# Blackwell order of information structures

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?

Its for all priors and all utility matrices. Its possible for $$p_1,u_1$$ and $$p_2,u_2$$ to rank $$S$$ and $$S'$$ different.