Mathematical Social Sciences
11 (1986) 19% 199
195
North- Holland
NOTE
ON LEXICOGRAPHIC
PROBABILITY
RELATIONS
Uzi SEGAL
Department
of Econornrcs,
Unrversrty of Toronto,
Toronto MSS IAI,
Canada
Commumcated
by F.W. Roush
Received 10 December 1984
Revrsed 6 May 1985
In thus note 1 propose a defmnron for lextcographic probabihty relatrons and discuss the question, under what conditrons 1s a probabihty relatron lexicographic.
Key words: Lexrcographrc
probabrhty
relatron;
trght probabtlity
relatton.
1. Introduction
In this note I examine conditions under which a probability relation on a set of
events is lexicographic.
Chipman (1971) discussed the question of the minimal ordinal CTsuch that there exists an order-preserving
function from an ordered set A
(in his; paper A is fR with any order on it) to lRa with the lexicographic order on it.
Since there always exists an ordinal /? such that there is an order-preserving
function
from A to the lexicographically
ordered lRP (see Chipman), one can define an order
as lexicographic if there is no order-preserving
function from A to R.
According to this definition, a countable set cannot be ordered lexicographically.
Consider, however, the following example. Let Q be the set of all the finite unions
of ra tional intervals [a, p) in [0,2). Q is closed under complementation
and finite intersections. Define on Q a relation 2 as follows: for every A, BE Q, A Z B iff
11(~1n[O,l))>~(Bft[O,l))
or
p(AfI[O,l))=p(BfI[O,l))
and
p(An[l,2))>
~/t/In[l, 2)) (p denotes the Lebesgue measure). Q is countable, but seems to be lexicographic. Indeed, there exists no probability function P on Q such that A 2 B iff
I’( 4) ZIP(B) (see Section 2).
A natural definition of lexicographic orders seems therefore as follows. An order
I? on a set X is lexicographic if there exist two orders R, and R2 on X such that for
every x,y~ X, xRy iff xRly, but not yR,x; or xR,y, yR,x, and xR,y. (R, and R,
may themselves be lexicographic orders.)
This definition may encompass too much. For example, the decimal writing of
real numbers induces on them an apparent lexicographic order. It seems, therefore,
that if one wishes to avoid defining too simple orders as lexicographic, yet does not
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U. Segai / Lexicographic
196
probablhty
relations
want to restrict the set of lexicographic orders too narrowly, one must use some properties of the space itself, as, for example, its being a vector space (see Hausner,
1954), or a space of sets, as below.
In Section 2 I define lexicographic probability relations. In Section 3 I present
conditions implying that a tight probability relation is lexicographic. The examples
of Section 4 prove that these are sufficient, but not necessary conditions.
2. Definitions
Let S be a set of states of the world and let Q 5 2’ be closed under complementation and finite intersections.
Elements of Q are called events, denoted by A, B, C
etc. Let Z be a binary relation on Q. A > B iff A 2 B but not B 2 A, and A-B iff
A2B and B2A. A denotes S\A.
Definition. A relation 2 on Q c 2’ is called a probability relation if
(a) 2 is complete and transitive;
(b) for every AEQ, A20 and S>0;
(c) for every A,B,CEQ
such that A(IC=BflC=O,
A2B iff AUCkBUC.
Proposition.
Let 2
and AflC=BflD=O,
be a probability relation on Q and let A, B, C, DE Q. If A -B
then CLD iff AUC2BUD.
Proof. A-B#(A\B)U(C\(BUD))-(BUC)\(AUD);
C2De
(BUC)\(AUD)
eAUCZBUD.
hence,
L (D\C)U(B\(AUC))
0
Definition. A probability relation 2 on Q is lexicographic if there exists an event
A >0 such that:
(1) ,?i includes an infinite number of non-equivalent
events;
(2) for every B, CE Q, BLC iff Bela > Cola, or Bfl&--Cola
and BiTA
CnA.
Lemma 1. A probability relation 2 on Q is lexicographic iff there exists A > 0 satisfying (l), such that
(3) for every B;C c A, B > Ce B > CUA.
Proof.(2)~(3):LetB,C5AsuchthatB>C.BnA=B>C=(CUA)nA,henceby
(2), B > CUA.
(3)*(2):
Let B2C. (Bfl&UA2B1CkCflA;
hence, BnAkCnA.
Assume
I/, Segal / Lexrcographlc probability relatrons
that BnA-CfiA.
(BnA)U(BnA)=B2C=(CfM)U(CflA).
tion, BnA z cnA.
lfBnA>cnA,
thenBLBnA>(cnA)uA2C.
Ifsnii-cnA
Cfl A, then by the proposition,
BZ C. 0
111the next section
I
will use the following
definitions
197
By the proposiandBnAZ
and theorem,
taken from
Savage (1972):
Definition. Let Q c 2’ be a space
it. P: Q-+ [0, l] is a probability
P(A) + P(B) - P(A fll?). P strictly
agrees with 2 iff A Z B * P(A) 2
of events and let 2 be a probability relation on
function if P(O)=O, P(S)= 1, and P(AUB)=
agrees with 2 iff A 2 B H P(A) 1 P(B). P almost
P(B).
Definition. A probability relation Z on Q c 2’ is fine if for every A > 0 there exists
a partition of S, with all its elements less probable than A. A probability relation
is tight if for every A >B there exists C> 0 such that BflC=O
and A > BUC.
Theorem.
If 2 on Q is fine, then there is one and only one probability function
almost agreeing with 2.
I.>
3. Lexicographic
If
2
is fine and tight, then this function strictly agrees with
probability relations
Obviously, if L is lexicographic, then it cannot be represented by a strictly agreeing probability
function. In this section I demonstrate
conditions under which a
probability relation which cannot be represented by a strictly agreeing probability
function is lexicographic.
Lemma 2. Let 2 be a tight probability relation on Q and let P be an almost agreeing
probability function on Q. If there exists A > 0 satisfying (I), such that
(4) P(A) = 0, and for every DE Q, P(D) = 0 implies A 2 D, then 2 is lexicographic.
Proof. Let A > 0 satisfying (1) and (4), and let B, CC A, B > C. By the definition,
HUA > CUA. Since 2 is tight, there exists D>0 such that (CUA)flD=O
and
P(B)=P(BUA)>P(CUAUD)=
HUA > CUAUD.
By (4),
P(AUD)>O.
P(C) + P(A UD)> P(C) = P(CUA); hence, B > CUA. By Lemma 1, 2 is lexicographic.
0
This lemma
note.
cannot
be reversed.
A counter-example
appears
at the end of this
U. &gal / Lexicographic probability reMions
198
Definition. A set of events Z= {Ai : ie I} is a chain if for every i, jc I, i# jl it
follows that A,5Ai or Aj($Ai. 2 is a maximal chain of events which have a certain
property if for every event B$Z which has this property, ZU {B} is not such a
chain.
Lemma 3. Let 2 be a probability relation on Q. IfP almost agrees with 2, then
there exists a maximal chain of events with probability 0.
Proof. Denote by r the set of all the chains of events with probability 0. Elements
of f are partially ordered by the inclusion relation. Let {Zj :jE J} be a chain of
elements out of r. Obviously, U jezZj is an upper boundary for this chain. By the
Lemma of Zorn (see Halmos, 1960), there exist in rmaximal elements; hence, there
exists a maximal chain of events with probability 0.
Obviously, more than one such chain may exist. Cl
Theorem. Let 2 be a tight probability relation on Q. If in one of the maximal chains
of events with probability 0 there exists a maximal event (containing all the events
of this chain), then 2 is lexicographic.
Proof. Let A be a maximal event of a maximal chain of events with probability 0.
2 is tight; hence, there exists no B > A such that P(B) = 0. The theorem now follows
from Lemma 2. 0
4. Examples
The question arises whether there must always exist a maximal event with prob ability 0. If so, then by the theorem a tight probability relation which can be represented by an almost agreeing probability function (but not by a strictly agreeing
probability function) must be lexicographic. The answer to this question is not
necessarily affirmative.
Example. S= [0, a~), Q is the set of all the finite unions of intervals [a, /3) in S
(8~ 00). Define on Q a probability relation Z as follows. A >B iff &A)>p(B), or
p(A) =p(B) = 00 and &I)>&).
2 is tight. Obviously, the only one probability function P on Q is given by
P(A) = 0, dA)<=+
1, ~(A)=oc
Of course, there is no maximal event with probability 0, and 2 is not lexicographic. Indeed, let A EQ such that p(A <- and let BCa such that 0<&3)1
p(A). B > 0, but 0UA 2 B. By Lemma 1, 2 is not lexicographic.
I/. Segal / Lexwographrc
probabllrty
relatrons
199
Denote this probability relation on [0,00) by 2 *. The following example presents
a tight and lexicographic probability relation which can be represented by an almost
agreeing
probability function, but does not include any maximal events with probability 0. In other words, Lemma 2 cannot be reversed.
Example. S= [- 1, oo), Q is the set of all the finite unions of intervals [Q, p) in S.
A 2 B iff A fl [0,00) > *Bn [0, a), or A n [0,m) -*Bn[O,m)
and &lfl[-1,O))r
p(Bn
I - 1,w.
Acknowlegements
I am grateful to R. Holzman,
for their comments.
J. Mirrlees,
J. Sobel, and especially
M.E. Yaari,
References
J.C Chipman
On the lexicographic representation
of preference ordering, in: J.C. Chlpman,
L.
Horowitz, M.K. Richter and H.F. Sonnenschem,
eds., Preference, Utlhty and Demand (Academic
Press, New York, 1971).
P.RHaImos, Naive Set Theory (Van Nostrand, Princeton, 1960).
M Hausner, Multidimensional
utihtles, m: R.M. Thrall, C.H. Coombs, and R.L. Davis, eds., Decision
Processes (Wiley, New York, 1954).
L.1 Savage The Foundations
of Statistics, second printing (Dover, New York, 1972).