Approval as an Intrinsic Part of Preference

The collective decision making problem can be conceived as the aggregation of a vector of utility functions whose informational content depends on the assumptions made about the cardinality and interpersonal comparability of individual preferences. To be more explicit, we consider a non-empty set

N

of individuals and a non-empty set

A

of alternatives. Letting

U

(

A

) be the set of real-valued “utility functions” defined over

A

, we model the problem through an aggregation function

f

:

U

(

A

)

N

→ 2

A

\{Ø} The assumptions about the cardinality and interpersonal comparability of individual preferences are formalized by partitioning

U

(

A

) into information sets, while requiring

f

to be invariant at any two vector of utility functions which belong to the same information set. At one extreme, one can assume the existence of an absolute scale over which the utilities of individuals are measured and compared. This assumption partitions

U

(

A

) into singleton information sets, hence imposing no invariance over

f

. At the other extreme, one can rule out any kind of cardinal information and interpersonal comparability, in which case an information set consists of the elements of

U

(

A

) which are ordinally equivalent, i.e., induce the same ordering of alternatives for every individual.

1

When cardinality and interpersonal comparability are ruled out, the problem can be modeled through an aggregation function

f

:

W

(

A

)

N

→ 2

A

\{Ø} where

W

(

A

) is the set of weak orders (i.e., complete and transitive binary relations) over

A

. We refer to this as the

Arrovian model

(Arrow 1950, 1951).