3 Preference Relations

In rational choice theory, preferences are about making comparisons. When we say that a decision-maker “prefers red wine,” it means they rank red wine above the other available options. For example, they might prefer red wine over white wine.

Suppose we have a set of alternatives, called \(X\). A decision maker’s preference over these alternatives can be described using three relations on \(X\):

Strict Preference (\(P\)): If the decision maker strictly prefers one option over another, we write \(x \mathrel{P} y\).
Indifference (\(I\)): If the decision maker is indifferent between two options, we write \(x \mathrel{I} y\). This means they see no difference between \(x\) and \(y\).
Non-comparability (\(N\)): If the decision maker cannot compare two options, we write \(x \mathrel{N} y\). This means that lack any preference between \(x\) and \(y\) since the options cannot be compared.

Assumption 1: For any two alternatives \(x\) and \(y\) in \(X\), exactly one of the following is true:

\(x \mathrel{P} y\): 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 \(y\).

This assumption means that there are only four ways a decision maker can compare any two alternatives: strict preference for one over the other, indifference between them, or inability to compare them.

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 \(P \subseteq X \times X\) 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 true that \(y \mathrel{P} x\).

This makes sense because it would be irrational for a decision maker to pay money to trade \(y\) for \(x\) and then also pay money to trade \(x\) for \(y\). Note that assuming that the strict preference relation \(P\) is asymmetric implies that it is irreflexive: for any \(x\), it is not the case that \(x \mathrel{P} x\).

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 \(N\) completely describe the preferences of a single decision maker:

Assumption 1: Suppose that \(X\) is a set. 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, we assume that 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 \(N\).

Assumption 2: Suppose that \(P\subseteq X\times X\) represents the decision maker’s strict preferences on a set \(X\). 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 asymmetry implies that the strict preference relation is irreflexive: for all \(x\), it is not the case that \(x \mathrel{P} x\).

While strict preferences represent situations where one option is clearly favored over another, decision makers often face choices where they either see two options as equally desirable or cannot compare them at all. For these situations, we need to introduce assumptions about how indifference and non-comparability work in the context of decision making.

While strict preferences represent situations where one option is clearly favored over another, decision makers often face choices where they either see two options as equally desirable or find them impossible to compare. To address these scenarios, we impose the following constraints on the relations representing the decision maker’s indifference and non-comparability.

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 (for all \(x\in X\), \(x\mathrel{I} x\)) and symmetric (for all \(x, y\in X\), if \(x\mathrel{I} y\), then \(y\mathrel{I}x\)), and
\(N\) is symmetric (for all \(x, y\in X\), if \(x\mathrel{N} y\), then \(y\mathrel{N}x\)).

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; and
\(N\) is symmetric.

3.1 Exercises

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 \(c\).