4 Transitivity
An important assumption in many rational choice models is that the decision maker’s preferences on a set
- the decision maker’s strict preference relation
is transitive, - the decision maker’s indifference relation
is transitive, and
- the decision maker’s non-comparability relation
is transitive.
4.1 Transitivity of Indifference and Non-Comparability
There are valid reasons to question the assumption that the decision maker’s indifference relation and non-comparability relation are transitive.
Example 4.1 (Transitivity of Indifference) Suppose that you are indifferent between a curry with
Given this, your preferences would look like this:
According to transitivity, if you are indifferent between
Example 4.2 (Transitivity of Non-Comparability) Suppose you are unable to compare having a job as a professor with having a job as a programmer. Additionally, you cannot compare having a job as a programmer with having a job as a professor plus an extra $1,000.
More formally, let
Setting aside the issues raised in Example 4.1 and Example 4.2, we assume the following:
- Transitivity of Indifference and Non-Comparability
-
Suppose that
represents a decision maker’s indifference relation and that represents a decision maker’s non-comparability relation. We assume that and are both transitive.
- For all
, if and , then . - For all
, if and , then .
4.2 Transitivity of Strict Preferences
While some experiments raise doubts about whether transitivity accurately describes people’s strict preferences1, it is still common to assume that a decision maker’s strict preference is transitive.
There are two ways in which a decision maker’s strict preference
- Lack of Strict Preference: There are
such that and , but (i.e., and are incomparable). - Preference Cycles: There are
such that , , and .
To justify the assumption that a strict preference relation is transitive, we must argue that both of the above situations are irrational. In the next section, we will explain how to rule out cycles in a decision maker’s strict preferences. The first situation can be ruled out with an additional assumption about the decision maker’s preferences (see Chapter 5).
4.2.1 Ruling out Cycles
A cycle (of length 3) in a relation
Argument 1
I do not think we can clearly say what should convince us that a [person] at a given time (without change of mind) preferred
to , to and to . The reason for our difficulty is that we cannot make good sense of an attribution of preference except against a background of coherent attitudes…My point is that if we are intelligibly to attribute attitudes and beliefs, or usefully to describe motions as behaviour, then we are committed to finding, in the pattern of behaviour, belief, and desire, a large degree of rationality and consistency. (Davidson 2001, 237)
Argument 2: The Money-Pump Argument
For an item
- Strict Preference Guides Choice: If
, then the decision maker will always take when is the only alternative. - Willingness to Pay: If
, then there is some such that for all , if and only if .
- Separation of Items and Money: The items and money are separable and the decision maker prefers more money to less:
- For all
and , we have that if and only if ; and - for all
and , if , then .
- For all
Now, suppose Ann has a cycle in her strict preferences over the set
Suppose Ann currently has item
- Since
, by Assumption 1, Ann will accept an offer to trade for plus pay $1. After the trade, she has and has paid $1. - Now, suppose she is offered the chance to trade
for plus pay $1. Since , by Assumption 1, she will accept the offer. She now has and has paid . - Suppose she is offered the chance to trade
for plus pay . Since , by Assumption 1, she will accept the offer. Now she has and has paid $3.
But Ann started with
Ann can avoid such a money-pump scenario by ensuring that there are no cycles in her strict preferences.
4.3 Exercises
Suppose that
. Which of the following relations are transitive? If the relation is not transitive, explain why.True or False: The Money-Pump argument shows that a rational decision maker’s strict preferences must be transitive.
Suppose that
. Which of the following relations are transitive? If the relation is not transitive, explain why.This relation is transitive.
This relation is not transitive since
, but .This relation is transitive.
This relation is not transitive since
, but . . This relation is transitive.This relation is not transitive since
, but .This relation is not transitive since
, butThis relation is not transitive since
, but
True or False: The Money-Pump argument shows that a rational decision maker’s strict preferences must be transitive.
This is false. The Money-Pump argument shows that a rational decision maker’s strict preferences cannot contain a cycle. We need an additional assumption to rule out situation in which a decision maker strictly prefers
to and to , but cannot compare and .
See A. Tversky’s classic paper (Tversky 1969) and Regenwetter, Dana, and Davis-Stober (2011) for a critique of these experiments.↩︎