Wednesday, May 02, 2007

Non-transitivity with no irrationality.

Can there be cases of non-transitivity in choice with no irrationality? Michael Mandler (A difficulty choice in Preference Theory) argues that it is possible.

First of all let us define non transitivity.

If x, y, and z are three goods, we have a failure of transitivity if

y ≥ z

x ≥ y


z > x

what is the interpretation of the symbols ≥ (weakly preferred to) and > (preferred to)?

Mandler distinguishes two possibilities here: an interpretation according to which preference corresponds to judgments of welfare level; and one in which it corresponds to choice in forced or unforced circumstances. Here I shall discuss the second type of interpretation. The first interpretation, Mandler argues, creates a problem with completeness: it is not plausible to expect a rational agent to be able to say, of every two alternatives, if he is better off with a as with b, or with b as with a, or as much well off with a as with b. Why? a and b may reflect complex situations, for example a policy choice having a certain impact on the environment and a certain impact on comfort, and the individual may not be able to assess as better, worse or indifferent different combinations of comfort and environmental value: he may not know what to choose.

This problem is avoided by imagining that we force i the subjects to make a choice, for example, we may imagine that unless the agent choses a good or another, we give him an electric shock. In such situations, choices can always be elicited.

The proposed interpretation of the symbols according to the forced choice interpretation is the following:

a > b : in all choice situations in which a and b are available, the subject consistently chooses a

a b : in at least some circumstances, the subject chooses a (but in other he may choose b).

Is the interpretation of a ≥ b meaningful and coherent with our idea of rationality? Yes: to see why it is enough to think about situations in which subject have a status quo bias. The situation above may be one in which the subject tends not to exchange a good b with another good a , when he has the good b, but in certain situations may choose the good a when b is available.

It is crucial for the example that this point sounds clear. Let us therefore make an example. Suppose that

x = car

y = bike

z = scooter

We may interpret the expression z > x as “every time the subject is offered the alternative between a scooter and a car, the subject chooses a scooter”. Notice that with this interpretation of the symbols, it is very hard not to ascribe a welfare meaning to the choice behavior in question, namely that the subject consider himself better off with a scooter than with a car.

The important step in Mandler's reasoning is that no such welfarist interpreation is forced on us if we interpret a ≥ b in terms of forced choice.

Imagine a subject who has some concern for both the environment and her own comfort. This subject may have inherited from her brother an old but well-functioning scooter. Let us also assume that he could sell the scooter and buy a bike with the same money. We may imagine this subject to show status quo bias: she may not sell the scooter to buy a new bike. Yet, we may imagine that this subject moves to another city, in which he faces a situation not at all different from the first. He may have the same amount of money to spend, and decide to buy a bike instead of the scooter. Notice that the subject need not have changed her mind. She may still ascribe a higher value in terms of environmental cleanness to the bike, and a higher comfort value to the scooter. We may imagine such a subject as one that has not got a clear idea of the exact trade-off between environmental value and comfort value. This subject is not necessarily irrational. We may express the preference of this value as

y ≥ x under the forced choice interpretation. Hence y ≥ z would signify: “there are circumstances in which the subject would choose a bike over an available alternative of a scooter.”

Notice that the difference with the interpretation in terms of welfare judgment: the subject does not need to think of himself as equally well off with the scooter or with the bike: he need not have no settled idea on the matter.

Now let us tell a story about a rational woman who has non-transitive preferences. The subject who has the preferences

y ≥ z : the subject weakly prefers bike to scooter

x ≥ y : the subject weakly prefers car to bike


z > x : the subject strongly prefers scooter to car

This subject may be not irrational. He may always prefer using a scooter to using a car, whenever given the possibility. This behavior could be interpreted as judging the scooter to be better than the car both in terms of comfort and in terms of environmental impact. This subject may also weakly prefer the bike to the scooter, meaning that in certain circumstances of (forced) choice, she may decide to take a bike instead of the scooter. And there may be circumstances in which the subject does not select the bike over the scooter, even if she has the opportunity to: as we have seen, the subject may have inherited the scooter from his brother, and show status quo bias.

What is the significance of these examples? The standard demonstration of transitivity in rational choice theory is based on the idea that a subject with non-transitive preference could be brought to ruin by exchange, as in the famous money pump example. A subject who strongly prefers apples to pears, bananas to apples, and pears to bananas, starting with pears may be willing to pay 10 c to get apples, then 10 c to get bananas, and eventually other 10 c to get pears. Hence the subject would come back to the status quo with a net loss of 30 c; then the process may start again, and in a sufficient long time the subject will be ruined.

But with weak preference interpreted in terms of forced choice, the money pump may not even get started. For the subject in the example may show status quo bias and refuse to exchange the scooter she inherited with a bike. That is to say, although the bike is not chosen from the set {bike, scooter}, when the scooter is the status quo,, it may well be chosen when the bike itself is the status quo. And similarly, the car may well be chosen from the set {car, bike} when the car is the status quo. Since the agent must always choose the scooter from the set {scooter, car} (and she can be said to be better off with a scooter than with a car), we have z > x (that is, there are no circumstances under which a car is chosen when there is a scooter available.) (See Mandler “A Difficult Choice in Preference Theory”, in Varieties of Practical Reasoning, MIT Press: p 397). As Mandler writes “Transitivity is therefore violated, and the case for the rational necessity of completeness and transitivity fails. Status quo bias and other discordant evidence have been widely interpreted, by both economists and others, as a strong repudiation of the standard economic model of rationality. And status quo bias indeed contradicts the standard model. But ... the phenomenon is not a sign of irrationality in the sense that status quo bias puts agents in harm's way. Hence, it is not any thesis about the prevalence of genuinely rational behavior that must be overturned; it is the economic account of rationality that must give way.” (ibidem).

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HarryB said...
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