Conditional Preference Orders and their Numerical Representations

1. Introduction

In decision theory, the normative framework of preference ordering classically requires the completeness axiom. Yet, there are good reasons to question the idea of completeness as famously pointed out by Aumann [2]: "Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable."

For instance, certain decisions that an individual is asked to make might involve highly hypothetical situations, which he or she will never face in real life. Thus, he or she might not be able to reach an “honest” decision in such cases. On the other hand, some decision problems might be extremely complex, too complex for intuitive “insight,” and our individual might prefer to make no decision at all in these problems.

Aumann's remark, supported by empirical evidence, triggered intensive research in terms of interpretation, axiomatization and representation of general incomplete preferences, see [3,,,,,,] and the references therein. It suggests that the lack of information underlying a decision making is a natural source of incompleteness. Here, we suggest a framework for formalizing this phenomenon of contingent decision making and its representation.

2. Conditional Sets

As mentioned in the introduction, we model the contingent information, conditioned on which a decision maker ranks prospective outcomes, by an algebra of events, denoted as A. For technical reasons, we assume that it is a 𝛔-algebra with a probability measure defined on it. The inclusion of two events is then to be understood in the almost sure sense.

A set X – which in the present context describes acts – is a conditional set of A if it allows for conditioning actions: X → X|A for each event A ∈ A, which satisfy consistency and aggregation properties.

3. Arguments and Conclusions

Our approach to utility theory proposes that the decision-making process is not simply a matter of selecting from an exogenously given set of alternatives, but rather involves a process of considering context-specific factors, and making decisions on the basis of this additional information.

A striking advantage of our approach is that it provides a powerful framework for modeling a variety of contexts in which a decision maker might face uncertainty, from financial investments to health, or even environmental choices.

In conclusion, the conditional version of classical decision theory tools outlined here, including the representation results of Debreu and Rader, and von Neumann and Morgenstern, offer flexibility and capability of capturing complex decision scenarios in a more realistic, and arguably, rational manner.