Invited ReviewDecision-theoretic foundations of qualitative possibility theory☆
Introduction
The expected utility criterion for decision under uncertainty was the first to receive axiomatic justifications both in terms of probabilistic lotteries [36] and in terms of preference between acts [30]. These axiomatic frameworks have been questioned later, challenging some of the postulates leading to the expected utility criterion, on the basis of systematic violations of these postulates (e.g., [1], [17]). For instance Gilboa [19] and Schmeidler [31] have advocated lower and upper expectations expressed by Choquet integrals attached to non-additive measures, sometimes corresponding to a family of probability measures (see also [20], [29]). In this paper, we propose axiomatic justifications for two qualitative criteria, an optimistic and a pessimistic one whose definitions only require finite linearly ordered scales. The pessimistic criterion can be viewed as a refinement of the Wald criterion, where uncertainty is expressed in a qualitative way and is captured in the framework of possibility theory [13], [15], [44].
Section snippets
Background on qualitative possibility theory
A possibility distribution π on a set of possible worlds or states S is a mapping from S to a bounded, linearly ordered valuation set (L,>). This ordered set is supposed to be equipped with an order-reversing map denoted by nL, that is, a bijection of L on itself such that if α>β∈L, then nL(β)>nL(α). Let 1 and 0 denote the top and the bottom of L, respectively. Then nL(0)=1 and nL(1)=0. In the numerical setting, L=[0,1], and function nL is generally taken as 1−·. Here, it is only assumed that L
Qualitative counterparts of expected utility
Generally, decisions are made in an uncertain environment. In the Savage framework [30], the consequence of a decision depends on the state of the world in which it takes place. If S is a set of states and X a set of possible consequences, the decision-maker has some knowledge of the actual state and some preference on the consequences of his decision. Here, a belief state about which situation in S is the actual one, is supposed to be represented by a normalized possibility distribution π from
The axiomatics of Savage for expected utility
The weak point of the above axiomatic justification of qualitative utility theory is that the uncertainty theory (here possibility theory) is part of the set of assumptions. While this approach is natural when uncertainty is captured by objective probabilities, as done by von Neumann and Morgenstern, it is more debatable for subjective uncertainty. On the contrary Savage has proposed a framework for axiomatizing decision rules under uncertainty where both the uncertainty function and the
Properties of possibilistic utility
One of the key postulates of Savage is the sure thing principle which expresses, roughly speaking, that if f is preferred or is equivalent to g and these two acts result in identical consequences on a subset B⊆S, then if f and g are modified in the same way on B, the two modified acts remain ordered as f and g.
However, two acts may be found equivalent just because they have identical and extreme (very good or very bad) likely consequences on B⊆S, while one act would be strictly preferred to the
Act-driven axiomatization of possibility theory and qualitative utility
In this section it is shown that the pessimistic and optimistic possibilistic utilities can be axiomatized in the style of Savage, just like expected utility. The main difference is that a finite setting is enough to prove the results. In a first step, we point out a general framework for describing many families of monotonic set-functions in terms of acts, thus providing a practically testable framework for many non-probabilistic uncertainty theories. Namely, by asking a decision-maker to rank
Concluding remarks
One strong assumption has been made in this paper, which is that uncertainty levels and utility levels are commensurate. This is already a consequence of the first axiom of Savage. An attempt to relax this assumption has been made in [8]. These authors point out that working without the commensurability assumption leads to a decision method based on uncertainty representations connected to non-monotonic reasoning. Unfortunately, that method also proves to be either very little decisive or to
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This work was done while the author was preparing a PhD at IRIT.