Stochastic Choice

* in conjunction with §5 of Microeconomics Foundation I (Kreps 2013)

yedlu, Fall 2024

Stochastic (Random) Choice

  1. Binary Random Choice Rule
  • \(X \neq \emptyset\) be the universe of alternatives.
  • \(\rho: X \times X \to [0,1]\) be the binary random choice rule if \(\rho(x,y) + \rho(y,x) = 1\) for all \(x,y \in X\).
  1. Random Choice Rule
  • \(P(X)\) be the nonempty finite subset of \(X\).
  • \(\rho: X \times P(X) \to [0,1]\) be the random choice rule if \(\sum_{x \in A} \, \rho(x, A) = 1\) for all \(A \in \mathcal{A}\).
    • Notation \(\rho(x, A = \{x,y\})\) is equivalent to \(\rho(x,y)\).

Random Utility and Random Preference

Random Preference Model

  • \(X = \{x_1, x_2, ..., x_n\}\) be the set of finite choices
  • \(\mu\) be the probability over strict rankings on \(X\)
    • \(\rho (x_i, A) = \mu\{\succ: \forall x_j \in A \setminus \{x_i\} \quad x_i \succ x_j\}\)

Random Utility Model (RUM)

  • \(X = \{x_1, x_2, ..., x_n\}\) be the set of finite choices
  • (\(\Omega, \mathcal{F}, \mathbb{P}\)) be the probability space (triple). A quick recap:
    • Sample space \(\Omega\): a set of all possible outcomes.
    • Event space \(\mathcal{F}\): a set containing events. An event is a set of outcomes in sample space.
    • Probability function \(\mathbb{P}: \mathcal{F} \to [0,1]\): assign a probability to each event. Specifically, \(\mathbb{P}\{\Omega\} = 1\)
  • \(U_1, U_2, ..., U_n: \Omega \to \mathbb{R}\) be random utilities (r.v.) with \(\mathbb{P}\{\omega \in \Omega: U_i(\omega) = U_j(\omega), i \neq j\} = 0\).
  • \(\rho(x_i, A) = \mathbb{P}\{\omega \in \Omega: \forall j \in A \quad U_i(\omega) \geq U_j(\omega)\}\)
    • \(\mathbb{P}\{\omega \in \Omega: U_A(\omega) > U_B(\omega) > U_C(\omega)\} = \mu\{A \succ B \succ C\}\)
  • \(U: \Omega \to \mathbb{R}\) be the random utility function. Its cumulative distribution function (cdf) is: \[\begin{aligned} F(t) & = \mathbb{P}\{\omega \in \Omega: U(\omega) \leq t\} \\ & = \mathbb{P} \{U \leq t\} \end{aligned}\]

Block & Marschak Proposition (1960)

\(\rho\) is a random preference model if and only if (iif) \(\rho\) is a random utility model. The Block & Marschak Inequalities states necessarily that:

If \(\rho\) is a RUM, then:

\[q(x,A) = \sum_{B \supseteq A} (-1)^{|B \setminus A|} \rho(x,B) \geq 0\]

E.g. 1: Given \(X = \{x,y,z\}\) and \(A = \{x,y\}\), \[\begin{aligned} q(x,A) & = \sum_{B \supseteq A} (-1)^{|B \setminus A|} \rho(x,B) \\ & = (-1)^{|X \setminus A|} \rho(x,X) + (-1)^{|A \setminus A|} \rho(x,A) \\ & = (-1)^{|\{z\}|} \rho(x,X) + (-1)^{|\emptyset|} \rho(x,A) \\ & = - \rho(x,X) + \rho(x,A) \geq 0 \\ \\ & \Rightarrow \rho(x,A) \geq \rho(x,X) \end{aligned}\]

Intuition: the possibility of choosing an option cannot become greater when new choices emerge. This implies that every \(\mu \in [0,1]\) and \(\sum \mu = 1\).

E.g. 2: Given \(X = \{x,y,z,w\}\) and \(A = \{x,y\}\). \[\begin{aligned} q(x, A) &= \sum_{B \supseteq A} (-1)^{|B \setminus A|} \rho(x,B) \\ &= (-1)^{|A \setminus A|} \rho(x,A) + (-1)^{|\{x,y,z\} \setminus A|} \rho(x,\{x,y,z\}) \\ & \quad + (-1)^{|\{x,y,w\} \setminus A|} \rho(x,\{x,y,w\}) + (-1)^{|\{x,y,z,w\} \setminus A|} \rho(x,\{x,y,z,w\}) \\ &= \rho(x,X) - \rho(x,\{x,y,z\}) - \rho(x, \{x,y,w\}) + \rho(x,A) \geq 0 \\ \\ &\Rightarrow \rho(x,A) - \rho(x, \{x,y,w\}) \geq \rho(x,\{x,y,z\}) - \rho(x,X) \end{aligned}\]

Intuition: if we use \(\mu\) in random preference model to denote the results above:

\[\begin{aligned} \rho(x,A) - \rho(x, \{x,y,w\}) &\geq \rho(x,\{x,y,z\}) - \rho(x,X) \\ \mu\{z \succ w \succ x \succ y\} + \mu\{w \succ z \succ x \succ y\} & \\ +\mu\{w \succ x \succ z \succ y\} + \mu\{w \succ x \succ y \succ z\} &\geq \mu\{w \succ x \succ y \succ z\} \end{aligned}\]

This would hold if \(\mu \in [0,1]\).

Falmagne’s Theorem (1978)

\(\rho\) is a RUM iif it satisfies the Block-Marschak inequalities (sufficiently).

Interpretation of the Block-Marschak inequalities: \(q(x,A)\) is the probability that:

  1. options in \(X \setminus A\) are ranked above \(x\), and

  2. \(x\)’s ranking is \(|X \setminus A| + 1\).

Example: Consider \(X = \{x,y,z\}\) and \(A = \{x,y\}\). Then \[\begin{aligned} q(x,A) &= \rho(x, A) - \rho(x,X) \\ &= \mu\{z \succ x \succ y\} + \mu\{x \succ z \succ y\} + \mu\{x \succ y \succ z\} \\ & \quad - \mu\{x \succ z \succ y\} + \mu\{x \succ y \succ z\} \\ &= \mu\{z \succ x \succ y\} \end{aligned}\]

This is:

  1. options in \(X \setminus A\) are ranked above \(x\) (\(z\)), and

  2. \(x\)’s ranking is \(|X \setminus A| + 1 = 1 + 1 = 2\).

Some Random Utility Models

1. Logit

  • Form: \(U_i = v_i + \epsilon_i\), where \(v_i \in \mathbb{R}\) and \(\epsilon_i \sim\) Gumbel (iid.)
  • Gumbel Distribution: - \(\mathbb{P} \{\epsilon_i \leq t\} = F(t) = \exp[-\exp(-t)]\) - Gumbel cdf: \(F(s) = e^{-e^{-s}} = \mathbb{P}\{\epsilon \leq s\}\) - Gumbel pdf: \(f(s) = \frac{d\,F(s)}{d\,s} = e^{-s} \cdot e^{-e^{-s}}\)

Then, under logit,
\[\begin{aligned} \rho(x_i, A) & = \mathbb{P}\{(\forall j \in A, j \neq i) \quad U_i > U_j\} \\ & = \mathbb{P}\{(\forall j \in A, j \neq i) \quad v_i + \epsilon_i > v_j + \epsilon_j\} \\ & = \mathbb{P}\{\epsilon_j < (v_i - v_j) + \epsilon_i\} \\ & = \int_{-\infty}^{+\infty} \Pi_{j \neq i} \, F(v_i - v_j + s) f(s)ds \\ & = \int_{-\infty}^{+\infty} \Pi_{j \neq i} e^{-e^{-(v_i - v_j + s)}} \cdot e^{-s} \cdot e^{-e^{-s}} ds \\ & = \frac{e^{v_i}}{\sum_{j \in A} e^{v_j}} \end{aligned}\]

2. Luce

\(v: X \to (0, \infty)\) \(\rho(x,A) = \frac{v(s)}{\sum_{y \in A} v(y)}\)

Directly Observable and Testable Transitivity

  • In deterministic models, transitivity is a single notion.
    • e.g. in preference relations, transitivity is \(x \succ y\) and \(y \succ z \Rightarrow x \succ z\)
  • In random choice models, there are different strengths of transitivity.

If \(\rho(a,b) \geq 1/2\) and \(\rho(b,c) \geq 1/2\), then there exists

  • weak transitivity if \(\rho(a,c) \geq 1/2\)
  • moderate transitivity if \(\rho(a,c) \geq \min\{\rho(a,b), \rho(b,c)\}\)
    (He and Natenzon (2020))
  • strong transitivity if \(\rho(a,c) \geq \max\{\rho(a,b), \rho(b,c)\}\)
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