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

  1. 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).

  1. 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:

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

  2. 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