Here’s my own class note from first two weeks of Professor Kamenica’s Topics in Informational Economics course.

Math Prelims: Comparing Two Distributions

Second-Order Stochastic Dominance (SOSD)

Definition.
Given $\mathbf{X}, \mathbf{Y} \in \mathcal{X}^n$, we say that $\mathbf{X}$ second-order stochastically dominates $\mathbf{Y}$, written as $\mathbf{X} \ge_{sosd} \mathbf{Y}$, , if for every increasing and concave function $u : \mathbb{R}^n \to \mathbb{R}$,

$$ > \int u(x)\, dF_{\mathbf{X}}(x) \ge \int u(x)\, dF_{\mathbf{Y}}(x). > $$

Mean-Preserving Spread (MPS)

Definition.
Given $\mathbf{X}, \mathbf{Y} \in \mathcal{X}^n$, we say that $\mathbf{X}$ and $\mathbf{Y}$ are equal in distribution, denoted $\mathbf{X} \overset{d}{=} \mathbf{Y}$, iff. $F_{\mathbf{X}}(\cdot) = F_{\mathbf{Y}}(\cdot)$ for their CDFs.

Definition.
Given $X, Y \in \mathcal{X}^n$, we say $X$ is a mean-preserving spread of $Y$, denoted

$$ > X \ge_{mps} Y, > $$

if there exists some $Z \in \mathcal{X}^n$ such that

$$ > X \overset{d}{=} Y + Z, \mathbb{E}[Z|Y] = 0. > $$

Equivalently, $F_X$ is a mean-preserving spread of $F_Y$.

Proposition.
Given $X, Y \in \mathcal{X}^n$,

$$ > X \ge_{mps} Y \Longleftrightarrow \int f(x) dF_X(x) \ge \int f(x) dF_Y(x) > $$

for every convex function $f : \mathbb{R}^n \to \mathbb{R}$.

Relationship Between SOSD and MPS

Theorem.
Suppose $X, Y \in \mathcal{X}^n$. Then

$$ > \mathbb{E}[X] = \mathbb{E}[Y] \ \text{ and } \ X \le_{sosd} Y \Longleftrightarrow X \ge_{mps} Y. > $$

Detour: One Dimensional Case

Theorem.
Given $X, Y \in \mathcal{X}^1$,

$$ > X \ge_{mps} Y \Longleftrightarrow \mathbb{E}[X] = \mathbb{E}[Y]\text{ and } \int_{-\infty}^{x} F_X(t) dt \ge \int_{-\infty}^{x} F_Y(t) dt, \ \forall x. > $$

Blackwell’s Theorem

Decision Problems

Fix a finite state space $\Omega$.

A decision problem is a triplet $(\mu, A, u)$ where:

  1. $\mu \in \Delta(\Omega)$ — the decision maker’s (DM’s) belief about $\Omega$;
  2. $A$, a compact set — the DM’s choice set;
  3. $u : A \times \Omega \to \mathbb{R}$ — the DM’s state-contingent utility (assumed continuous).

Value Function

Define the value function:

$$ V_{(A,u)}(\mu) = \max_{a \in A} \mathbb{E}_\mu[u(a,\omega)]. $$

Theorem.
For all $(A, u)$, the value function $V_{(A,u)}$ is convex.

Theorem.
Given any convex function $f : \Delta(\Omega) \to \mathbb{R}$,
there exists a decision problem $(A, u)$ such that

$$ > V_{(A,u)} = f. > $$

(NOTE: This shows that any convex function over beliefs can be represented as a decision problem with some utility and action set.)

Experiment

Definition (Experiment).
An experiment $\mathcal{E}$ is a Markov kernel that maps a prior belief $\mu \in \Delta(\Omega)$
to a random posterior $\tilde{\mu} \in \Delta(\Omega)$ satisfying the Bayes plausibility condition:

$$ > \mathbb{E}_{\tilde{\mu} \sim \lang \mathcal{E}|\mu \rang}[\tilde{\mu}] = \mu. > $$

We write this random posterior as

$$ > \langle \mathcal{E} \mid \mu \rangle \in \Delta(\Delta(\Omega)), > $$

i.e., $\langle \mathcal{E} \mid \mu \rangle$ is the distribution over posteriors induced by experiment $\mathcal{E}$ when the prior is $\mu$.

Note: this revelation approach abstracts away the underlying signal space.

Blackwell Order

Definition.
Given a decision problem $(\mu, A, u)$, the value of an experiment $\mathcal{E}$ is $W_{(\mu, A, u)}(\mathcal{E}) = \mathbb{E}_{\tilde{\mu} \sim \lang \mathcal{E}|\mu\rang} \left[ V(\tilde{\mu}) \right]$.

Definition.
Given two experiments $\mathcal{E}$ and $\mathcal{E}'$, $\mathcal{E}$ is (Blackwell) more informative than $\mathcal{E}'$, denoted $\mathcal{E} \ge_B \mathcal{E}'$, if

$$ > W_{(\mu, A, u)}(\mathcal{E}) > \ge > W_{(\mu, A, u)}(\mathcal{E}') > \quad \forall (\mu, A, u). > $$

Induced Mean-Preserving Spread (IMPS) Order

Definition.
Given $\mathcal{E}, \mathcal{E}'$, $\mathcal{E}$ is above $\mathcal{E}$ in the induced mean-preserving spread order, denoted $\mathcal{E} \ge_{\mathrm{IMPS}} \mathcal{E}'$, if

$$ > \langle \mathcal{E} | \mu \rangle > \ge_{\mathrm{MPS}} > \langle \mathcal{E}' | \mu \rangle > \quad \forall \mu. > $$

Garbling Order

Definition.
Given experiments $(\mathcal{E}, Y)$ and $(\mathcal{E}', Y')$, say that $\mathcal{E}'$ is a garbling of $\mathcal{E}$, denoted $\mathcal{E} \ge_g \mathcal{E}',$ if there exists a stochastic kernel $g : Y \times Y' \to [0,1]$ such that:

  1. $\displaystyle \sum_{y' \in Y'} g(y, y') = 1 \quad \forall y \in Y;$
  2. $\displaystyle \mathcal{E}'(y'|\omega) = \sum_{y \in Y} g(y, y') \mathcal{E}(y|\omega).$

Blackwell’s Theorem (1951)

Theorem (Blackwell 1951).
The following are equivalent for any experiments $\mathcal{E}, \mathcal{E}'$:

  1. $\mathcal{E} \ge_B \mathcal{E}'$
  2. $\mathcal{E} \ge_{\mathrm{IMPS}} \mathcal{E}'$
  3. $\mathcal{E} \ge_g \mathcal{E}'$

An interesting Convex Support Condition

Proposition.
If $\mathcal{E} \ge_B \mathcal{E}'$, then

$$ > \mathrm{co}\big(\mathrm{supp}(\langle \mathcal{E} | \cdot \rangle)\big) > \supseteq > \mathrm{supp}(\langle \mathcal{E}' | \cdot \rangle). > $$

Some Examples to Play Around

Example 1 (Location Experiment)

$$ \Omega \subseteq \mathbb{R}, \quad Y \subseteq \mathbb{R} $$$$ \mathcal{E}: y \sim \mathcal{N}(\omega, \sigma^2), \qquad \mathcal{E}': y' \sim \mathcal{N}(\omega, {\sigma'}^2) $$

Claim: $\mathcal{E} \ge_B \mathcal{E}' \iff \sigma \le \sigma'.$

Example 2

$$ \mathcal{E}: y = \omega + \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, \sigma^2) $$$$ \mathcal{E}': y' = \omega + \varepsilon', \quad \varepsilon' \text{ with } \mathrm{Var}(\varepsilon') > 0 $$

Claim: $\mathcal{E} \not\ge_B \mathcal{E}'$ and $\mathcal{E}' \not\ge_B \mathcal{E}$.

Example 3

$$ \mathcal{E}: y = \omega + U, \quad U \sim \mathrm{Unif} \left(-\tfrac{1}{2}, \tfrac{1}{2}\right) $$$$ \mathcal{E}': y' = \omega + U', \quad U' \sim \mathrm{Unif} \left(-\tfrac{1}{2x}, \tfrac{1}{2x}\right), \ x > 1 $$

Claim: $\mathcal{E} \not\ge_B \mathcal{E}'$,
and $\mathcal{E}' \ge_B \mathcal{E} \iff x \in \mathbb{N}$ (e.g., $x = 1.5$ fails).