1  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 (e.g., they prefer red wine over white wine, red wine over beer, red wine over lemonade, etc.).

Suppose that \(X\) is the set of available alternatives—i.e., the items on the menu. A decision maker’s preference over these alternatives is described using the following three relations on \(X\):

  1. Strict Preference (\(P\)): If the decision maker strictly prefers one option over another, we write \(x \mathrel{P} y\).

  2. Indifference (\(I\)): If the decision maker is indifferent between two options, we write \(x \mathrel{I} y\). This means that they see no difference between \(x\) and \(y\).

  3. Non-comparability (\(N\)): If the decision maker cannot compare two options, we write \(x \mathrel{N} y\).

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

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

This assumption means that a decision maker’s opinion about two alternatives falls into exactly one of the following categories: 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 \(y\) (denoted \(x\mathrel{P}y\)), then the decision maker would pay some non-zero amount money to trade \(y\) for \(x\). Under this interpretation, 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 \(x \mathrel{P} x\).

While strict preferences represent situations where one option is clearly favored over another, decision makers sometimes face choices where they either see two options as equally desirable or cannot compare them at all. For these situations, we impose the following minimal constraints on the indifference and non-comparability relations for a decision maker:

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:
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:

  1. 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;
  2. \(P\) is asymmetric;
  3. \(I\) is reflexive and symmetric; and
  4. \(N\) is symmetric.

1.1 Exercises

  1. 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 \(c\).

      Show Answer
      • \(P=\{(a,b), (a,c), (b,c)\}\)
      • \(I=\{(a,a), (b,b), (c,c)\}\)
      • \(N=\varnothing\)

    2. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and striclty prefers \(c\) to \(a\).

      Show Answer
      • \(P=\{(a,b), (c, a), (b, c)\}\)
      • \(I=\{(a, a), (b, b), (c, c)\}\)
      • \(N=\varnothing\)

    3. The decision maker strictly prefers \(a\) to \(c\), strictly prefers \(b\) to \(c\), and is indifferent between \(a\) and \(b\).

      Show Answer
      • \(P=\{(a, c), (b, c)\}\)
      • \(I=\{(a, a), (b, b), (c, c), (a, b), (b, a)\}\)
      • \(N=\varnothing\)

    4. The decision maker strictly prefers \(a\) to \(b\), strictly prefers \(b\) to \(c\), and cannot compare \(a\) and \(b\).

      Show Answer
      • \(P=\{(a, c), (b, c)\}\)
      • \(I=\{(a, a), (b, b), (c, c)\}\)
      • \(N=\{(a, b), (b, a)\}\)

    5. The decision maker strictly prefers \(a\) to \(b\), is inidfferent between \(b\) and \(c\), and cannot compare \(a\) and \(c\).

      Show Answer
      • \(P=\{(a, b)\}\)
      • \(I=\{(a, a), (b, b), (c, c), (b, c), (c, b)\}\)
      • \(N=\{(a, c), (c, a)\}\)

  2. Construct the preference relations \(P\), \(I\), and \(N\) for a decision maker ranking three restaurants where they love restaurant \(A\), hate restaurant \(C\), but have never tried restaurant \(B\) (so cannot compare it to the others).

    Show Answer
    • \(P=\{(A, C)\}\)
    • \(I=\{(A, A), (B, B), (C, C)\}\)
    • \(N=\{(A, B), (B, A), (B, C), (C, B)\}\)

  3. True or False: It is possible for a decision maker to be both indifferent between \(x\) and \(y\) (\(x\mathrel{I} y\)) AND unable to compare \(y\) and \(x\) (\(x\mathrel{N} y\)).

  4. True or False: It is possible for a decision maker to strictly prefer \(x\) to \(y\) (\(x\mathrel{P} y\)) AND be unable to compare \(y\) and \(x\) (\(y\mathrel{N} x\)).

  5. What additional properties should be imposed on the relations \(P\), \(I\), and \(N\) to ensure that they represent a rational decision maker’s preferences?

    Show Answer

    We will discuss this in the next chapters.