Greek To Me

1.0 Introduction
I previously described, in an abstract way, a model in which individuals choose rationally even though they may not have a complete transitive preference relation. In that post, I relied heavily on a paper by S. Abu Turab Rizvi. Searching on some of Turab Rizvi's references, I stumbled upon Jeanne Peijnenburg's doctoral thesis, Acting Against One's Best Judgement: An Enquiry into Practical Reasoning, Dispositions and Weakness of Will. Reading some of this thesis inspired me to revisit my model by presenting a somewhat more concrete example.

2.0 Background
I learned a new word from Peijnenburg's thesis. Acting against one's own best judgement is called "akrasia". Peijnenburg shows that discussion of being divided in mind goes back, at least, to debates among Socrates, Plato, and Aristotle. She provides some amusing quotes about akrasia:

"I do not do what I want to do but what I hate... What happens is that I do, not the good I will to do, but the evil I do not intend." -- Romans 7:15 and 7:19

"The mind orders the body and is obeyed. But the mind orders itself and meets resistance." - Augustine

"Two souls, alas, do dwell within this breast" - Goethe

"Faust complained that he had two souls in his breast. I have a whole squabbling crowd. It goes on as in a republic." -- Otto von Bismarck

3.0 The Example
Consider an individual choosing among three actions. This person foresee an outcome for each action. For my purposes, it is not necessary to distinguish between an action and the outcome the individual believes will result from the action. Accordingly, let A, B, and C denote either the three actions or the three outcomes, depending on context.

3.1 Tastes
Suppose that the individual cares about only three aspects of the outcome. For example, if the action is obtaining an automobile of one of three brands, one aspect of the outcome might be the fuel efficiency obtainable from the car. Another might be the roominess of the car interior. And so on.

In the example, the individual has preferences among these three aspects of the outcomes, but not over the outcomes as a whole. "Preferences" are here defined as in neoclassical economics, that is, as a total order. Let the individual order the actions under each aspect as shown in Table 1. For example, under the first aspect, this person prefers A to B and B to C. Since a total order is transitive, one can conclude that this individual prefers A to C under the first aspect. The individual prefers C to A, however, under either of the other two aspects. (This example has the structure of a Condorcet voting paradox, but as applied to an individual.)

Table 1: Preferences Over Aspects of Outcomes

Aspect	Preference Over Aspect
1st	A > B > C
2nd	B > C > A
3rd	C > A > B

3.2 The Choice Function
The individual is not necessarily confronted with a choice over all three actions. Mayhaps only two of the three needed automobile dealers have franchaises in this person's area. The specification of the example is completed by displaying possible choices for each menu of choice with which the individual may be confronted. That is, I want to specify a choice function for the example:

Definition: A choice function is a map from a nonempty subset of the set of all actions to a (not necessarily proper) subset of that nonempty subset.

The domain of a choice function is then the set of all nonempty subsets of the set of all actions. Informally, the value of a choice function is the set of best choices on a menu of choices with which an agent is confronted. (The above definition is a variation on the one I gave in my previous post.)

Table 2 gives the choice function for this example. The first three rows show that in a menu consisting of exactly one action, the individual chooses that action. In a menu consisting of exactly two actions, the individual is willing to choose only one of those actions. And in a menu with three actions, the individual is willing to choose any of the three.

Table 2: The Choice Function

Choices on the Menu	Best Choice(s)
{A}	{A}
{B}	{B}
{C}	{C}
{A, B}	{A}
{A, C}	{C}
{B, C}	{B}
{A, B, C}	{A, B, C}

3.3 The Conditions of Arrow's Impossibility Theorem
I intend the above example as an illustration of application of Arrow's impossibility theorem to a single individual. The choice function given above is compatible with the conditions of Arrow's impossibility theorem:

• No Dictator Principle: For each aspect, some menu exists in which the choice function specifies a choice in conflict with preferences under that aspect. For example, the choice from the menu {A, C} conflicts with the individual's preferences under the first aspect of the outcomes.
• Pareto Principle: This principle is trivially true in the example. No menu with more than one choice exists in which preferences under all aspects specify the same choices. So the choice function cannot be incompatible with the Pareto principle when it applies, since it never does apply.
• Independence of Irrelevant Alternatives: I think this principle is also trivially true.

In compatibility with Arrow's impossibility theorem, the existence of a single preference relation is not possible for the above choice function. A preference relation applies to all possible pairs of actions, and it must be transitive. But a transitive relation cannot be constructed for the three menus consisting of exactly two actions. So I have defined a choice function, but preferences (one total order) does not exist.

4.0 Conclusions
Neoclassical economists tend to equate rationality with the existence of a unique preference relation for an individual. In other words, rationality for an individual is identified with the existence of one total order (that is, a complete and transitive binary relation) over a space of choosable actions. The example suggests this point of view is mistaken. An orthodox economist can either assert that the individual in the example is not rational or accept that he has been learning and teaching error.

A choice function is a generalization of preferences, as neoclassical economists understand preferences. If such preferences exist for an individual, then a choice function exists for that individual. But individuals can have choice functions without having such preferences, as is demonstrated by the above example. It is up to those asserting the existence of preferences to state their special-case assumptions, to show that models with those assumptions can provide falsifiable predictions about society, and to provide empirical evidence. The evidence from experimental economics, though, is systematically hostile to neoclassical economics. The phenomenon of menu-dependence is particularly apposite here.

So much for prattle about competitive markets yielding efficient outcomes.