<p>Once again, I will refer to the setting of <a href="https://onlinelibrary.wiley.com/doi/epdf/10.3982/TE1808" rel="nofollow noreferrer">Bergemann and Morris (2016)</a> and write here the payoff formula of player <span class="math-container">$i$</span> from the perspective of the information designer. The payoff formula is the following:</p>
<p><span class="math-container">$$V_i(a,t,\theta)=\sum_{a_{-i},t_{-i},\theta}\psi(\theta)\pi(t_{i},t_{-i}|\theta)\sigma((a_{i},a_{-i})|(t_{i},t_{-i}),\theta)u_i((a_{i},a_{-i}),\theta)$$</span></p>
<p>where <span class="math-container">$\Theta$</span> is the state space and <span class="math-container">$\theta$</span> its typical element, <span class="math-container">$\psi$</span> is the full support common prior, <span class="math-container">$\pi:\Theta\to\Delta(T)$</span> the type probability function, whete <span class="math-container">$T$</span> is the space of types, <span class="math-container">$\sigma:T\times\Theta\to\Delta(A)$</span> the decision rule and <span class="math-container">$u_i:A\times\Theta\to\mathbb{R}$</span> the payoff of player <span class="math-container">$i$</span>. The decision rule is the tool used by the mediator to propose action profiles to the players that are willing to follow and hence they will play a correleted strategy, that is the so-called obedience criterion which needs to be satisfied by <span class="math-container">$\sigma$</span> in order to  define a BCE.</p>
<p><span class="math-container">$\textbf{Question:}$</span> In case we want to define <span class="math-container">$\sigma$</span> as a recommendation in mixed (or behavioral) strategies what is going to change in the formulation of <span class="math-container">$\sigma$</span>, <span class="math-container">$V_i$</span> and in the definition of the Bayesian correlated equilibrium? I would appreciate if someone could write explicitly the formual in this case.</p>

CC BY-SA 4.0
2021-12-10T02:05:04.317Z
Look at $\sigma$, it maps to $\Delta(A)$, so it is already a recommendation in mixed strategies. Unless I'm misunderstanding your question?
Walrasian Auctioneer
CC BY-SA 4.0
2021-12-10T08:41:26.880Z
@WalrasianAuctioneer well true story...the formula as it seems is defined in such a way that a recommendation could be a mixed strategy...and even in restricting the case to pure strategies all we need is to simply put $\sigma=1$ at the recommended profle of strategies, right?
Hunger Learn
CC BY-SA 4.0
2021-12-10T22:49:56.413Z
I don't get why $a$, $t$, and $\theta$ are arguments of $V$.
Michael Greinecker
CC BY-SA 4.0
2021-12-11T00:08:39.037Z
@HungerLearn a pure strategy recommendation is just the dirac measure on it, so you're correct.
Walrasian Auctioneer
CC BY-SA 4.0
2021-12-16T21:40:54.937Z
As interpreted, $\sigma$ indeed only makes pure action recommendations. But you can think of the action set $A_i$ of agent $i$ as the set of all mixed strategies over some primitive set of pure actions so that $\sigma$ then indeed recommends to each agent a mixed strategy. Of course, we would then have to allow for infinite action sets, which is not covered in the paper. However, the definitions don't really change, except that you have to replace summation with integration.
axelniemeyer
CC BY-SA 4.0
2021-12-20T11:37:12.420Z
@axelniemeyer thank you for your answer!
Hunger Learn
<microeconomics><mathematical-economics><game-theory><nash-equilibrium><bayesian-game>
What if Bergemann and Morris setting used mixed (or bbehavioral) actions instead of pure actions as reccomendations?
Hunger Learn
# What if Bergemann and Morris setting used mixed (or bbehavioral) actions instead of pure actions as reccomendations?

(Asked by: username_0 on 2021-12-09)

Once again, I will refer to the setting of [Bergemann and Morris (2016)](https://onlinelibrary.wiley.com/doi/epdf/10.3982/TE1808) and write here the payoff formula of player $i$ from the perspective of the information designer. The payoff formula is the following:

$$V\_i(a,t,\theta)=\sum\_{a\_{-i},t\_{-i},\theta}\psi(\theta)\pi(t\_{i},t\_{-i}|\theta)\sigma((a\_{i},a\_{-i})|(t\_{i},t\_{-i}),\theta)u\_i((a\_{i},a\_{-i}),\theta)$$

where $\Theta$ is the state space and $\theta$ its typical element, $\psi$ is the full support common prior, $\pi:\Theta\to\Delta(T)$ the type probability function, whete $T$ is the space of types, $\sigma:T\times\Theta\to\Delta(A)$ the decision rule and $u\_i:A\times\Theta\to\mathbb{R}$ the payoff of player $i$. The decision rule is the tool used by the mediator to propose action profiles to the players that are willing to follow and hence they will play a correleted strategy, that is the so-called obedience criterion which needs to be satisfied by $\sigma$ in order to define a BCE.

$\textbf{Question:}$ In case we want to define $\sigma$ as a recommendation in mixed (or behavioral) strategies what is going to change in the formulation of $\sigma$, $V\_i$ and in the definition of the Bayesian correlated equilibrium? I would appreciate if someone could write explicitly the formual in this case.

**Tags:** microeconomics, mathematical-economics, game-theory, nash-equilibrium, bayesian-game

> **Comments:**
> **username_1**: Look at $\sigma$, it maps to $\Delta(A)$, so it is already a recommendation in mixed strategies. Unless I'm misunderstanding your question?  
> **username_0**: @username_1 well true story...the formula as it seems is defined in such a way that a recommendation could be a mixed strategy...and even in restricting the case to pure strategies all we need is to simply put $\sigma=1$ at the recommended profle of strategies, right?  
> **username_2**: I don't get why $a$, $t$, and $\theta$ are arguments of $V$.  
> **username_1**: @username_0 a pure strategy recommendation is just the dirac measure on it, so you're correct.  
> **username_3**: As interpreted, $\sigma$ indeed only makes pure action recommendations. But you can think of the action set $A_i$ of agent $i$ as the set of all mixed strategies over some primitive set of pure actions so that $\sigma$ then indeed recommends to each agent a mixed strategy. Of course, we would then have to allow for infinite action sets, which is not covered in the paper. However, the definitions don't really change, except that you have to replace summation with integration.  
> **username_0**: @username_3 thank you for your answer!  

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## 0 answers

