Mathematical Social Sciences 38 1999 11–20
Qualitative probabilities on l-systems
Jiankang Zhang
Department of Economics , Social Science Centre, University of Western Ontario, London, N6A 5C2,
Canada Received 1 March 1998; received in revised form 1 July 1998; accepted 1 August 1998
Abstract
The class of unambiguous events in a state space is naturally modeled as a l-system. Given a binary ‘likelihood’ relation K defined on a l-system, this paper provides necessary and sufficient
conditions such that K can be represented numerically by a convex-ranged, countably additive probability measure.
1999 Elsevier Science B.V. All rights reserved.
Keywords : Qualitative; l-systems
1. Introduction
Given a state space S and a binary relation K on a class of events or subsets of S, a number of papers have described necessary and sufficient conditions in order that K
admit numerical representation by a probability measure. For a finite state space see Kraft et al. 1959; Scott 1964; for an infinite state space, see Savage 1954; Villegas
1964; Fishburn 1970, 1986; Chateauneuf 1985 where the representing probability measure is convex-ranged. The cited results help to axiomatize decision theories in
which preference is based on probabilities and these represent beliefs about likelihoods of events.
In all of the above studies, it is assumed that is a s -algebra. This a priori restriction is problematic for the following reason: The Ellsberg 1961 Paradox and related
evidence show that many decision-makers do not attach probabilities to some events, namely to ‘ambiguous’ ones. In other words, probabilities are assigned only to
‘unambiguous’ events. But the collection of such events is often not a s -algebra because, as illustrated shortly, it is not closed with respect to intersections. On the other
hand, it is intuitive that the collection of unambiguous events is closed with respect to
E-mail address: jzhang2julian.uwo.ca J. Zhang 0165-4896 99 – see front matter
1999 Elsevier Science B.V. All rights reserved.
P I I : S 0 1 6 5 - 4 8 9 6 9 8 0 0 0 3 5 - 3
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
