The impossibility of rational consumer choice

Abstract

In this paper we show that a rational consumer choice along the lines traditionally suggested might lead to paradoxical results if one considers multidimensional goods, which incorporate a series of incommensurable aspects. Thereby, we explore the similarity between the resulting paradox and Kenneth Arrow’s well known Impossibility Theorem. Based on these considerations we suggest a solution for the former problem along the lines of Herbert Simon and Amos Tversky, which might—if driven to its extreme—even provide a unique and arguably rational solution for consumer choice among multidimensional goods. Eventually, we argue that the resulting framework poses a potentially useful starting point for further developing an evolutionary theory of consumer choice.

Appendix: Arrow’s theorem and the impossibility of consumer choice: an analogy

This section is devoted to a formal treatment of the aforementioned topics. We are going to define a rational ordering process and set down several axioms this process should satisfy. The treatment mirrors the discussion in Arrow (1950, 1951), but is presented completely independently. We aim to show that the problem of collective choice as posed by Arrow is formally equivalent to the problem of consumption choice among multidimensional goods. Table 2 provides a first overview by showing some basic terminological similarities between these two approaches.

Table 2 Some basic structural analogies between Arrow’s impossibility theorem and the problem of consumption choice among multidimensional goods

Linear, transitive orderings

We will demand that the ordering $\succcurlyeq_{d}$ of the products within each product dimension d is linear and transitive. Linearity states that for any two products P and Q we either have

$$ P \succcurlyeq_{d} Q~~~\mbox{or}~~~ Q \succcurlyeq_{d} P. $$

This excludes the possibility that two products are incomparable within said dimension. Furthermore, we shall assume transitivity.

Rational ordering process and axioms

Assume now we are given a set of products and dimensions in which those products are ranked following a linear and transitive order relation. How would a rational consumer rank the products according to their desirability based on their dimensional rankings alone? A rational ordering function will be any process satisfying certain axioms which, based on the dimensional orderings $\succcurlyeq_{d_{1}}, \succcurlyeq_{d_{2}}, \dots$ alone, extracts an ordering $\succcurlyeq$ of the products reflecting their overall desirability.

Axiom 1

(Unrestricted domain) Although somehow already inherent in our discussion, we explicitly state that the rational ordering process should yield a product ordering for any possible orderings within the dimensions.

Axiom 2

(Account for ordinal superiority) So far, any universally applicable process is regarded as a rational ordering process, we therefore need to make certain that the dimensional orderings are reflected in the overall ranking. Suppose we are given m products in d dimensions and are given the dimensional rankings and a rational ordering process assigns the overall ranking

$$ P_1 \succcurlyeq P_2 \succcurlyeq P_3 \succcurlyeq \dots $$

Assume now furthermore that from this we create a second set of dimensional rankings by choosing one product P and setting up the new dimensional rankings by using the old rankings and moving the product P to the left or leave it at the same place while, at the same place, leaving all other interproduct relations the same. This corresponds to a manufacturer improving the product in some dimensions while leaving it untouched in other dimensions. Then, in the overall product ranking, any rational ordering process $\succcurlyeq$ should rank the improved product P superior to all products to which it was superior before.

Axiom 3

(No prejudices) This axiom is in somewhat the same spirit as the last axiom and states that for any two distinct products P and Q, there are dimensional rankings such that

$$ P \succcurlyeq Q~~~\mbox{ and other rankings yielding }~~~Q \succcurlyeq P. $$

Axiom 4

(Independence of irrelevant alternatives) If the rational ordering process assigns an overall ranking $\succcurlyeq$ to a given set of products based on their dimensional rankings $\succcurlyeq_{d}$ and if then furthermore one product turns out to be unavailable and is then removed from the dimensional rankings, then any rational ordering process applied to the reduced dimensional relations should yield the same overall ranking as before with the single change of the unavailable product being removed.

The surprising result is now that although all these conditions are very natural within this framework, there is only a simple type of rational order function satisfying all these conditions.

Theorem 1

(Impossibility of rational order functions; Arrow 1950) Suppose we are given at least three products and a rational order process satisfying all the conditions above. Then this process is monomanical, i.e. there exists a ‘dictatorial’ dimension such that the process assigns to each possible set of dimensional relations an overall rating, which is identical to the product rating within this dimension.

Proof

We deduce the result from Arrow’s Impossibility theorem. The linearity and transitivity conditions imply Arrow’s axioms 1 and 2; the universality condition implies Arrow’s condition 1 [unrestricted domain], the second and the third axiom imply Arrow’s conditions 2 [positive association of social and individual values] and 4 [citizens’ sovereignty], while the axiom of independence of irrelevant alternatives is the same in both settings. Arrow’s theorem then implies that the social process must be dictatorial, which corresponds to our definition of monomanical. □

Remark (Two products)

(Two products) The assumption that at least three products be presented is a necessary one: there is a rational ordering process if we are only given two different products P and Q ordered in d dimensions according to $\succcurlyeq_{1}, \dots, \succcurlyeq_{d}$ given by saying that product P is ranked superior to Q if it is ranked superior to Q in more or the same number of dimensions than Q is ranked superior to P. In the latter case both products are equal.