Saturday, March 28, 2009

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
AspectPreference Over Aspect
1stA > B > C
2ndB > C > A
3rdC > 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 MenuBest Choice(s)
{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.


YouNotSneaky! said...

'So much for prattle about competitive markets yielding efficient outcomes. '

What in the world does this have to do with competitive markets or even efficiency?

Robert Vienneau said...

"Efficiency", as used in the first fundamental theorem of welfare, is only defined if people have preferences, that is, one total order over the set of allocations.

YouNotSneaky! said...

Right, so your post has nothing to do with efficiency since it makes it impossible to define in the first place.

YouNotSneaky! said...

In other words competitive markets will not yield efficient outcomes (since efficiency is undefinable) but neither will any other social arrangements. So you could've just as well written:
"So much for prattle about socialism yielding efficient outcomes"
"So much for prattle about small scale husbandry and subsistence farming yielding efficient outcomes"
"So much for prattle about techno utopia yielding efficient outcomes"

Neverfox said...


Yes, he could have but none of those statements have the pride of place of being "fundamental theorems of welfare economics". Perhaps when they achieve that status, Robert will end his essays with them.

YouNotSneaky! said...

Yeah, but that's only because the socialists lost the Socialist Calculation Debate (empirically and theoretically). It's worth remembering that in the first half of 20th century the debate really was conducted in those kinds of terms - of relative efficiency of capitalism vs. socialism. For Oskar Lange or Abba Lerner the General Equilibrium system was just a series of equations that the central planner should solve in order to be efficient. It was only when subsequent events made the claim of socialism as more efficient seem absurd, did the socialists drop efficiency as a criteria and effectively said "let's talk about something else".

Or think of this in a different ways. Austrian economists tend to insist that the market is always "good", no matter what kind of real life inefficiencies it might produce because .... well, because they've defined their terms in a way that efficiency is basically undefinable unless you equate it with the market outcome. Robert's doing a similar thing here. He's defining things so that the very concept of efficiency is not possible and then, hop, skip, jump, to the conclusion that the market is not efficient.

I'll take my boring neoclassical theory where there is a clear definition of efficiency and it's possible for markets to sometimes produce efficient outcomes and sometimes produce inefficient outcomes. At least that way one's not ruling out one possibility by assumption and holding one's thumb on the scales.

Robert Vienneau said...

Nor do I "conclu[de] that the market is not efficient." And anybody aware of the history of the left can point to some arguing in any period for communism or socialism on other grounds than "efficiency".

James said...

Nor do I "conclu[de] that the market is not efficient."

Fair point. You only dimiss the opposing view as "prattle."

On a more serious note, if you intend posts like this as attacks on mainstream economics, you missed. Mainstream economists don't really believe that their models are true with a capital t. They only believe that their models are less bad than alternative models at generating predictions that resemble reality. So well crafted special cases, numerical examples, and the like won't really do what I think you are trying to do.

You can certainly demonstrate that mainstream economics fails according to other standards of your own choosing.

Here, you've shown that if a person lacks marginal rates of substitution across the aspects by which outcomes are judged, the person lacks transitive preferences across the full set of outcomes. No argument.

But when you embed such a trivial claim in such a lengthy post, you risk giving the reader the impression that you've shown something significant.