Indifference and incompleteness distinguished by rational trade

a r t i c l e i n f o a b s t r a c t JEL classification: D01 D11 Keywords: Incomplete preferences Indifference Sequential rationality Revealed preference Incommensurability We use an agent's strict preferences to define indifference and incompleteness relations that identify the sequences of trades that are rational to undertake. If an agent makes sequences of trades of options labeled indifferent, the agent will never be led to an inferior outcome, but trades of options where no preference judgments obtain can lead to diminished welfare. For one-shot choices, in contrast, there can be no behavioral distinction between indifference and incompleteness. Applications include: an isomorphism for incomplete preferences that indicates when weak and strict preferences contain interchangeable information, a characterization of the (possibly incomplete) preference relations consistent with a one-shot choice function, and an equivalent definition of incompleteness that relies on the philosophical theory of incommensurability. It is often said that indifference and incompleteness are behaviorally indistinguishable. Both an agent who is indifferent between two options and an agent who cannot rank the options can rationally choose either one. And if indifference and incompleteness lead to the same choice behavior, the entire concept of incompleteness would seem to be superfluous. The trouble with these views is that they look at each of an agent's choices in isolation. When decisions are strung together, the terrain looks different. Indifference and incompleteness relations can then demarcate which sequences of trades are rational: the options that we will label 'indifferent' can always be traded in either direction without leading an agent to a dispreferred final outcome, while trades of the options we label 'unranked' can make an agent worse off. (We use 'unranked' as the adjective to describe incompleteness: x and y are unranked if x is not weakly preferred to y and y is not weakly preferred to x.) We call these indifference and incompleteness relations 'behavioral' because they are determined solely by properties of an agent's strict preferences , which can be inferred from choice behavior, and because they make testable predictions. Let the notation x ∼ * y mean not x y and not y x. It can be inappropriate to view ∼ * as an indifference relation; even when is transitive, the relations and ∼ * taken together need not form a transitive weak preference relation. Instead, we define the behavioral indifference relation ∼ B by …