1.1. RATIONALITY AND PROBABILITY OF CHOICE
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amount of money to exchange a for b, another amount to exchange c for a,
and a third amount to exchange b for c. Thereby, the person loses money but
ends up with the same alternative. Obviously, the circular argument a » b,
6 » c, and c » a can be used recursively and may be extended to more than
three alternatives.
B.
A.
C.
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Figure 1.1: Examples of inconsistent choices: A. Preference reversal; B.
Intransitive triple; C. Intransitive preferences between three and four objects.
On the other hand, if we think of choices in terms of preference and indifference,
written as a £ b, b £ c, and c 2 a, the phenomenon of intransitive
choices may be interpreted as indifference between alternatives. This can be
modeled by an equivalence relation between indifferent alternatives, written
as ab, b - c, and c - a (Definition A.2.3). However, it has been shown
that indifference is not necessarily transitive, and therefore not an equivalence
relation. The economist Armstrong (1939) was one of the first to argue that
indifference is not necessarily transitive. He discussed the following example:
Suppose a boy is indifferent about receiving as a gift a pony or a bicycle. He
will undoubtedly prefer the bicycle with a bell attached to the bicycle without
a bell. But he is still likely to be indifferent about the bicycle with bell
and the pony. Intransitive indifference in the form of limited discriminatory
ability of the decision-maker has been incorporated into deterministic and
probabilistic models.
Another axiom that has been associated with rational choice and that has
implications for the representation on ordinal scale is asymmetry. If somebody
prefers a over b, he or she should not prefer b over a. Again, the phenomenon
of preference reversals might be considered as transitive or intransitive
indifference, but only a few preference reversals between alternatives
