Consider a model where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world.
The state of the world has support $\mathcal{V}$.
When DM chooses action $y\in \mathcal{Y}$ and the state of the world is $v\in \mathcal{V}$, she receives the payoff $u(y,v)$.
Let $P_V\in \Delta(\mathcal{V})$ be the DM's prior.
The DM also processes some signal $T$ with support $\mathcal{T}$ distribution $P_{T|V}$ to refine his prior and get a posterior on $V$, denoted by $P_{V|T}$, via the Bayes rule.
Let $S\equiv \{\mathcal{T}, P_{T|V}\}$ be called "information structure".
A strategy for the DM is $P_{Y|T}$. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior, $P_{V|T}$.
Question: consider two information structures, $S$ and $S'$. We can compare them by using Blackwell Theorem which says that $S$ is more informative than $S'$ if the maximal expected payoff under $S$ is al least equal to the maximal expected payoff under $S'$. Is this correct? If yes, then it seems to me that I can rank any information structure using this criterion. Hence, why the Blackwell order is a partial order?
