Rational Choice and Revealed Preference

Abstract

According to the currently dominant view, the choice behaviour of an agent is construed to be rational if there exists a preference relation R such that, for every set S of available states, the choice therefrom is the set of R-optimal points in S.' There are at least two alternative definitions of R-optimality-R-maximality and R-greatestness. On the one hand, an x in S is said to be R-maximal in S if there exists no y in S which is strictly preferred to x in terms of R. On the other, an x in S is said to be R-greatest in S if, for all y in S, x is at least as preferable as y in terms of R. The former viewpoint can claim its relevance in view of the prevalent adoption of the concept of Pareto-efficiency in the theory of resource allocation processes.2 The latter standpoint is deeply rooted in the well-developed theories of the integrability problem, revealed preference and social choice. The difference between these two definitions of rational choice is basically as follows. Any two states in a choice function which is R-maximal rational are either R-indifferent or R-incomparable, while any two states in a choice function which is R-greatest rational are R-indifferent. A condition for rational choice has been put forward in terms of the R-greatestness interpretation of optimality (Hansson [4] and Richter [9, 10]). In this paper, a condition for rational choice in its R-maximality interpretation will be presented. The condition in question will, in a certain sense, synthesize both concepts of rationality, because it can be seen that the rational choice in terms of R-maximality is rational in terms of R-greatestness as well. The role of various axioms of revealed preference and congruence in the theory of rational choice will also be clarified. In this kind of analysis, special care should be taken with the domain of the choice function. It was Arrow [1] who first suggested that the demand-function point of view would be greatly simplified if the range over which the choice functions are considered to be determined is broadened to include all finite sets . This line of enquiry was recently completed by Sen [13].' It is true, as was persuasively discussed by Sen [13, Section 6], that there is no convincing reason for our restricting the domain of the choice function to the class of convex polyhedras representing budget sets in the commodity space. At the same time, however, it should be admitted that there exists no specific reason for our extending the domain so as to include all finite sets. This being the case, no restriction whatsoever will be placed on the domain of the choice function in this paper except that it should be a non-empty family of non-empty sets. In Section 2, our conceptual framework will be presented. The main results will be stated in Section 3, proofs thereof being given in Section 4. In Section 5 we will present some examples which will negate the converse of our theorems. Finally, Section 6 will be devoted to comparing our results with the Arrow-Sen theory, on the one hand, and the Richter-Hansson theory, on the other.