possible realizations of
the received signals. In this context, a belief formation pr
ocess is a function
f
:
X
T
Ñ
∆
p
Ω
q
.
Our main consistency property, in addition to the strict wei
ghted aver-
aging property, is the
stationarity
property. A belief formation process is
stationary if a belief induced by a set of received signals an
d their timing
is the same as the belief induced by a constant shift of timing
s of the same
received signals. More precisely:
Definition 9.
A
stationary
belief formation is a function
f
:
X
T
Ñ
∆
p
Ω
q
such that
f
pp
A, T
A
`
c
qq “
f
p
A, T
A
q
,
for
A
P
X
̊
,
T
A
:
A
Ñ
N
, and
c
P
N
.
The next proposition characterizes stationarity belief-f
ormation processes
satisfying the strict weighted averaging property.
Proposition 1.
Let a rich and stationary belief formation process
f
:
X
T
Ñ
∆
p
Ω
q
satisfy the strict weighted averaging property. Then, there
exist a
unique discount factor
q
P p
0
,
8q
and a unique (up to multiplication by a
positive number) weight function
w
:
X
Ñ
R
``
, such that for all
p
A, T
A
q P
X
T
:
f
p
A, T
A
q “
ř
x
P
A
q
T
A
p
x
q
w
p
x
q
f
p
x
q
ř
x
P
A
q
T
A
p
x
q
w
p
x
q
.
(4.3)
As a consequence of the representation, under the assumptio
n of the
proposition, the weight over a received signal
x
P
A
can be separated into
two separate factors. One is the intrinsic value of the signa
l, captured by
w
p
x
q
. The other one is the role of timing, captured by
q
T
A
p
x
q
. Moreover, the
only discounting that captures the role of the timing is the e
xponential form.
If
q
“
1, the timing is not important. Hence, the belief formation p
rocess
only considers the intrinsic value of each signal. However,
when
q
‰
1, the
belief formation process places relatively more (
q
P p
0
,
1
q
) or less (
q
P p
1
,
8q
)
weight on a signal received closer to the time of the predicti
on.
4.3. Bayesian Updating.
Let
p
X, X
̊
Y tHuq
be the measure space of
events, where
X
has a finite number of disjoint events. The space of events
captures the information structure of the belief formation
process. Similarly,
by considering the set Ω
“ t
1
, . . . , n
u
, we denote
p
Ω
,
2
Ω
q
as the measure
space of states of nature, where 2
Ω
is the set of subsets of the set Ω. For any
probability distribution
d
P
∆
p
Ω
q
and any subset of the state of nature
B
P
12
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
Ω, let
d
p
B
q
denote the probability of
B
which is induced by the distribution
d
. Hence,
d
p
B
q “
ř
ω
P
B
d
p
ω
q
.
Definition 10.
A belief formation process
f
:
X
̊
Ñ
∆
p
Ω
q
is
Bayesian
, if
there exists a probability measure
P
on the space
p
Ω
ˆ
X,
2
Ω
ˆ
X
q
, such that
for every
A
P
X
̊
and
B
P
2
Ω
we have:
`
f
p
A
q
̆
p
B
q “
P
p
B
ˆ
A
q
P
X
p
A
q
(4.4)
where,
P
X
is the marginal probability distribution of
P
over
X
.
The right-hand side of the previous equation is the conditio
nal probability
of
B
given
A
. Therefore, a Bayesian belief formation process
f
behaves as
a Bayesian updater: by observing an event
A
in her information structure
X
̊
, her prediction about the probability of the true state bein
g in a subset
B
P
Ω comes from the Bayes rule. To put it differently,
`
f
p
A
q
̆
p
B
q
is equal
to the conditional probability
P
p
B
|
A
q
.
Our next proposition shows that our strict weighted averagi
ng axiom is
the necessary and sufficient condition for a rich belief forma
tion process to
be Bayesian.
Proposition 2.
A rich belief formation process is Bayesian if and only if
it satisfies the strict weighted averaging property.
Note that the richness condition is crucial. Otherwise, as s
hown in Ex-
ample
3
, there are cases where a belief formation process satisfies t
he strict
weighted averaging axiom, but it is not a Bayesian updater. W
e will now
present a more general version of Proposition
2
by adding the strong rich-
ness condition and weakening the strict weighted averaging
condition to the
weighted averaging property. In the more general version, i
t is possible to
have zero probability events. The belief formation process
behaves as a
Bayesian updater, even conditional on observing a zero prob
ability event.
To capture the idea, we need the following definition.
Definition 11.
A class of functions
t
P
A
|
P
A
: 2
Ω
ˆ
X
̊
Ñ r
0
,
1
s
, A
P
X
̊
u
is
a
conditional probability system
if it satisfies the following properties:
(1) For every
A
P
X
̊
such that
A
‰ H
,
P
A
is a probability measure on
Ω
ˆ
X
with
P
A
p
Ω
ˆ
A
q “
1.
(2) For every disjoint events
A
1
, A
2
P
X
̊
and for every
C
P
Ω
ˆ
X
, we
have:
P
A
1
Y
A
2
p
C
q “
P
A
1
Y
A
2
p
Ω
ˆ
A
1
q
P
A
1
p
C
q `
P
A
1
Y
A
2
p
Ω
ˆ
A
2
q
P
A
2
p
C
q
In this definition, the probability measure
P
Ω
represents a prior probabil-
ity measure, and
P
A
represents a posterior (conditional) probability prob-
ability given the event
A
. Therefore, for any set
B
P
Ω,
P
A
p
B
ˆ
A
q
is the
conditional probability of
B
given
A
. Moreover, for any two events
A
2
Ă
A
1
in
X
̊
,
P
A
1
p
Ω
ˆ
A
2
q
is the conditional probability of event
A
2
given
A
1
.
The first property of Definition
11
requires that the support of the posterior
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
1
3
probability conditioned on an event
A
is contained in
A
. The second prop-
erty requires that, conditional on the event
A
1
Y
A
2
, the Bayes updating
rule should be satisfied even if the prior probability of
A
1
Y
A
2
is zero.
Definition 12.
A belief formation process
f
:
X
̊
Ñ
∆
p
Ω
q
is
rational-
izable by a conditional probability system
t
P
A
|
P
A
: 2
Ω
ˆ
X
̊
Ñ
r
0
,
1
s
, A
P
X
̊
u
if every
A
P
X
̊
and
B
P
2
Ω
we have:
`
f
p
A
q
̆
p
B
q “
P
A
p
B
ˆ
A
q
.
(4.5)
By adding the strong richness condition, the next theorem sh
ows that
the weighted averaging axiom is the necessary and sufficient c
ondition for
rationalizing a belief formation process by a conditional p
robability system.
Proposition 3.
A strongly rich belief formation process is rationalizable
by a conditional probability system if and only if it satisfie
s the weighted
averaging axiom.
Remark
2
.
Shmaya et al.,
2007
considers the problem of characterizing the
updating rules
(in our context the belief formation processes) that appear
to be Bayesian. By providing an example, they show that their
soundness
condition
, our strict weighted averaging condition, is not a sufficient
con-
dition for an updating rule to behave as a Bayesian updater. H
owever, we
show that the strict weighted averaging condition is the nec
essary and suffi-
cient condition as long as the belief formation process sati
sfies our richness
condition.
5. Average Choice Functions
In this section, the set of features is a subset of
R
n
. We interpret each
feature as a choice object. The interpretation of the aggreg
ation rule is a
decision-maker that selects a choice randomly from a menu of
choice objects.
We model the decision-maker as an
average choice function
that associates
with any menu of choice objects, an average choice (mean of th
e distribu-
tion of choices) in the convex combination of choice objects
. The average
choice is easier to report and obtain rather than the entire d
istribution
8
.
However, except for the case where elements of a menu are affine
ly indepen-
dent, average choice does not uniquely reveal the underlyin
g distribution of
choices.
First, using our main representation, we show that it is poss
ible to
uniquely extract the underlying distribution of choices as
long as the av-
erage choice function satisfies the weighted averaging axio
m.
Then, we illustrate two applications of the result. In one ap
plication, we
consider the class of average choice functions that can be ra
tionalized by a
Luce rule
,
i.e.
, a stochastic choice function that satisfies the
independence
of irrelevant alternatives
axiom (IIA) proposed by Luce,
1959
. We show
8
Check Ahn, Echenique, and Saito,
2018
for the complete discussion on merits of
average choice.
14
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
that the average choice functions satisfying the strict wei
ghted averaging
axiom are exactly the ones that can be rationalized by a Luce r
ule. More
generally, we show that the class of average choice function
s satisfying the
weighted averaging axiom is the same as the class of average c
hoice functions
rationalizable by a
two-stage Luce
model proposed by Echenique and Saito,
2018
.
In the second application, we consider continuous average c
hoice func-
tions. First, we show that any continuous average choice fun
ction under the
weighted averaging axiom is rationalizable by a Luce rule. T
his means that
there is no continuous average choice function that is ratio
nalizable by a
two-stage Luce rule but not with a Luce rule.
Then, we illustrate a connection of our result with the one by
Kalai and
Megiddo,
1980
, regarding the impossibility of an average choice function
to
satisfy both the
path independence
axiom and continuity.
5.1. Set up.
In this section,
X
is a nonempty subset of
R
n
, which is not
a subset of a line. For any
A
Ď
R
n
, we denote by Conv
p
A
q
the set of all
convex combinations of vectors in
A
.
Definition 13.
An aggregation rule
f
:
X
̊
Ñ
R
n
is called an
average
choice function
, if for any (menu of choices)
A
P
X
̊
,
f
p
A
q P
Conv
p
A
q
.
One of the goals of this section is to present a connection bet
ween our
weighted averaging condition and the Path Independent, Luc
e, and two-
stage Luce choice models. The following is a corollary of the
orems
2
and
4
,
Corollary 3.
Let an average choice function
f
:
X
̊
Ñ
Conv
p
X
q
be strongly
rich. The following statements are equivalent:
(1)
The average choice function
f
satisfies the weighted averaging con-
dition.
(2)
There exists a unique weak order
ě
on
X
and a unique weight func-
tion
w
:
X
Ñ
R
``
, up to multiplication over equivalence classes of
the weak order such that for every
A
P
X
̊
:
f
p
A
q “
ř
x
P
M
p
A,
ě
q
w
p
x
q
x
ř
x
P
M
p
A,
ě
q
w
p
x
q
“
ÿ
x
P
M
p
A,
ě
q
̈
̊
̋
w
p
x
q
ř
x
P
M
p
A,
ě
q
w
p
x
q
̨
‹
‚
x.
(5.1)
Moreover, if the average choice function
f
satisfies continuity and the
weighted averaging condition, the weight function
w
is continuous and the
weak order
ě
is the equivalence order. In this case, for every
A
P
X
̊
:
f
p
A
q “
ř
x
P
A
w
p
x
q
x
ř
x
P
A
w
p
x
q
“
ÿ
x
P
A
̈
̋
w
p
x
q
ř
x
P
A
w
p
x
q
̨
‚
x.
(5.2)
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
1
5
5.2. Luce Rationalizable Average Choice Functions.
The following
definitions are standard definitions in the context of indivi
dual decision-
making.
Definition 14.
A
stochastic choice
is a function
ρ
:
X
̊
Ñ
∆
p
X
q
, such
that
ρ
p
A
q P
∆
p
A
q
for any
A
P
X
̊
.
For an average choice function
f
:
X
̊
Ñ
Conv
p
X
q
and a menu
A
P
X
̊
,
f
p
A
q P
Conv
p
A
q
. Therefore, there exists a stochastic choice
ρ
:
X
̊
Ñ
∆
p
X
q
(which may not be unique) that rationalizes the average choi
ce function
f
,
i.e.
,
f
p
A
q “
ř
x
P
A
ρ
p
x, A
q
x
, where
ρ
p
x, A
q
is the probability of selecting the
element
x
from the menu
A
.
One appealing form of a stochastic choice function is the one
that sat-
isfies
Luce’s IIA
,
i.e.
, the probability of selecting an element over another
element is independent of any other element. Luce,
1959
shows that sto-
chastic choices that satisfy the IIA axiom are in the form of L
uce rules.
Definition 15.
A stochastic choice
ρ
:
X
̊
Ñ
∆
p
X
q
is a
Luce rule
if there
is a function
w
:
X
Ñ
R
``
, such that:
ρ
p
x, A
q “
w
p
x
q
ř
y
P
A
w
p
y
q
.
Furthermore, if
w
is continuous, then
ρ
is a continuous Luce rule.
Definition 16.
An average choice function
f
is rationalizable by a stochastic
choice
ρ
, if for all
A
P
X
̊
:
f
p
A
q “
ÿ
x
P
A
ρ
p
x, A
q
x.
Furthermore, if there exists a Luce rule that rationalizes t
he average choice
function
f
, then
f
is
Luce rationalizable
.
By considering our Theorem
1
and corollary
3
, a choice
f
has a Luce form
representation,
i.e
,
f
p
A
q “
ř
x
P
A
p
w
p
x
q
ř
x
P
A
w
p
x
q
q
x
if and only if it satisfies the strict
weighted averaging condition. As a result:
Corollary 4.
An average choice function is Luce rationalizable if and only
if it satisfies the strict weighted averaging condition. Mor
eover, the Luce
rule that rationalizes the average choice function is uniqu
e.
Furthermore, an average choice function is continuous Luce r
ationalizable
if and only if it is continuous and satisfies the strict weight
ed averaging
condition.
In the Luce model, the decision-maker selects each element o
f a given
menu with a strictly positive probability. However, this is
not a plausible
assumption in many situations. The decision-maker may alwa
ys select a
better choice between two alternatives. We model this behav
ior by a two-
stage Luce model. Echenique et al.,
2018
introduces the two-stage Luce
16
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
model. In this model, there exists a ranking order and a weigh
t function
over elements. A decision-maker choosing from a menu only se
lects the
highest-ordered elements from the menu. The probability of
the selection
of each highest-ordered element is related to the weight ass
ociated with the
element. Formally:
Definition 17.
A stochastic choice
ρ
:
X
̊
Ñ
∆
p
X
q
is a
two-stage Luce
rule
if there are a function
w
:
X
Ñ
R
``
and a weak order
ě
over elements
of
X
, such that:
ρ
p
x, A
q “
#
w
p
x
q
ř
y
P
M
p
A,
ě
q
w
p
y
q
if
x
P
M
p
A,
ě
q
,
0
otherwise
.
(5.3)
Given
A
P
X
̊
, the decision-maker only selects the elements in
M
p
A,
ě
q
,
that are the highest-ordered elements of
A
. She chooses each element of
M
p
A,
ě
q
with a probability associated with its weight.
By considering our Theorem
2
, any average choice function under the
weighted averaging axiom is rationalizable by a two-stage L
uce rule.
Corollary 5.
A strongly rich average choice function is two-stage Luce
rationalizable if and only if it satisfies the weighted avera
ging axiom. More-
over, the two-stage Luce rule that rationalizes the average c
hoice function is
unique.
Remark
3
.
Under the continuity condition, using Theorem
4
implies that
both the two-stage Luce model and Luce model are equivalent.
The next
section discusses this observation.
5.3. Continuous Average Choice Functions.
In this section, we con-
sider the class of continuous average choice functions sati
sfying the weighted
averaging condition. First, we reinterpret our corollary
3
as an impossibility
result. This means that no continuous average choice functi
on is rationaliz-
able by a two-stage Luce model but not by a Luce model. Then, we
show
the connection with the impossibility result by Kalai and Me
giddo,
1980
,
regarding the impossibility of a choice function satisfyin
g both the path
independence and continuity.
Plott,
1973
extensively studies choice functions under the path indepe
n-
dence axiom. Plott’s notion of path independence requires a
choice from the
union of two disjoint menu
A
Y
B
, to be the choice between the choice from
A
and the choice from
B
. Using this axiom, the choice from any menu can be
recursively obtained by partitioning the elements of the me
nu into disjoint
sub-menus. Then, the choice from the whole menu would be the c
hoice from
the choices of each sub-menu. In our setup, for an average cho
ice function
f
, we have:
Definition 18.
f
satisfied the
(path independence)
condition if
f
p
A
Y
B
q “
f
pt
f
p
A
q
, f
p
B
quq
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
1
7
for all
A, B
P
X
̊
such that
A
X
B
“ H
.
The path independence condition is stronger than our weight
ed av-
eraging condition. In other words, any average choice funct
ion under
Plott’s notion of path independence satisfies the weighted a
veraging con-
dition. More precisely, given a choice function
f
:
X
̊
Ñ
Conv
p
X
q
and
two disjoint menus
A, B
P
X
̊
, under the path independence condition,
f
p
A
Y
B
q “
f
pt
f
p
A
q
, f
p
B
quq
. By the definition of average choice functions,
f
pt
f
p
A
q
, f
p
B
quq P
Conv
p
f
p
A
q
, f
p
B
qq
, which shows that the choice function
f
satisfies the weighted averaging axiom.
Continuity is an appealing property of an average choice fun
ction. It
specifies that by replacing an element of a menu with another e
lement close
to it, with respect to the norm of
X
, the average choice of the new menu is
close to the average choice of the previous menu. Kalai and Me
giddo,
1980
shows that there is no average choice function that satisfies
both path inde-
pendence axiom and continuity. Here, we reinterpret the res
ult of corollary
3
to show a more general result for average choice functions.
Corollary
3
states that, for a strongly rich continuous average choice f
unc-
tion
f
:
X
̊
Ñ
Conv
p
X
q
satisfying the weighted averaging condition, there
exists a unique weight function
w
:
X
Ñ
R
``
such that for any
A
P
X
̊
:
f
p
A
q “
ÿ
x
P
A
p
w
p
x
q
ř
x
P
A
w
p
x
q
q
x.
There are two important observations regarding the represe
ntations
above.
First, through discussions in Section
5.2
, the representation shows that
any continuous average choice function that is rationaliza
ble by a two-stage
Luce model is also rationalizable by a Luce model. Second, si
nce the function
w
is strictly positive, the average choice of any menu should b
e in the relative
interior of the convex hull of members of the menu.
As a result, our impossibility result specifies that for an av
erage choice
function that satisfies the weighted averaging condition, i
t is impossible to
satisfy the continuity condition and also to have a choice fr
om a menu that
is on the relative boundary of the elements of the menu. We sum
marize the
observation in the following corollary
9
.
Corollary 6.
If
X
is a nonempty convex subset of a vector space that con-
tains at least three non-collinear points, then an average c
hoice function
9
To see the connection between our corollary
6
and the result in Kalai and Megiddo,
1980
, it is enough to consider a menu with three non-collinear mem
bers. Kalai and
Megiddo,
1980
, Thm. 1 shows that the average choice of a path independent av
erage
choice function from any menu is the average choice of the ave
rage choice function from a
sub-menu of two members of the menu. This shows that the avera
ge choice from a menu
with three non-collinear members is on the line segment conn
ecting two of the member of
the menu. As a result, the choice should be on the relative bou
ndary of the menu. That
is why it cannot satisfy continuity.
18
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
f
:
X
̊
Ñ
X
that satisfies the weighted averaging condition cannot be bo
th
continuous and contains a menu
A
P
X
̊
, with
f
p
A
q P B
r
p
Conv
p
A
qq
.
6. Extended Pareto Aggregation Rules
This section demonstrates an application of section
3
in the social choice
problems. In this domain, each feature represents a prefere
nce ordering of
individuals over a set of alternatives. Each preference ord
ering satisfies the
axiom of Von-Neumann and Morgenstern,
1944
. The role of an aggrega-
tion rule is to associate with each coalition of individuals
another vN-M
preference ordering over the set of alternatives.
An appealing property of an aggregation rule, in this contex
t, is to satisfy
the
extended Pareto
axiom. Shapley et al.,
1982
introduced the extended
Pareto. It specifies that, if two disjoint coalitions of indi
viduals, each prefers
an outcome over another outcome, then the union of the coalit
ions also
should prefer the same outcome over the other one. Moreover,
if one of
them strictly prefer one outcome over the other one, then the
union of the
coalitions should also strictly prefer the same outcome ove
r the other one.
First, we show that under a normalization of cardinal utilit
ies of indi-
viduals and a minor richness condition, aggregation rules u
nder the strict
weighted averaging (weighted averaging) axiom are exactly
aggregation rules
under the
extended Pareto
(
extended weak Pareto
) axiom.
Following the equivalence, we use our main representation r
esult as a
technical tool to pin down the representation of the extende
d Pareto aggre-
gation rules. We show that the only possible extended Pareto
aggregation is
to have a positive weight over each individual in the society
. Then, the ag-
gregated preference ordering of a given group of individual
s is the weighted
sum of their preference ordering.
The representation can be considered as a multi-profile vers
ion of the the-
orem by Harsanyi,
1955
on Utilitarianism. Harsanyi considers a single profile
of individuals and a variant of Pareto to get the Utilitarian
ism. However,
in our approach, we partition a profile into smaller groups. T
hen, we ag-
gregate the preference ordering of these smaller groups usi
ng the extended
Pareto. Hence, we get the Utilitarianism through this consi
stent form of
aggregation. As a result, in our representation, the weight
associated with
each individual appears in all sub-profiles that contain her
.
10
In Section
6.3
, we extend our result on extended Pareto aggregation rules
to the class of
generalized social welfare function
. Unlike our previous model,
individuals may have different preference orderings. Theref
ore, the domain
of the generalized social welfare function is a set of all diffe
rent groups (with
all possible sizes) of individuals, with each individual ha
ving all different
possible preference orderings. Our definition of generaliz
ed social welfare
10
Similar to the discussion of Weymark,
1991
regarding the debate of Sen-Harsanyi,
our result is better to be interpreted as a representation ra
ther than a justification of the
utilitarianism.
AGGREGATION OF MODELS, CHOICES, BELIEFS, AND PREFERENCES
1
9
function extends the standard definition used by Arrow,
1963
, in which the
domain is a set of fixed-length profiles of individuals.
For a technical reason, we restrict the set of vN-M preferenc
es to those
in which all of them strictly prefer one fixed lottery to anoth
er fixed one.
We show that the only possible extended Pareto generalized s
ocial welfare
functions are the ones that associate a positive number to ea
ch individual’s
preferences (unlike the previous section, in which each wei
ght depends on
both the individual and the whole profile), and it associates
each coalition
with the weighted sum of their cardinal utility using the wei
ght associated
to their preferences.
The important observation is that,
each positive weight in the representa-
tion is independent of the other individuals in any profiles
. The weight only
depends on each individual and her own preference ordering.
Our representation above has a positive nature, compared to
the claims
by Kalai and N. Schmeidler,
1977
and Hylland,
1980
that the negative con-
clusion of Arrow’s theorem holds even with vN-M preferences
. Moreover,
the representation provides an answer to the main concern of
Borgers and
Choo,
2017a
; Borgers and Choo,
2017b
regarding the correctness of the main
theorem of Dhillon,
1998
.
Dhillon,
1998
by considering a set of axioms, other than the ones by
Arrow, provides one of the first axiomatizations of relative
utilitarianism as
a possibility result. However, Borgers et al.,
2017a
shows a counterexample
to their representation. Our representation fixes the error
using our variant
of the extended Pareto axiom and our restricted domain of the
generalized
social welfare function.
Finally, adding the anonymity and the weak IIA axiom of Dhill
on,
1998
gives us the relative utilitarianism as one possible choice
of the weight func-
tion. However, the primary concern of our paper is to show tha
t
the weighted
averaging of preferences is the only generalized social wel
fare function that
respects extended Pareto
. The possible choices of weights are not our focus
in this paper.
6.1. Set up.
Let the set
M
“
t
0
,
1
, . . . , m
u
and
L
“
tp
p
1
, . . . , p
m
q|
ř
m
i
“
1
p
i
ď
1
, p
i
ě
0
u
. A lottery
p
P
L
associates the
probability
p
i
to the prospect
i
P
M
zt
0
u
and 1
́
ř
m
i
“
1
p
i
to the prospect 0.
A
vN-M preference
over the set
L
is a preference relation that satisfies
the axioms of Von-Neumann et al.,
1944
as defined below
11
.
Definition 19.
We say that
R
is a vN-M preference over the set
L
if it is
a weak order and if there exists a
u
P
R
m
, known as a utility, such that for
any
x, y
P
L
,
xRy
if and only if
u
̈
x
ě
u
̈
y
where “
̈
” represents the inner
product in
R
m
. Moreover, the (unique)
ray
U
“ t
αu
|
α
ą
0
u
contains all
normalized affine utilities that represent the vN-M preferen
ce
R
. We write
11
If
R
is a vN-M preference over the set
L
, then, by the vN-M theorem, there exists
an affine representation of the preference
R
. For notational convenience, we normalize all
affine representations to have the value 0 over the prospect 0.
20
HAMED HAMZE BAJGIRAN, HOUMAN OWHADI
R
for the set of all vN-M preferences over
L
and
R
for the strict part of the
preference
R
P
R
.
Let
X
“ t
1
, . . . , n
u
represent the set of all agents and
X
̊
be the set of
all finite subsets of
X
. Write
R
X
for the X-Fold Cartesian product of
R
.
Every
R
X
P
R
X
defines a
preference profile
of the set of agents over the set
of lotteries.
Definition 20.
A
group aggregation rule
on X is a function
f
:
X
̊
Ñ
R
,
that associates with every coalition of agents
A
P
X
̊
a vN-M preference
f
p
A
q P
R
.
An rational property of group aggregation rules is that when
ever two
disjoint coalitions,
e.g.
A, B
P
X
̊
, both prefer a lottery
x
to another lottery
y
, then their union,
A
Y
B
, also prefers the lottery
x
to the lottery
y
.
Definition 21.
A group aggregation rule
f
:
X
̊
Ñ
R
satisfies the
ex-
tended Pareto property
if for all disjoint coalitions of agents
A, B
P
X
̊
,
and for all lotteries
x, y
P
L
,
x f
p
A
q
y, x f
p
B
q
y
ñ
x f
p
A
Y
B
q
y
(6.1)
x
f
p
A
q
y, x f
p
B
q
y
ñ
x
f
p
A
Y
B
q
y
(6.2)
Our last condition requires the existence of two lotteries i
n the set of
lotteries, in which all agents strictly prefer one over the o
ther.
Definition 22.
A group aggregation rule
f
:
X
̊
Ñ
R
satisfies the
minimal
agreement condition
if there exist two lotteries
x, x
P
L
such that for every
agent
i
P
X
,
x
f
p
i
q
x
.
Remark
4
.
Let a group aggregation rule
f
:
X
̊
Ñ
R
satisfy both the
minimal agreement and extended Pareto axiom. Given two agen
ts
i, j
P
X
,
by applying the strict part of the definition of the extended P
areto axiom, we
have
x
f
pt
i, j
uq
x
. Similarly, for every coalition of agents
A
P
X
̊
, recursively
using the strict part of the extended Pareto axiom, we deduce
x
f
p
A
q
x
.
Remark
5
.
Let the vector
v
P
R
m
be
x
́
x
, where
x, x
are the two lotteries in
the definition of the minimal agreement condition. Let
u
i
P
R
m
represent the
vN-M preference
f
p
i
q
. Hence,
x
f
p
i
q
x
if and only if
u
i
̈
v
ą
0. Therefore, the
definition of the minimal agreement condition is equivalent
to the existence
of a direction
v
P
R
m
such that for all
i
P
X
,
u
i
̈
v
ą
0. this interpretation
of
v
is exactly the role of
ν
in section
3
.
6.2. The Representation of Extended Pareto Group Aggregation
Rules.
In this section, we assume that the group aggregation rule
f
:
X
̊
Ñ
R
satisfies the minimal agreement condition. In particular, w
e assume that
all agents strictly prefer the lottery
x
P
L
over the lottery
x
P
L
. Considering
remark
5
, we define
v
“
x
́
x
as the direction that every agent agrees on. For
a coalition of agents
A
P
X
̊
, let the ray
U
A
represents the vN-M preference
f
p
A
q
. Let
H
:
“ t
u
P
R
m
|
u
̈
v
“
1
u
represent the normalization of utilities