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Game theory
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Preference Relations and Revealed Preference BINARY RELATIONS
For a given set X , let X x X denote the usud Cartesian p d a c t of all ordered pairs ( z , y ) , where both z and y are from X .
A binary relation B on the set X is formdly defined as a aubat of X x X - write B E X x X, and (t, y) E B if the o r d d pair ( z , y ) is in the relation B . Another, quicker way to m-te (z , y) E B is zBy , which can be read as ' z Bees y " or " z stands in the relation B to y." If (z,y) 4 B , I'll write "not zByn or tBy.
(8) Let X = {1,2,3) and B = {(1,1), (1921, (1,319 (2,319 (3,111. (b) Let X = all people in the world and I d B be the relation "&am at least one given name with.* (c) Let X = R (the red line, ranember) and let B be the nldian "greater or equal ton; that is, B =?. (d) Let X = R and let B be the relation: zBy if jz - yl> 1. (e) Let X = R and let B be the relation zBy if t - y is an integer multiple of 2.
There is a long list of properties that a given b i i d a t h might or might not have. The properties that will be importsat in this book are the following. A binary relation B on a set X is:
rt@xitx if zBz for all z E X; if z ~ z f o r d z E X; symmetric if zBy implies yBz; asymmetric if zBy implies y j z ; antisymmetric if zBy and yBz imply z = y ; tmnsitiue i f zBy and yBa imply zBz;
Refnence Rehations and Rrornlai P@ma 9
ncgaiidy imnsitm if zBY and yBz imply rib; complrtc or canccted if for all z, y E X, zBy or yBz (or both;
u n or s are never exclusive in this book unless s p d c d y mentioned); ureaklyumncctedifforall s,y E X , z = y or zBy or yBz; acyclic if zl Bz2, zzBzs,. . . z,-~ Bz, imply zl # z, .
Example (a) (above) is weakly connected, but nothing else. Exrunple (b) is reflexive and symmetric. Example (c) is dexive, antisymmet- ric, transitive, negatively transitive, complete, and w d y connected. Example (d) is idex ive , symmetric. Example (e) is reflexive, sym- metric, transitive.
PREFERENCE RELATIONS
In this section, we take up the following simple story. There is a set of items X , and Totrep ia willing to express his preferences among these items by making paired comparisons of the fonn: "I strictly prefer z to yn which is written z >- y. "Strict preferencen is a binary relation on X . Consider the following properties that this binary relation might possess:
(a) Asymndty - if z is strictly preferred to y, then y is not strictly preferred to z. (What do you think of this? Reasonable normatively? HOW about descriptively? Think of these questions for each of the following.)
(b) ~mifmrfmrfy - if z is strictly p r e f d to y and y is strictly pre- ferred to z , thexi z is strictly preferred to z .
(c) r+ti& - no z is strictly prefenwi to itself. (d) Nqatioe ttnnsitivity - if z is not strictly preferred to y and y is not strictly preferred to z, then z is not strictly preferred to z.
Negative transitivity ia a hard property to deal with intuitively in the form given, so let me develop an dtemative statement that is completely equivalent.
(2.1). A b i relation B is negstively transitive iff (if and only if) zBz implies that, for all y E X , zBy or yBz.
Pnwf. (Very pedantic.) The statement [M implies N] is the mne aa the statement [N or not M], thus [M implies N] is the uane as [not N implies not Mj. (The second equivalence is called contrapmition.) Thus [(zBz) implies (+By or yBz for aII y E X)] is the same as [{not(zBy or yBz for all y E X)) implies {not zBz)J which is [{there exists y E Z with ~8~ and y ~ t ) implies {zi)z)], whicb is negative transitivity.
Now bad; to F . Is negative transitivity reasonable? b it r e d to say that if z is strictly preferred to z, then for dl y either t irr h'ctly preferred to y or y is strictly preferred to z? As anormati* propaty, I think it is (barely) masonable. But as a descriptive property, 1 don't think it is reasonable. Suppose X = (0,oo) x (0, oo), whac t = (zl, t 2 ) E X is interpreted as the commodity b d e z l bottles of beer and 2 2 bottles of wine. Tot- (if his tastes are like my own) would certainly say that (10,lO) + (9,9). But -dm (15,6). Totrep might not be willing to say either that (10,lO) + (15,6) ap (15,6) + (9,9) - he might plead that comparisons d l e d for an too difficult for him to make.
Despite these difficulties with negative transitivity, it is standard to proceed assuming that + is asymmetric and negatively transitive.
Definition (2.2). A binary relation + on a set X is called a pnfnnrce relafion if it is asymmetric and negatively transitive.
Proposifia (23). If + is a preference relation, then >- is idexive, transitive, md acyclic.
Proof. (a) Asymmetry directly implies irdexivity. (b) Suppose z >- y and y + z. By negative transitivity (and La~ls (2.1)), z + y implies that either z >- y or t >- z . But z + y is impossible because y F z is assumed and + is asymmetric Thus z + z, which is transitivity; (c) If 21 >- t 2 , z ~ + 25,. . . , zn-1 + 2,. then by t-tivity + x,. Since >- is imflexive, this implies z l # z,. Thus + is acyclic.
When we are given a binary relation + that arprescs strict pnt- erence, we use it to define two other binary relations:
where w
e are using as shorthsnd for >= or for %
ot + ." The relation is d
ed
racllk pnfmne~, although it redly expresses the absence of
strict ptefetence The relation - is called intiifereme - it expr-
the absence of strict pteference in either direction, which is perhaps
not quite the same thing as active indifference.
Wf
Ml
(2.4). If + is a preference relation, then: (a) For d
2 and y, exactly one of z
+ y,y + z or z - y holds. (b) 2 is com
plete and tdtiv
e.
(c) - is reflexive, symm
etric, and transitive. (d) w
+z,z-y,y+
zimply
w+
y and r+
z.
(e) 2)-y ifTz+
yor z-y. (f) z
ky
and ytzim
ply
z-y.
Proof. (a) foII0w
s hm the definition of - and the fact that + is
asymm
etric. (b) By the asym
metry of +, either z
3 y or y 3 z (or both) for dl
z and y, thus 2 is complete. For transitivity of t
, note that this follow
s immediately from the negative transitivity of +.
(c) N is reflexive because + is irreilexive. - is sym
metric because the
definition of - is symm
etric. For transitivity, suppafe t
- y - z. Then z + y 3 z and z 3 y 3 t
.
By negative transitivity of +, z)(z+
z,or Z
-z. (d) If w
+ z - y, then by part (a) one of w + y or y - w
or y + w.
But y >- w
is impossible, since then trsnsitivity of + w
ould imply
y + 2. A
nd y - w is impossible, since then transitivity of - w
ould im
ply z - w
, cantradicting w + t. Thus w + y m
ust hold. The other part is pim
ilarIy done. (e) z 2 y iff y 3 z iff z + y or z - gt (by part (a)). (f) This is'im
mediate from the definitions of & and -.
Note w
ell the plot: Totrep expresses strict preferences, from
w
hich we define weak preferences and indifference. It is strict pref-
erence that is basic - Totrep is not being called upon to express any judgm
ents concerning weak preference or indifference, and he m
ight disagree w
ith our use of those terms to describe the negation of strict prdennce.
Another possible plot w
ould be to ask Totrep to ex
pm
weak
preferencar or preferepce or indifference. That is, the basic relation is
Prtfemuc Relations and Retm
lui Pqkencc 11
I 2
. This is a plot that is followed in many developm
ents of &dm the-
ory, ad
in the standard treatment it leads to the sam
e mathem
atical results:
Proposition 05). Given a binary relation 2' on a set X
, d&
taro new
binary relations hJ and -' from
2' by
Then if k' is complete and transitive, +' will be a preference relation.
Moreover, if w
e start with a binary relation k', define +' and -' aa
above from k', and then define k and - from
+I by
then 2' and w
ill agree, as will
and -. The proof is I&
as an exercise. So it doesn't matter w
hetha we ntart with a strict preference relation that is asym
metxic and negatively
transitive or with a weak preference relation that is com
plete and transitive - w
e end up in the same place. For reawns of intapretation
I prefer to take strict preference as being basic But it is a matter'of
personal taste, and most authors do it the other way.
REVEALED PREFERENCE THEORY
In the previous section, t6e story was that Totrep waa maU
q paired com
parisons between item
s in X. B
ut especially ftom a dt .
scriptive point of view, we would like to start w
ith an even more basic
concept - that of choices made rather than p
ref~~
llcc~
arpnsssd. That is, from
a descn'ptin point of view what we stx is an individ- ual's choice behavior - we have to connect that behavior as best we can w
ith his preferences which are neva directly e
xp
d.
The indi- vidual's choice behavior reveals his preferences, hence the nam
e of this subject: revealed preference thw
ry. This subject also hs, som nor-
mative justifications - taking preferences as given, how should choices
be made? B
ut this subject is of greatest interest from the d
dp
tive view
point.
To keep matters sim
ple, throughout this section I'll assume that
the choice set X is finite. E
epeddy if the application you are think- ing of is dem
and for consumption bundles or for any item
that is in- finitely divisible, this is not a very nice sim
plification. For nonempty
subsets of X, I'll
use notation such as A, B, etc.
The set of all nonem
pty subsets of X will be denoted P(X
).
DeFnition (2.6). A
choice fundia for a (finite) set X is a function
c : P(X) +
P(X
) such that for all A E X
, c(A) E A
.
The interpretation is: If Totrep is offered his choice of anything in the set A
, he says that any mem
ber of c(A) will do just h
e.
If Totrep's preferenas are given by the bi
i relation + (and by
the corresponding >- and -), it is natural to suppose that he &cm
s according to the d
e that from a set A
, anything that is undominufad
will be okay. In symbols, define a function c(-, +) : P(X
) by
It is dear that for any + , c(A, +) E A
, but it isn't dear whether
c(A, +
) # 0. Thus it isn't clear that c(-, +) is a choice function.
That will be som
ething to be investigated The otha questions to be looked at are:
(a) Fiom the norm
ative point of view: G
inn a relation >- (not nec- essarily a preference relation), w
hen is o(., +) a choice function? If + is a preference relation, w
hat propeaties does c(-, +) have? (b) R
om the descriptive point of view:
Givm
a choice function c, w
hen is there, a bi
i relation + such that c(-) =
c(-, +)? When is this b
ii
relation a pnference nlation? (NB
., this last question is the &
tit$ one, aa w
e're going to be building models w
here individuals are ausum
ed to be maximizing their prdermces according to some
preference relation.)
PIopwitia (2.7). It a binary relation + is acyclic, then c(., +) is a
choice function.
Proof. We need to show
that for A E P(X
), the set
is nonempty. Suppose it was em
pty - then for esch z E A then exids
a y E A such that y + z. Pick zl E A (A
is nonempty), and let
zl be zl 's "y ". Let z3 be zl's y ", and so on.
In 0th- words,
zl, 23, zj, . . . is a sequence of elemeats of A
where
... 2, + 2,-1 + ... + z1 +z1. Because A
is a finite set, there must exist some rn and n such that
z, =
z, and m
> n. But thie m
uld be a cyde, and + ia assum&
to be acyclic. The neassary contradiction is establishad.
Note the follow
ing instant oxolhy: If + is a prdercn~b A-
then <-,+) is a choice function. Also, we can strengthen (27) M
folhs.
Propwitia (2.8). For a binary &
ion +, c(., >-) is a choice function
ifT + is acyclic. Proving this is left ss an exercise.
Next we survey som
e properties of choice functions. Th dsb
sic axiomatic property of choice is H
outhakker's &om
of naacd
pnfereLlce.
Houfhakkds &om
(29). If z and y are both in A and B and if
z E c(A
) and y E dB), then z E c(B
).
In words, if z
is sometim
es chosen (&om A) w
hen y is availab15 then w
henever y is chosen and z is available, z is also chosen. H
outhskker's axiom is broken into two pieces by Sen:
Sen3 prop@ a (2.10). If z E B
G A and t E c(A
) , then z E c(B). Sen's paraphrase of this is: If the w
orld champion in som
e game C a
Pakistani, then he must slso be the cham
pion of Pakistan.
Sen's propnty @ (23U. If z, y E c(A), A
G B and y E c(B
), then z E c(B
).
Sen's paraphrase: If the world cham
pion in some game is s
Pakidmi,
then all champions (in this gam
e) of Pakistan are also world champi-
ons.
P
F R D- Y B ,.
- 8 .
CL 5' VJ' u * rn-
90 9 Y V
rn R ix 8
Pnfcnurc Relntions and Rmtnlcd Pnfam
u 17
A binary nIntion + is acydic if and only if
thncdtsachuicefundion c=c(.,+) for + w
hich implies that, but is not im
plied by e satisfics Sm
's a, and none of the previous three im
ply or are implied by
c safisfit5 Sm's p.
This is just an introduction to revealed prefmee theory - there
is a very large literature on the subject. Two im
portant questions, both related to the applications of these ideas to classical dem
and theory in m
icroemnom
ica, that we haven't discussed are:
What if X
is an infinite et? (Where did w
t use the finiteness of X in the developm
ent above?) For one approach to an answer to this question, see problem
8 of the next chapter.
Suppose we can't observe Totrep's choices from d
subsets of X . That
is, suppcwe c is defined only for a subset of P(X
). You can set? how this w
ould provide problems, especially if sets of the form
(2, y) are
not in the domain of c. A
nd you can see why this is a natural question
- espccidly if we have in m
ind a descriptive theory of choice. What
can be said in such caw?
PROBLEMS
(1) For eseh of the five examplea on page 7, show
that the b'
i
relation has precisely the list of properties that are &bed to it on
page 8 (froal the list of properties on pages 7 and 8).
(2) A binary &tion
E that b refiexive, symm
etric and transitive is called an q
uid
- relation. (For exam
ple, if >- is a preference relation, then - ia an equivalence relation; d
Pmposition (2.4)(c).)
Hen is an essy proof that if E
is symm
etric and transitive, then it is autom
aticdly dexive (thus reflexive could be deleted from
the list of properties): Fix = E
X and take some y such that zE
y. Then y
Et by symmetry, and hence xEz by transitivity.
Unfortuuately,
this easy proof is &m
. W
hy?
(3) Prove Proposition (2.5).
(4) Show that the properties (for a b
ii
relation) of asymm
etry ad
negative transitivity are independent. d
(5) Prow Proposition (2.8): For a binary relation + (on a finite set
X), c(., +
) is a choice function if and only if + is acyclic. In what sense is it im
portant here that X is finite?
(6) Prove that for any binary relation +, 4, +) dc&d
as in the display on page 12 satisfies Sen's a. (Is it im
portant here that X is finite? In w
hat sense is it important that + is acyclic, even though
you weren't told to assuxne acycliaty?)
(7) Give an exam
ple of a finite set X and an acyclic binary nlation
where c(-, +) does not satisfy Sen's property 8.
(8) In and around Propositions (2.13) and (2.14), I seem to get very
confused about whetha negative transitivity is dl I need to prow
H
outhskker's axiom, and thus asymmetry.
Unw
&
me. H
ave I im
plicitly used asymm
etry in the proof of (2.13). and if so, when? D
eal with the proof. I know (and so do you, if you did problem
(4)) that negative transitivity dasn't im
ply asymm
etry, and I don't want
an example of that - I w
ant to know what is going on in the pm
of.
(9) Give an exam
ple of a finite set X and a choice function q on
P(X
) that satisfies Sen's a but such that there is no binary &ion
+ such that c(., +) = c.
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