Comparisons of Distributions

We study random variables on $\mathbb{R}$ (aka lotteries). Each lottery can be identified with its cumulative distribution function (cdf):

$$ F(z) = \Pr(Z \leq z). $$

First-Order Stochastic Dominance (FOSD)

Definition. For two distributions characterized by their cdf $F, G$, $$ F \succeq_\text{FOSD} G \iff \int u(x) dF(x) \geq \int u(x) dG(x), \quad \forall \text{increasing } u. $$

Theorem (Characterization of FOSD). $$ F \succeq_\text{FOSD} G \iff F(x) \leq G(x) \quad \forall x. $$

Second-Order Stochastic Dominance (SOSD)

Now suppose we restrict to increasing and concave utility functions,

Definition. For two distributions characterized by their cdf $F, G$, $$ F \succeq_\text{SOSD} G \iff \int u(x) dF(x) \geq \int u(x) dG(x) \quad \forall \text{increasing, \textbf{concave} } u. $$

Theorem (Characterization of SOSD). $$ F \succeq_\text{SOSD} G \quad \iff \quad \int_{-\infty}^x F(t) dt \leq \int_{-\infty}^x G(t)dt \quad \forall x. $$

There’s another weaker characterization for SOSD:

Theorem (SOSD and mean-preserving spread) If $F$ and $G$ have the same mean, then $$ F \succeq_{\text{SOSD}} G \iff F \text{ is a mean-preserving spread of }G. $$

Experiments and Blackwell Dominance

State-Dependent Utility

Let $\Omega =$ state space, define state-dependent utility function $u : X \times \Omega \to \mathbb{R}$ (utility depends on both action $x$ and state $\omega$). A belief, denoted as $\mu$ is a probability distribution $\mu \in \Delta \Omega$.

An experiment $\mathcal{E}$ maps states $\omega \in \Omega$ into signals, which then update beliefs via Bayes’ rule. Take a revelation-principle minset — the formality of signals doesn’t really matter, what we care about is whatever affects decision making — hence we want to model the posterior belief and its ex-ante distribution induced by the experiment. Denote it as $\lang \mathcal E \mid \mu_0\rang \in \Delta \Delta \Omega$.

Proposition. (Corollary of Bayes’s rule) Fix a prior $\mu_0$, for any experiment $$ \mathbb E_{\mu\sim\lang \mathcal E\mid\mu_0\rang}[\mu] = \mu_0. $$

Definition. (Blackwell Dominance)

An experiment $\mathcal{E}_1$ Blackwell dominates $\mathcal{E}_2$ if, for every prior $\mu_0$, every action set $X$, and every state-dependent utility $u : X \times \Omega \to \mathbb{R}$, we have

Induced Value Function

Given $\mu$, the decision maker solves

$$ V(\mu) = \max_{x \in X} \mathbb{E}_{\omega \sim \mu}[ u(x,\omega) ]. $$

Lemma. $V(\cdot)$ is convex in $\mu$.

Equivalence

Claim. If for some interior prior $\mu_0$ we have $\lang \mathcal E_1 \mid \mu_0 \rang$ $\succeq_{\text{SOSD}} \lang \mathcal{E}_2 \mid \mu_0 \rang$, then for all prior $\mu$,

Theorem. $\mathcal{E}_1$ Blackwell dominates $\mathcal{E}_2$ iff $\lang \mathcal{E}_1 \mid \cdot \rang$ is a mean-preserving spread of $\lang \mathcal{E}_2 \mid \cdot \rang$, which is also equivalent to