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relations must present, besides transitivity and completeness, other properties of
continuity, monotonicity, substitutability and decomposability.
Continuity means that for any weight distributions
p
,
q
and
r
on C, with
p q
and
q r
, there is a real
C.
Monotonicity means that if
p
and
q
are weight distributions concentrated on {c
1
,
c
2
} (that means, such that
p
(c
1
)+
p
(c
2
)=
q
(c
1
)+
q
(c
2
) = 1), for a pair of criteria (c
1
,
c
2
), if c
1
c
2
and
p
(c
1
)>
q
(c
1
), then
p q
.
Substitutability means that if
p
(c
1
)=
q
(c
2
) and
p
and
q
assign the same value for
any other criterion c, then c
1
*
ʱ∈
[0, 1] such that
q
(c) =
ʱ
p
(c) + (1
− ʱ
)
r
(c) for any c
∈
c
2
implies
p
q
.
*
Decomposability employs the de
nition of
p
ω
: for any distribution
on D(C),
ω
p
ω
denotes the distribution on C determined by
X
p
x
c
ðÞ
¼
p
2
D
ðÞ
x
ðÞ
p
ðÞ:
c
Decomposability holds if and only if
x
1
¼
x
2
is equivalent to
p
x
1
¼ p
x
2
:
These properties are not as natural as they may seem to be. But to determine
weights for preference criteria, what is going to be useful from Expected Utility
Theory is the representation theorem asserting that for any set of criteria C and any
preference relation
on C for which there exists an extension
to D(C) with some
properties,
identi
es a unique distribution of weights u on C such that
and a unique distribution of weights u on D(C) such that, for
ˉ
1
concentrated in
c
1
and
ˉ
2
concentrated in c
2
,
This u is de
ned on D(C) by
X
m
u
ð
x
Þ
¼
1
x
ð
c
j
Þ
u
ð
c
j
Þ
for any
x
2
D
ð
C
Þ:
j¼
If there is an outcome c
0
such that
X
u
ð
c
0
Þ
¼
x
ð
c
j
Þ
u
ð
c
j
Þ;
j
this outcome c
0
may be seen as the certainty equivalent of
, in the sense that a
distribution of weights with the unique outcome c
o
has the same expected utility of
ˉ
.
When the outcomes in C have numerical values, besides computing the expected
utility we can compute the expected outcome
ˉ
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