Choices/Preference/Utility over Finite Outcomes

* in conjunction with §1 of Microeconomics Foundation I (Kreps, 2013)

Y. Eddie Lu, Fall 2024

Key Concepts Overview

Choices

Choices are observable actions taken when confronted with alternatives. Economists model choices using the following notations:

Notation	Definition	Example
\(X\)	Finite set of alternatives	\(X = \{a,b,c\}\)
\(P(X)\)	All non-empty subsets of \(X\)	\(P(X) = \{\{a\}, \{b\}, \{c\},\) \(\{a,b\}, \{a,c\}, \{b,c\}, \{a,b,c\}\}\)
\(B \subseteq P(X)\)	Set of feasible choice menus	\(B = \{\{a\}, \{a,b\}, \{a,b,c\}\}\)
\(c: B \to P(X)\)	Choice function: \(c(A) \subset A\)	\(c(\{a,b,c\}) = \{a,b\}\)

Preferences

Preferences reflect internal evaluations of choices, modeled abstractly:

Notation	Definition	Example
\(X\)	Set of alternatives	\(X = \{a,b,c\}\)
\(X \times X\)	Cartesian product: pairs of alternatives	\((a,b), (b,c), \dots\)
\(R \subseteq X \times X\)	Binary relation: \((a,b) \in R \iff a \succsim b\)	Preferences: \(a \succsim b, b \succsim c\)

Properties of \(R\)

• Complete: For all \(x, y \in X\), \(x \succsim y\) or \(y \succsim x\).
• Transitive: \(x \succsim y\) and \(y \succsim z \implies x \succsim z\).
• Rational: \(R\) is complete and transitive.

Choices from Preferences

A rational binary relation \(R\) generates a choice function \(c_R\):

\[ c_R(A) := \{x \in A \mid (\forall y \in A) \, (x,y) \in R\} \]

Proofs

Proposition 1

If \(R\) is rational, then \(c_R\) is a choice function.

Proof of Proposition 1

By mathematical induction. Need to show that \(c_R (A) \neq \emptyset\) for all \(A \subseteq X\).

1. Base Case (\(|A| = 1\)):
   Let \(A = \{x\}\). A complete \(R\) implies that \((x,x) \in R\), which in turn implies that \(c_R (A) = \{x\} \neq \emptyset\).

2. Inductive Step (\(|A| = n + 1\)):
   Assume true for \(|A| = n\). Pick any \(x \in A\). By assumption, \(c_R (A \setminus \{x\}) \neq \emptyset\).

   • By definition, there exists \(y \in A \setminus \{x\}\) such that \((y,z) \in R\) for all \(z \in A \setminus \{x\}\).
   • Since \(R\) is complete, either \((x,y) \in R\) or \((y,x) \in R\).
     • If \((x,y) \in R\), then transitivity implies \((x,z) \in R\) for all \(z \in A\), so \(x \in c_R(A)\).
     • If \((y,x) \in R\), then \(y \in c_R(A)\).

Thus, \(c_R (A) \neq \emptyset\) for all \(A \subseteq X\). \(\blacksquare\)

Proposition 2

If \(c\) satisfies WARP, then \(R_c\) is a rational preference relation.

Proof of Proposition 2

1. Completeness:
   Take any \(x, y \in X\).
   • If \((x,y) \notin R_c\), then \(x \notin c(\{x,y\})\).
     
   • By definition, \(y \in c(\{x,y\})\), so \((y,x) \in R_c\).
     Therefore, \(R_c\) is complete.
2. Transitivity:
   Suppose \((x,y) \in R_c\) and \((y,z) \in R_c\).
   • This implies \(x \in c(\{x,y\})\) and \(y \in c(\{y,z\})\).
     
   • Assume \((x,z) \notin R_c\). Then \(z \in c(\{x,z\})\) contradicts WARP.
     Hence, \((x,z) \in R_c\), and \(R_c\) is transitive. \(\blacksquare\)

Preferences Revealed by Choices

Revealed preference relation \(R_c\):

\[ R_c := \{(x,y) \in X \times X \mid x \in c(\{x,y\})\} \]

Weak Axiom of Revealed Preferences (WARP)

If \(x, y \in A \cap B\), \(x \in c(A)\), and \(y \in c(B)\), then \(x \in c(B)\).