Preference and Choice Lecture

Indifference in Choices: Refers to situations where an individual has no preference between options provided. An example is traveling to another store for a $5 discount on either a calculator or a stereo.

Transitivity Violation: If a choice leads to violating transitive preferences:

Define:

$x$: Travel for discount on calculator
$y$: Travel for discount on stereo
$z$: Buy both items at the first store

Choices may indicate x > z and z > y, but x < y shows inconsistency, violating transitivity.

Examples of Apparent Intransitivity

3. Mom-Dad-Child Preferences: A family chooses evening entertainment by majority voting among preferences:

Preferences: Mom: O > MR > MI; Dad: I > DO > DR; Child: R > C > CO.

Through majority voting, outcomes can cycle (e.g., O > R, R > I, I > O) leading to intransitive household preferences (Condorcet paradox).

5. Change of Tastes: Preference changes in addictive behaviors:

Initial preferences: $y$ (abstaining) < $x$ (smoking one cigarette) < $z$ (heavy smoking).

Once started smoking ($x$), becomes z > x > y reflecting changing tastes related to addiction.

This illustrates how earlier frameworks of preference may not capture future behavior.

Utility Functions

Utility Function Definition: A function $u: X \to \mathbb{R}$ representing preference relations such that:

If x > y, then $u(x) \geq u(y)$.

Ordinal vs. Cardinal Properties:

Ordinal: Preference order matters, e.g., $u(x)$ represents rank without absolute differences.
Cardinal: Magnitudes of differences imply preferences, sensitive to specific transformations.

Proposition 1.B.2:

A preference relation can be represented by a utility function only if the preferences are Rational (i.e., complete and transitive).

Choice Rules

Choice Structure: Reflects decision behavior, defined by a budget set $B$ and a choice function $C(\cdot)$. Consists of:

2. Nonempty subsets of alternatives: $B \in \mathbb{X}$, outlines all possible decision scenarios.

3. Choice correspondence: Assigns chosen elements from $B$ to $C(B)$.

Weak Axiom of Revealed Preference:

Definition: If $x$ is chosen over $y$ in $B$ and $y$ appears in a broader set $B'$, then the choice should remain consistent.

This implies observed choices in $C(B)$ should not contradict previous selections, enforcing consistency.

Revealed Preference Relation

Revealed preference $x \succeq y$ implies $x$ is preferred (or at least as good as) $y$ if there exists a budget set $B$ where $x$ was chosen.

Consistency Condition: If $x \succeq y$, $y$ cannot be preferred over $x$. Examples demonstrate satisfaction of this axiom and its implications for rational preferences.

Relationship Between Preference Relations and Choice Rules

Rational Preferences: Every rational preference can generate a choice structure that satisfies the weak axiom of revealed preferences.

Existence of Rationalizing Preferences: A proper choice structure can potentially have multiple rationalizing relationships.

Conclusion

The framework effectively connects preferences, decision-making, and choice behavior