8 Rational Preferences

We start with some definitions:

Definition 8.1 (Strict weak order) A strict weak order on a set \(X\) is a transitive and asymmetric relation on \(X\).

Definition 8.2 (Equivalence relation) An equivalence relation on \(X\) is a reflexive, transitive and symmetric relation on \(X\).

In a rational choice model, a standard assumption is that a decision maker’s preferences satisfies completeness (see Chapter 7) and transitivity (see Chapter 6).

Definition 8.3 (Rational preference) A rational preference on a set \(X\) is a pair \((P, I)\) where \(P\) is a strict weak order on \(X\), \(I\) is an equivalence relation on \(X\), and completeness is satisfied: for all \(x, y\in X\), exactly one of \(x \mathrel{P} y\), \(y \mathrel{P} x\) or \(x \mathrel{I} y\) is true.

Note that for a rational preference \((P, I)\), there is no \(x, y\in X\) such that \(x \mathrel{P} y\) and \(x \mathrel{I} y\) (i.e., a rational agent cannot both strictly prefer \(x\) to \(y\) and be indifferent between \(x\) and \(y\)). Using this fact and transitivity of \(P\) and \(I\), we have the following two key properties:

For all \(x, y, z\in X\), if \(x \mathrel{P} y\) and \(y \mathrel{I} z\), then \(x \mathrel{P} z\).
For all \(x, y, z\in X\), if \(x \mathrel{I} y\) and \(y \mathrel{P} z\), then \(x \mathrel{P} z\).

In many situations it is natural to assume that the decision maker is not indifferent about any of the items in a set \(X\) (i.e., the decision maker’s indifference relation is empty). In such a case, we can represent a decision maker by a single relation \(P\) (the decision maker’s strict preference relation) satisfying the following properties:

Definition 8.4 (Linear order) A linear order on a set \(X\) is a transitive, connected and asymmetric relation on \(X\).

8.1 Weak preference relation

Warning

This section contains more advanced material and can be skipped on a first reading.

Even if a decision maker is indifferent between some items, we can represent the decision maker’s rational preferences by a single relation.

Definition 8.5 (Derived weak preference relation) Suppose that \((P, I)\) is a rational preference on \(X\). The weak preference relation based on \((P, I)\) is defined as follows: \(R\subseteq X\times X\), where \(x \mathrel{R} y\) if and only if \(x \mathrel{P} y\) or \(x \mathrel{I} y\). If \(x \mathrel{R} y\), we say that “\(x\) is weakly preferred to \(y\)”.

It is not hard to see that if \(R\) is a weak preference relation based on a rational preference \((P, I)\), then \(R\) is a reflexive, transitive and connected relation.

We can also start with a weak preference relation and induce strict preference and an indifference relation.

Definition 8.6 (Rational weak preference) Suppose that \(X\) is a set. A rational weak preference on \(X\) is a relation \(R\subseteq X\times X\) that is reflexive, transitive and connected.

A key observation is that rational weak preferences can be used to represent a decision maker’s rational preferences.

Lemma 8.1 Suppose that \(R\subseteq X\times X\) is a reflexive and transitive relation. Define relations \(P_R\subseteq X\times X\) and \(I_R\subseteq X \times X\) as follows:

\(x \mathrel{P_R} y\) if and only if \(x \mathrel{R} y\) and not-\(y \mathrel{R} x\).
\(x \mathrel{I_R} y\) if and only if \(x \mathrel{R} y\) and \(y \mathrel{R} x\).

Then, \((P_R, I_R)\) is a rational preference on \(X\).

Proof. Clearly, there are no \(x, y\in X\) such that \(x \mathrel{P_R} y\) and \(x \mathrel{I_R} y\).

We prove that \(P_R\) is transitive: Suppose that \(x \mathrel{P_R} y\) and \(y \mathrel{P_R} z\). Then \(x \mathrel{R} y\), not-\(y\mathrel{R}x\), \(y \mathrel{R} z\) and not-\(z\mathrel{R}y\). Since \(x \mathrel{R} y\), \(y \mathrel{R} z\), and \(R\) is transitive, we have that \(x \mathrel{R} z\). To prove that \(x\mathrel{P_R} z\) we must show that not-\(z\mathrel{R} x\). Towards a contradiction, suppose that \(z \mathrel{R}x\). Then since \(z \mathrel{R}x\), \(x \mathrel{R} y\), and \(R\) is transitive, we have that \(z\mathrel{R}y\), which contradicts the assumption that not-\(z\mathrel{R}y\). Hence, \(x\mathrel{R}z\) and not-\(z\mathrel{R}x\), and so \(x \mathrel{P_R} z\). The proof that \(P_R\) is asymmetric is left as an exercise.

The proof that \(I_R\) is an equivalence relation is left as an exercise.

Finally, we show that \((P_R, I_R)\) satisfies completeness. Suppose that \(x, y\in X\). Since \(R\) is connected, we have either \(x\mathrel{R}y\) or \(y\mathrel{R}x\). There are three possibilities:

\(x\mathrel{R}y\) and not-\(y\mathrel{R}x\). In this case, \(x \mathrel{P_R} y\).
not-\(x\mathrel{R}y\) and \(y\mathrel{R}x\). In this case, \(y \mathrel{P_R} x\).
\(x\mathrel{R}y\) and \(y\mathrel{R}x\). In this case, \(x \mathrel{I_R} y\).

8.2 Exercises

Suppose \((P, I)\) is a rational preference on \(X\). Explain why the following are true:
1. For all \(x, y, z\in X\), if \(x \mathrel{P} y\) and \(y \mathrel{I} z\), then \(x \mathrel{P} z\).
2. For all \(x, y, z\in X\), if \(x \mathrel{I} y\) and \(y \mathrel{P} z\), then \(x \mathrel{P} z\).
Suppose that \(X=\{a,b,c,d\}\) and that \[R=\{(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (b,c), (c,b), (b,d), (c,d)\}\] represents a decision maker’s weak preference over the items in \(X\). Select all the statements that are true about the decision maker:
1. The decision maker strictly prefers \(a\) over \(b\).
2. The decision maker strictly prefers \(b\) over \(a\).
3. The decision maker is indifferent between \(a\) and \(b\).
4. The decision maker strictly prefers \(a\) over \(c\).
5. The decision maker strictly prefers \(c\) over \(a\).
6. The decision maker is indifferent between \(a\) and \(c\).
7. The decision maker strictly prefers \(b\) over \(c\).
8. The decision maker strictly prefers \(c\) over \(b\).
9. The decision maker is indifferent between \(b\) and \(c\).
Suppose that \((P, I)\) is a rational preference on \(X\) and \(R\) is the derived weak preference relation (see Definition 8.6). Suppose that \(x, y\in X\) and \(x\mathrel{R}y\) and \(y\mathrel{P}z\). Which of the following is true:
1. \(x\mathrel{P}z\)
2. \(z\mathrel{P}x\)
3. \(x\mathrel{R}z\)
4. \(z\mathrel{R}x\)
Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(b\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(c\)?
Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(b\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(c\)?

Ann strictly prefers \(a\) over \(c\).
Ann strictly prefers \(c\) over \(a\).
Ann is indifferent between \(a\) and \(c\).
Ann cannot compare \(a\) and \(c\).
There is not enough information to answer this question.

Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(c\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(b\)?
1. Ann strictly prefers \(a\) over \(b\).
2. Ann strictly prefers \(b\) over \(a\).
3. Ann is indifferent between \(a\) and \(b\).
4. Ann cannot compare \(a\) and \(b\).
5. There is not enough information to answer this question.
Suppose that Ann’s preferences are rational, Ann strictly prefers \(a\) over \(b\), and \(c\) is some alternative different from both \(a\) and \(b\). What else can you conclude about Ann’s preferences?
1. Ann strictly prefers \(a\) over \(c\) and she strictly prefers \(c\) over \(b\).
2. Ann strictly prefers \(a\) over \(c\) or she strictly prefers \(c\) over \(b\).
3. There is not enough information to answer this question.

Solutions

Suppose \((P, I)\) is a rational preference on \(X\). Explain why the following are true:
1. For all \(x, y, z\in X\), if \(x \mathrel{P} y\) and \(y \mathrel{I} z\), then \(x \mathrel{P} z\).
  
  Suppose that \((P, I)\) is a rational preference and for \(x, y, z\in X\) we have that \(x \mathrel{P} y\) and \(y \mathrel{I} z\). By completeness, exactly one of \(x \mathrel{P} z\), \(z \mathrel{P} x\) or \(x \mathrel{I} z\) is true. We show that both \(z \mathrel{P} x\) and \(x \mathrel{I} z\) lead to a contradiction leaving only \(x \mathrel{P} z\), as desired. If \(z \mathrel{P} x\), then since \(x \mathrel{P} y\) and \(P\) is transitive, we have that \(z \mathrel{P} y\). Since \(I\) is symmetric and \(y \mathrel{I} z\), we have that \(z \mathrel{I} y\). This is a contradiction since \(z \mathrel{P} y\) and \(z \mathrel{I} y\) cannot both be true. If \(z \mathrel{I} x\), then since \(y \mathrel{I} z\) and \(I\) is transitive, we have that \(y \mathrel{I} x\). Since \(I\) is symmetric, we have that \(x \mathrel{I} y\). This is a contradiction since \(x \mathrel{P} y\) and \(x \mathrel{I} y\) cannot both be true.
2. For all \(x, y, z\in X\), if \(x \mathrel{I} y\) and \(y \mathrel{P} z\), then \(x \mathrel{P} z\).
  
  Suppose that \((P, I)\) is a rational preference and for \(x, y, z\in X\) we have that \(x \mathrel{I} y\) and \(y \mathrel{P} z\). By completeness, exactly one of \(x \mathrel{P} z\), \(z \mathrel{P} x\) or \(x \mathrel{I} z\) is true. We show that both \(z \mathrel{P} x\) and \(x \mathrel{I} z\) lead to a contradiction leaving only \(x \mathrel{P} z\), as desired. If \(z \mathrel{P} x\), then since \(y \mathrel{P} z\) and \(P\) is transitive, we have that \(y \mathrel{P} x\). Since \(I\) is symmetric and \(x \mathrel{I} y\), we have that \(y \mathrel{I} x\). This is a contradiction since \(y \mathrel{P} x\) and \(y \mathrel{I} x\) cannot both be true. If \(z \mathrel{I} x\), then since \(x \mathrel{I} y\) and \(I\) is transitive, we have that \(z \mathrel{I} y\). Since \(I\) is symmetric, we have that \(y \mathrel{I} z\). This is a contradiction since \(y \mathrel{P} z\) and \(y \mathrel{I} z\) cannot both be true.
Suppose that \(X=\{a,b,c,d\}\) and that \[R=\{(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (a,d), (b,c), (c,b), (b,d), (c,d)\}\] represents a decision maker’s weak preference over the items in \(X\). Select all the statements that are true about the decision maker:
1. The decision maker strictly prefers \(a\) over \(b\): This is true since \((a,b)\in R\) and \((b,a)\not\in R\).
2. The decision maker does not strictly prefer \(b\) over \(a\): This is not true since \((b,a)\not\in R\) (but we do have \((a,b)\in R\)).
3. The decision maker is not indifferent between \(a\) and \(b\): This is not true since \((b,a)\not\in R\) (but we do have \((a,b)\in R\)).
4. The decision maker strictly prefers \(a\) over \(c\): This is true since \((a,c)\in R\) and \((c,a)\not\in R\).
5. The decision maker strictly prefers \(c\) over \(a\): This is not true since \((c,a)\not\in R\) (but we do have \((a,c)\in R\)).
6. The decision maker is not indifferent between \(a\) and \(c\): This is not true since \((c,a)\not\in R\) (but we do have \((a,c)\in R\)).
7. The decision maker does not strictly prefer \(b\) over \(c\): This is not true since \((b,c)\in R\) and \((c, b)\in R\).
8. The decision maker does not strictly prefer \(c\) over \(b\): This is not true since \((b,c)\in R\) and \((c,b)\in R\).
9. The decision maker is indifferent between \(b\) and \(c\): This is true since \((b,c)\in R\) and \((c,b)\in R\)
Suppose that \((P, I)\) is a rational preference on \(X\) and \(R\) is the derived weak preference relation (see Definition 8.6). Suppose that \(x, y\in X\) and \(x\mathrel{R}y\) and \(y\mathrel{P}z\). Which of the following is true:
1. (\(\checkmark\)) \(x\mathrel{P}z\)
2. \(z\mathrel{P}x\)
3. \(x\mathrel{R}z\)
4. \(z\mathrel{R}x\)
We must have \(x\mathrel{P}z\).

Since \(x\mathrel{P}y\), we have either \(x\mathrel{P}y\) or \(x\mathrel{I}y\). If \(x\mathrel{P}y\), then by transitivity of \(P\), we have that \(x\mathrel{P}z\). If \(x\mathrel{I}y\), then using the argument given in the answer to question A., we must have that \(x\mathrel{P} z\). In both cases, \(xPz\).
Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(b\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(c\)?

Ann strictly prefers \(a\) over \(c\).

Let \((P, I)\) be Ann’s rational preferences. Since \(P\) is transitive and we have that \(aPb\) and \(bPc\), then \(aPc\).
Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(b\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(c\)?
1. (\(\checkmark\)) Ann strictly prefers \(a\) over \(c\).
2. Ann strictly prefers \(c\) over \(a\).
3. Ann is indifferent between \(a\) and \(c\).
4. Ann cannot compare \(a\) and \(c\).
5. There is not enough information to answer this question.
Ann strictly prefers \(a\) over \(c\).

Let \((P, I)\) be Ann’s rational preferences. Since \(P\) is transitive and we have that \(aPb\) and \(bPc\), then \(aPc\).
Suppose that Ann’s preferences are rational and that Ann strictly prefers \(a\) over \(c\) and she strictly prefers \(b\) over \(c\). What can you conclude about Ann’s preference of \(a\) and \(b\)?
1. Ann strictly prefers \(a\) over \(b\).
2. Ann strictly prefers \(b\) over \(a\).
3. Ann is indifferent between \(a\) and \(b\).
4. Ann cannot compare \(a\) and \(b\).
5. (\(\checkmark\)) There is not enough information to answer this question.
There is not enough information to answer this question.
Suppose that Ann’s preferences are rational, Ann strictly prefers \(a\) over \(b\), and \(c\) is some alternative different from both \(a\) and \(b\). What else can you conclude about Ann’s preferences?
1. Ann strictly prefers \(a\) over \(c\) and she strictly prefers \(c\) over \(b\).
2. (\(\checkmark\)) Ann strictly prefers \(a\) over \(c\) or she strictly prefers \(c\) over \(b\).
3. There is not enough information to answer this question.
Ann striclty prefers \(a\) over \(c\) or she strictly prefers \(c\) over \(b\).

Suppose that \((P, I)\) represents Ann’s rational preferences. By completeness, we have either \(aPc\), \(cPa\), or \(a I c\). Also by completeness, we have either \(bPc\), \(cPb\), or \(b I c\). This gives a total of 9 possible situations. The following list describes all the possible situations that can arise and what they imply about Ann’s preference between \(a\) and \(c\) and between \(c\) and \(b\):
1. \(aPc\) and \(bPc\): In this case we have that \(bPc\)
2. \(aPc\) and \(cPb\): In this case we have that \(aPc\) and \(cPb\)
3. \(aPc\) and \(bIc\): In this case we have that \(aPc\)
4. \(cPa\) and \(bPc\): This is impossible since this implies that \(bPa\)
5. \(cPa\) and \(cPb\): In this case we have that \(cPb\)
6. \(cPa\) and \(bIc\): This is impossible since this implies that \(bPa\)
7. \(aIc\) and \(bPc\): This is impossible since this implies that \(bPa\)
8. \(aIc\) and \(cPb\): In this case we have that \(cPb\)
9. \(aIc\) and \(bIc\): This is impossible since this implies that \(bPa\)
The only possibilities are that either Ann strictly prefers \(a\) over \(c\) or she strictly prefers \(c\) over \(b\).