Rational preferences

This note collects the definitions used in Tool 04. It builds on note 02, where relations and their properties are defined, and on note 03.

Preference relations

A consumer chooses among the elements of a set $X$ of bundles. A preference relation is a relation $\succsim$ on $X$, and $x \succsim y$ reads “$x$ is at least as good as $y$”. The preference relation is called rational when it is

• complete: for all $x, y \in X$, $x \succsim y$ or $y \succsim x$; and
• transitive: for all $x, y, z \in X$, if $x \succsim y$ and $y \succsim z$, then $x \succsim z$.

From $\succsim$ we define two further relations:

Relation	Definition	Reads
strict preference $x \succ y$	$x \succsim y$ and not $y \succsim x$	$x$ is better than $y$
indifference $x \sim y$	$x \succsim y$ and $y \succsim x$	$x$ and $y$ are equally good

Two bundles are incomparable when neither $x \succsim y$ nor $y \succsim x$ holds. Incomparability is not indifference: indifference means that both comparisons hold, and incomparability means that neither does.

If $\sim$ is an equivalence relation, its equivalence classes are called indifference classes.

Pictures of a preference relation

A preference relation on a finite set can be given in either of the forms used in Tool 03.

• As a directed graph. There is an arrow from $x$ to $y$ when $x \succsim y$, and a loop at $x$ when $x \succsim x$. Arrows in both directions between $x$ and $y$ mean $x \sim y$.
• As a matrix. The entry in row $x$, column $y$ is 1 when $x \succsim y$, and 0 otherwise.

Choice sets

A menu is a nonempty subset $B \subseteq X$ of bundles that are available. The choice set of $B$ consists of the bundles in $B$ that are at least as good as everything in $B$:

$$C(B) = \{x \in B : x \succsim y \text{ for every } y \in B\}.$$

The choice set may contain several bundles, or none at all.

Utility representations

A function $u : X \to \mathbb{R}$ represents $\succsim$ when, for all $x, y \in X$,

$$x \succsim y \quad\text{if and only if}\quad u(x) \ge u(y).$$

A utility function carries only order information. If $u$ represents $\succsim$ and $f : \mathbb{R} \to \mathbb{R}$ is strictly increasing, then $f \circ u$ represents $\succsim$ as well, since $f(u(x)) \ge f(u(y))$ exactly when $u(x) \ge u(y)$. So the values $u = 4, 1, 3$ and $u = 100, 0, 7$ represent the same preferences on three bundles.

Lexicographic preferences. On bundles $x = (x_1, x_2)$ and $y = (y_1, y_2)$, the lexicographic preference compares the first coordinates and uses the second only to break a tie:

$$x \succsim y \quad\text{when}\quad x_1 > y_1, \;\text{ or }\; x_1 = y_1 \text{ and } x_2 \ge y_2 .$$

The name comes from the way a dictionary orders words: by the first letter, then by the second.

Test these definitions in Tool 04. Some of its exercises ask for a utility function that does not exist; showing why is part of the exercise.