12 J
. Zhang Mathematical Social Sciences 38 1999 11 –20
complements and disjoint unions. Thus, as argued in Zhang 1997; Epstein and Zhang 1997, a l-system is a more appropriate mathematical structure for modeling the
collection of unambiguous events. This paper provides necessary and sufficient conditions such that the binary relation
K on a l-system can be represented by a convex-ranged, countably additive probability
measure. To illustrate the failure of to be an algebra, consider the following example taken
from Zhang 1997: There are 100 balls in an urn and a ball’s color may be black B, red R, grey G or white W. The sum of black and red ball is 50 and the sum of
black and grey ball is also 50. One ball will be drawn at random. It is intuitive that the unambiguous events are given by
D
5 f, B,R,G,W , B,G , R,W , B,R , G,W ,
1.1 h
h j h
j h j h
j h jj
where each of these events has the obvious probability. Observe that hB,Gj and hB,Rj are
in , but that hBj5hB,GjhB,Rj is not in .
2. Preliminaries
2.1. l-systems Let S be a nonempty set. Say that a nonempty class of subsets of S is a l-system, if
l.1 S [;
c
l.2 A[ ⇒
A [; and l.3 A [, n51,2, . . . and A A 55, ;i ±j
⇒ A [.
n i
j n
n
These properties are intuitive if we interpret as the collection of all events that are assigned probability by the decision-maker. For example, if she can assign probability to
c
A, then it is natural for her to assign the complementary probability to A . For l.3, if she can assign probabilities to the disjoint events A and B, then it is natural for her to assign
the sum of these probabilities to AB. On the other hand, when these events have a nonempty intersection, the union may be ambiguous. Equivalently, as we have seen, the
intersection of two unambiguous events may be ambiguous.
We have the following lemma see Billingsley, 1986:
Lemma 2.1. A nonempty class of subsets of S is a l-system if and only if
l.19 5, S[; l.29 A, B[ and AB
⇒ B
\A[; and
l.39 A [ and A A , n51,2, . . . ,
⇒ A [
.
n n
n1 1
n n
A function
J . Zhang Mathematical Social Sciences 38 1999 11 –20
13
p: A →
[0,1] is a countably additive probability measure over if:
P.1 p550, pS 51; and P.2 p A 5
o pA , ;A A 55, for all i ±j.
n n
n n
i j
Denote by S, , p a l-system probability space. A probability p is convex-ranged if for all A[ and 0,r ,1, there exists B , A, B [, such that pB 5rpA. A
probability measure p is nonatomic if for all A[ with pA.0, there exists B , A, B [ such that 0,pB ,pA. It is well known that p is convex-ranged if and only if it
is nonatomic when is a s -algebra. However, it is an open question whether such equivalence is valid on a l-system.
2.2. Qualitative probabilities Let K be a binary relation . K is a qualitative probability if
Q.1 K is a weak order reflexive, complete and transitive; Q.2 AK5 for all A[;
Q.3 S s5; and Q.4 AC 5B C 55 implies [AKB
⇔ ACKB C].
A countably additive probability measure p on represents K if AKB
⇔ pA pB ,
for all A and B in .
`
The sequence hA j
in K-converges to A[ if for any events A , A in with
n n 51
A a Aa A, there exists an integer N such that A a A a A, whenever n N.
n
3. Main theorem
