### Discussion Paper

A Generalized Unification Theorem for Choice Theoretic Foundations: Avoiding the Necessity of Pairs and Triplets

## Abstract

This paper is concerned with the axiomatic foundation of the revealed preference theory. Many well-known results in literature rest upon the ability to choose over budget sets that contains only 2 or 3 elements, the situations which are not observable in real life. In order to give a more realistic approach, this paper shows that many of the famous consistency requirements, such as those proposed by Arrow, Sen, Samuelson etc., are equivalent if the domain of choice function satisfies some set of theoretical properties. And these properties, unions and inclusions for example, are proposed in a way that gives observability.

## Comments and Questions

see attached file

May be the author have not made it clear in the paper, the referee could have misunderstood or overlooked one interesting point. That is Assumption 1 is actually a weaker assumption than what (s)he called the "up to three elements" assumptions. Therefore Assumption 1 is easier to verify, or more ...[more]

... plausible than the the "up to three elements" assumption.

My apologies for not making this clear in my report: I do know that the "up to three elements condition" implies Assumption 1 (by taking B_1 to be the set of all singleton sets).

Yet that a condition is "satisfied" is different from it being "verified". A condition is ...[more]

... satisfied if it is true; it is verified if we can assign a true/false value to it. The condition "my wife loves me" is satisfied but not verified if she does love me but I am unsure if she does (say we just quarreled). Suppose I obtain a data set of a consumer's choice, with a collection of budgets (out of a finite consumption set) and choices from each budget. It is fairly straight-forward to write a computer program to check whether the "up to three elements condition" is satisfied. The computer program for checking Assumption 1 would be much less straight-forward and would take a much longer time to run. In this sense I regard Assumption 1 harder to verify.

I also understand that the probability that Assumption 1 is satisfied cannot be smaller than the probability that the "up to three elements" condition is satisfied. Yet if the author criticizes the the "up to three elements condition" as "unlikely to be observed" (p. 1) and "unrealistic" (p. 9), it is only fair to ask how likely we will observe situations in which Assumption 1 is satisfied but the "up to three elements condition" is not. Indeed, if the majority of the "observable" instances of Assumption 1 also satisfies the "up to three elements condition", then the critique the author makes against the literature applies to this paper as well! (This is NOT to suggest that the author should go about using his/her combinatorial theory to find out the proportion of budget domains satisfying each condition -- the probability distribution need not be uniform.) For the sake of argument let's say that it is not easy to observe all budget sets containing exactly two or three elements. But is it easier to observe all three-set unions of budget triangles? In this sense I do not see how Assumption 1 is more plausible.

Thanks to referee's reply. The comments are very valuable. It is true that the paper was not clear why some assumptions are better in as in the referee's opinion. Maybe the following arguments would help making it more precise.

First, it is not whether WARP is true on "budgets ...[more]

... containing up to three bundles" that the paper is trying to generalize upon, but whether we should believe that there are situations where people choose from "budgets containing up to 3 bundles". It seems even if we can afford only 2 goods, a bottle of milk and a loaf of bread, and then there are immediately infinitely many bundles we can choose to consume. The budget now would contain all combinations of milk and bread out of the bottle and the loaf. So it is perhaps reasonable to say that when people choose from a budget, the budget would most likely contain a large number of bundles. Because we want to maintain and support theories built upon revealed preference theory, so we propose assumption 1 and show that the rationalizable conditions are equivalent is not dependent upon choosing from budgets containing up to 3 elements. In this way, we hope the theory would not be undermined by thought experiments such as "milk and bread" above.

Secondly, it is natural to ask whether the assumption 1 is true. In answering this questions, we consider the collection "of all

finite discretised budget triangles, claiming that unions of budget triangles can be observed in price cut or wholesale situations (p. 5)." We do this simply to provide a way to think of the evidences to support the assumptions. We do not suggest that this would verify the assumptions since we cannot really "show the assumption is true" and it is beyond the reach of this paper. However, we do hope that thought experiments such as the "milk and bread" example above, would suggest that unions of budgets do give a budget.

Finally, we think that it is nice to know some deep results on rationalizability do not dependent upon the choosing from budgets containing up to 3 bundles. We do not know a way to test every assumption, but as Sen (1973) pointed out, whether some assumptions are true of not should be taken on faith. And it would be nice to avoid potential paradoxes in assumptions, and we think a weaker assumption would be better theoretically. We understand that such theoretical results in the paper and other pure theoretical papers may not be applicable (at least not immediately), but its value would lie in avoiding paradoxes.

Again, the comments from the referee are very genuine and helpful. The paper has not made many of the above arguments. And we do not claim the above arguments are comprehensive. The paper was written in a pure theoretical way and it was not intended to address any practical methods. The word "plausible" is hence not "experimentally testable", but avoiding a relatively strong assumption in thought experiments.

see attached file

Many thanks to the referee, the comments were well made. We would try to improve the paper and try to address issues in both referee's reports.