Why be Consistent? A Critical Analysis of Consistency Requirements in Choice Theory

Abstract

Consider the following very general and entirely conventional description of a choice problem. There is a set X of conceivable end states; these are finite in number and mutually exclusive. Some non-empty subset of X, which I shall denote by S, is the feasible set; it is from this set that a choice must be made. A plan or procedure for choosing can be represented by a choicefunction C( * ) that assigns a chosen set C(S) to every logically possible feasible set S-that is, to every non-empty subset of X. The chosen set is a non-empty subset of S. To borrow a term of Schwartz's, C(S) is the set of choosable end states (any one of which, according to the rule, may be chosen), and S C(S) is the set of non-choosable end states (none of which may be chosen). Any choice function is to be understood as referring to a given environment; its purpose is to dictate how choices should change as the feasible set changes, everything else remaining constant. To say that a choice function exists is to say that some choice is possible in every contingency; but this says nothing about the rationality or consistency of these choices. Consistency conditions are imposed on choice functions as a means of ruling out patterns of choice that are held to be irrational. In this paper I shall discuss a very weak property of consistency, which I shall call minimal consistency.