Choices/Preference/Utility over Infinite Outcomes

yedlu, Fall 2024

Recap: Finite Chocices

  • \(X = \mathbb{R}^n_+\) be finite choices
  • \(c: P(X) \to X\) be the choice function
  • \(\succsim\) be the preference relation
  • \(U: \mathbb{R}^n_+ \to \mathbb{R}\) be the corresponding utility function.

What if Choices Become Infinite

Different Levels of Finite

  • Finite: there exists a bijection \(X \leftrightarrow \{1, ..., n\}\).
    • each \(x \in X\) is paired with exactly one number in \(\{1, ..., n\}\)
    • each number in \(\{1, ..., n\}\) is paired with exactly one \(x \in X\)
    • nobody is left unpaired
  • Countably Infinite: there exists a bijection \(X \leftrightarrow \mathbb{N} = \{1, 2, ...\}\). Below are number sets by their finiteness
    • \(\mathbb{Z} = \{..., -1, 0, 1, ...\}\) is countably infinite
    • \(\mathbb{Q} = \{a/b: a \in \mathbb{Z}, b \in \mathbb{N}\}\) is countably infinite
    • \(\mathbb{R}\) is not countably infinite

Countable Infinite

Proposition. \(\succsim\) be a rational preference and \(X\) be a countable set \(\Rightarrow\) there exists \(U: X \to \mathbb{R}\) representing \(\succsim\).

Proof (Click to Expand)

Consider:

  • a countable choice set: \(X = \{x_1, ..., x_n, ...\}\).
  • a weight indicator: \(d_j = \frac{1}{2^j}\).

We would have \[\begin{aligned} \Sigma_{j=1}^{\infty} d_j = \Sigma_{j=1}^{\infty} \frac{1}{2^j} = \lim_{n \to \infty} [\Sigma_{j = 1}^n \frac{1}{2^j}] = 1 \end{aligned}\]

We can define the utility function \(U: X \to \mathbb{R}\) as \[\begin{aligned} U(z) = \Sigma_{\{j \in \mathbb{N}: (z, x_j) \in \succsim\}} d_j \end{aligned}\]

\(\blacksquare\)

Incountable Infinite

It is not always true that there exists a utility function \(U: X \to \mathbb{R}\) to represent a rational \(\succsim\).

E.g. Bowser Jr.’s Lexicographic Preferences (\(X = \mathbb{R}^2_+\)).

We would need some other properties that \(\succsim\) must satisfy to make the utility function possible.

Continuous Preference Relation

Definition: for all pairs \(x,y \in X\) with \(x \succ y\), we can always find \(\epsilon > 0\) such that: - \(|x - x'| < \epsilon\) - \(|y - y'| < \epsilon\)

implies \(x' \succ y'\).

Intuition: a strictly preferred pair should remain their preference status even under a sligtest change in any direction. We would not expect a rational person to freak out if a slightest change takes place.

Utility Representation

Proposition. If \(\succsim\) is complete, transitive, and continuous, then there exists a continuous \(U: \mathbb{R}^n_+ \to \mathbb{R}\) that represents \(\succsim\).

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