Indifference, indecision, and coin-flipping
Introduction
The foundations of theories of economic choice are often discussed in terms of three relations: strict preference, indifference, and weak preference, the last of which is the union of the first two: It is usually proposed that these relations completely order the set of possible options, , so that an agent either strictly prefers any given option to any other given option, or is indifferent between them.
If these relations do not completely order a set, that does not mean that some pair of options has is no relation; rather, its relation is a sort of complement: But there is some challenge in giving an interesting empirical content to the supposition of such a relation.
Given an interpretation of indifference as equal valuation, a natural candidate for this complement would be some sort of indecision about relative valuation, such that the individual was neither prepared to say that one choice were better than another nor that they were equally good. But it is actually not immediately clear how behavior under such indecision would differ from that under indifference, except for utterance. For example, were an agent told that she would be given if she did not actively request , then she would end up with the default if she were eitherindifferent or undecided. And, while an undecided person may later come to the decision that one was indeed better than another, a change of mind is also possible with indifference. It does not seem much to matter to the economist whether the agent says “I don’t know” or “I don’t care”. If such indecision does not produce choices different from those of equal valuation, then one might as well interpret indifference as the union of the two.
However, this article will identify two relations distinct from strict preference which correspond to meaningfully distinct choice behavior. One of these relations will have some intuitive correspondence to indecision about relative valuation.
Section snippets
A problem of ordinary interpretation, and an observable distinction
When the question is asked of how an agent makes a choice between two things, and , between which she is indifferent, a stock reply is that she “flips a coin”. There are at least two problematic aspects to this reply.
The first is that it quite fails to answer the question asked, but presumes that every choice between and may be replaced with a choice amongst , and , where is a lottery between and . The ability to make such replacements is rarely if ever explicit or
Preliminary
The conceptual foundations of a theory of choice are often expressed principally in terms of preference relations. In this article, however, foundations will be laid in terms of choice functions. The operationalization of the classic relations and of any proposed additional relation is in the choices that result; and, while choice functions are imperfectly observable, they are observable less indirectly than are preferences.
However, this article does not have the same sort of ambitions with
Significance of the model
The model functions as a sort of proof of concept for an operationalization of preferences as an incomplete preordering (by the union of strict preference with equi-indifference). Differences, beyond utterance, are observable between paralysis which is ended by inclusion of an option of a “coin flip” and that which is not. Plausible propositions imply intuitively appealing properties to relations defined in terms of these observable behaviors, as well as other properties which are themselves at
Areas for possible future work
I have presumed that the outcome of a lottery can be described without reference to the lottery, but some might prefer an alternative conceptualization in which a state of the world intrinsically includes the means by which it was effected, so that one who chooses to “flip a coin” has chosen a different ultimate outcome. The model herein could accommodate that conceptualization largely by no more than a reinterpretation of notation, but the lottery equalities (5), (6), (7) would have to be
Conclusion
A bit more than fifty years ago, Savage wrote
There is some temptation to explore the possibilities of analyzing preferences among acts as a partial ordering, that is, in effect, to replace part 1 of the definition of a simple ordering by the very weak proposition , admitting that some pairs of acts are incomparable. This would seem to give expression to introspective sensations of indecision or vacillation, which we may be reluctant to identify with indifference. My own conjecture is that
Acknowledgments
I am grateful to Mark Machina, Steven Raphael, Anthony Gamst, and Walter J. Savitch for stimulating discussions on these issues, and to Christina J. van Leeuwen for support during the writing of this paper. Any inadequacies of logic or of exposition are of course my responsibility.
References (9)
- et al.
Expected utility without the completeness axiom
Journal of Economic Theory
(2004) - et al.
Indifference or indecisiveness? choice-theoretic foundations of incomplete preferences
Games and Economic Behavior
(2006) Incomplete preferences and rational intransitivity of choice
Games and Economic Behavior
(2005)Indifference and incompleteness distinguished by rational trade
Games and Economic Behavior
(2009)
Cited by (1)
Formal qualitative probability
2021, Review of Symbolic Logic