Journal of Mathematical Economics 29 Ž1998. 255–269
Variations on the measure representation
approach
Marco LiCalzi
)
Department of Applied Mathematics, UniÕersity of Venice, Dorsoduro 3825r E, 30123 Venezia, Italy
Received 1 July 1996; accepted 19 March 1997
Abstract
The measure approach represents a preference relation over functions by the measure of
their epigraphs Žor hypographs.. This paper proves a measure representation theorem for a
class of increasing functions and shows how its proof can be modified to yield another
measure representation theorem for functions of bounded variation. q 1998 Elsevier
Science B.V.
JEL classification: D81
Keywords: Preference relation; Representation; Measure; Increasing functions; Functions of bounded
variation
1. Introduction and theme
The measure approach to the representation of preferences over lotteries was
proposed by Segal Ž1989; first version 1984. as a generalization of the rank-dependent model initiated by Quiggin Ž1982. and Schmeidler Ž1989; first version 1982..
According to Ža version of. the rank-dependent model, a monetary lottery X on
the positive reals with cumulative distribution F can be evaluated by the functional
RDEU Ž X . s
H uŽ x . dŽ g ( F . Ž x . ,
Ž 1.
where g : w0,1x ™ w0,1x is strictly increasing and onto. This form reduces to the
)
Corresponding author. Fax: q39-41-5221756; e-mail: [email protected].
0304-4068r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.
PII S 0 3 0 4 - 4 0 6 8 Ž 9 7 . 0 0 0 1 9 - 0
256
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
special case of expected utility when the probability transformation function g is
the identity, but it still separates attitudes to outcomes, modeled through u, and
attitudes to probabilities. modeled through g.
Segal Ž1989. pointed out that the rank-dependent model may be interpreted as if
the underlying preference relation K over lotteries is represented by a Žproduct.
measure of the epigraphs of the cumulative distribution functions. In fact, assume
without loss of generality that uŽ0. s 0 and note that the two increasing functions
u and g are defined respectively on the outcomes axis and on the probability axis,
so that we can compute the measure of the rectangle w x 1 , x 2 . = w p 1 , p 2 . by the
product w uŽ x 2 . y uŽ x 1 .x P w g Ž p 2 . y g Ž p 1 .x.
Consider a lottery X with a finite number of possible outcomes 0 F x 1 F PPP
F x n . Let F Ž x 0 . s 0 and Ž1. reduces to
n
RDEU Ž X . s Ý Ž g ( F . Ž x i . y Ž g ( F . Ž x iy1 . u Ž x i . ,
is1
which is precisely the sum of the measures of the rectangles w0, x i . = w F Ž x iy1 .,
F Ž x i ... The union of these rectangles constitutes the epigraph of F truncated from
above at the level p s 1.
Following this interpretation, a natural extension of the rank-dependent model
is to represent the preference relation K over lotteries by a general Žnot
necessarily product. measure of the truncated epigraphs. This idea was pursued in
Segal Ž1989., Green and Jullien Ž1988. and Chew and Epstein Ž1989. but the
statements of their representation theorems were incorrect, as shown by the
counterexamples in Wakker Ž1993.. Later on, Wakker Ž1993a. gave a thorough
discussion of the difficulties related to the measure representation approach and
correct proofs were offered in Segal Ž1993., Chew et al. Ž1993. and Chateauneuf
Ž1996: first version 1990.. Recently, Chew and Wakker Ž1996. generalized the
representation from lotteries to acts.
The simple idea of a measure representation may be applied for preferences
over many relevant classes of functions. See Chateauneuf Ž1985. and Lehrer
Ž1991. for a very general approach, closely related to the theory of qualitative
probability. Among the possible examples, we mention: Ži. cumulative distribution
functions or, alternatively, probability densities; Žii. cumulated cash flows or,
alternatively, gainrloss processes; Žiii. income distribution functions on the real
line or alternatively, concentration curves; and Živ. investment or, alternatively,
consumption profiles for a single good over time.
Most of these examples, however, refer to functions that are not necessarily
increasing and therefore require versions of the measure representation theorem
different from the ones available in the literature. So far, instead, the measure
representation approach has been mainly concerned with preferences over lotteries
represented by their cumulative distribution functions, i.e. with bounded, increasing and right-continuous functions.
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
257
The main purpose of this paper is to prove a similar result for a more general
class of functions as one of the many possible ‘ variations’ on the theme of the
measure representation approach.
The paper is organized as follows. The first variation is a theorem for the
measure representation of preferences over increasing functions. The second
variation presents some technical comments on this result and prepares for the
third and last variation, which shows how to modify the proof given for increasing
functions and get a measure representation theorem for preferences over functions
of bounded variation.
2. Variation 1: Increasing functions
We begin with some notation. Let F s F,G, . . . 4 be the set of all increasing
and right-continuous functions defined on w0,1., taking values in w0,1., and
different from the constant zero function. Given a function F in F , we denote by
x, xX , . . . the elements of its domain and by y, yX , . . . the elements of its image.
Suppose that there is a preference relation Ži.e., a complete weak order. K on
F satisfying the following three properties:
Continuity. The preference relation K is continuous in the topology of the weak
convergence. That is, for any pair of functions F and G, suppose that the
sequence of functions Fn 4 weakly converges to F Ži.e., FnŽ x . converges to F Ž x .
at each continuity point x of F .: then Fn K G for all n implies F K G and G K Fn
for all n implies G K F.
Strict monotonicity. The preference relation K is strictly monotone with respect to
the pointwise order. That is, for any pair of functions F and G, suppose that
F Ž x . G GŽ x . for all x and there exists at least some x 1 such that F Ž x 1 . ) GŽ x 1 .;
then F % G, where % is defined as usual.
Independence Ž w.r.t. the graph.. The preference relation K is independent of
common pieces of the function graphs. That is, for any segment S in w0,1. and any
quadruple F, F X , G and GX , suppose that F Ž x . s GŽ x . and F X Ž x . s GX Ž x . on S
and that F Ž x . s F X Ž x . and GŽ x . s GX Ž x . on w0,1. _ S; then F K G if and only if
F X K GX .
Recall that a finitely additive and extended positive real-valued function m on
an algebra A of subsets of a set X is called a measure on X. A measure m on X
is said to be continuous from aboÕe if, for all sequences of sets A n in A such that
A n xB, m Ž A n .x0. Whenever A is a s-algebra, a measure m on X is countably
additive if and only if it is continuous from above. A Žcountably additive. measure
m on X is said to be s-finite if there exists a sequence of sets A n in A such that
m Ž A n . - ` for all n and D n A n s X.
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M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
We will prove a measure representation theorem for K . Intuitively, its content
is the following. Denote by R the rectangle w0,1. = w0,1.. The graph of each
function F in F is contained in R and uniquely defines its Žtruncated. hypograph
Fˆ s Ž x, y . g R : y - F Ž x .4 . We can put the set of functions F in a one-to-one
relation with the set Fˆ of their truncated hypographs, so that each function F in
F corresponds to its hypograph Fˆ in Fˆ. Continuity, strict monotonicity, and
independence Žw.r.t. the graph. will be shown to imply the existence of a measure
ˆ
m on R such that F K G if and only if Fˆ is bigger than G.
If we knew m to be finite, this statement would be formalized as
F K G if and only if m Ž Fˆ . G m Ž Gˆ . .
Ž 2.
However, it turns out that we can only prove that m is s-finite. Therefore. we
need a stronger formulation that implies Ž2. when m is finite. Thus, we will show
that continuity, strict monotonicity, and independence Žwith respect to the graph.
imply the existence of a Žcountably additive and s-finite. measure m on R such
that F K G if and only if m Ž Fˆ _ Gˆ . G m Ž Gˆ _ Fˆ ..
Alternatively, Segal Ž1993. suggests the following Žequivalent. formulation. For
any z g w0,1., let F z s F g F : F Ž x . G z for x ) 1 y z 4 be a subset of functions
in F and Fˆ z the set of their truncated hypographs. Let Z s Ž1 y z,1. = w0, z ..
Denote by m z the measure on R defined by m z Ž A. s m Ž A _ Z .. For all z,
continuity, strict monotonicity, and independence Žw.r.t. the graph. imply the
existence of a Žcountably additive and finite. measure m z on R such that m z
represents K on F z ; that is, for every F and G in F z , F K G if and only if
m z Ž Fˆ . G m z Ž Gˆ ..
Some more definitions are necessary before we state the theorem. A curÕe C in
R is the image of a continuous bijective 1 function f : w0,1. ™ R; note that a curve
does not necessarily belong to F. A curve C in R is increasing if Ž x 1 , y 1 . g C
implies that the intersection between C and the northwest region Ž x 2 , y 2 . g R : x 2
- x 1 and y 2 ) y 14 is empty. A rectangle is any set w x 1 , x 2 x = w y 1 , y 2 . in R such
that x 1 - x 2 and y 1 - y 2 ; note that a rectangle is the product of two intervals
closed on the left and open on the right and that by definition it has nonempty
interior.
The following characterization theorem is very similar to Theorem 1 in Segal
Ž1993., but our setting has some advantages that will be discussed in the second
variation. The proof is in three parts. For the first two Žeasier. parts, we follow
Segal Ž1993.. For the harder proof of the third part, we exploit an argument similar
to the one put forth in Green and Jullien Ž1988. and Chew and Epstein Ž1989. that
has also been used in the literature on rank-dependent models.
1
The standard definition of curve does not require bijectivity. We impose it because, by Netto’s
theorem, this rules out the possibility of space-filling curves. See for instance Theorem 1.3 in Sagan
Ž1994..
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
259
Theorem 1. The following three statements are equiÕalent:
1. A preference relation K on F satisfies continuity, strict monotonicity and
independence w.r.t. the graph.
2. There exists a Ž countably additiÕe and s-finite . measure m on R such that, for
all z,
Ž a. the preference relation induced by K on F z is represented by m z ;
Ž b . m z is countably additiÕe and finite;
Ž c . m assigns strictly positiÕe measure to any rectangle in R;
Ž d . m assigns zero measure to any increasing curÕe in R.
3. There exists a measure m on R as in Proposition 2 satisfying Ž c . and Ž d . and
such that F K G if and only if m Ž Fˆ _ Gˆ . G m Ž Gˆ _ Fˆ ..
Proof. We prove that Proposition 2 implies Proposition 3, which in turn implies
Proposition 1, which in turn implies Proposition 2.
Proposition 2 implies Proposition 3. Let F and G be in F. By definition of F ,
there exists ´ ) 0 such that lim x ≠ 1 F Ž x . ) ´ and lim x ≠ 1 GŽ x . ) ´ . Therefore, by
right-continuity, there exists ´ X ) 0 such that min F Ž x ., GŽ x .4 ) 0 for x G 1 y ´ X .
For z s min ´ , ´ X 4 , then, F and G are in F z and thus
F K G if and only if m z Ž Fˆ . G m z Ž Gˆ . if and only if m Ž Fˆ _ Z . G m Ž Gˆ _ Z . .
Since Fˆ _ Z s Ž Fˆ _ Gˆ . j wŽ Fˆ l Gˆ . _ Z x, this implies by additivity of m that F K G
if and only if m Ž Fˆ _ Gˆ . G Ž Gˆ _ Fˆ .. I
Proposition 3 implies Proposition 1. Strict monotonicity follows from Ž c . and
independence w.r.t. the graph from the measure representation. It remains to prove
that K is continuous in the topology of the weak convergence. It suffices to show
that, for all sequences Fn weakly converging to F, m Ž Fˆn . converges to m Ž Fˆ ..
Let Sn s Ž Fˆn j Fˆ . _ Ž Fˆn l Fˆ . be the symmetric difference between the hypographs of F and Fn . Let Tn s D `ksn Sk . Note that Tn 4 is a decreasing sequence
of sets: let T s lim n Tn s F nTn . Since < m Ž Fˆn . y m Ž Fˆ .< F m Ž Sn . F m ŽTn ., we only
need to show that lim n ™` m ŽTn . s 0. This will follow if we prove that m ŽT . s 0.
In fact, since m is countably additive and Tn4 is a decreasing sequence 2 of sets,
continuity from above of m implies that lim n m ŽTn . s m ŽT .; see for instance
Theorem 10.2.ii in Billingsley Ž1986..
Thus, it remains to show that m ŽT . s 0. Let F ) be the northwest boundary of
ˆ
F: that is, F ) s Ž x, y . g clŽ Fˆ . : xX - x and yX ) y implies Ž xX , yX . f Fˆ4 . Since
F ) is an increasing curve, m Ž F ) . s 0. We show that T ; F ) and therefore
m ŽT . s 0.
Suppose by contradiction that T o F ) . Then there exists some point Ž xX , yX . in
T which is not contained in F ) . There are two possible cases: either yX ) F Ž xX . or
2
By the s-finiteness of m , it also holds that m ŽT1 . -`.
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M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
yX - lim x ≠ x X F Ž x .. Moreover, since Ž xX , yX . is in T s F nTn s F n D `ksn Sk s
lim sup Sk , there exists a subsequence Žwhich we index by n as the original
sequence. Fn4 such that Ž xX , yX . g Sn .
ˆ Thus, by definition of Sn , Ž xX , yX . g Fˆn and
If yX ) F Ž xX ., then Ž xX , yX . f F.
X
lim x x x X FnŽ x . ) y for all n. Since functions in F are increasing and right-continuous, there exists ´ ) 0 such that F Ž x . - w F Ž xX . q yX xr2 - yX - FnŽ x ., for all
x g w xX , xX q ´ .. As the interval w xX , xX q ´ . must contain a continuity point of F,
Fn cannot be weakly converging to F.
ˆ thus Ž xX , yX . f Fˆn ; and
Similarly, if yX - lim x ≠ x X F Ž x ., then Ž xX , yX . g F;
X
lim x ≠ x X FnŽ x . F y for all n. Then there exists ´ ) 0 such that FnŽ x . F yX wlim x ≠ x X F Ž x . q yX xr2 - F Ž x ., for all x g Ž xX y ´ , xX x. Again, the interval Ž xX y
´ , xX x must contain a continuity point of F and Fn cannot be weakly converging to
F. I
Proposition 1 implies Proposition 2. This part of the proof is organized in four
steps that are conceptually simple, but somewhat cumbersome due to the number
of definitions involved. An overview of the proof is the following. First, we prove
the existence of a real-valued functional V defined on a subset Fr of simple
functions in F such that V Ž F . G V Ž G . if and only if F K G for all F and G in
Fr . Second we show that for any z this functional induces a representing measure
m on the algebra Arz generated by the hypographs of the functions in Fr z s Fr l
F z and that m is s-finite and satisfies Žc. of Theorem 1; that is, for any z a
measure representation holds for Fr z. Third, we check that m extends to the
s-algebra generated by the hypographs of all functions in F and that this
extension is s-additive. Fourth, we prove that for any z this extension provides a
measure representation for all functions in F z and satisfies Žd. of Theorem 1.
Step 1. Existence of a representing functional on Fr .
A function F in F is said to be simple if its image is a finite subset of w0,1..
Note that the image of a simple function in F cannot be the singleton 04 . Using
the indicator function 1 P4 , a simple function can be written as a finite sum
F Ž x . s Ý nis1 F Ž x i . P 1w x i , x iq1 .4 with 0 F x 1 F x 2 F PPP F x n F 1 ' x nq1 with
at least one inequality holding strictly. The ranked n-tuple of points x 1 , x 2 , . . . , x n
used in this representation is called a nth dimensional basis for F and is denoted
by x. A simple function admits an infinite number of bases; however, there always
exists a smallest one that we call its minimal basis.
We say that a simple function F in F is simpler if its minimal basis contains
only rational numbers of the type kr2 n for some n s 2,3, . . . and k s 1, . . . ,2 n y 1.
Let Fr be the set of all simpler functions in F and Fr n be the subset of all
simpler functions such that their minimal basis contains only rational numbers of
the type kr2 n for k s 1, . . . ,2 n y 1 and F Ž2 ny 1r2 n . ) 0. Note that Fr n ; Fr m
for all n - m and that Fr s lim n Fr n s D n Fr n.
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
261
Consider a class Fr n and an arbitrary element F in Fr n. For each k s 1, . . . ,2 n
y 1, let y k s F Ž kr2 n . and put each function F in Fr n in a one-to-one relation
with the Ž2 n y 1.-tuple y which lists its image points 0 F y 1 F y 2 F PPP F y 2 ny1
- 1 Žpossibly with repetitions and with at least one weak inequality holding
strictly.. Denote by Irn the set of these rank-ordered Ž2 n y 1.-tuples of reals
representing the functions in Fr n. Note that the two Ž2 n y 1.-tuples consisting
respectively only of 0’s and 1’s do not belong to Irn. These would represent the
extreme alternatiÕes which are respectively inferior and superior to any element in
Irn ; thus Irn contains no extreme alternatives.
The restriction of the preference relation K to Fr n defines another preference
relation K n on Irn that inherits the Žequivalent of the. properties of K . First, K n
is a complete weak order. Second, it is continuous with respect to the product
topology on Irn. Third, K n is strictly increasing in the following sense: write
yyk a for y in Irn with the k th coordinate y k replaced by a in w0,1.; then a ) b
implies yyk a % yyk b for all a and b in w0,1.. Fourth, K n satisfies coordinate
X
X
independence: yyk a K n yyk
a if and only if yyk b K n yyk
b.
By Theorems 3.2 and 3.3 in Wakker Ž1993a., then, there exist Ž2 n y 1. strictly
increasing and continuous real-valued functions Vkn : w0,1. ™ R Žfor k s 1,2, . . . ,2 n
y 1. such that V n Žy. s Ý k Vkn Ž y k . represents K n on Irn. Moreover, the functions
Vnk . are jointly cardinal; that is, they can be replaced by Wkn 4 if and only if there
exist real a 1 , . . . , a 2 ny 1 and a positive b such that Wkn s a k q b Vkn for all k.
Through the one-to-one relation between Fr n and Irn , the functional V n : Irn ™ R
also represents the preference relation K on Fr n. It remains to prove that there
exists a functional representing K on Fr . Since Fr s D n Fr n , this will be
obtained by ‘pasting’ together the representations for each Fr n
Each set Irn has Žmore than. one representing functional Vn for K n . However,
since Fr n ; Fr m for n - m and K n agrees with K m on Fr n , we can use the
degrees of freedom provided by joint cardinality to normalize the functions Vkn 4
for all n s 2,3, . . . and k s 1, . . . ,2 n y 1 so that: Ži. Vkn Ž0. s 0, and hence
V m Ž0. s V n Ž0. for m ) n; and Žii. if m ) n,
2 ny1
Ý
2 my1
Vkn
Ž y. s
ks1
Ý
Vkm Ž y .
for all y g w 0,1 .
ks1
so that whenever y n is an Ž2 n y 1.-tuple with y k s y for all k, V m Žy. s V n Žy..
Let w n be the real-valued functional on 0,1r2 n,2r2 n , . . . ,Ž2 n y 1.r2 n 4 = w0,1.
defined by
k
k
w n Ž 0, y . s 0 and w n
, y s Ý Vsn Ž y .
2n
ss1
ž
/
for all n s 2,3, . . . and k s 1,2, . . . ,2 n y 1. Note that
2 ny1
Ý
ks1
w
n
ž
k
2n
n
/ ž
, yk y w
ky1
2n
2 ny1
, yk
/
s
Ý Vkn Ž yk . s V n Ž y .
ks1
Ž 3.
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
262
represents K n on Fr n. Moreover, by the above normalization, w n is positive with
w n Ž x,0. s 0 for all n and all x in 0,1r2 n,2r2 n, . . . ,Ž2 n y 1.r2 n4 .
Denote by Q the set of rational numbers of the type kr2 n for some n s 2,3, . . .
and k s 1, . . . 2 n y 1. After the above normalization,
wn
k
, y sw m
ž /
2
n
k
ž /
2n
,y ,
for all m ) n and thus, applying the definition of w n for all n, we obtain a
positive real-valued functional w defined on Q = w0,1. such that: Ži. the form
given in Ž3. represents K on Fr ; Žii. w Ž x, 0. s 0 for all x in Q; and Žiii.
w Ž0, y . s 0 for all y in w0,1.. I
Step 2. A measure representation exists for Fr z.
Given z, we wish to show that there exists a representing measure m defined
on the algebra Arz generated by the hypographs of the simpler functions in Fr z.
Observe that the hypograph of a simple function is the union of a finite number of
rectangles. For a simpler function, these rectangles are of the form w x, xX . = w y, yX .
with x s Ž k 1r2 n1 . and xX s Ž k 2r2 n 2 .. Without loss of generality, we take n1 s n 2
s n. These rectangles will be called eÕenly rational. Denote by Ar the algebra
generated by all evenly rational rectangles and by Arz the algebra generated by the
differences between all evenly rational rectangles and the southeast corner Z s
Ž z,1. = w0, z .. Note that Ar s lim z ≠ 1 Arz s D z Arz.
The basic tool to construct a measure m on Arz is the positive functional w on
Q = w0,1.. It satisfies the following property, known as total strict positivity of
order 2 or strict supermodularity:
w Ž xX , yX . y w Ž x , yX . y w Ž xX , y . q w Ž x , y . ) 0,
Ž 4.
for all x - xX in Q and y - yX in w0,1.. In fact, for x s Ž k 1r2 n . - Ž k 2r2 n . s xX ,
w Ž x X , yX . y w Ž x , yX . y w Ž x X , y . y w Ž x , y .
k2
s
k1
k1
Ý Vsn Ž ys . y Ý Vsn Ž ys .
ss1
ss1
ss1
k2
s
k2
Ý Vsn Ž yXs . y Ý Vsn Ž yXs . y
Ý
ssk 1q1
ss1
k2
n
Vs Ž
yXs
.y
Ý
Vsn Ž ys .
ssk 1q1
k2
s
Ý
Vsn Ž yXs . y Vsn Ž ys . ) 0,
ssk 1q1
where the inequality follows from strict increasingness of V n.
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
263
For every Ževenly rational. rectangle w0, x . = w0, y . in Arz , define its measure by
n z Ž w x 1 , x 2 . = w y1 , y 2 . . s w Ž x 2 , y 2 .
y w Ž x 1 , y 2 . y w Ž x 2 , y1 . q w Ž x 1 , y1 . .
Ž 5.
This gives a finite measure nz on Arz characterized by the two-dimensional
distribution function w Ž x, y . s n z Žw0, x . = w0, y ... Moreover, for any z - w, Arz ;
A w and n z Ž A. s n w Ž A. if A is in Arz. Therefore, applying the definition of nz
for all z, we obtain a measure m on Ar .
By Ž4., m is strictly positive on any rectangle in R. Moreover, since for any
increasing Žcountable. sequence z n4 converging to 1 we have m Žw0, z n x = w z n ,1..
- `, the measure m is also s-finite.
It remains to prove that m z represents K on Fr z for any z. This follows easily
because the hypograph Fˆ of each function F in Fr z is an element of Arz and the
representing functional given in Ž3. computes m z Ž Fˆ .. I
Step 3. Extension of m to a countably additiÕe measure on a s-algebra.
The extension of m from Ar to the s-algebra generated by Ar is a standard
procedure: see for instance Billingsley Ž1986., Theorem 12.5. It is based on the
Caratheodory extension theorem, that we state in a version especially convenient
for our setting; see for instance Aliprantis and Border Ž1994., Theorem 8.40.
Lemma 2. Let m be a measure continuous from aboÕe on an algebra A of subsets
of R. Then m extends to a countably additiÕe measure on the s-algebra generated
by A. MoreoÕer, if m is s-finite on A, this extension is unique.
Denote by A the algebra generated by all Žnot necessarily evenly rational.
rectangles. The s-algebra generated by Ar and A is the same, namely the Borel
s-algebra B for R. By the first part of Lemma 2, proving that there exists a
countably additive extension of m to B requires only to show that m is
continuous from above or, more simply, that w is continuous Žseparately. in each
argument.
Continuity of w Ž x, y . with respect to y g w0,1. follows from continuity of
Vkn Ž y . for all n and k. Continuity with respect to x g Q is obvious for y s 0
because w Ž x, y . s 0 for all x and follows by continuity of K if y ) 0. In fact,
suppose by contradiction that x n converges to x but w Ž x n , y . does not converge
to w Ž x, y .. Without loss of generality, assume that there exists z ) x such that
max n x n - z. Define Fn s y P 1w x n ,1.4 and F s y P 1w x,1.4 . Then Fn weakly
converges to F, but m z Ž Fˆn . s w Ž z, y . y w Ž x n , y . does not converge to m z Ž Fˆ . s
w Ž z, y . y w Ž x, y .. Since Fn4 and F are in Fr z and m2 represents K on Fr z , this
would contradict continuity of K .
The following implication of the continuity of w will be useful. A subset
w x, xX . = y4 in R is called a horizontal segment; similarly, a subset x 4 = w y, yX .
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
264
in R is called a Õertical segment. The Žfrom now, extended. measure m assigns
zero measure to any horizontal or vertical segment. For instance, the measure of
the horizontal segment w x, xX . = y4 is
m Ž w x , xX . = y . s lim
m Ž w x , xX . = w y, yX . .
X
yxy
X
X
s lim
w Ž x , y . y w Ž x X , y . y w Ž x , yX . y w Ž x , y . s 0
X
yxy
and a similar derivation holds for vertical segments. I
Step 4. For all z, a measure representation exists for F z.
Given z, we wish to show that the measure m z represents K on F z. First, we
check that Fˆ z is contained in B. Note that the hypographs of all continuous
function Ževen those not in F z . are contained in B. Since each function F in F z
is increasing, it has at most a countable number of discontinuities. Therefore its
hypograph Fˆ is the union of at most a countable number of hypographs of
continuous functions; hence, Fˆ belongs to B.
Take now F in F z _ Fr z. Observe that F is the limit Žin the topology of the
weak convergence. of some sequence Fn4 in Fr z. Therefore, by continuity of K ,
it should be assigned the measure lim n m z Ž Fˆn . for the measure representation to
hold. Hence, we need to show that
m z Ž Fˆ . slim m z Ž Fˆn . ,
Ž 6.
n
for any sequence Fn converging to F.
Given any such sequence, note that FnŽ x . may fail to converge at F Ž x . only in
a Žpossibly empty. countable set D s x 1 , x 2 , . . . 4 of discontinuity points of F. If
D is empty, then Fˆ s lim n Fˆn and Ž6. holds by continuity of m. If D is not empty,
for any x k in D let Õ k the vertical segment x k 4 = w0,1.. Then
< m z Ž Fˆ . ylim m z Ž Fˆn . < F m z Ž j k Õ k . s Ým z Ž Õk . s 0
n
k
and again Ž6. follows.
There remains only to prove that m satisfies Žd. of Theorem 1. First, note that
any increasing curve C is in B because it can always be identified with the
northwest boundary of an appropriate function F in F. Next, suppose by
contradiction that there exists an increasing curve C such that m Ž C . ) 0. Given F
in F corresponding to C, choose a rectangle Z s Ž z,1. = w0, z . such that Z l C s
B and let Fn4 be a sequence in F converging from above to F Žin the topology
of the weak convergence. and such that C ; Fˆn for all n. Since C l Fˆ contains at
most vertical segments that have zero measure, we can assume without loss of
generality that C l Fˆ s B. Then we have that m z Ž Fˆn _ Fˆ . G m z Ž C . ) 0 s m z Ž Fˆ _
Fˆn .. But this implies that m z Ž Fˆn . s m z Ž Fˆn _ Fˆ . q m z Ž Fˆ l Fˆ n . cannot converge to
m z Ž Fˆ . s m z Ž Fˆ _ Fˆn . q m z Ž Fˆ l Fˆn ., which is impossible. I
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
265
It is worth pausing on the main difference between our proof of Theorem 1 and
the proofs of similar results given in Green and Jullien Ž1988. and Chew and
Epstein Ž1989.. Our construction of the representing functional on F r in Step 1
partitions the x-axis instead of the y-axis. This simple trick, analogous to the
intuition that makes Lebesgue integration more general than Riemann integration,
will be crucial during our third variation.
We close this variation making two observations. First, the representing measure m is unique up to a ratio scale; that is, up to a positive scale factor. Second,
as shown by the counterexamples in Wakker Ž1993., m may not be absolutely
continuous with respect to the Lebesgue measure l on R and, conversely, l may
not be absolutely continuous with respect to m.
3. Variation 2: Technical remarks
Theorem 1 may be generalized in many directions. This section considers and
comments on some of these possibilities. Recall that the functions in F satisfy the
following four properties: they are Ži. increasing; Žii. right-continuous; Žiii. defined
on w0,1. and taking values in w0,1.; and Živ. different from the constant zero
function. Leaving for the next section a relaxation of Ži., let us examine here how
Žii. – Živ. might be weakened.
We begin with right-continuity. Intuitively, its purpose is to exclude the
possibility of two functions F % G in F with their hypographs Fˆ and Gˆ
coinciding everywhere except for an Žat most. countable set of vertical segments.
In this case, the measure m could not represent F % G because of property Žd..
Obviously, we might substitute left-continuity for right-continuity.
More generally, we could start with a set F X of functions containing F and
define an equivalence relation ; on F X such that F ; G if and only if
clŽ Fˆ . s clŽ Gˆ .. Then each right-continuous function in F is the representative
element of a Ždistinct. equivalence class. Assuming that K on F X is consistent
with ; , Theorem 1 would yield a representing measure for K on F Xr; . All in
all, however, right-continuity Žor left-continuity. appear the most reasonable
assumptions.
Properties Žiii. and Živ., instead, have the purpose of excluding front F the two
extreme alternatives, namely the constant zero and one functions. This is necessary
in order to apply Theorem 3.3 from Wakker Ž1993a. during Step 1 of the proof
that Proposition 1 implies Proposition 2. The two properties could be removed if
the second-order Archimedean axiom for K described in Section 3.3 of Wakker
Ž1993a. would be added. However, as we find this axiom difficult to interpret, we
prefer to avoid it and impose some restrictions on the domain and the range of the
functions in F.
These restrictions must be such that F contains no extreme alternatives.
Among the many possibilities, our choice to set the domain and the range both
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M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
equal to w0,1. is only a convenient normalization. In fact, we could substitute 3 any
pair of clopen Žand bounded from below. intervals w a1 ,b 1 . = w a2 ,b 2 . in R 2 and, in
particular, a i s 0 and bi s q` for i s 1,2. This gives the class of positive
Žpossibly unbounded. real-valued functions defined on Rq. As these are often used
to model cumulated cash flows, for instance, Theorem 1 may be applied as it is to
yield a measure representation for Žpositive and increasing. cumulated cash flows.
All the measure representation theorems so far presented in the literature are
inspired by the rank-dependent model and refer to Žtruncated. epigraphs. Theorem
1, instead, is based on Žtruncated. hypographs. An obvious comment is that this is
a consequence of the direction in which strict monotonicity holds. More interestingly, however, there is a simple way to turn Theorem 1 into a representation
result based on epigraphs.
For any increasing function F in F , define its Žgeneralized. inverse w by
w Ž y . s inf x : F Ž x . ) y4 . Let F be the set of all generalized inverses of F and
let KX be the preference relation on F induced by K . Note that the strict
monotonicity of K for F is increasing with respect to the pointwise order on F ,
while the strict monotonicity of KX for F is decreasing. Continuity and independence with respect to the graph, instead, carry over naturally without modifications. Therefore, the measure representing K on F via hypographs also represents KX on F via epigraphs and the two approaches are equivalent.
Our last comment concerns the possibility to get another Žmore natural.
measure representation for functions taking both positive and negative values. We
present one example that should suffice to clarify the issue. Let F be the set of all
increasing and right-continuous functions defined on w0,1., taking values in wy1,1.
and different from the constant function F Ž x . s y1. Suppose that there exists a
preference relation K on F satisfying continuity, strict monotonicity and independence w.r.t. the graph. Then there exists a measure m on R s w0,1. = wy1,1.
that represents K .
Fix some z w0,1. and define m z on R _ Z as above. Decompose R into the
union of the two disjoint sets Rqs w0,1. = w0,1. and Rys w0,1. = wy1,0.. Define
a finite signed measure n z on R by
n z Ž A . s m z Ž A l Rq . y m z Ž Ry_ A . ,
for any m-measurable set A. Since m z is finite, we have
m z Ž Fˆ . s m z Ž Fˆ l Rq . q m z Ž Fˆ l Ry .
s m z Ž Fˆ l Rq . q m z Ž Ry . y m z Ž Ry_ Fˆy . s n z Ž Fˆ . q m z Ž Ry . .
Since m z Ž Ry. is constant, we can represent K on R _ Z by the signed measure
n z : that is, by the difference between the measure of the hypograph of the positive
part Fqs max F,04 Žtruncated between 0 and 1. and the measure of the epigraph
of the negative part Fys min F,04 Žtruncated between y1 and 0..
3
It suffices to map continuously w0,1. into w a i ,bi ..
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
267
4. Variation 3: Functions of bounded variation
This section shows how the proof of Theorem 1 can be generalized to obtain a
measure representation for functions of bounded variation. The crucial ingredient
is having constructed the representing functional on F r in Step 1 by partitioning
the x-axis instead of the y-axis. We will point out during the proof the key point
where this is used.
Unless explicitly mentioned, we keep the same notation as above. Let F be
the set of all right-continuous functions of bounded variation F defined on w0,1.,
taking values in w0,1., and such that lim x ≠ 1 F Ž x . ) 0. Assume that K is a
preference relation on F.
The theorem that we present in this section is the analog of Theorem 1, except
for Žd. which requires the following definition. A curve C in R is not backward
bending if its intersection with any vertical segment in R is either a singleton or a
vertical segment. Note that any increasing curve is also not backward bending.
Theorem 3. The following three statements are equiÕalent:
1. A preference relation K on F satisfies continuity, strict monotonicity and
independence w.r.t. the graph.
2. There exists a Ž countably additiÕe and s-finite . measure m on R such that, for
all z,
Ž a. the preference relation induced by K on F z is represented by m z ;
Ž b . m z is countably additiÕe and finite;
Ž c . m assigns strictly positiÕe measure to any rectangle in R;
Ž d . m assigns zero measure to any curÕe in R which is not backward
bending.
3. There exists a measure m on R as in Proposition 2 satisfying Ž c . and Ž d . and
such that F K G if and only if m Ž Fˆ _ Gˆ . G m Ž Gˆ _ Fˆ ..
Proof. Again, we prove that Proposition 2 implies Proposition 3. which in turn
implies Proposition 1, which in turn implies Proposition 2. The proof that
Proposition 2 implies Proposition 3 carries over identically from Theorem 1 and
thus we omit it.
Proposition 3 implies Proposition 1. Assume that K is continuous in the topology
of pointwise convergence and proceed as in the proof of Theorem 1 until when it
ˆ that is, F )
remains to show that m ŽT . s 0. Let F ) be the upper boundary of F;
X
ˆ
is the set of all pairs Ž x, y . in clŽ F . such that: Ži. either x - x and yX ) y implies
Ž xX , yX . f F 4 , or Žii. xX ) x and yX ) y implies Ž xX , yX . f Fˆ4 . Then F ) is not a
backward bending curve and m Ž F ) . s 0. We show that T _ F ) is a subset of a
Žpossibly empty. countable union of vertical segments, so that m ŽT _ F ) . s 0.
Then m ŽT . s m ŽT _ F ) . q m ŽT l F ) . s 0.
Suppose that T _ F ) is not empty Žotherwise the claim follows immediately..
Let Ž xX , yX . be a point in T which is not contained in F ) . There are two possible
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M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
cases: Ži. yX ) max F Ž xX ., lim x ≠ x X F Ž x .4 ; or Žii. yX - min F Ž xX ., lim x ≠ x X F Ž x .4 .
Denote by Fn4 the subsequence for which Ž xX , yX . g Sn .
ˆ Thus, Ž xX , yX . g Fˆn and
If yX ) max F Ž xX ., lim x ≠ x X F Ž x .4 , then Ž xX , yX . f F.
X
X
X
X
X
FnŽ x . ) y for all n. Then F Ž x . - y - FnŽ x . for all n implies that xX must be a
discontinuity point of F.
ˆ Thus, Ž xX , yX . f
Similarly, if yX - min F Ž xX ., lim x ≠ x X F Ž x .4 , then Ž xX , yX . g F.
X
X
X
X
X
Fˆn and FnŽ x . F y for all n. Then FnŽ x . F y - F Ž x . for all n implies that xX
must be a discontinuity point of F.
Recall that each function of bounded variation can be written as the difference
of an increasing function and a decreasing function and thus it has at most a
countable number of discontinuities. Denote by D the set of discontinuity points
of F. We have just shown that any point Ž xX , yX . contained in T _ F ) must be such
that xX g D. Therefore, if we let Õ k denote the vertical segment corresponding to
each x k in D, it follows that T _ F ) ; D k Õ k . I.
Proposition 1 implies Proposition 2. Proceed as in the proof of Theorem 1 up until
the definition of Irn. Note that the Ž2 n y 1.-tuple y which is to be put in a
one-to-one relation with each function F in Fr n does not necessarily satisfy
0 F y 1 F y 2 F PPP y 2 n y 1 - 1 because F may not be increasing. Instead, we can
only say that yi g w0,1. for all i. Therefore, let Irn the set of all Žnot necessarily
rank-ordered. Ž2 n y 1.-tuples of reals representing the functions in Fr n. Again. the
restriction of K to Fr n defines a preference relation K n on Irn that inherits the
properties of K .
Thus, instead of Wakker’s, we can call upon the additive utility theorems of
Debreu Ž1960. and Gorman Ž1968. – here is the key point – to deduce the
existence of Ž2 n y 1. strictly increasing and continuous real-valued functions Vkn
such that V n Žy. s Ý nVkn Ž y k . represent K n on Irn. The rest of the proof goes on
unchanged until Step 4, where we need to check that Fˆ w is contained in B. This
follows because each function in F is of bounded variation and thus has at most a
countable number of discontinuities. Now, we can resume the argument in the
proof of Theorem 1 up until to when we need to show that Žd. holds. Note that any
curve C which is not backward bending is in B because it can be identified with
the upper boundary of some appropriate function F in F. The rest of the
argument goes through unchanged. I
We close this paper by noting a corollary of Theorem 3 that relies on some of
the technical remarks in our second variation. For a - 0, let F be the set of all
real-valued and right-continuous functions F of bounded variation on w a,q `.
such that F Žq`. s lim x ™q` F Ž x . ) a. For F in F , denote by Fˆq the Žtruncated. hypograph of Fq and by Fˆy the Žtruncated. epigraph of Fy. Finally, given
F and G in F , let z - min F Žq`., GŽq`.4 and say that F and G are in F z.
Corollary 4. The following two statements are equiÕalent:
1. A preference relation K on F satisfies continuity, strict monotonicity and
independence w.r.t. the graph.
M. LiCalzir Journal of Mathematical Economics 29 (1998) 255–269
269
2. There exists a Ž countably additiÕe and s-finite . measure m on R that satisfies
Ž c . and Ž d . of Theorem 3 and, for all functions F, G in F z , defines another
(countably additiÕe and finite) measure m z such that
F K G if and only if m z Ž Fˆq . y m z Ž Fˆy . G m z Ž Gˆq . y m z Ž Gˆy . .
Acknowledgements
I wish to thank Erio Castagnoli and an anonymous referee for their helpful
comments. Marco Barlotti and Guido Rossi alerted me about the subtleties of
space-filling curves.
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