<p>In <a href="https://doi.org/10.1093/restud/rdw052" rel="nofollow noreferrer">Kamenica and Gentzkow (2017)</a> as well as in <a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/TE1808" rel="nofollow noreferrer">Bergemann and Morris (2016)</a> the notion of  <a href="https://www.jstor.org/stable/2236332" rel="nofollow noreferrer">Blackwell</a> 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.</p>
<p>In Kamenica and Gentzkow environment, there are <span class="math-container">$n-$</span> senders indexed by <span class="math-container">$i$</span>, there exist a finite state space <span class="math-container">$\Omega$</span> and <span class="math-container">$\omega$</span> is the typical element of the state space. All senders share a commom prior <span class="math-container">$\mu_0$</span> about the state of the world and each of them chooses a singan <span class="math-container">$\pi_i\in \Pi_i$</span> where <span class="math-container">$\Pi=\times_{i}\Pi$</span> is the information environment, then if <span class="math-container">$\pi\in\Pi$</span> is a profile of strategies (signals that are chosen by the senders) let <span class="math-container">$&lt;\pi&gt;$</span> denote the distribution of beliefs of a Bayesian with prior <span class="math-container">$\mu_0$</span> who observes the realization of all signals in <span class="math-container">$\pi$</span>. So,  if <span class="math-container">$\pi$</span> is a Nash equilibrium, then <span class="math-container">$&lt;\pi&gt;$</span> denotes the equilibrium outcome and we define the distribution of posteriors as <span class="math-container">$\tau = &lt;\pi&gt;$</span>. In other words, <span class="math-container">$\tau$</span> is a feasible outcome if there exists <span class="math-container">$\pi\in\Pi$</span> such that <span class="math-container">$\tau = &lt;\pi&gt;$</span>.</p>
<p>Kamenica and Gentzkow quote that a distribution of posterior beliefs <span class="math-container">$\tau \succeq \tau^{'}$</span> if <span class="math-container">$\tau$</span> is a mean preserving spread of <span class="math-container">$\tau^{'}$</span> in which case this means that <span class="math-container">$\tau$</span> is more informatiove than <span class="math-container">$\tau^{'}$</span>.</p>
<p>From my understanding the relation of Blackwell order with the Stochastic dominance order should be something like the following</p>
<p><span class="math-container">$$\tau \succeq \tau^{'} \Leftrightarrow \text{$\tau \leq_{SOSD} \tau^{'}$ ? or $\tau \geq_{SOSD} \tau^{'} $}? \tag{1}$$</span></p>
<p>because <span class="math-container">$\tau$</span> is a mean preserving spread of <span class="math-container">$\tau^{'}$</span>, and this means that the latter must be less informative with respect to the former which part of <span class="math-container">$(1)$</span> is true?</p>

CC BY-SA 4.0
2023-10-09T14:36:25.817Z
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.
Michael Greinecker
CC BY-SA 4.0
2023-10-09T15:14:36.780Z
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?
Oliver Queen
CC BY-SA 4.0
2023-10-09T16:57:18.163Z
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.
Michael Greinecker
<microeconomics><mathematical-economics><game-theory><bayesian-game><expected-utility>
What is the relation between Blackwell's order and Stochastic Dominance order?
Oliver Queen
# What is the relation between Blackwell's order and Stochastic Dominance order?

(Asked by: username_0 on 2023-10-09)

In [Kamenica and Gentzkow (2017)](https://doi.org/10.1093/restud/rdw052) as well as in [Bergemann and Morris (2016)](https://onlinelibrary.wiley.com/doi/abs/10.3982/TE1808) the notion of [Blackwell](https://www.jstor.org/stable/2236332) 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?

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, expected-utility

> **Comments:**
> **username_1**: 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.  
> **username_0**: 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?  
> **username_1**: 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.  

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## 0 answers

