5 Preference Relations

Preferences in rational choice theory are understood comparatively. So, if a decision maker “prefers red wine”, then this means that the decision maker prefers red wine to the other available alternatives (e.g., red wine more than white wine).

5.1 Strict Preference, Indifference, and Incomparability

Let \(X\) be a set of alternatives. A decision maker’s preference over \(X\) is represented by the following relations on \(X\):

\(P\subseteq X\times X\): for \(x, y \in X\), \(x\mathrel{P} y\) means that the decision maker strictly prefers \(x\) to \(y\).
\(I\subseteq X\times X\): for \(x, y \in X\), \(x\mathrel{I} y\) means that the decision maker is indifferent between \(x\) and \(y\).
\(N\subseteq X\times X\): for \(x, y \in X\), \(x\mathrel{N} y\) means that the decision maker cannot compare \(x\) and \(y\).

The first assumption is that the the relations \(P, I\), and \(N\) represent the subjective preferences of a single decision maker:

Assumption 1: For all \(x, y \in X\), exactly one of \(x \mathrel{P} y\), \(y\mathrel{P} x\), \(x \mathrel{I} y\), or \(x \mathrel{N} y\) is true.

Thus, there are four distinct ways a decision maker can compare alternatives \(x\) and \(y\):

\(x \mathrel{P} y\): the decision maker strictly prefers \(x\) to \(y\).
\(y \mathrel{P} x\): the decision maker strictly prefers \(y\) to \(x\).
\(x \mathrel{I} y\): the decision maker is indifferent between \(x\) and \(y\).
\(x \mathrel{N} y\): the decision maker cannot compare \(x\) and \(y\).

There are additional constraints that we will impose on the relations \(P, I\), and \(N\). The intended interpretation of a strict preference is that if the decision maker strictly prefers \(x\) to \(y\) (i.e., \(x\mathrel{P}y\)), then the decision maker would pay some non-zero amount money to trade \(y\) for \(x\). Under this interpretation, it is clear that a minimal constraint on \(P\) is that it is asymmetric:

Assumption 2: Suppose that \(P\subseteq X\times X\) represents the decision maker’s strict preferences. We assume that \(P\) is asymmetric: for all \(x, y\in X\), if \(x \mathrel{P} y\), then it is not the case that \(y \mathrel{P} x\) (written not-\(y \mathrel{P} x\)).

It is clearly irrational for a decision maker to pay some money to trade \(y\) for \(x\) and, at the same time, be willing to pay additional money to trade \(x\) for \(y\). Note that since asymmetry implies that strict preference relation is irreflexive: for all \(x\), it is not the case that \(x \mathrel{P} x\).

Example

Examples of asymmetric strict preferences on the set \(X=\{a, b, c\}\) include:

\(\{(a, b), (a, c)\}\): \(a\) is strictly preferred to \(b\) and strictly preferred to \(c\), but there is no strict preference one way or the other between \(b\) and \(c\).
\(\{(a, b), (b, c), (a, c)\}\): \(a\) is strictly preferred to \(b\) and strictly preferred to \(c\), and \(b\) is strictly preferred to \(c\).
\(\{(a, b), (b, c)\}\): \(a\) is strictly preferred to \(b\) and \(b\) is strictly preferred to \(c\), but there is no strict preference one way or the other between \(a\) and \(c\).
\(\{(a, b), (b, c), (c, a)\}\): \(a\) is strictly preferred to \(b\), \(b\) is strictly preferred to \(c\), and \(c\) is strictly preferred to \(a\).

In the first example, the decision maker does not have a preference over all the elements of \(X\). In particular, the decision maker does not have a strict preference one way or the other between \(b\) and \(c\). That is, it is not the case that \(b P c\) and it is not the case that \(c P b\).

Definition 5.1 (No Strict Preference) Suppose that \(P\) is an asymmetric relation on \(X\). Define a relation \(\simeq \subseteq X \times X\) as follows: For all \(x, y\in X\),\[x \simeq y\text{ if and only if not-}x P y\text{ and not-}y P x.\]

It is not hard to see that for any asymmetric relation \(P\) on \(X\),

exactly one of \(x \mathrel{P} y\), \(y \mathrel{P} x\), and \(x\simeq y\) is true.
\(\simeq\) is symmetric for all \(x, y \in X\), if \(x \simeq y\) then \(y\simeq x\); and
\(\simeq\) is reflexive: for all \(x\in X\), \(x \simeq x\).

In many situations, it is convenient to decompose the \(\simeq\) relation further. Given a strict preference \(P\) on \(X\) for a decision maker and items \(x, y\in X\), there are two reasons why \(x \simeq y\):

The decision maker is indifferent between \(x\) and \(y\). In this case, we write \(x \mathrel{I} y\).
The decision maker cannot compare \(x\) and \(y\). In this case, we write \(x\mathrel{N} y\).

Assumption 3: Suppose that \(I\subseteq X\times X\) represents the decision maker’s indifferences and \(N\subseteq X\times X\) represents the decision maker’s non-comparabilities. We assume that \(I\) is reflexive and symmetric, and that \(N\) is symmetric.

Notation

There is no settled notation for strict preferences and indifference. In some texts, you might see \(\succ\) instead of \(P\) representing a strict preference and \(\sim\) instead of \(I\) representing an indifference relation.

Putting everything together, a decision maker’s preferences on \(X\) is represented by three relations \(P\subseteq X\times X\), \(I\subseteq X\times X\) and \(N\subseteq X\times X\) satisfying the following minimal constraints:

For all \(x, y \in X\), exactly one of \(x \mathrel{P} y\), \(y\mathrel{P} x\), \(x \mathrel{I} y\) and \(x \mathrel{N} y\) is true.
\(P\) is asymmetric
\(I\) is reflexive and symmetric.
\(N\) is symmetric.

5.2 Exercises

Suppose that \(P\subseteq X\times X\) is an asymmetric relation and \(\simeq\) as defined in Definition 5.1.
1. Explain why \(\simeq\) is symmetric.
2. Explain why \(\simeq\) is reflexive.
Suppose that \(X=\{a, b, c\}\). Give the relations that represent the following decision makers:
1. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and striclty prefers \(a\) to \(c\).
2. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and striclty prefers \(c\) to \(b\).
3. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and is indifferent between \(a\) and \(b\).
4. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and cannot compare \(a\) and \(b\).
5. The decision maker strictly prefers \(a\) to \(b\), is inidfferent between \(b\) and \(c\), and cannot compare \(a\) and \(c\).