<p>Under the prior, the defendant is more likely to be innocent than guilty. So, without any additional information, the defense attorney in your example need not provide any information. A lot of signals are optimal here.</p>
<p>If the prosecutor and the defense attorney compete, one equilibrium is full revelation from both. No individual deviation from either sender changes anything here.</p>

CC BY-SA 4.0
2024-03-16T18:57:04.210Z
I understand what you say and I also come to the same conclusion. If we are in this case with the above numbers, then the judge - defense attorney problem seems to be satisfied in the exact same way with the judge - prosecutor one. The DA sends the signal $i$ with if the posterior belief of $I$ is equal to $1$ and the signal $g$ if the posterior belief of the accused to be either innocent or guilty is equally likely. The question that arises is, how did Kamenica and Getzkow solved the problem so as to reach to the  optimal signal $\pi(i|I)=4/7$, $\pi(i|G)=0$, $\pi(g|I)=3/7$, $\pi(g|G)=1$?
Oliver Queen
CC BY-SA 4.0
2024-03-16T19:00:59.740Z
Intuitively, we could say that 

$$\pi(i|I)+\pi(g|I)=1\tag{1}$$ and

$$\pi(i|G)+\pi(g|G)=1\tag{2}$$

If the state is $G$, then form $(2)$ we could claim that always it is optimal for the prosecutor to send the signal $g$ and hence $\pi(g|G)=1$ and $\pi(i|G)=0$, but how do we colcude to $\pi(i|I)=4/7$ and $\pi(g|I)=3/7$?
Oliver Queen
CC BY-SA 4.0
2024-03-17T00:02:37.803Z
Presumably, K&G solved the problem using concavification.
Michael Greinecker
CC BY-SA 4.0
2024-03-17T09:43:38.940Z
Sure this is what they do, but I do not understand....where does this property takes place in the solution? Do you understand it?
Oliver Queen
CC BY-SA 4.0
2024-03-17T10:13:46.443Z
I'm not sure what "this property" refers to.
Michael Greinecker
CC BY-SA 4.0
2024-03-17T10:20:13.463Z
Sorry let me re-state.....where does this concavification takes place in the solution?
Oliver Queen
CC BY-SA 4.0
2024-03-17T12:17:02.890Z
Draw the graph of the value function. Draw the concavification. The value of the concavification at the prior will be a convex combination of two beliefs, and the signals should induce these two beliefs. Bayes plausibility gives the right weights on the signals.
Michael Greinecker
CC BY-SA 4.0
2024-03-17T12:22:12.793Z
Ok. Is there another way in case someone won't use the graphical approach? To solve it algebraically? Cause in the case where two or more senders exist then drawing a single graph that contains the value functions of all of them starts to become difficult
Oliver Queen
CC BY-SA 4.0
2024-03-17T12:26:28.607Z
Having more senders is a different model, one I did not look closely at. But in the single sender case one can also solve the problem by doing linear programming on the space of Bayes correlated equilibria.
Michael Greinecker
CC BY-SA 4.0
2024-03-17T13:58:20.277Z
So, `` by doing linear programming on the space of Bayes correlated equilibria", i believe you mean that it is the exact same approach with Begemann and Morris, but instead of the optimal decision rule, I will solve for the optimal $\tau$ as this is defined in the sender's problem above
Oliver Queen
CC BY-SA 4.0
2024-03-17T14:09:53.930Z
As for the competition, I am sure that you know about the join and the meet of two different partitions of a set. The finite partitions are the signals chosen by the senders and if we take the join of the partitions and then the set of all signals becomes a lattice. On a lattice structure the join is a way to somehow ``add" signals together.
Oliver Queen
CC BY-SA 4.0
2024-03-17T14:18:28.593Z
If you solve the optimal obedient decision rule, you have also found an optimal information structure: The recommended action is slways a suitable signal.
Michael Greinecker
CC BY-SA 4.0
2024-03-15T23:48:33.190Z
2024-03-15T23:48:33.190Z
Michael Greinecker
<p>Based on Kamenica and Gentzkow example about the prosecutor and the judge. The utilities of the sender and the receiver are <span class="math-container">$v(\alpha,\omega)=\alpha$</span> and <span class="math-container">$u(\alpha,\omega)=-(\alpha-\omega)^2$</span>.</p>
<p>The judge is the receiver and the prosecutor is the sender, where <span class="math-container">$\omega =\{I, G\}$</span> and the <span class="math-container">$\mu_0(G) = 0.3$</span>. The action set is <span class="math-container">$A = \{0,1\}$</span> where <span class="math-container">$0 : = \text{ acquit}$</span> and <span class="math-container">$1 : = \text{ convict}$</span>. For any <span class="math-container">$\omega$</span>, the prosecutor's utility is <span class="math-container">$v(1,\omega)= 1$</span> and <span class="math-container">$v(0,\omega)=0$</span> and the judge's utility is <span class="math-container">$u(0,I)=1=u(1,G)$</span> and <span class="math-container">$u(1,I)=0=u(0,G)$</span>. The expected utility of the judge and the prosecutor are the following:</p>
<p><span class="math-container">$$u(\alpha,\omega)=0.3\times 1 + 0.7\times 1 = 1\text{ and }v(\alpha,\omega)= 0.3\times 1 + 0.7\times 0 = 0.3$$</span></p>
<p>The prosecutor chooses a signal space <span class="math-container">$S=\{i,g\}$</span> and a distribution <span class="math-container">$\pi$</span> over <span class="math-container">$S$</span>, that is <span class="math-container">$\pi(i|I)=4/7$</span>, <span class="math-container">$\pi(i|G)=0$</span>, <span class="math-container">$\pi(g|I)=3/7$</span>, <span class="math-container">$\pi(g|G)=1$</span> and hence by the law of total probability</p>
<p><span class="math-container">$$\tau(i) = \pi(i|G)\mu_0(G)+\pi(i|I)\mu_0(I)=0.4 \text{ and } \tau(g) = \pi(g|G)\mu_0(G)+\pi(g|I)\mu_0(I)=0.6$$</span></p>
<p>Hence, we have that</p>
<p><span class="math-container">$$\mu_i(I) = \frac{\pi(i|I)\mu_0(I)}{\pi(i)}=1,\quad \mu_i(G) = \frac{\pi(i|G)\mu_0(G)}{\pi(i)}= 0$$</span></p>
<p>and</p>
<p><span class="math-container">$$\mu_g(I) = \frac{\pi(g|I)\mu_0(I)}{\pi(g)}=\frac{1}{2},\quad \mu_g(G) = \frac{\pi(g|G)\mu_0(G)}{\pi(g)}= \frac{1}{2}$$</span></p>
<p>By <span class="math-container">$u$</span> and <span class="math-container">$v$</span> we have that the optimal response of the receiver is equal to <span class="math-container">$\alpha^{\star}(\mu_s) = \mathbb{E}_{\mu_s}(\omega)$</span> and thus sender's value function is</p>
<p><span class="math-container">$$\hat{V}(\mu_s)=\mu_s(G)v(\alpha^{\star}(\mu_s(G)),G)+\mu_s(I)v(\alpha^{\star}(\mu_s(I)),I),\text{ for all $s\in\text{Supp}(\tau)$}$$</span></p>
<p>and therefore the sender's problem is simplified to</p>
<p><span class="math-container">$$\text{max}_{\tau\in\Delta(\Delta(\Omega))} \{ \tau(i) \hat{V}(\mu_i) + \tau(g) \hat{V}(\mu_g) \}\text{ s.t. $\tau(i)\mu_i+\tau(g)\mu_g = \mu_0$ for all $\omega$}\tag{S.P.}$$</span></p>
<p>From the above  numbers and if we substitute them for every signal <span class="math-container">$s=\{i,g\}$</span>, the payoff of the sender is</p>
<p><span class="math-container">$$\tau(i)\hat{V}(\mu_i)=\tau(i)\left( \mu_i(G)v(0,G)+\mu_i(I)v(0,I)\right)=0.4\times \left(0\times 0+ 1\times 0 \right)=0$$</span></p>
<p>and</p>
<p><span class="math-container">$$\tau(g)\hat{V}(\mu_g)=\tau(g)\left( \mu_g(G)v(1,G)+\mu_g(I)v(1,I)\right)=0.6\times \left(\frac{1}{2}\times 1+ \frac{1}{2}\times 1 \right)=0.6$$</span></p>
<p>and <span class="math-container">$0.4 \times 1+ 0.6 \times(1/2) = 0.7$</span>  which is the prior of innocence and <span class="math-container">$0.4 \times 0+ 0.6 \times(1/2) = 0.3$</span> which is the prior of guilty.</p>
<p>To wrap up the prosecutor will send the signal <span class="math-container">$i$</span> only if she is <span class="math-container">$100\%$</span> sure that the accused is guilty (based on the posterior belief of <span class="math-container">$I$</span>) and the signal <span class="math-container">$g$</span> if the posterior belief of the accused to be either innocent or guilty is equally likely.</p>
<p>Although they do not present how they find the values for <span class="math-container">$\pi$</span> etc, i assume that the idea starts going backwordly and solving the problem having as constraints that <span class="math-container">$\pi(g|I)+\pi(i|I) = 1$</span> as well as <span class="math-container">$\pi(g|G)+\pi(i|G) = 1$</span>.</p>
<p>My question is how do I solve the problem if the sender's role change and we have isntead of a prosecutor a defense attorney where her utility will be <span class="math-container">$v(\alpha, \omega)=1-\alpha$</span> and what becomes when both the defense attorney and the prosecutor comptete? (In the <span class="math-container">$2017$</span> paper where they talk about competition, I fail to understand their game when there exist two players, a defense attorney and a prosecutor, check in <a href="https://chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://web.stanford.edu/%7Egentzkow/research/BayesianCompRich.pdf" rel="nofollow noreferrer">here</a>.</p>

<microeconomics><mathematical-economics><game-theory><bayesian-game><competition>
Judge prosecutor example of Kameinica and Gentzkow
Oliver Queen
# Judge prosecutor example of Kameinica and Gentzkow

(Asked by: username_0 on 2024-03-15)

Based on Kamenica and Gentzkow example about the prosecutor and the judge. The utilities of the sender and the receiver are $v(\alpha,\omega)=\alpha$ and $u(\alpha,\omega)=-(\alpha-\omega)^2$.

The judge is the receiver and the prosecutor is the sender, where $\omega =\{I, G\}$ and the $\mu\_0(G) = 0.3$. The action set is $A = \{0,1\}$ where $0 : = \text{ acquit}$ and $1 : = \text{ convict}$. For any $\omega$, the prosecutor's utility is $v(1,\omega)= 1$ and $v(0,\omega)=0$ and the judge's utility is $u(0,I)=1=u(1,G)$ and $u(1,I)=0=u(0,G)$. The expected utility of the judge and the prosecutor are the following:

$$u(\alpha,\omega)=0.3\times 1 + 0.7\times 1 = 1\text{ and }v(\alpha,\omega)= 0.3\times 1 + 0.7\times 0 = 0.3$$

The prosecutor chooses a signal space $S=\{i,g\}$ and a distribution $\pi$ over $S$, that is $\pi(i|I)=4/7$, $\pi(i|G)=0$, $\pi(g|I)=3/7$, $\pi(g|G)=1$ and hence by the law of total probability

$$\tau(i) = \pi(i|G)\mu\_0(G)+\pi(i|I)\mu\_0(I)=0.4 \text{ and } \tau(g) = \pi(g|G)\mu\_0(G)+\pi(g|I)\mu\_0(I)=0.6$$

Hence, we have that

$$\mu\_i(I) = \frac{\pi(i|I)\mu\_0(I)}{\pi(i)}=1,\quad \mu\_i(G) = \frac{\pi(i|G)\mu\_0(G)}{\pi(i)}= 0$$

and

$$\mu\_g(I) = \frac{\pi(g|I)\mu\_0(I)}{\pi(g)}=\frac{1}{2},\quad \mu\_g(G) = \frac{\pi(g|G)\mu\_0(G)}{\pi(g)}= \frac{1}{2}$$

By $u$ and $v$ we have that the optimal response of the receiver is equal to $\alpha^{\star}(\mu\_s) = \mathbb{E}\_{\mu\_s}(\omega)$ and thus sender's value function is

$$\hat{V}(\mu\_s)=\mu\_s(G)v(\alpha^{\star}(\mu\_s(G)),G)+\mu\_s(I)v(\alpha^{\star}(\mu\_s(I)),I),\text{ for all $s\in\text{Supp}(\tau)$}$$

and therefore the sender's problem is simplified to

$$\text{max}\_{\tau\in\Delta(\Delta(\Omega))} \{ \tau(i) \hat{V}(\mu\_i) + \tau(g) \hat{V}(\mu\_g) \}\text{ s.t. $\tau(i)\mu\_i+\tau(g)\mu\_g = \mu\_0$ for all $\omega$}\tag{S.P.}$$

From the above numbers and if we substitute them for every signal $s=\{i,g\}$, the payoff of the sender is

$$\tau(i)\hat{V}(\mu\_i)=\tau(i)\left( \mu\_i(G)v(0,G)+\mu\_i(I)v(0,I)\right)=0.4\times \left(0\times 0+ 1\times 0 \right)=0$$

and

$$\tau(g)\hat{V}(\mu\_g)=\tau(g)\left( \mu\_g(G)v(1,G)+\mu\_g(I)v(1,I)\right)=0.6\times \left(\frac{1}{2}\times 1+ \frac{1}{2}\times 1 \right)=0.6$$

and $0.4 \times 1+ 0.6 \times(1/2) = 0.7$ which is the prior of innocence and $0.4 \times 0+ 0.6 \times(1/2) = 0.3$ which is the prior of guilty.

To wrap up the prosecutor will send the signal $i$ only if she is $100\%$ sure that the accused is guilty (based on the posterior belief of $I$) and the signal $g$ if the posterior belief of the accused to be either innocent or guilty is equally likely.

Although they do not present how they find the values for $\pi$ etc, i assume that the idea starts going backwordly and solving the problem having as constraints that $\pi(g|I)+\pi(i|I) = 1$ as well as $\pi(g|G)+\pi(i|G) = 1$.

My question is how do I solve the problem if the sender's role change and we have isntead of a prosecutor a defense attorney where her utility will be $v(\alpha, \omega)=1-\alpha$ and what becomes when both the defense attorney and the prosecutor comptete? (In the $2017$ paper where they talk about competition, I fail to understand their game when there exist two players, a defense attorney and a prosecutor, check in [here](https://chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://web.stanford.edu/%7Egentzkow/research/BayesianCompRich.pdf).

**Tags:** microeconomics, mathematical-economics, game-theory, bayesian-game, competition


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## 1 answers

### Answer by username_1 on 2024-03-15. Score: 1

Under the prior, the defendant is more likely to be innocent than guilty. So, without any additional information, the defense attorney in your example need not provide any information. A lot of signals are optimal here.

If the prosecutor and the defense attorney compete, one equilibrium is full revelation from both. No individual deviation from either sender changes anything here.

> **Comments:**
> **username_0**: I understand what you say and I also come to the same conclusion. If we are in this case with the above numbers, then the judge - defense attorney problem seems to be satisfied in the exact same way with the judge - prosecutor one. The DA sends the signal $i$ with if the posterior belief of $I$ is equal to $1$ and the signal $g$ if the posterior belief of the accused to be either innocent or guilty is equally likely. The question that arises is, how did Kamenica and Getzkow solved the problem so as to reach to the  optimal signal $\pi(i|I)=4/7$, $\pi(i|G)=0$, $\pi(g|I)=3/7$, $\pi(g|G)=1$?  
> **username_0**: Intuitively, we could say that   $$\pi(i|I)+\pi(g|I)=1\tag{1}$$ and  $$\pi(i|G)+\pi(g|G)=1\tag{2}$$  If the state is $G$, then form $(2)$ we could claim that always it is optimal for the prosecutor to send the signal $g$ and hence $\pi(g|G)=1$ and $\pi(i|G)=0$, but how do we colcude to $\pi(i|I)=4/7$ and $\pi(g|I)=3/7$?  
> **username_1**: Presumably, K&G solved the problem using concavification.  
> **username_0**: Sure this is what they do, but I do not understand....where does this property takes place in the solution? Do you understand it?  
> **username_1**: I'm not sure what "this property" refers to.  
> **username_0**: Sorry let me re-state.....where does this concavification takes place in the solution?  
> **username_1**: Draw the graph of the value function. Draw the concavification. The value of the concavification at the prior will be a convex combination of two beliefs, and the signals should induce these two beliefs. Bayes plausibility gives the right weights on the signals.  
> **username_0**: Ok. Is there another way in case someone won't use the graphical approach? To solve it algebraically? Cause in the case where two or more senders exist then drawing a single graph that contains the value functions of all of them starts to become difficult  
> **username_1**: Having more senders is a different model, one I did not look closely at. But in the single sender case one can also solve the problem by doing linear programming on the space of Bayes correlated equilibria.  
> **username_0**: So, `` by doing linear programming on the space of Bayes correlated equilibria", i believe you mean that it is the exact same approach with Begemann and Morris, but instead of the optimal decision rule, I will solve for the optimal $\tau$ as this is defined in the sender's problem above  
> **username_0**: As for the competition, I am sure that you know about the join and the meet of two different partitions of a set. The finite partitions are the signals chosen by the senders and if we take the join of the partitions and then the set of all signals becomes a lattice. On a lattice structure the join is a way to somehow ``add" signals together.  
> **username_1**: If you solve the optimal obedient decision rule, you have also found an optimal information structure: The recommended action is slways a suitable signal.  

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