4  Rational Preferences

A rational preference is any preference relation (Chapter 1) that satisfies two key properties: transitivity (Chapter 2) and completeness (Chapter 3).

Definition 4.1 Suppose that \(X\) is a set. A pair \((P, I)\) is a rational preference on \(X\) provided that \(P\subseteq X\times X\) and \(I\subseteq X \times X\), such that

1. \(P\) is asymmetric and transitive.
2. \(I\) is reflexive, symmetric, and transitive.
   
3. Completeness: For all \(x, y\in X\), exactly one of \(x \mathrel{P} y\), \(y\mathrel{P} x\) or \(x \mathrel{I} y\) is true.

A relation that is asymmetric and transitive is known as a strict weak order, while a relation that is reflexive, symmetric, and transitive is called an equivalence relation. Thus, an alternative way to define a rational preference \((P, I)\) is to specify that \(P\) is a strict weak order and \(I\) is an equivalence relation.

An important feature of rational preferences is that defining the strict preference relation \(P\) alone is sufficient, as the indifference relation \(I\) can be inferred under the assumption of Completeness: If neither \(x \mathrel{P} y\) nor \(y \mathrel{P} x\) holds, then the decision maker must be indifferent between \(x\) and \(y\) (\(x \mathrel{I} y\)).

4.1 Exercises

1. Suppose that \(X=\{a,b,c\}\) and that \((P, I)\) is a rational preference on \(X\). Assume that \(a\mathrel{I} b\) and \(c \mathrel{P}b\). Can you infer the decision maker’s preference about \(a\) and \(c\)?

   Show Answer

   Yes. By completeness, exactly one of the following must be true: \(a\mathrel{P} c\), \(c\mathrel{P} a\), or \(a\mathrel{I} c\). We argue as follows:

   1. It is not the case that \(a\mathrel{P} c\): Towards a contradiction, suppose that \(a\mathrel{P} c\), then since \(c \mathrel{P}b\) by transitivity, we have that \(a\mathrel{P} b\). But this contradicts the assumption that \(a\mathrel{I} b\). So, it is not the case that \(a\mathrel{P} c\).

   2. It is not the case that \(a\mathrel{I} c\): Towards a contradiction, suppose that \(a\mathrel{I} c\), then since \(a\mathrel{I} b\) by symmetry and transitivity, we have that \(a\mathrel{I} c\). But this contradicts the assumption that \(a\mathrel{P} c\). So, it is not the case that \(a\mathrel{I} c\).

   By completeness, we must have that Since \(c \mathrel{P} b\) and \(b \mathrel{I} a\), we have \(c\mathrel{P} a\).

2. Suppose that \(X=\{a,b,c\}\) and that \((P, I)\) is a rational preference on \(X\). Assume that \(b \mathrel{P} c\) and \(a \mathrel{P} c\). Can you infer the decision maker’s preference about \(a\) and \(c\)?

   Show Answer

   No. Both of the following are consistent with the given preferences: \(a\mathrel{P} b\) and \(a\mathrel{I} b\).

3. Suppose that \(X=\{a, b, c\}\) and the strict preference relation \(P\) is given by \(P=\{(a, b), (b, c), (a, c)\}\). What is the indifference relation \(I\) that makes \((P, I)\) a rational preference on \(X\)?

   Show Answer

   The indifference relation \(I\) is given by \(I=\{(a, a), (b, b), (c, c)\}\).

4. Suppose that \(X=\{a, b, c\}\) and the strict preference relation \(P\) is given by \(P=\{(b, c), (a, c)\}\). Can you find an indifference relation \(I\) that makes \((P, I)\) a rational preference on \(X\)?

   Show Answer

   The indifference relation \(I\) is given by \(I=\{(a, a), (b, b), (c, c), (a, b), (b, a)\}\).

5. Suppose that \(X=\{a, b, c\}\) and the strict preference relation \(P\) is given by \(P=\{(a, c)\}\). Can you find an indifference relation \(I\) that makes \((P, I)\) a rational preference on \(X\)?

   Show Answer

   No such indifference relation exists. The problem is that we cannot have both \(a\mathrel{I} b\) and \(b\mathrel{I} c\): By transitivity, we have \(a\mathrel{I} c\), which contradicts the assumption that \(a\mathrel{P} c\).