<p><span class="math-container">$\textit{Question 1:}$</span> As explained by Nav89, compactness of the strategy sets and continuity of the payoff function is needed to ensure equilibrium existence.</p>
<p><span class="math-container">$\textit{Question 2:}$</span> I am not sure whether I truly understand your question. The assumption that the set of signals is finite is for convenience: finite probability distributions are easy to work with. A probability measure is nothing but a probability distribution. So given that I get signal <span class="math-container">$x_i$</span>, I will have some belief about the signals of others <span class="math-container">$x_{-i}$</span>. As defined in Section 2.2., in a game extended by an information structure, players condition their behavior in the game on the signal they get. These signals thus act like a coordination device (e.g., if the sun shines, we do XYZ...).</p>
<p><span class="math-container">$\textit{Question 3:}$</span> A communication mechanism is another way of allowing players to coordinate in the game. The players send messages to the mechanism, and then the players again observe signals on which they condition their behavior (think, I see that in NYC the sun shines, you see that in LA it rains, and the mechanism will send us some information about what everyone said).</p>
<p>So if we send messages <span class="math-container">$(t_1,t_2)$</span>, then the mechanism sends signals according to a lottery <span class="math-container">$l(t_1,t_2)$</span>. Here, <span class="math-container">$l(t_1,t_2)(x)$</span> is the probability that signal profile <span class="math-container">$x$</span> is sent by the mechanism in this lottery.</p>
<p>Now, suppose we don't send a single message <span class="math-container">$t_1$</span> or <span class="math-container">$t_2$</span> to the mechanism, but each of us sends a mixed message <span class="math-container">$\tau_1$</span> and <span class="math-container">$\tau_2$</span> ---probability distributions over messages---to the mechanism. Then, <span class="math-container">$l(\tau_1,\tau_2)(x)$</span> is the probability that signal <span class="math-container">$x$</span> is sent back by the mechanism if I send mixed message <span class="math-container">$\tau_1$</span> and you send mixed message <span class="math-container">$\tau_2$</span>.</p>
<p>So if <span class="math-container">$\tau_i(t_i)$</span> is the probability that <span class="math-container">$i$</span> sends <span class="math-container">$t_i$</span> in the mixed message <span class="math-container">$\tau_i$</span>, then</p>
<p><span class="math-container">$$l(\tau_1,\tau_2)(x)=\sum_{t_1}\sum_{t_2} \tau_1(t_1)\tau_2(t_2)l(t_1,t_2)(x).$$</span></p>

CC BY-SA 4.0
2021-09-16T16:48:02.057Z
Ok, my only comment is the following...you say $l(t)(x)$, but the mechanisms says $l(t)(y)$ i think you used $x$ instead of $y$ just for your own convinience, right? If so, no need to make changes, if not then explain me what am I missing? 

Thank you in advance!
Nav89
CC BY-SA 4.0
2021-09-16T16:50:14.623Z
Yes sorry: $x$ should be $y$ in question 3. The sets $X$ and $Y$ can be different but the idea is the same.
axelniemeyer
CC BY-SA 4.0
2021-09-16T16:52:07.273Z
Ok no worries. Thanks for your help!
Hunger Learn
CC BY-SA 4.0
2021-09-16T16:34:28.467Z
2021-09-16T16:34:28.467Z
axelniemeyer
<p><span class="math-container">$\textit{Question 1:}$</span> The assumption of  compactness is often assumed in order to guarantee existence of a solution to a problem. It is standard and as you have seen, the only thing that you need, the weakest of the assupmtions is the the finite set of strategies <span class="math-container">$S=\Pi_{i\in I}S_i$</span> (which is not necessarily finite) and the continuous function <span class="math-container">$g$</span>. So these two assumptions are enough to quarantee the existence of a solution to the problem.</p>
<p><span class="math-container">$\textit{Question 2:}$</span> So the information strucutre is <span class="math-container">$\mathcal{I}=((X^i)_i,\mu)$</span>, where <span class="math-container">$X_i$</span> denotes the finite set of strategies of player <span class="math-container">$i$</span> and <span class="math-container">$X=\Pi_{i\in I} X_i$</span>, whre this probability measure is defined from <span class="math-container">$X$</span> to <span class="math-container">$\Delta(X)\subset[0,1]^I$</span>. More precisely, it holds that</p>
<p><span class="math-container">$$\mu:X\to \Delta(X)\subset [0,1]^I$$</span></p>
<p>if <span class="math-container">$x=(x^i,x^{-i})\in X$</span> is drawn (which means this is a lottery) according to <span class="math-container">$\mu$</span>, then player <span class="math-container">$i$</span> is informed only about his coordinate <span class="math-container">$x^i$</span>. Suppose, <span class="math-container">$I=2$</span>, and <span class="math-container">$X^1=\{\text{buy}, \text{sell}\}$</span> and <span class="math-container">$X^2=\{\text{hold}, \text{sell}\}$</span>, then a drawn according to <span class="math-container">$\mu$</span> for both playes could be <span class="math-container">$\mu(x^1,x^2)=\left((1/2,1/2),(3/4,1/4)\right)$</span>, so player <span class="math-container">$1$</span> will be infored about <span class="math-container">$x^1$</span> which means, buy with probability <span class="math-container">$1/2$</span> or sell with probability <span class="math-container">$1/2$</span> and player <span class="math-container">$2$</span> will be informed about <span class="math-container">$x^2$</span> which means hold with probablility <span class="math-container">$3/4$</span> or sell with <span class="math-container">$1/2$</span>.</p>
<p><span class="math-container">$\textit{Question 3:}$</span> Well a communucation mechanism is defined as a triplet <span class="math-container">$\mathcal{C}=((T^i)_i, (Y^i)_i, l)$</span> but lets make it clear what is this mechanims. Think about it like a device that takes an input this is <span class="math-container">$T^i$</span> and in returt it gives an outpout that is <span class="math-container">$Y^i$</span> and the function <span class="math-container">$l:T\to\Delta(Y)$</span> is again a lottery or better a probability distribution according to which you are taking back the message <span class="math-container">$y\in Y$</span>. In other words this signal function <span class="math-container">$l$</span> tells you that if you give a message whatever this is, I can give you back an estimation, or better a probability distributuon on which you will know that something will happen with probabolity <span class="math-container">$l(t)$</span>. Suppose again that <span class="math-container">$I=2$</span>, <span class="math-container">$t=(t^1,t^2)=\left(\underbrace{(\text{buy}, \text{sell})}_{t^1},\underbrace{(\text{hold}, \text{sell})}_{t^2}\right)$</span>. So the players decide to send this message <span class="math-container">$t$</span> in the mechanism and this in returns gives back to them <span class="math-container">$y\in Y$</span> which is, say <span class="math-container">$l(t)=\left(\underbrace{(2/3(\text{buy}),1/3(\text{sell}))}_{y^1},\underbrace{(2/3(\text{hold}),1/3(\text{sell}))}_{y^2}\right)$</span> and it is easy to understand what is the meaning of <span class="math-container">$y$</span> now. Even furhter, instead of <span class="math-container">$T$</span>, you can generalize the set of finite messages sent to the communication mechanism to a mixed set of messages (generaliztaion of the <span class="math-container">$l$</span> function) s.t <span class="math-container">$l:\mathcal{T}\to\Delta(Y)$</span>, where <span class="math-container">$l(\tau)(y)$</span> is again this lottery according to which the mechanism will return you <span class="math-container">$y$</span>, so again it is a probability distribution or better and estimation. Thus  <span class="math-container">$l(\tau)(y)$</span> denotes something like a product of vectors that give <span class="math-container">$(\tau \times l(t))(y)$</span>. In other words this \tau is also a probability distribution on the space of finite messages so it can also serve as the probability measure and this is why <span class="math-container">$l(\tau)(y)=\mathbb{E}_{\tau}l(t)(y)$</span></p>

CC BY-SA 4.0
2021-09-15T16:53:55.707Z
Oh thanks but it will take some time to check this....
Hunger Learn
CC BY-SA 4.0
2021-09-15T16:56:16.703Z
@HungerLearn no worries....I want other people here to take a look to check if I am mistaken or not....this is a difficult understanding... so maybe others could check here
Nav89
CC BY-SA 4.0
2021-09-16T11:06:18.073Z
well I am not sure about the last par $l(\tau)(y)$, but it seems to me as a product of the vectors from the probability distribution with the signal $y$ which comes back from the mechanism to every player. If anyone knows somthing please i would like to know it and of course endorse my answer! Thank you in advance!
Nav89
CC BY-SA 4.0
2021-09-15T16:52:59.753Z
2021-09-15T17:11:39.543Z
2021-09-15T17:11:39.543Z
Nav89
<p>The paper of <a href="https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&amp;rep=rep1&amp;type=pdf" rel="nofollow noreferrer">Olivier Gossner in Security Protocols in 1998</a> has some definitions that confuse me too much. I will cite here these definitions and my questions and I hope someone is familiar with these notions.</p>
<p><span class="math-container">$\textit{Question 1:}$</span> <span class="math-container">$I$</span> is a finite set of players and and <span class="math-container">$G=((S^i)_i,g)$</span> is a compact game, that is given by a compact set of strategies <span class="math-container">$S^i$</span> for each player <span class="math-container">$i$</span> and by a continuous payoff function <span class="math-container">$g:S=\times S^i \to \mathbb{R}^{I}$</span>. Also the mixed set of strategies is defined as <span class="math-container">$\Sigma^i=\Delta(S^i)$</span> which is a standard way in game theory, but  why do we need the notion of compactness from topology?</p>
<p><span class="math-container">$\textit{Question 2:}$</span> The information structure <span class="math-container">$\mathfrak{I}=((X^i),\mu)$</span> is given by a finite set of signals <span class="math-container">$X^i$</span> for each <span class="math-container">$i$</span> and by a probability measure <span class="math-container">$\mu$</span> over <span class="math-container">$X$</span>. When <span class="math-container">$x$</span> is drawn
according to<span class="math-container">$\mu$</span>, player <span class="math-container">$i$</span> is informed about the coordinate <span class="math-container">$x^i$</span>. Why do we need to define the information structure as a measure set which is finite and what does it mean that we now the probability measure?</p>
<p><span class="math-container">$\textit{Question 3:}$</span> A communication mechanism is a triple <span class="math-container">$\mathfrak{C}=((T_i)_i, (Y_i)_i , l )$</span>, where <span class="math-container">$T_i$</span> is <span class="math-container">$i's$</span> finite set of messages, <span class="math-container">$Y_i$</span> is <span class="math-container">$i's$</span> finite set of signals, and <span class="math-container">$l: T\to \Delta(Y)$</span> is the signal function. When <span class="math-container">$t$</span> is the profile of messages sent by the players, <span class="math-container">$y\in Y$</span> is drawn according to <span class="math-container">$l(t)$</span> and player <span class="math-container">$i$</span> is informed of <span class="math-container">$y_i$</span>. <span class="math-container">$\mathfrak{T}_i=\Delta(T_i)$</span> represents the set of mixed messages for player <span class="math-container">$i$</span> and <span class="math-container">$l$</span> is extended to <span class="math-container">$\mathfrak{T}$</span> by <span class="math-container">$l(\tau)( y)=\mathbb{E}_{\tau} l(t)( y)$</span>. I am totally lost in this point. What is this <span class="math-container">$\tau$</span> probability measure and what is the meaning of <span class="math-container">$l(t)(y)$</span>, does this mean <span class="math-container">$l(t,y)$</span>? I have never seen this symbolism <span class="math-container">$l(t)(y)$</span> again. Apparently, the way that the communication mechanism is defined comes from the measure theory, but how did he end up with the <span class="math-container">$l$</span> function defined under a <span class="math-container">$\tau$</span> probability measure?</p>
<p>I also struggle to understand the definitions <span class="math-container">$2.1$</span> to <span class="math-container">$2.5$</span> but I will stop here in order to find some help with the basic. Thank you in advance!</p>

CC BY-SA 4.0
2021-09-15T16:00:01.210Z
I will try to give an explantation, however if anyone of the knows something more...please reply
Nav89
CC BY-SA 4.0
2021-09-15T16:06:31.307Z
Ok give it a shot...
Hunger Learn
<microeconomics><mathematical-economics><game-theory><probability><information>
Olivier Gossner - Secure Protocols or How Communication Generates Correlation
Hunger Learn
# Olivier Gossner - Secure Protocols or How Communication Generates Correlation

(Asked by: username_0 on 2021-08-02)

The paper of [Olivier Gossner in Security Protocols in 1998](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.685.4204&rep=rep1&type=pdf) has some definitions that confuse me too much. I will cite here these definitions and my questions and I hope someone is familiar with these notions.

$\textit{Question 1:}$ $I$ is a finite set of players and and $G=((S^i)\_i,g)$ is a compact game, that is given by a compact set of strategies $S^i$ for each player $i$ and by a continuous payoff function $g:S=\times S^i \to \mathbb{R}^{I}$. Also the mixed set of strategies is defined as $\Sigma^i=\Delta(S^i)$ which is a standard way in game theory, but why do we need the notion of compactness from topology?

$\textit{Question 2:}$ The information structure $\mathfrak{I}=((X^i),\mu)$ is given by a finite set of signals $X^i$ for each $i$ and by a probability measure $\mu$ over $X$. When $x$ is drawn
according to$\mu$, player $i$ is informed about the coordinate $x^i$. Why do we need to define the information structure as a measure set which is finite and what does it mean that we now the probability measure?

$\textit{Question 3:}$ A communication mechanism is a triple $\mathfrak{C}=((T\_i)\_i, (Y\_i)\_i , l )$, where $T\_i$ is $i's$ finite set of messages, $Y\_i$ is $i's$ finite set of signals, and $l: T\to \Delta(Y)$ is the signal function. When $t$ is the profile of messages sent by the players, $y\in Y$ is drawn according to $l(t)$ and player $i$ is informed of $y\_i$. $\mathfrak{T}\_i=\Delta(T\_i)$ represents the set of mixed messages for player $i$ and $l$ is extended to $\mathfrak{T}$ by $l(\tau)( y)=\mathbb{E}\_{\tau} l(t)( y)$. I am totally lost in this point. What is this $\tau$ probability measure and what is the meaning of $l(t)(y)$, does this mean $l(t,y)$? I have never seen this symbolism $l(t)(y)$ again. Apparently, the way that the communication mechanism is defined comes from the measure theory, but how did he end up with the $l$ function defined under a $\tau$ probability measure?

I also struggle to understand the definitions $2.1$ to $2.5$ but I will stop here in order to find some help with the basic. Thank you in advance!

**Tags:** microeconomics, mathematical-economics, game-theory, probability, information
> **Comments:**
> **username_1**: I will try to give an explantation, however if anyone of the knows something more...please reply  
> **username_0**: Ok give it a shot...  

---

## 2 answers

### Answer by username_2 on 2021-09-16. Score: 1 | Accepted answer

$\textit{Question 1:}$ As explained by Nav89, compactness of the strategy sets and continuity of the payoff function is needed to ensure equilibrium existence.

$\textit{Question 2:}$ I am not sure whether I truly understand your question. The assumption that the set of signals is finite is for convenience: finite probability distributions are easy to work with. A probability measure is nothing but a probability distribution. So given that I get signal $x\_i$, I will have some belief about the signals of others $x\_{-i}$. As defined in Section 2.2., in a game extended by an information structure, players condition their behavior in the game on the signal they get. These signals thus act like a coordination device (e.g., if the sun shines, we do XYZ...).

$\textit{Question 3:}$ A communication mechanism is another way of allowing players to coordinate in the game. The players send messages to the mechanism, and then the players again observe signals on which they condition their behavior (think, I see that in NYC the sun shines, you see that in LA it rains, and the mechanism will send us some information about what everyone said).

So if we send messages $(t\_1,t\_2)$, then the mechanism sends signals according to a lottery $l(t\_1,t\_2)$. Here, $l(t\_1,t\_2)(x)$ is the probability that signal profile $x$ is sent by the mechanism in this lottery.

Now, suppose we don't send a single message $t\_1$ or $t\_2$ to the mechanism, but each of us sends a mixed message $\tau\_1$ and $\tau\_2$ ---probability distributions over messages---to the mechanism. Then, $l(\tau\_1,\tau\_2)(x)$ is the probability that signal $x$ is sent back by the mechanism if I send mixed message $\tau\_1$ and you send mixed message $\tau\_2$.

So if $\tau\_i(t\_i)$ is the probability that $i$ sends $t\_i$ in the mixed message $\tau\_i$, then

$$l(\tau\_1,\tau\_2)(x)=\sum\_{t\_1}\sum\_{t\_2} \tau\_1(t\_1)\tau\_2(t\_2)l(t\_1,t\_2)(x).$$

> **Comments:**
> **username_1**: Ok, my only comment is the following...you say $l(t)(x)$, but the mechanisms says $l(t)(y)$ i think you used $x$ instead of $y$ just for your own convinience, right? If so, no need to make changes, if not then explain me what am I missing?   Thank you in advance!  
> **username_2**: Yes sorry: $x$ should be $y$ in question 3. The sets $X$ and $Y$ can be different but the idea is the same.  
> **username_0**: Ok no worries. Thanks for your help!  

---

### Answer by username_1 on 2021-09-15. Score: 0

$\textit{Question 1:}$ The assumption of compactness is often assumed in order to guarantee existence of a solution to a problem. It is standard and as you have seen, the only thing that you need, the weakest of the assupmtions is the the finite set of strategies $S=\Pi\_{i\in I}S\_i$ (which is not necessarily finite) and the continuous function $g$. So these two assumptions are enough to quarantee the existence of a solution to the problem.

$\textit{Question 2:}$ So the information strucutre is $\mathcal{I}=((X^i)\_i,\mu)$, where $X\_i$ denotes the finite set of strategies of player $i$ and $X=\Pi\_{i\in I} X\_i$, whre this probability measure is defined from $X$ to $\Delta(X)\subset[0,1]^I$. More precisely, it holds that

$$\mu:X\to \Delta(X)\subset [0,1]^I$$

if $x=(x^i,x^{-i})\in X$ is drawn (which means this is a lottery) according to $\mu$, then player $i$ is informed only about his coordinate $x^i$. Suppose, $I=2$, and $X^1=\{\text{buy}, \text{sell}\}$ and $X^2=\{\text{hold}, \text{sell}\}$, then a drawn according to $\mu$ for both playes could be $\mu(x^1,x^2)=\left((1/2,1/2),(3/4,1/4)\right)$, so player $1$ will be infored about $x^1$ which means, buy with probability $1/2$ or sell with probability $1/2$ and player $2$ will be informed about $x^2$ which means hold with probablility $3/4$ or sell with $1/2$.

$\textit{Question 3:}$ Well a communucation mechanism is defined as a triplet $\mathcal{C}=((T^i)\_i, (Y^i)\_i, l)$ but lets make it clear what is this mechanims. Think about it like a device that takes an input this is $T^i$ and in returt it gives an outpout that is $Y^i$ and the function $l:T\to\Delta(Y)$ is again a lottery or better a probability distribution according to which you are taking back the message $y\in Y$. In other words this signal function $l$ tells you that if you give a message whatever this is, I can give you back an estimation, or better a probability distributuon on which you will know that something will happen with probabolity $l(t)$. Suppose again that $I=2$, $t=(t^1,t^2)=\left(\underbrace{(\text{buy}, \text{sell})}\_{t^1},\underbrace{(\text{hold}, \text{sell})}\_{t^2}\right)$. So the players decide to send this message $t$ in the mechanism and this in returns gives back to them $y\in Y$ which is, say $l(t)=\left(\underbrace{(2/3(\text{buy}),1/3(\text{sell}))}\_{y^1},\underbrace{(2/3(\text{hold}),1/3(\text{sell}))}\_{y^2}\right)$ and it is easy to understand what is the meaning of $y$ now. Even furhter, instead of $T$, you can generalize the set of finite messages sent to the communication mechanism to a mixed set of messages (generaliztaion of the $l$ function) s.t $l:\mathcal{T}\to\Delta(Y)$, where $l(\tau)(y)$ is again this lottery according to which the mechanism will return you $y$, so again it is a probability distribution or better and estimation. Thus $l(\tau)(y)$ denotes something like a product of vectors that give $(\tau \times l(t))(y)$. In other words this \tau is also a probability distribution on the space of finite messages so it can also serve as the probability measure and this is why $l(\tau)(y)=\mathbb{E}\_{\tau}l(t)(y)$

> **Comments:**
> **username_0**: Oh thanks but it will take some time to check this....  
> **username_1**: @HungerLearn no worries....I want other people here to take a look to check if I am mistaken or not....this is a difficult understanding... so maybe others could check here  
> **username_1**: well I am not sure about the last par $l(\tau)(y)$, but it seems to me as a product of the vectors from the probability distribution with the signal $y$ which comes back from the mechanism to every player. If anyone knows somthing please i would like to know it and of course endorse my answer! Thank you in advance!  

---

