43
Internat. J. Math. & Math Sci.
VOL. ii NO.
(1988) 43-46
RATIONAL CHOICE FUNCTION DERIVED FROM A FUZZY PREFERENCE
JIN BAI KIM
Department of Mathematics
West Virginia University
Morgantown, W. V. 26506
KERN O. KYMN
Department of Economics
West Virginia University
Morganto, W. V. 26506
(Received September 7, 1984 and in revised form April 22, 1987)
ABSTRACT.
lar
(see
ftmctions
We shall prove that every fuzzy rational choice function is fuzzy reguRichter [6, p. 36] ), count the total number of the fuzzy rational choice
on a set. of four elements and consider a semigroup of all fuzzy ratlonal
choice functions on a set.
KEY WORDS
AND PHRASES.
choice
regular
I.
fuzzy
function
rational
fuzzy preference
fuzzy transitive
choice function
fuzzy
1985 AMS CLASSIFICATION .ER 03E72
semigroup.
We
INqT{ODUCTION.
fuzzy binary relation
Fuzzy relation
have
ntroduced a rational choice function derived from a
preference (see [2], [3], [4]). We shall establish two theorems (Theoand 2) which are motivated from the following theorems:
fuzzy
rems
THEOREM
4
(Richter
[6]).
There exists a total rational choice which is not
[6]).
There exists a rational choice which is not total
transitive rational.
THEOI
6
(Richter
rational.
We
find that the number of all fuzzy rational choice functions on a set X
c,
d} of four elements is equal to 57751 (see [2].
We
note
that
{a, b,
We shall consider a semigroup.
in [4] there is a beautiful counting formula of the total number of
all final choice functions on a finite set,.
2.
Let
DEFINITIONS AND THEOREMS.
X
be
a
finite set with more than two elements.
For definitions of a choice
function on X and a fuzzy binary relation (R, r) on X, we refer to [2] and [3].
Let (R, r) be a fuzzy relation X and let a e X.
p. 38].
0| and R, (a)= {x R(a)" r(a,x) _->
X" aRx and rta,x)
}
We
define
function
a
for
(0,1)].
h as follows" Let a A _c X. Then
We add that ha(0)= O, the empty set. Note that h
R a(a).
h (A) iff A
is in general, not a choice function. Let h be a choice function on X. If there
exists a fuzzy relation (R, r) on X such that h,= h, then we shall say that h is
[2,
DEFINITION
Define
R(a)=
Ix
J.B. KIM and K.O. KYMN
44
fuzz. l’ational and (R, r)’attonalzes h.
NOTATION I. We denoto by F(X) the set of all fuzzy binary relations on X. We
and C(:<. Z
denotes lh, s,l ,I" all -hoic,, fmctons h on X. I.et
define Z
2
e use (x,)-I e R td x R .hen rl..,y) g O. I.1. h a t’lX, Z)
(R, r) e F(X).
a choice I’ction on X.
DEFINITI
{R,
exists
F(X)
(R, r) s l.Faltsillx. (tol, reflexive).
if (R,r} is reflexive, t}ta] tnd trsittve,
uF(B) such that (R,
if there exists {R, r)
I,t
h is fuzz)’ regular
ion
is
fuzzy trsitive.
a fuzzy ratiorml choice t’mction on X.
h
a
(R,r)
is t’eglar.
r}
We shall prove the followira theorem.
DDI 1. Every f,,zzy rat ional :hoce fct
PF.
(R,r) rationalizes h}.
e F(X}"
{(R, r)
ks said to Ix, fuzzy t[’ansitixe (Iol, reflexive} if there
in aF(h) such that
r)
regular
is
t’ine F(h}
h
2.
Then F(h) is non-
Supse that {R, r) is not trsi{r(x,y)
O" x,ye X} for (R, r). We ea find a sitivie
fine {r}
tire.
1
{r i, where k is a sitive integer. We define
such that %
nr
n+k
a fuzzy relation IS, s) a follows" If r(x,t} # O, then we t s(x,yl:r{x,y},
empty
d
let
r{x,y):O
if
relation
on
hs.
en
c
ss
{R,
r)
Then h= h,.
F{h).
then
we put s(x,y}:%.
X.
We
sho
It is clear that {S,s} is a trsitive fuzzy
h,:hs.
that
To
sho
this,
that
assm
ue
h
We
hs(A)=C.
non-empty set % sh that B
h
1
g B. Then {,x} S for all x A, s(a,x}
>k
d hence s{a,xl #
In view of it}
[ {r}, it is clear that
e A,
B. A
a
here a a B. is eontiets
s(a,x}=r{a,x) for all x
e
b
silu prmf for b
C bris a tietion. efore B:C
B
there
,
h=hs =h.
exists
that
e
is proves
a
C d a
e
eorem
(A)
.
1.
Every fuzzy mtiol choice fmetion h on X is fuzzy tol.
2.
t h
a fuzzy rtioI choice fraction on X. en there exists
For x, y e X a x # y, it is clear that h {x,y}f{x,y}.
smh that h=h.
1
m we Mve that either r{x,y) $1 or r{y,x)
erefore {R,r) is toI.
2
is proves eor 2.
F.
r)
Y
foIlo fr
Every
1.
fuzzy
mr
mtiol
choice fetion is
reDIsr. e
2.
A SI.
We gin ith the followia definition.
F(X)
DIRIIS 3.
t {R, r}
3.
tol
if r(a,b) # 0
e@letely 1 if
r(b,al
there
,
a fuzy relation.
0 fo 11 ,b
exists
(R,r)
e
X.
{R, r} is elely
A choi ftion h is fuzzy
smh tt h,=h
a F{X}
(R, r} is
h is fuzzy cpletely relar if em exists {R, r} smh t
h=h is fuzzy eDl a fuzzy eapletely tol.
We Mve coir a smgroup in [2] a [4]. We note by @{X} the set of 11
e@letely
tol.
empletely
re, at
hve
hh c_h
that
fzy mtiol choice fetio on X.
u
,
h,h
e (X}.
we
By
Mve
4-( i
the
folloia
rem,
3.
(X}
fore
a
sgm er
the bi
omtion
-
2 ],
defin by
if h e(X), then there exists (P,p) smh tt h=h
(P, p) is
epletely toni.
F.
It is clear that the bi omtion is ssmiative. It is aIso
p
clear tMt
u@ R {or(R, r I} is regular a cpletely tol. ttia P U
We
note
relar
tht
a
RATIONAL CHOICE FUNCTION DERIVED FROM A FUZZY PREFERENCE
,
hlA
s
A
that.
hlA)
s.
m.
Since
hlA)
x
.
t:o
.t,’
Thi shows that a
A.
te defn ion of h. e fntion 1). e prove
-. ror-empy, to asse that A
md ],]
part :!
xt,t
h
l[lat
1
r(a,x)
f]
1o: xplo
s
to
shot.
that:
not
a
fuzzy
chotc.e
x on
thoagt
he tho I,
for
for all
m
’lha I)t’ov’s The,,rem
t-l.
The
rat onai
!
and hnc’e pa,x)
nvtA)
From rla,xl=Inax Ipta,-., qta,?.l It lrll.)t:s
A.
e
x
a
no-(ml)ly
45
th,-- r-om)s te sot fction,
h,
and
Let
IR,
h0
are N)th fuzzy rational
.hoees on X.
Lt
1.
X=la,
){O,q)=lqla,a}=q{b,b)-qlc,c}=l,
l_.
e
Then
we
lst the
for
d
is
a
in
otto
hlAI,
qla,b)=qla,e)-
a fuzzy relation IP, pl such that
memr
that
n X..a necessary d sufficient con-
t,t
X s thai for every non-empty subset
A such that rla,xl
1
-ondt,t tvlds for IR,ri.
Let
a
tt
such that ra,x
hl,41
X.
on
such
k
eorem
_a
in
a
fotion
choice
proves
s
s
a
uzz3 fetal
e supso that the
PF.
there
not
s
a -hotoe ftmc’t,n r
to
h
be a
of X there exists at least
that
qlb,a)=q(’,al=qlo,bl-, qlb,cl-j-,
there
and
4
foIlo-’trN theorem.
NODI 4. Let Ir, r
dttion
c
rlb,al=rc,al=rle,bl
1
that
prove
c
t,
1
rla,bt=rla,c’l=rlb,cl=
,4
s
1
r
lrt=hor
a
.
for alI x c
#
d asse
Then A i
definition of h. Thus
all x
A.
X. Then for each A #
Supse h s a ’t,te
1
a
hr 14 Irom dc,h ;- obt, atn that ria,x}
hl Is
a
0 there
This
]A
4.
OK
cHOICES ON la,h,c,dl. Let X
a set of n
ntr of all Iuzzy rar tol choice fcttons on X by
hrx) In).
In [2] we show that hr(x) t3)
93. In ths stion we oce
that hrx) 4)
57751.
shall prove this in a serrate r. A .)ustification of hr(x) (4)
57751 ne several ges.
4.
We
elents.
iZZ$ ]iN
denote
the
REFERIENCES
[I] K. J. Arrow, Social Choice and Individml Value (Wiley, New York, 19631.
[2] Jin
37-43.
B. Kim, Fuzzy Rational Choice Functions, FVZZ SETS AND SSTEN I011983),
[3] Jin B. Kim and Kern O. Kymn, Rational Choice and Gain Functions Derived From a
Fuzzy Relation, ECONOMICS LEqUEIq 1311983), 113-116.
[4] Jin B. Kim, Final Choice Functions, EICN LZTFERS 14(1984), 143-148.
[5] Jin
519-522.
B.
Kim,
A
Certain
Matrix
Semigroup,
Mathematics
Japonica 22(1978),
[6] M. K. Richter, Rational Choice, in" J. S. Chipman, L. Huvicz, M. K. Richter,
and H. F. Sonnenschein, Eds., Preferences, Utility and Demand (Harcourt Brace
Jovanovich, New York, 1971).