# What is the relation between Blackwell's order and Stochastic Dominance order?

In Kamenica and Gentzkow (2017) as well as in Bergemann and Morris (2016) the notion of Blackwell comparioson of experiments is used to compare different information structures. I am trying to find the equivalence between Blackwells order and stochastic dominance order. Let me give a bit notation of the Kamenica and Gentzkow environment.

In Kamenica and Gentzkow environment, there are $$n-$$ senders indexed by $$i$$, there exist a finite state space $$\Omega$$ and $$\omega$$ is the typical element of the state space. All senders share a commom prior $$\mu_0$$ about the state of the world and each of them chooses a singan $$\pi_i\in \Pi_i$$ where $$\Pi=\times_{i}\Pi$$ is the information environment, then if $$\pi\in\Pi$$ is a profile of strategies (signals that are chosen by the senders) let $$<\pi>$$ denote the distribution of beliefs of a Bayesian with prior $$\mu_0$$ who observes the realization of all signals in $$\pi$$. So, if $$\pi$$ is a Nash equilibrium, then $$<\pi>$$ denotes the equilibrium outcome and we define the distribution of posteriors as $$\tau = <\pi>$$. In other words, $$\tau$$ is a feasible outcome if there exists $$\pi\in\Pi$$ such that $$\tau = <\pi>$$.

Kamenica and Gentzkow quote that a distribution of posterior beliefs $$\tau \succeq \tau^{'}$$ if $$\tau$$ is a mean preserving spread of $$\tau^{'}$$ in which case this means that $$\tau$$ is more informatiove than $$\tau^{'}$$.

From my understanding the relation of Blackwell order with the Stochastic dominance order should be something like the following

$$\tau \succeq \tau^{'} \Leftrightarrow \text{\tau \leq_{SOSD} \tau^{'} ? or \tau \geq_{SOSD} \tau^{'} }? \tag{1}$$

because $$\tau$$ is a mean preserving spread of $$\tau^{'}$$, and this means that the latter must be less informative with respect to the former which part of $$(1)$$ is true?

• Usually, second-order stochastic dominance is defined for one-dimensional distributions. But $\tau\succeq\tau'$ is equivalent to $\int f~\mathrm d\tau\leq \int f~\mathrm d\tau'$ for every concave function. Oct 9 at 14:36
• Well, the distrubutions that are defined in Kamenica and Gentzkow are n-dimensional distributions, with $n\geq 2$ so the Blackwell order is a better way of representation than the Stochastic dominance order and that is why they do not choose to use latter ordering, right? Oct 9 at 15:14
• Depending how you define it, there are versions of the order known as convex order, dilation order, Choquet order... There is no hope for a single all-encompassing name and, thanks to the equivalence theorems, no need. Oct 9 at 16:57