<p>Its for all priors and all utility matrices. Its possible for <span class="math-container">$p_1,u_1$</span> and <span class="math-container">$p_2,u_2$</span> to rank <span class="math-container">$S$</span> and <span class="math-container">$S'$</span> different.</p>

CC BY-SA 4.0
2020-07-15T15:31:52.137Z
2020-07-15T15:31:52.137Z
user28714
<p>Consider a model where a decision maker (DM) has to choose action <span class="math-container">$y\in \mathcal{Y}$</span> possibly without being fully aware of the state of the world.</p>
<p>The state of the world has support <span class="math-container">$\mathcal{V}$</span>.</p>
<p>When DM chooses action <span class="math-container">$y\in \mathcal{Y}$</span> and the state of the world is <span class="math-container">$v\in \mathcal{V}$</span>, she receives the payoff <span class="math-container">$u(y,v)$</span>.</p>
<p>Let <span class="math-container">$P_V\in \Delta(\mathcal{V})$</span> be the DM's prior.</p>
<p>The DM also processes some signal <span class="math-container">$T$</span> with support <span class="math-container">$\mathcal{T}$</span> distribution <span class="math-container">$P_{T|V}$</span>  to refine his prior and get a posterior on <span class="math-container">$V$</span>, denoted by <span class="math-container">$P_{V|T}$</span>, via the Bayes rule.</p>
<p>Let <span class="math-container">$S\equiv \{\mathcal{T}, P_{T|V}\}$</span> be called &quot;information structure&quot;.</p>
<p>A strategy for the DM is <span class="math-container">$P_{Y|T}$</span>. Such a strategy is optimal if it maximises his expected payoff, where the expectation is computed using the posterior, <span class="math-container">$P_{V|T}$</span>.</p>
<p>Question: consider two information structures, <span class="math-container">$S$</span> and <span class="math-container">$S'$</span>. We can compare them by using Blackwell Theorem which says that <span class="math-container">$S$</span> is more informative than <span class="math-container">$S'$</span> if the maximal expected payoff under <span class="math-container">$S$</span> is al least equal to the maximal expected payoff under <span class="math-container">$S'$</span>. 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?</p>

<decision-theory><bayesian-game>
Blackwell order of information structures
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# Blackwell order of information structures

(Asked by: username_0 on 2020-07-15)

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?

**Tags:** decision-theory, bayesian-game


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## 1 answers

### Answer by username_1 on 2020-07-15. Score: 3 | Accepted answer

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.


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