We consider a class of relations which includes irreflexive preference relations and interdependent preferences. For this class, we obtain necessary and sufficient conditions for representation of the relation by two numerical functions in the sense of aPreference; Continuous representation; Pseudotransitivity; Biorders;

Estévez Toranzo, Margarita

García Cutrín, Javier

Hervés Beloso, Carlos

López López, Miguel A.

English

Research Papers in Economics

~'~~~'-----'-~---------
Working  paper  93-42  Departamento de  Economla 
Economy  Series  17  Universidad Carlos III de Madrid 
December  1993  Calle Madrid,  126 
28903  Getafe  (Madrid) 
Fax  (34  1)  624-9875 
A NOTE ON REPRESENTATION OF PREFERENCES 
Margarita  Estevez  Toranzo,  Javier Garcla Cutrln,  Carlos￿ 
Herves  Beloso  and  Miguel  A.  L6pez  L6pez  •￿ 
Abstract,  _ 
We  consider  a  class  of  relations  which  includes  irreflexive 
preference  relations  and  interdependent  preferences.  For  this 
class,  we  obtain  necessary  and  sufficient  conditions  for 
representation of the relation by two numerical functions  in the 
sense  of  a  ~  x  if and  only if u(a)  <  vex). 
Key  words￿ 
Preference,  continuous  representation,  pseudotransitivity,￿ 
biorders,  countable  bounded preferences.￿ 
·Estevez  and Garcla,  Departamento de Matematicas,  Universidad de 
Vigo;  Herves,  Departamento  de  Economla,  Universidad  Carlos  III 
de  Madrid  and  L6pez,  Departamento  de  Algebra,  Universidad  de 
Santiago de Compostela.  The  work of M.  Estevez,  C.  Herves and M. 
L6pez  is  partially  supported  by  Research  grant  PS-0042  from 
D.G.I.C.Y T.  (Ministerio de Educaci6n de Espana)  and the work of 
J.  Garcla  by  XUGA-30001A92  (Xunta  de Galicia). 1  Introdllctiol1 
This work conce1'1lS  the existence of a llumerical representation for a class of 
relations which  includes il'l'eflexi \'e  preference relations and interdependent 
preferences. 
As it is  well  known, if X  is a connected and separable topological space, 
continuous  preference orderings on  X  always  have  utility  representations. 
(Eilenberg  (1941)).  The assumption of  connectedness  is  not  necessary  in 
the setting of metric spaces; this fact is  a consequence of a  result of Debreu 
(1954)  which establishes that if X  is  perfectly separable, every  continuous 
preference ordering is  representable by  an  utility  function.  However,  as  it 
was noted by  Monteiro (1987). the above results may not be useful in infinite 
dimensional spaces because we  lack.  ill  gelleraL the separability of the space: 
he  proved tbat a  continuous preferellce relation on a  path connected space 
bas  Cl  continuous utility representation if and only if it is countably bounded. 
If the  preference  is  given  by  an  asymmetric  binary  relation,  it  is  not 
possible to have an  utility representation but it  has been suggested by some 
authors  that  a  preference relation  on  a  set  X  could  be represented  with 
the help of two real valued  functions u and v,  where v( x)  and u(x)  can be 
interpreted  as  the lower  and  upper  bounds of  the utility  perceived  of  the 
objet l' (see Fishbul'1l  (1973)  and DoigllOll  et  al.  (1984)). 
TIle  representatioll  Gy  two  numerical  functions  generalizes the classical 
utility function  theory.  because it  allo\l's  the relation  ""","  (absence of strict 
prdel'ellce) llOt  to be tronsiti ve.  \\'llicll  seems more in  accordance with eco-
nomic  choices,  since  J:  "'"  y  may  correspond  not  only  to  true indifference 
between  :1'  and  y.  but also  to an  inability  to  cboose  between  them.  This 
type of relations are pseudotransiti\'es; that is, if J:  --<  y  ~  y'  --<  z then x  --<  z. 
Pseudotransitivity implies tbe transitivity of the strict preference --<  but does 
not  imply the transitivity of the indifference.  Bridges (1983) is  the first inter-
ested in  the existence of a cOlltinuous representation in  the case of a  binary 
preference relation  defined  on  a  closed  convex  set of Rn and Chateauneuf 
(1987)  gi res,  in  terms of stl'Ollg  separability,  necessary  and sufficient  con-
ditions for  the existence of representation of a  pseudotransitive preference 
relation by  two continuous functiolls in  a connected topological space. 
In the recent literature about walrasian equilibrium (see Florenzano (1990), 
Yannelis  (1991))  it is  frequent  to consider  interdependent preferences (the 
1 2 
preference of an  agent  depend  of the choice of the rest of the agents)  or 
preferences with extel'llalities.  For  example, if we  consider a set of n agents 
and Xi  represents the consumption set of the agent i, the preferences of the 
agent i are given  by  a  correspondence Pi  defined on the cartesian product 
n 
of the consumption sets A = I1X  j  into Xi'  If £1=  (al,''',an )  E  A, Pi(a) 
i=1 
is  interpreted as the set of consumptions x  E  Xi  which  the agent i  prefers 
(strictly) to a, when the consumption of the agent k -=I- i  is  ak. It is  usual to 
consider a condition of irreftexivity expressed by aj  rt  Pi(£1)  (or even more, ai 
is not in  the convex hull of h(£L)).  Formally, this class of relation is  a subset 
-<  ofAx Xi, that is  -<=  {(£L,J'):.l'  E Pi(a)}  =  {(a,x);a -<  x}.  We have an 
analogous situation  if  a  factor  is  tlJe  consumption set of an agent and the 
rest of the factors represents the set of possible externalities.  Although these 
preferences don't have all  utility representation, in  some cases they can be 
represented by two numerical functions. 
To  formalize these ideas, if  A and  X  are topological spaces, we say that 
the relation  "-<" C A x  X  has a  numerical represelltation if  there exist two 
functions u  :  A ---t R. t'  :  X  ---t 1R  such  that a  -<  x  if and only if  u(a) < 
v( J').  This kind of representation is  useful, for  instance, to characterize the 
optima.]  allocations: the allocation a E A is optimal if and only if the "more 
prefered" set  P(a) = {.r  E X;a -<  .r}  is  empty and it occurs when u(a) is  an 
upper bouud of r. 
III  this  work  we  consider the class of relations, between two  topological 
spaces, which verify  et  property of irreflexivity which generalizes the ordinary 
ineftexivity  for  a  binary  relation on a  set;  this  class  includes  the class  of 
interdependent preferences 01'  preferences  with externalities.  For this class, 
wc  give necessary conditions for  the existence of a continuous numerical rep-
resentation and we  pro\'e that these conditions are sufficient in  the setting of 
patl}  cOllllecled  topological spaces. 
Definitions and notations 
A relation between two sets A and X is a subset P ofAx  X. When (£1,3.')  E P, 
we  write a  -<  x.  The notation  J'  j  a signifies that (a, J;)  rt  P.  We say that 
the relation is  representable by  two functions u  : A  ---t /R,  v  : X  ---t R  if 3 
a  -<  x  is  equivalent to u(a) < v(x).  The relation is  said a  biorder if for  all 
a,b E A, 3.:,y  E X, the property 
[a  -<  x, b -<  y]  ==::;.  a -<  y or b -<  x  holds 
That property can be also expressed by 
If a  relation is  representable by  two functions is, obviously, a  biorder.  A 
relation  bet,veen A and X  induces relations  :::51  on  A and  :::52  on X  on the 
natural way 
a :::51  b if and only if z  :::5  a implies z  :::5  b; a, b E  A, z E X 
.1'  :::52  y if and only if  y  ~  c implies.1':::5 c;x,y E X,c E  A 
The relations  :::51,:::52  are preorders on  A  and .X  respectively.  They are 
complete  preorders  if  and  only  if  -<  is  a  biorcler.  In  this  case,  the strict 
relations associated to :::51,:::5l  are defined by 
a  -<1  b if there exists :1'  E X  such that a -<  :1.'  :::5  b; a, bE A 
.1'  -<2  Y if there exists a E  A such that  :1.'  :::5  a -<  y; x, Y E X 
If A  and  X  are topological spaces, the relation  -<  is  continuous  if the sets 
(ao.--t)  =  {.1'  E  X;ao  -<  :r}  and (-,J'o) =  {a  E  A;a -<  xo}  are open in X 
and  A.  respecti\'e!y, for all  (lu E  A,xu E X 
A preorder :::5  on a set )' is  countably bOlludecl  if there is some countable 
subset n of }' such that fur all yE}' there exist b1, b2  E  B  with b]  :::5  y  :::5  b2• 
Results 
In  this  work  we  consider the class of relations  "-<'le  A x  X  verifying the 
following property: 
For all a  E  A,  there exists  X a  E  X  such that X a  :::5  a  and for  all x  EX, 
there exists aJ.'  E A such that x  :::5  a x .  [1] 
This class of relations includes the class of binary irrefiexive relations on 
Cl  set and the class of interdependellt preference relations. 
3 Proposition 1  If then,  exists a "epresentation u, v of --<,  then 
1)  u  is  a  pseudoutility fol'  --<I,  thal  is a  --<1  b  :=:::}  u(a)  <  u(b)  for  all 
a, IJ  E A 
2)  'V  is  a  ]!seudoutility  for  --<2,  that  is  a:  --<2  Y  :=:::}  v(x) < v(y)  for  all 
,T,yEX. 
. 'J)  t1u,  complete preol'ders :SI,  :S2  are  countably bounded. 
Proof. It is clear, from our definition of:s, :SI, :S2, that the parts (1) and (2), 
are true.  Let us  prove the part (3).  To show that :SI  is countably bounded, 
WE'  can SUppOSE'  that u is  a  bounded function; if we  denote 1= inf u(a) and 
aEA 
oS'  =  supu(a), we have two possibilities: 
uEA 
a) if S =  I1lax  uta). tllere exists a",  E  A such that S =  u(UAJ)  and then 
aEA 
ti(o)  :s  S =u(a.\J): frolll  the part (1)  we  have a :SI  aM, that is,  :SI  is  upper 
coulltably bounded. 
b)  if u  has llOt  maximum, for each n  E  1\T  we  can  choose  Sn  E  A such 
that  tl(~n)  2:  S  _..!..  If a  E  A,  there exists no  E  N  such  that  u(a)  :s 
11 
S - ~  :s  U(8  no  );  then, a :SI  sno  and :SI  is  upper countably bounded by  the 
nu 
set  {"'Ii} nEl";' 
In  the analogous way.  we  call also  havE'  either am E  A with I =  u(am )  or 
a cOlllltably set  {11/}  to [>1'O\'e  that ::SI  is  lower countaGI)' bounded. Then, ::SI 
is  countabl,v bounded. 
Proposition 2  If A and X  connected spaces and the  nlatioll --<  t'uifying (U'( 
/1)  is  nprescntablt  by  two  continuous functions H, t',  thfll --<1,  --<2  and  -<  an 
continuous. 
Proof.  WC'  proye that -<1  is  contilluous.  If b E (a, -)1' there exists  ,1'  E  X 
slIdl  that  (I  -<  :1'  :S  b.  For condition [1].  there is  :l'u  E X such that :1"u  :::  a and 
lllen  l'(,l'u)  ::;  u(u) <u(J') ::;  u(IJ).  Since u is  continuous and X  is  connected, 
it  follows  thal tllere exists': E X  sucb tbat 
V(Xu ) S u(a) < v(z) < v(x) S u(b) 
From continuity of H, one deduces the existence of a  neighborhood U of 
b such that u(c)  > v( z)  if c E U.  Hence a  -<  z  ~  c or equivalently a  --<1  c if 
c E U. 
4 We  have shown that (a, - h is open.  In  the analogous way  we  can prove 
that (-,  a) I is  open and then it is  proved that -< I is  continuous.  The proof 
of continuity of -<2  and -<  is  similar. 
Our aim is  to prove that the necessary conditions above established are 
also sufficieut in  tile settiug of path connected topological spaces.  The idea 
is  to prove  that there exists a  connected and separable subset of  A  where 
the relation is  representable and to show  after that this representation can 
be extended.  If A' C  A and X' eX, the restricted relation -<'=-<  nA' x X' 
iuduces  -<;  aud  -<~,  Note that in  general  -<;  and  -<~  do not coincide with 
-<I  nA' x A' and -<2 nx'xX', respectively, as we see in the following example. 
Example 3  Let  A  ==  X  ==  [1 \10]  and  a  -<  :to  if and  anly  if a2 <  x.  Let 
A'  =  X'  =  [1,3]:  we  Iwt't  '2  -<I  :3  because  the7'e  exists  J'  E  A  such  that 
'2  -<  :1'  ~  3 (any:1'  such that 4 < .1'  :s  ~)  but  we  don't  have  :2  -<;  3. 
Proposition 4  Let  "-< "c A x X  ([  biordel'  ve1'ifying  the  condition [l) such 
that  -<.  :::S I,  :::S'l  art  continuous.  if  A' C A and X' C  X  an connected and A' 
bound...  :::SI  and X I  bounds  ~2,  thclI.  -<I =-<;,  -<2=-<~ 
Proof.  Let  us  remark that the relation  -<'  verifies  also  the condition  [1]: 
since  -<  verifies  the condition  [1]  and ,X'  bounds  ~2\  for  each  a  E  A' there 
exist  .1'a  E ,\', J'~  E X' such that 
:1' I  -< _2  '  .1' U  -<  a ::::::}  J"u  -< _ a (J  -
It is  similar to pJ'Ove  that for  eadl .1'  E X', there is  (/~.  E A' such that :1.'  ~  a~. 
Let  us  sho\\'  110\\' that a -<1  b <===>  (/  -<;  b for all a, bE A'.  If a  -<1  b;  there 
exists  .1'  E  X  such  that  (/  -<  .1'  :::S  b:  since X' bounds  ::S2..1'  ~2  J"  for  some 
J,I E X I  and then (/  -<  .1'  :::S2  .1"  ::::::}  a -<  ,1" 
1'1'0111  condition [1]  we  have  J'~  E X' such  that x~  :::S  Q  -<  X  ====>  .<  -<2  :Z:. 
Moreo\'er if z E X' is  such that z 1:  (a,-),:::  ~ a -<:1.'  ====>.z  -<2  x 
Then,  X'  =  [(a, -)  n X']  u [(-, J'b n X'];  that is,  X' is  the union  of 
t\\'o  lion empty open sets.  Since X' is  connected, there is  y' E X' such that 
y' E (a. -)  n(-,  J')' ====>  a  -<  y'  -<2  l'  ~ b::::::}  a  -<  y'  ~  b  ====>  a -<;  b. 
It follows  that -<I =-<;  and in  the similar way  it is  showed that  -<2==-<~' 
Note that if X' = X  (respectively A'  ==  A) then  -<1  is  always identical to 
-<;  (respectively  -<2=-<~)' 
5 Theorem 5  Let A be a path connected space,  X  a connected space and  "-< "c 
A x X  verifying the  con,dition [1].  There  exists a contin'uo-us  1'e]J1'esentation 
It, v  of -<  if and only lj 
a)  -<  is a biordt",. 
b)  tin complete pnol'del's  :::S 1,:::S2  are  countably bounded, 
c)  t!lt  relations -<,  -<1,  -<2  art  conlinuous. 
Mor'tOvfl',  whtn such  a rtprtstnlalion e:risls,  then exists one such that u  and 
L'  artutildy functions for,  repectively.  tht complete  p1'eorders  :::SI  and :::S2' 
Proof. The necessa.ry conditions ha\'e already been proved.  Let us now turn 
to the sufficiency part. 
The space A is path connected; then, by using a result of Monteiro (1987), 
t.here exists a connected and separable subset of A, A', which bounds :::SI  (Let 
a E  X; if {:2'n}JlEI:\  is  a.  count.able set which bounds j1, for each n  ~  1, let fn 
be a path connecting a  to J: n .  'vVe  define}' = U I" [0,1];  the set Y  bounds 
nE [j\; 
jl and it  is  patll connected, then it  is  connected; moreover, it  is  separable 
uecause if A  is  a  countable dense subset of [0,1],  the set  U fn(A) is dense 
nEDV 
in }',) 
The relation:::S1  is continuous and :::S1=j'\; by applying the classical result 
of  Eilenberg  (19.1 1),  there exists u'  :  A'  ---t  [0, 1]  such  that a  :::S~  b {::::::} 
v'(a)::; u'(b)  fOI all  a,b EA', t.hat is  v' is  a  continuous utility representation 
f'  -I' -<' 01 
The functioll  It'  can  be extended  to  A:  for  each a E  A,  the sets {a'  E 
04';0' jl a}  al1C]  {a'  E  A';0:51 o'}  alC'  non empty (A' bounds jd and closed 
sets on  ,4',  Since A'  is  connecteeL there is  a E  A'  such that a ""1  a.  Let us 
dd1ue u(a) =  a'(a),  We have, for all a.b EA, 
a j1 b <=? a j; b<=? v'(a)  ::;  u'(h) ~ u(a) ::;  u(b) 
that  is,  '/I  is an utility representation for  :::SI.  It is  not difficult to prove that 
for all  0  E  JR.,  the sets  {a  E A;u(a) ~ o}  and  {a  E  A;u(a)  ~  o}  are closed; 
tlwlJ  1I  is continuous. 
I\o\\'.  we define v  : X  ---t [0,1]  by 
a  i r then  i~  not a E A such that a -<  x 
V(.l·)= { sup{ll(a),a E A,a -<  l:}  in othe1'  case 
(j 4 
The pair u, v  is  a  representation of -<  on A x X:  let a  E  A, x  E  X  such 
tllat li  -<  J';  from condition [1],  J:  ~  ax  from some ax  E A and then a  -<1  ax ' 
Moreover, if c E  A  is  such that c  rt.  (+-,x),a -<  x  ~  c  ===>  a  -<1  c.  Then, 
A = (+-, x) n (a, -+ ) I; that is,  A  is  the union of two non empty open sets; 
since A is connected, there is  bE A such that a -<1  b -<  x  ===>  u(a) < u(b)  s: 
v(x) ===:}  u(a) < v(;r). 
Reciprocally, if x  ::S  a, there are two possibilities:  if there is not b E A such 
that b -<  :2'  ===:}  v(:r) = o:=}  V(.1:)  s:  u(a);  if there is  bE A such that b -<  x, 
we  have b -<:2'  ~  a ===:} b -<1  a:=} u(b) < u(a) for all  b -<  x ===:}  v(x) s:  u(a). 
The function  1.'  is  a  continuous utility representation for  ~2:  let x  ::S2  Y; 
there are two possibilities: if there is  not a E A such that a -<  x  ===>  v(x) = 
o===:} v(:I:)  s:  v(y); if there exists a E A such that a -<  :1'  ===>  a -<  x  ::S2  Y ===:} 
a  -<  y  ===:}  v(J')  s:  v(y).  Reciprocally,  if Y  -<2  x,  there exists  a  E  A  such 
that  y  ~  a  -<  .1'  ===:}  1'(Y)  ~  u(o.)  < t'(;r)  ===:}  1.'(Y)  < v(x),  Moreover, v  is 
continuous  becausE'  the sets {:r  EX;  v( x) > Q}  a.nd  {:1'  EX;  v( x) < Q}  are 
open for  all  (I E 81. 
Final renlarks 
\\'('  remark  that  our  theorem  generalizes  the  Chateauneuf's  result  in  the 
sense that we  allow relations bet\\'een two different spaces A and X  and even 
in  the case  where A  = X. we  don't require the relation  be irreftexive.  By 
otller side. our cllCiracterization is in terms of the property of being countably 
bounded instead of the property of strong separahility used by  Chateauneuf. 
\Ye  remark tllal  strong separability implies that the associate preorders are 
countably  bounded and  in  general  they  are  not equivalents.  However,  as 
a  consequence of tllC  result,  we  see  that  ill  the hypothesis of the theorem 
botb  pl'Operties  are equi\'iilents.  In  infinite dimensional spaces,  the strong 
separability  cellI  be difficult  to  test  and  sometimes  the  property  of  being 
countably bounded can be easy to test; tllis is  the case when the spaces are 
compact 01'  a-compact.  As  a  consequence, we  see that if A is  a  a-compact 
and path connected space, every biorder in  A x X  \vith continuous associate 
preorders is  representable by  two continuous functions.  As  a particula.r case, 
every complete and continuous preorder in  A is  representahle by a cOlltinuous 
utility function, 
i References 
[1]￿  Bridges D.S., 1983, "A J\umerical Representation of Preferences with In-
transitive Indifference", Journal of AJathematical Economics, 11,  pp.  25-
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