MEASURE REPRESENTATIONS FOR EVENTS IN A PARTIAL
ORDER: AN AXIOMATIZATION
By
Roger A. Blau, Richard H. Shachtman, Thomas S. Wallsten
School of Business Administration
Department of Biostatistics
Curriculum in Operations Research and Systems Analysis
L. L. Thurstone Psychometric Laboratory
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 967
DECEMBER 1974
MEASURE REPRESENTATIONS FOR EVENTS IN A PARTIAL
ORDER: AN AXIOMATIZATION
ABSTRACT
Currently all the axiom systems for probability representation include the
assumption that the ordering of events is total.
exist for which this condition is too strong.
events A , A in an algebra
2
l
A for
However, empirical systems
In particular, there may exist
which the ordering is not connected; i.e.,
it is not possible to judge with complete confidence whether A $A A or
2
l
One approach to deriving representations for such sets proceed as follows:
assume the existence of a uniformly distributed random variable which induces
a a-field C containing ordered events C., i
1.
intervals are formed, S ..
1.J
~o
<
0
A<
-0
I.
For each A
1
~S
A the triple (S,
~S'
E
A, half-open
C.) for all i, j such that i < j, where
J
is an ordering between the event A and the events C., C.
each A E
j,
= (C.1
~
J
E
C.
Consider for
D), where S is a collection of S .. , for all i and
1J
is a total order to be read as "not more confident that" and 0 is a con-
catenation operation.
Plausible axioms about the triple lead to a distribution
of probabilities lPA for each A
E
A.
MEASURE REPRESENTATIONS FOR EVENTS IN A PARTIAL
ORDER: AN AXIOMATIZATION
3
1
2
ROGER A. BLAU , RICHARD H. SHACHTMAN , THOMAS S. WALLSTEN
1.
Introduction and Background
Consider a decision maker (DM) at a race track trying to decide whether
horse A , A or A is more likely to win.
l
2
3
He may have sufficient knowledge to
judge that A is at least as likely to win as A (A
2
l
l
or nothing about A ,
3
A
3
~A
Ai or Ai
~A
~A
A ), but know little
2
Therefore he is unable to decide conclusively that either
A3 , i
= 1,
2.
One approach to providing a solution for the
DM's dilemma is to invoke the "principle of insufficient reason", Savage [5J,
which instructs the DM to behave as if Ai
claiming that A
l
-A
-A
A ,
3
This would entail, however,
A , which the DM may not be willing to do.
2
We are forced
to conclude that the set {A , A , A } is partially ordered, and therefore not
l
2
3
capable of being represented by a unique probability measure,
All current axiom systems sufficient to insure a probability representation over an algebra of sets (e.g"
[4]).
Chapter 6
in DeGroot [lJ, Fine [2J, Luce
Assume that the ordering between pairs of events is total, i.e"
reflexive, connected and transitive,
Generally, each system also has four
additional axioms concerning properties of the algebra, which in conjunction
with the first condition yield the required measure.
lGraduate School of Business Administration and Curriculum in Operations
Research and Systems Analysis.
2Department of Biostatistics and Curriculum in Operations Research and
Systems Analysis,
3L. L. Thurstone Psychometric Laboratory
2
The above example may be taken as prototypical of many empirical or
practical contexts in which a subjective probability representation is desired,
but cannot be constructed, because the events are partially, but not totally
ordered.
Our goal is to develop a set of axioms sufficient to yield a
numerical representation of
a partial ordering on
nected.
A.
(A,
~A)'
That is,
where
~A
A is an algebra of sets and
~A
is
is reflexive and transitive, but not con-
In addition, we want this representation to be of such a form that a
class of probability measures can be derived, each element of which is consistent with the partial ordering.
Our solution to the problem of numerically representing (A,
achieved in two stages.
~A)
is to be
The first step consists of exhibiting a set of axioms
sufficient to insure the existence of a probability distribution over a probability space defined specifically for each A
€
set of probability distributions indexed over
A,
A.
Hence, we end up with a
{FA:
A
€
A}.
The second
stage of the work consists of developing rational techniques for integrating
the collection of probability distributions {FA:
ability distributions over
A.
A
€
A} into specific prob-
The present paper is concerned with the first
stage.
To obtain the desired individual representation for any A
€
A, we need a
set of axioms that are somewhat different in nature from those normally employed,
but are empirically reasonable nevertheless.
First of all, we will incorporate
into the notion of a partial ordering the idea, alluded to in the opening
example, of differing levels of knowledge about events.
Given sufficient know-
ledge about a pair of events A and A , the DM knows definitely that either
l
2
Al $A A2 or A2
~A
at all whether Al
AI·
~A
Given no knowledge about A or A , the DM cannot judge
l
2
A2 or A2
~A
AI·
However, given at least some knowledge
3
about A and A , the DM will have more confidence either in the statement
2
l
A < A or in the statement A < A.
1 ~A 2
2 ~A 1
Consequently, rather than assuming either that events are comparable or
that they are not, we postulate a totally ordered continuum of confidence
along which statements of the form A
l
~A
A are ordered.
2
a pair of events from A are ordered with respect to
~A
Conceptually, then,
if the DM has
"complete confidence" in one of the two possible orders.
They are unordered
if he has less than complete confidence in either of the statements.
In addition, we will assume the existence of a random variable which has
a uniform distribution on the interval [O,lJ.
It should be noted that the
existence of such a random variable can be proven given a set of purely qualitative axioms.
This has been done by Villegas [6J.
is identical to SP
Furthermore, this assumption
s used by DeGroot [lJ in his development of a subjective
probability representation.
However, our use of this axiom differs somewhat from DeGroot's in a
manner now to be specified.
The existence of the uniform random variable
).
DeGroot
there exists aCE e such that A ~ C.
We will
implies the existence of a totally ordered class of events
assumes that for every A E
A,
assume, instead, that for every A E
(e,
5
e
o
A and
all C , C , C , C E
k
i
j
l
e,
such that
Ci. <eCj and Ck <e Cl' the half-open intervals, C <oASOC and Ck <oA5 Cl
j
0
i
C,Jn
be formed.
Making use of the confidence property assumed for the ordering
of p<lirs of events from
A, it is reasonable to assume that the DM has more
confidence either in the statement C
i
<oA~OCj
or in the statement C
k
<oA~oCl'
fn other wOI'ds, the DM
lS
uncertain of the likelihood ordering of A with respect
to the other events in
A,
but for example, is more confident that A falls into
the interval (C , CjJ than into the interval (C , C J.
i
1
k
4
The notions outlined above will be developed formally in the next section.
roY' cilch A
E
A,
the set of all half-open intervals {(C. , C.J}, together with a
l
]
confidence ordering on the set, and a concatenation operation different from
the union operation, will lead to what we call a comparison algebra.
Two
;:Iructur'dl axioms, in addition to the two assumptions discussed above, will be
i ntroduc('cl.
'lbout
I
The four axioms are then used to prove a variety of properties
he ;;ystem.
The representation theorem is proved in Section 3.
We show that the
algebra, with the properties developed in Section 2, is an open, extensive
structure, as defined by Krantz, Luce, Suppes, and Tversky [3J.
the representation proved by them applies to our structure.
Therefore,
It is then a small
step to show that the representation is, in fact, a unique, finitely additive
probability measure.
The probability measure, it should be kept in mind, is not over
exists independently for each A
E
A.
A,
but
The final section notes the remaining
task for obtaining a class of probability measures consistent with the original
partial ordering.
2.
The Structural Development of a Comparison Algebra
In order to establish a representation for A E
A,
a comparison structure
is required, similar to that developed from qualitative considerations by
Villegas [6J and employed by DeGroot [lJ.
Axiom I:
We therefore state:
There exists a uniform random variable.
The existence of a uniform random variable implies the existence of a totally
ordered collection of events
CC,
< ) distinct from
-C
CA,
~A).
Hence for any
collection {Cj}~=l c C and the binary relation ~C' there exists a permutation
5
(m , ... ,m ) of (l, ... ,n) such that
1
n
<
-C
C
m
n
The connectedness of the ordering allows us to index the collection
closed interval I in the real line 1R;
i. e., for I c 1R
C = {C:
a
~A
Since we have assumed that
to compare an A
E
C with a
a
E
I}.
is a partial order, it makes little sense
A directly with a C.
E
J
C, as was done in [1] and [6].
Instead
we will compose the events from A and C into new events which are totally
ordered as follows.
Definition 2.1:
Let A
E
A and C., C.
1
E
J
C such that C. < C., i < J'; i,J'
-C
1
J
E
I.
We define the half-open intervals
where < and < are a pair of likelihood orderings between any ordered pair
o
-0
C., C.
1
J
E
C and any A
E
A.
Definition 2.2:
SeA) = {S .. :
Definition 2.3:
Let ~S(A) be an ordering on SeA).
Axiom II:
1J
(S(A), ~S(A»
c., C.
1
E
J
C,
i < j; i,j
E
n.
is totally ordered.
Note that SeA) is a collection of sets, each of which is itself a triple.
each set, S .. , represents a likelihood comparison between A, which is only
1J
partially ordered in
A,
and each of C. and C., where C. <
1
J
1
-C
C ••
J
We desire a
6
mPdsurn of the likelihood that A is, in fact, in the interval (C., C.J.
1
till'
~S(A)'
trilJle Sij is ordered by
Thus,
J
which represents the confidence ordering
on half-open intervals discussed 1n Section 1.
The measure obtained for SeA)
trom the ordering and the axioms will induce a probability distribution over
tile
interval [O,lJ for a given A.
When there is no confusion, the dependence on A will be suppressed.
One motivation for Definitions 2.1, 2.2, and 2.3 arises from imagining
tllcit there is mass associated with the events S ...
1J
I"cometrically by the following:
interval on JR and let i,j
let {C :
k
k
E
I
This can be illustrated
= [a,
bJ} correspond to an
I with i < j.
E
[Figure I goes hereJ
The shaded area beneath the curve in Figure 1 represents the density
associated with the event S .. := (C. <0 A <0 C. ).
1J
1
-
J
The ordering for the S .. ' s in
1J
conjunction with our axioms will determine a measure like the one indicated
by the shaded area.
In this sense, a particular ordering on S deter-
m1nes a particular curve for a given A.
The equivalence and strict ordering relations for all the orders <
-A'
~c' ~o
for S and
~S
are defined in the usual way and denoted by -
and < with the appropriate sUbscripts where necessary.
For unions of the events in S to be well-defined, and to facilitate
subsequent developments, it is useful to construct an element definition of
the sets S ..
1J
E
S.
For Sij := (C i <oA:SoC j ), where Ci ' C
j
Definition 2.4:
and A
E
A,
C, i < j; i,j
E
the event S .. is composed of a collection of elementary events
{(C ' AC )}, a
a
a
E
1J
E
I, i < a < j.
I
Figure 1:
Hypothetical density over SeA)
7
Therefore S ..
1J
= (C.1
fusion with other A's in
A,
U
{(C , A )}.
C
.
ex
i <ex< J
ex
we use A instead of A
C
ex
< A -0
< C. )
a
J
=
When there is no con-
.
ex
Definition 2.4 can be understood geometrically by imagining the element
(C , A ) to represent the line segment originating at the point C
ex
a
ex
on the
abscissa and extending vertically to the curve as indicated in Figure 2.
[Figure 2 goes here]
Definition 2.4 provides a basis for constructing the union operation.
={(C,A):
i < ex < j}
U
= {(C , A):
Y
Y
YEO, j]
U
ex
ex
k <
8 <
t}
(k, tJ}.
As presently defined, the union operation on S is not closed.
lD
For example,
equation (2.1), if j < k or t < i, then the right hand side is not in
S;
i.e.,
is not a half-open interval.
Hence we will define a more general binary operation that is closed on
S.
Furthermore this operation will allow constructions sufficiently strong
to satisfy axioms needed in Section 3.
Prior to introducing this binary operation, however, further structure
must be imposed on S in the form of an additional axiom.
"atomless" property for
S.
We will assume an
That is, for the example illustrated in Figure 1,
the curve generated by the binary orderings
~o
and
of an absolutely continuous distribution function.
~S
is the density function
This means that for the
likelihood ordering in S, A is never associated with any particular C. with a
J
positive mass.
Figure 2:
Illustration of element definition of S ..
lJ
8
S; i.e., S0 n Sij
Let S0 be the null set for
Definition 2.5:
either Sk~
Sij is an atom of
(S,
= S0 for all i,j E I, i < j.
~S) if for any Sk~ E
S with
Sk~ ~S Sij'
-S S0 or Ski -S Sij'
Axiom III:
S contains no atoms.
Henceforth, Axioms I, II and III will be assumed.
Theorem 2.6:
Proof:
(C.
1
/\
Sk~
For all Sij and
E S, there exists Shm E S such that
We will show that there exists a C E
m
k
<
0
A <
C ) where i A k
m
-0
equal to {(C , A):
y
y
i
A
k <
= min
C such that (Sij
-S
(i, k) and the right hand side is
Y ~ m}.
T8 ~S (Sij U Sk~)} where T
8
Let 8* = sup{8 E I:
U Sk~)
The ~;c t of 8' s is non-empty and the supremum exists.
= (C i
Let
r,~':
/\ k <0 A SO C8 )·
= inf{ r,
E I:
Sij U Ski ~S Tr,}; then 8* < r,*
Suppose 8* < r,*; then there exists
there exists C
E
e
Slnce
s* <
u
e
<
r*.
'"
C and T
e
E
eE
S such that
Contradiction.
Hence, uS*
We finish by selecting m = 0* and h = i /\ k.
Coro 11 ary 2.7:
(i)
(8*, r,*) and by Axioms I and III,
= ","
r
A
and To~': -S (S ij u S
)
k~'
II
Let S .. E S.
lJ
For all a E I such that a < i, there exists bEl with b
~
j such
9
(ii)
for all bEl such that b > j, there exists a E I with a > i such
The more general binary operation can be introduced now.
Definition 2.8:
For any Sij' Sk£ E S we define a binary operation, 0, on S by
where Shm is given by Theorem 2.6.
We note that Shm is not necessarily a uniquely determined set.
However,
we can, by convention, uniquely determine the result of the composition, 0,
by using the hand m defined in the proof of Theorem 2.6.
It should also be
pointed out that 0 is commutative, as evidenced by the proof of Theorem 2.6.
If (i, j) n (k, £) ~ 0, then Sij 0 Sk£
Corollary 2.9:
where j v £
~
Lemma 2.10:
= Sij
U
Sk£
= Si
A
k, j v £'
max (j, £).
S ..
1J
0
S~
~
= S1J
.. ,
for all S .. E
1J
S.
We will construct the universal set, Sn' n = n(A), where
for all S .. E S.
1J
Let y _ inf{a E I:
T
-
inf{B
E
I:
in I, where T
Lemma 2.11:
S ..
~S (C
1J-
= T(A).
C E C}.
a
<
yo
A <
-0
y exists and is in I.
We can then define
CQ)' for all S .. E S}, which exists and is
~
1J
10
There exists 0 ~ sup{n
o = O(A).
S .. <S (C < A < C ) , for all S ..
1J n 0
-0 T
1J
I:
E
E
S}
E
I, with
By Axiom III:
Theorem 2.12:
(i)
(ii)
( iii)
( iv)
o < T.
there exists S
S
o <S
-
(ii)
1J
:Ss
S
OT'
S.
E
for all S ..
1J
E
S.
S0 <S S OT
Corollary 2.13:
(i)
S ..
OT
S ..
1J
For all S ..
1J
D S OT
S .. u S
1J
Corollary 2.14:
OT
E
S,
= SOT
= S OT
Srl - S OT is unique and 0, T are independent of the order of
computation.
Note that
-S
is an equivalence relation on
which are equivalent
Neither (S,
m
-s
to any S .. using
1J
S
and we may construct sets
D.
nor (S, u) is an algebra:
the former does not contain
complemerrts and the latter does not contain finite disjoint unions.
Proceeding
to a collection of finite disjoint unions would yield an algebra, but is not as
ddvuntageous to us as working with a collection of equivalence classes, [SJ.
The latter is introduced in the next section with an operation,
the equivalence classes, yielding the algebra ([SJ,
G~,
defined on
D*).
1'0 complete our system, we introduce a fundamental axiom.
Axiom IV:
For our purposes, there is no advantage in using the convention of having other
axioms imply the above likelihood property.
11
3.
A Probabil ity Representation
It is convenient to collect and restate the axioms introduced in Section
2, which will be assumed in this section.
Axiom I:
There exists a uniformly distributed random variable
(and thus a totally ordered class of events (C,
Axiom II:
(S, :::S)
S
==
is totally ordered, where
{s .. :
C;
C. , C.
E
(c. <0
A <
-0
l
lJ
S .. lJ
Axiom III:
~C)).
J
l
i < j ; i, j
E
r}
and
c. ) .
J
S contains no atoms.
In as much as the events A
E
A are
not totally ordered with respect to
their likelihood, there does not exist a unique probability measure over these
events.
However, we shall show that under our assumptions, for each A E
A there
exists a unique probability measure over the events in S(A).
Theorem 3.1:
ll'A
[r'offl
For each A
E
A, there exists a unique order-preserving function
S(A) into the unit interval [0,1] such that (Srl(A)' S(A), TI'A)
is a
finitely additive probability space.
In order to prove Theorem 3.1, it is first necessary to construct an
ordered algebra based on S(A).
Suppressing A in the notation, let [S .. J denote
lJ
the equivalence class defined by the equivalence relation -S that includes S .. ,
lJ
and let [SJ be the set of all equivalence classes, excluding [S0 J .
Definition 3.2:
The operation
G~
is defined on [SJ x [SJ as
12
= [8
where Si'j' 0 Sk'~' = Smln"
Sm'n l
E
for some Si'j'
[SmnJ, such that Si'j' n Sk'~'
E
mn
J,
[SijJ, Sk'~'
E
[Sk~J and
= S0'
Using Definition 2.8 and Corollary 2.7, the following lemma can be proved:
Lemma 3.3:
([SJ, 0*) is an algebra.
= {([SijJ,
Consider, [TJ
Si'j'
E
[SijJ, Sk'~'
E
E
(a,T).
S~
~*
E
= S0}'
We now show that [SJ
= SOT' and Axiom III assures us that a < T.
By Axioms I and III, C
s
and ([SaeJ, [SeTJ)
If we let
S0 <S Sij' S0 <S Sk~ and there exists
[Sk~J with Si'j' n Sk'~'
and [TJ are non-empty.
any e
[Sk~J):
E
C; SaS' SST
E
S.
Thus [SasJ
Choose
E
[SJ
[TJ.
be the induced total order on [SJ, the proof of Theorem 3.1
essentially involves demonstrating that the six axioms stated in Secti?n 3.4
of Krantz et al. [3J for an open extensive structure with no essential maximum
are satisfied by the quadruple ([SJ,
~*,
[TJ, 0*).
[3J, which proves the existence of a function~:
[So .J, [SknJ
1]
N
( i)
[S
E
k~
We then invoke Theorem 3.3,
[SJ ~ R+ such that for all
[SJ,
J
<~':
~
[S .. J iff
1J
~([SknJ)
rurthermore, if another function
N
~'
<
-
~([S ..
1J
J);
satisfies (i) and (ii), then there exists
a > 0 such that, for all nonmaximal [S .. J E [SJ,
1J
~'([S ..
1J
J) =
a~([S
.. J).
1J
The method of proving Theorem 3.1 by introducing equivalence classes and
demonstrating that the resulting system is an open extensive structure with no
essential maximum is similar to that employed in the proof of Theorem 5.2 in [3J.
13
Given
Q
set of five axioms, they prove the existence of a unique, order preserv-
ing prohalJility measure for a set of totally ordered events in a specified
However, although the general approach for the present system and
algebra.
their system is similar, the system axioms, and thus the specifics of the
foY'
proofs differ considerably.
We now restate from [3J the six axioms for an open extensive structure
with
nu
essential maximum, in notation appropriate to the present system.
all [S .. J, [SkoJ, [S J
1J
x,
mn
1.
([SJ,
2.
(Associativity).
E
For
[SJ,
1S a total order.
-::'~':)
then ([SkOJ, [S
x,
I f ([SijJ, [Sk£J)
mn
J)
E
E
[TJ and ([SijJ D~" [Sk£J, [SmnJ)
[TJ, ([S .. J, [SkoJ 0* [S J)
1J
x,
mn
E
E
[TJ,
[TJ, and
[S .. J 0* ([S J 0* [S J) <* ([S .. J 0* [S J) 0* [S J.
1J
k£
mn
1J
k£
mn
J.
(Monotonicity and commutativity).
[Sk£ J
[S
mn
<~':
If ([S .. J, [S J)
1J
mn
[So .J, then ([S J, [SkoJ)
1J
mn
x,
E
[TJ
E
[TJ
and
and
J [}': [SkoJ <* [S .. J 0* [S J.
x,
1J
mn
If [Sk£J <* [SijJ, then there exists [SpqJ
(Solvability).
[S ..
1J
5.
(Positivity) .
Ei •
(Archimedean) .
[s.
If ([S .. J, [SkoJ)
1J
x,
[Sk£J
E
D~':
J.
then [S .. J <* [S .. J 0* [Sk o]'
1J
1J
x,
. ] is a standard sequence if [5.
1n ,J n
[So
.
J
1n-l,J n - l
[TJ,
[SJ such that
Every strictly bounded standard sequence is finite, where
. J, ... , [5.
11,J l
E
E
. ] =
1n ,J n
[So . J, for n > 2, and it is strictly bounded if for some
11 ,J l
-
[SJ and for all LSI
. J in the sequence LSI . J <* [Sk£J.
n,J n
n,J n
14
The following eight lemmas demonstrate that Axioms 1-6 for an open
~*,
extensive structure are satisfied by ([SJ,
Lemma 3.4:
A
~*)
([SJ,
is a total order.
[TJ,
D*).
(Axiom 1 obtains.)
stronger version of Axiom 2 can be shown, where associativity holds
as an equivalence relationship.
Lemma 3.5:
If ([SijJ, [SkQ,J)
([SkoJ, [S
~
[SijJ
m'n'
J)
D* [S mn J) E [TJ and
([SijJ D* [SkQ,J) D* [SmnJ·
1J
~*
1
S
n S , , = S0'
ac
mn
E
Suppose i' < k' < m'.
(Axiom 2 obtains.)
[SijJ, Sk'Q,'
Also Sac
E
[SkQ,J, and
S Si'j' D Sk'Q,"
D,
By Theorem 2.6 and the convention
Sac
[Sld J
[j:'"
[S
[Sad J
"V"J';
[S .. J G" ([S kQ, J
1J
mn
J)
E
Furthermore, ([S .. J
[TJ.
1J
..
[S
[}'~
mn
J) .
G'~
[SkQ,J )
G'~
[S
mn
]
and [S
mn
Proof.
]
If ([SijJ, [SmnJ)
[}~
~
Select S.,.,
1 J
Suppose j' < m'.
P such that Sk'Q,'
exist r
~*
[SkoJ
G~
[S .. J
1J
II
[TJ and [SkQ,J $* [SijJ, then ([SmnJ, [SkQ,J)
[S
mn
J.
E
[TJ
(Axiom 3 obtains.)
[S .. J, S, , E [S J, where S.,., n S, ,= S0'
1J
mn
mn
1 J
mn
E
If Sk'Q,'
S
E
-~',
The result is shown in a similar manner
when the other possible orders of i', k', and m' are considered.
Lemma 3.6:
and
.. J, and
= Si'c = Sab D Sbc where Sab E [S 1J
Consider Sac D Sm, n , = Sac D Sc d = Sa d' where Sc d E [S mn J.
.. ],
Sbc n Scd = S0' it follows that ([SkQ,J, [S mn J) E [TJ and ([8 1J
for the operation
..
J
= S0'
[S mn J such that S.,.,
n Sk'o'
1 J
~
Since Sab n
[TJ, then
E
~
By assumption, there exists S.,.,
E
D* [SkQ,J, [Smn J)
[TJ, ([S .. J, [SkoJ
E
D* ([SkQ,J D* [Smn J )
Proof.
S
mn
[TJ and ([SijJ
E
S.,
1
P
.
E
[SkQ,J, and j' < k', by Corollary 2.7, there exists
By Axiom IV, p < j'.
and s such that SmIn'
~S
S.,
J r
~S
Again by Corollary 2.7, there
S
and by Axiom IV, s < r.
ps
Since
15
= S0'
Sill' n Sps
([Smn], [Sk.Q,J)
E
T.
Furthermore, Sk'Q,' 0 SIn'n'
Sm'n' [I Sk,nl
~S S.,
~S S'I
~S S'I"
N
1 S
1 r
1 J
=
IJ SmI n I' by the commutativity of O.
SimiLlr arguments can be made for other orderings of the primed subscripts.
II
A otronger version of Axiom 4 can be shown, where solvability holds
as an equivalence relationship.
Lemma 3.7:
([SknJ, [8
N
Proof:
If [Sk.Q,J
pq
J)
E
<~':
[So .J, then there exists [S
1J
[TJ and [Sk.Q,J 0* [S J
pq
Suppose S.
1
f"
J
E
[So .J, Sk,n,
1J
C'
.)
pq
[S ..
.LJ
=
J
C'
.)
ri
f'
..
E
[SJ such that
(Axiom 4 obtains.)
1J
By Corollary 2.7
N
Si'r and by Axiom IV, r
~
j'.
Consider
= S.1 , r
0 Srj , , so that
A similar argument can be used when k ' < 1. I • II
Thus, ([Sk.Q, ], [S J)
pq
[Sk.Q,J 0'" [S pq J.
",";,t:.
~S
[So .J.
J
[SknJ and i' < k'.
E
N
there exists r such that Sk'.Q,I
~*
pq
E
[TJ and S. I"
1
J
Lemma 3.8:
olJ taLIl:; . )
P!'...Q_o}_:
Clearly [SijJ :s~': [SijJ D~': [SkQ,J·
~;("l('c:t Si1i'
Tnll:;!
E
[SijJ, Skl.Q,I
Lemma 3.9:
[SkJl,J such that Si'jl n Sk'.Q,'
Contradiction.
For all n = 2
m
and all S ..
S, where SflI <S S.. , there exists a
E
~
1J
=
1J
{T~~)}n
0.=1
1J
of S .., such that
1J
~ S T(B)
ij f or a 11 a., B = 1 , 2 , ••. , n.
Proof.
o
Then Sk'.Q,I
II
partition (mutually exclusive and exhaustive),
1,(.0..)
1]
= S0'
However in that case, [Sk.Q, J i [SJ, and thus
be (?qui valent to S0'
(ISijJ, [Sk.Q,J) i [TJ.
E
Suppose [SijJ D~': [SkQ,J ~~': [SijJ·
= sup{a.
Suppose m = 1 (n = 2) and let y
inf{B
I:
E
SBj ~S SiB}'
suppose that y < o.
exists Si8
im~lying
E
E
y
~
o.
Then there exists 8
E
(y,o).
Clearly,
I:
s.10. ::;S Sa.J.} and
We claim that
y = 0,
By Axioms I and III, there
S with 8 > y, implying that S8j <S Si8' and 8 < 0,
that SiB <S SBj' which is a contradiction.
for
16
Thus, T ~:)
lJ
= S.
1y
= T ~ :) .
S S .
YJ
Suppose the lemma is true for n
lJ
(8)
there exists a partition
{T~~)}2m
a, B, = 1, 2, ... , 2 m.
The lemma can be shown to be true for 2
=i
ing the case m
lJ
to subdivide each
likely sets.
II
Lemma 3.10:
For all Sij' Sk~ E
n an d a par t 1Ot'lon
Proof:
(a)
of S .. such that T..
lJ
lJ
0.=1
0
S,
T~~),
lJ
where S0
= 1,
a
<S
~s
n = 2
Ti~) -S Ti~)
for all
m+l
by consider-
m
Sij' S0
<$
Sk~' there exists an
~
u.
-- 1 , .
",
By Corollary 2.7, there exists a p such that Sk~
m
From Lemma 3.9, for all n = 2 , there exists a partition of S .. ,
1J
such that
i.e. ,
... , 2 , into two equally
f S ij , {T(.o..)}n
S k~'
lJ 0.=1 such that T(.a.)
1J <
~S
Assume i < k.
T..
lJ
= 2m ,
for all a,
B=
1, 2, ... , n.
n
•
-S
Sip'
{T~~)}n
1J 0.=1'
Thus associated with each
m
is a partition whose first element we denote by T . . ( ). Either there
1,J n
m
exists an n = 2 which yields a partition where T . . ( ) $$ S. or not. Suppose
1,J n
1p
m
it is not true, i.e., for all n = 2 ,
s.1p <S
(3.1 )
t,
Note that as m
j(n) ~ and since by Axiom IV, p < j(n), the limit of j(n),
say q, is in I and is such that p
p = q, then T.
1q
T.1,J. ( n ).
~
q.
Therefore there exists S.
1p
= S. , which contradicts (3.1).
1p
Suppose p < q.
~$
If
T ..
1q
By Lemma 3.8,
T. can be partitioned into S. , and S , , i < q' < q, where S. , ~S Sq'q'
1q
1q
q q
1q
P < q', then this contradicts the fact that q is the limit of j(n).
If
On the
other hand, if q' < p, a partition of S .. has been found such that
1J
T~~)
= S.1q ,<-S
1J
Lemma 3.11:
S. , contradicting (3.1).
lp
The proof for k < i is similar.
II
Every strictly bounded standard sequence is finite, where a
standard sequence is defined in the statement of Axiom 6.
(Axiom 6 obtains.)
17
Proof:
Ejther every strictly bounded standard sequence is finite or there
exists dt least one which is infinite.
{[So
]
LS.
r
Assume the latter case; Le., suppose
. J}'X)
is strictly bounded, i.e., there is some [Sk~J E [SJ such that
,-J
r=l
-r
. J <~': [Sk~J for all r. Select SkI ~, E [Sk~J, S., ., E [So . J. By
,
ll,J l
ll,J l
lr' J r
(a)
}
n
.
.
,
f S
Lemma 3.10, there exists an n such that { T , ,~' a=l lS a partltlon 0
k' ,~'
k
(a)
dnd Tk, ~I ~S S., ""
,
I I ,J 1
S."
] ,J_."
E
rr
[S.
.
lr' J r
J,r
r = 1, ... , n - 1.
Select
a = 1,
= 1,
...
, n
l,such that S'I .,
ll,J l
Thus, by Lemma 3.6,
(1)
Tk'~'
(2)
o Tk'~'
(2)
<
0 S., . I
~S Tk'~'
ll,J l
<
.,
'11 0 S.,
~S S."
ll,J 1
ll' J 1
~
By
'11
S S."
l2,J 2
incluction, we are led to the following contradiction:
Therefore [So
"J is not strictly bounded for all r, i.e., there exists some
lr' J r
III ~3Uch thilt [Sk~J ~}: [Si
. J. By Lemma 3.7, [Sk~J ~~': [Si . J for all r > m.
m,J m
r,J r
Ilence, every stI'ictly bounded standard sequence is finite. II
We now proceed with the proof of Theorem 3.1.
By Lemmas 3.4 through 3.8
and 3.11, ([SJ, $*, [TJ, 0*) is an extensive structure with no essential maximum.
Thus by Theorem 3.3, in [3J, there is a positive-valued ratio scale
~
on
18
For any S ..
1J
1P(S .. ) =
1J
S, let
E
1ji([S .. J)
1J ,
1ji([Sn J )
0
1P(S,,) = 1 and i f [S .. J
~.
<~':
if
..
SQj <S S1J
if
SQj
[S,,], 0 < 1P(S .. ) < 1.
1J
~
S ..
S 1J
1P is a unique finitely
1J~.
ddditive probability representation, for if another function 1P'
tjJ'([S .. J) :: 1P'(S .. ) would be a representation of ([S],
1J
tjJ'
= atjJ
1J
and 1ji'([SnJ)
= 1ji([SnJ) = 1,
4.
Iji'
= Iji
Subsequent Work:
:s~':,
existed, then
[TJ, D·';).
Since
= 1P. II
and 1P'
A measure on A
As we previously indicated, our approach to representing (A, :SA) consists
of two stages.
The first step, contained in this paper, has been to derive an
indiv idual probability representation 1PA for each A
E
A.
An individual 1PA
does not assign a unique probability to A but reflects a relationship
A and the class C; i.e., 1P A is defined on S(A).
between
If there were to exist j
E
and a sequence {i } c I, i < j, such that i t j and:
n
n
n
lim 1PA (S. .) = lim 1PA (C
i
n~
1n J
n~
<
n
o
A <
~o
C.) = 1,
J
then we could construct a probability value for the individual A.
probability measure on
Let U be a
C; i.e., U measures sets which are inverse images of a
uniform random variable.
A probability value for A, say Q(A), is defined by
Q(A) = \.l(C.).
]
However, it will not be possible to derive a probability value Q(A) in
this manner, because the limit defined in the previous paragraph will be zero
when the atomless property applies.
In this case other techniques must be
I
19
employed to obtain a unique value Q(A) from the probability representation W .
A
Regardless of the method employed, care will have to be taken in deriving
Q(A) for all A
E
A, so that the resulting representation of A will itself be a
probability measure.
That is, the representation must satisfy the three basic
Kolmogorov axioms concerning nonnegativity, boundedness, and finite additivity.
Any set of Q(A) for all A
ability measure on C.
E
A can
meet the first two requirements, given a prob-
But constraints will have to be imposed on the Q(A) in
order to insure that the set is finitely additive.
Thus, the second stage of the initial representation problem is to integrate the W for all A
A
measure over
E
A in
such a manner that the result is a probability
A, and is consistent with
~A'
a first order approximation to each Q(A), A
derived distribution W , A
A
clxioTTls.
E
For example, we may wish to derive
E
A,
which "best agrees with" the
A, constrained only to satisfy the Kolmogorov
We then have the problem of providing and justifying a definition of
"best agrees with" and of specifying the constraints in a workable manner.
There exist a variety of numerical techniques for reducing the W for all
A
A (
A to
a probability representation of
(A,
These will be examined in a subsequent paper.
~A)'
each with its own motivation.
20
ACKNOWLEDGEMENTS
'I'll i,; report is being issued simultaneously as Biostatistics , Institute
() [ ~;t
,1
Ii,; t
ic:~
Mimeo Series report number 967, Operations Research and Systems
Anary,; is Technical Report number 74-3, and as an L. L. Thurstone Psychometric
l.ahorcJlof'y Report number 136.
Supporting agencies are cited below.
!\ul',er fl. Blau, School of Business Administration and Curriculum in Operations
!\('sC'drcll and Systems Analysis.
01
Supported in part by the Business Foundation
NorLh Carolina.
Richard Ii. Shachtman, Department of Biostatistics and Curriculum in Operations
Researcll and Systems Analysis.
Supported in part by National Institute of
Child Health and Human Development Grant number 5-ROl-HD072l4 and National
Institutes of Health, Health Services Research Grant number 5-TOl-HS00045.
Thomas S. Wallsten, L. L. Thurstone Psychometric Laboratory, University of
North Carolina, Chapel Hill, N. C.
Supported in part by Public Health Service
!\(';3earch Crant MH-I0006 from the National Institute of Mental Health and in
p,11't by Science Development Grant No. GU-2059 from the National Science
foundation to UNC.
Thanks also to Richa1'd Haden for a critical reading.
21
REFERENCES
1.
DeGroot, M.H.:
1970.
Optimal Statistical Decisions, McGraw-Hill, New York,
2.
Fine, T.: A Note on the Existence of Quantitative Probability, Ann.
Math. Statist. 42 (1971) 1182-1186.
3.
Krantz, D.H., Luce, R.D., Suppes, P., & Tversky, A.: Foundations of
Measurement. Vol. I: Additive and Polynomial Representations,
Academic Press, New York, 1971.
4.
Luce, R.D.: Sufficient conditions for the existence of a finitely
additive probability measure, Ann. Math. Statist. 38 (1967)
780-786.
s.
Savage, L.J.:
G.
Villegas, C.: On qualitative probability a-algebras, Ann. Math Statist.
35 (1964) 1787-1796.
The Foundations of Statistics, Wiley, New York, 1954.