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) < v(x)

Estévez Toranzo, Margarita

García Cutrín, Javier

Hervés Beloso,Carlos

López López, Miguel A.

English

Universidad Carlos III de Madrid e-Archivo

~'~~~'-----'-~---------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 Introd llctiol1 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 tronsi ti 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 separabili ty, 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(a m ) 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(X u ) 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 con ti nuous 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 cell I 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. 25-42. [2] Chateauneuf A.. 1987, "Continuous Representation of a Preference Re-la.tion on a Connected Topological Space", Journal of Mathematical Eco-nomics. 16, pp. 139-146, [:3] Debreu G. I 1~')-1 ... Hepre::;entation of a Preference Ordering by a Nu-merical Function" in Mathemati.cal Economics: Twenty papC7'S of Gerad Dfbnll, Cambridge Uni\'ersity Press. pp. 10,5-110. [4] Doignoll J,P" Ducamp A" Falmagne J.C., 1984, "On Realizable Biorders and the Biorder Dimension of a Relation", Journal of Alathematical Psycholoy!}, vol. 28. 1. pp, 29-6'2. [:j] EilclllJl'rg S" 1!:lll, "Ordered Topological Spaces", Amtrican Journal of AJatlltllluf icl:'. 63. pp, JY-.{5. [6] Fi::;IJl)llrll P.C., lY70, "Intransiti\'e Indifference with Unequallndiffel'ence Intel'\'als", JOlL/'Nal of Mathtllwti.cul Psychology. 7, pp. 144-179. [7] flol'cnzano I\L 198~. "On the I\on-emptiness of the Core of Cl. Coali-tional Production Ecollomy without Ordered Preferences", Journal of Mathematical Analysis und Applications, 141, pp. 484-490. [8] FlorcllzelllO ~1.. 1990. "[dge\\'oltb Equilibria. fuzzy Core. and Equilib-ria of Cl Pruduct iOlI bUlIUIII.1 \\'itlJUul Ordered Preferences", Journal of !\Jut/It tuaticul .-\ I/aly,..,i,~ (tnd .-\ppliCClf iOI/,'), 153, 1. pp. 18-36. [9] I\lonteiro. P.h .. 1987,,"Sollle Hesultsol1 the Existence of Utility Func-tions on Path Connected Space::;", Journal of Mathematical Economics, 16, pp. 147-156. [10] Peleg 8., 1970, "Utility Functions for Partially Ordered Topological Spaces", L'coIW71ldrica, :38. ]JfJ. 93-96. ---------------------------------[11] Yannelis, N.C., 1991, "The Core of an Economy without Ordered Pref-erences", in Equilibriul/l. l'htory in Infinilf Dimensional Spaces ed. by t\1.A. Khan and N.C. Yannelis. Spl'ingel'-Verlag. pp. 102-123. 

A J\umerical Representation of Preferences with Intransitive Indifference&quot;,

Continuous Representation of a Preference Rela.tion on a Connected Topological Space&quot;,

dge\\'oltb Equilibria. fuzzy Core. and Equilibria of Cl Pruduct iOlI bUlIUIII.1 \\'itlJUul Ordered Preferences&quot;,

I 1~')-1 ... Hepre::;entation of a Preference Ordering by a Numerical Function&quot; in Mathemati.cal Economics: Twenty papC7'S of Gerad Dfbnll,

lY70, &quot;Intransiti\'e Indifference with Unequallndiffel'ence Intel'\'als&quot;,

On Realizable Biorders and the Biorder Dimension of a Relation&quot;,

On the I\on-emptiness of the Core of Cl. Coalitional Production Ecollomy without Ordered Preferences&quot;,

P.h .. 1987,,&quot;Sollle Hesultsol1 the Existence of Utility Functions on Path Connected Space::;&quot;,

A note on representation of references

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