<p>One way to represent the game tree (assuming that the congressman observes the expert's choice):</p>

<p>The game tree is drawn using LaTeX with <a href="https://ctan.org/pkg/istgame" rel="nofollow noreferrer"><code>istgame</code></a> package.</p>

<p><a href="https://i.sstatic.net/89w9f.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/89w9f.jpg" alt="enter image description here"></a></p>

<pre><code>\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ <span class="math-container">\begin{matrix} #1 \end{matrix}</span> }

<span class="math-container">\begin{istgame}[scale=2,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntm(0)[chance node]{Nature}
  \istb{w}[r]  \istb{w'}[r]         \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntm(1)(0-1)&lt;-170&gt;{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\istrootcntm(2)(0-2)&lt;-10&gt;{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\cntmdistance{15mm}{20mm}
\istrootcntm(3)(1-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(4)(1-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntm(5)(2-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(6)(2-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfoset(3)!.55!(5){Congressman}
\xtCInfoset[dashed](4)!.45!(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}</span>

\end{document}
</code></pre>

<p>If you want an arc type continuum of actions:</p>

<p><a href="https://i.sstatic.net/ou4We.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ou4We.jpg" alt="enter image description here"></a></p>

<pre><code>\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ <span class="math-container">\begin{matrix} #1 \end{matrix}</span> }

<span class="math-container">\begin{istgame}[scale=2]%,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntmA(0)[chance node]{Nature}
  \istbA(.865){w}[r] \istbA(.865){w'}[r]     \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntmA(1)(0-1)&lt;-170&gt;{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\istrootcntmA(2)(0-2)&lt;-10&gt;{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\cntmdistance{15mm}{20mm}
\istrootcntmA(3)(1-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(4)(1-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntmA(5)(2-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(6)(2-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfosetO(3)(5){Congressman}
\xtCInfosetO[dashed](4)(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}</span>

\end{document}
</code></pre>

CC BY-SA 4.0
2019-12-01T07:40:05.787Z
2019-12-01T08:39:54.287Z
2019-12-01T08:39:54.287Z
InSung Cho
<p>I must represent in extended (tree) form the following Game Theory problem:</p>
<blockquote>
<p>A committee of experts must recommend a policy <span class="math-container">$a_1\in \mathbb R$</span> to Congress. Congress will then choose a policy <span class="math-container">$a_2\in \mathbb R$</span>. The result of this policy is uncertain, determined by <span class="math-container">$x = a_2 + w$</span>, where <span class="math-container">$w$</span> is random variable uniformly distributed between <span class="math-container">$[0,1]$</span>, chosen by the &quot;nature&quot; player. The experts know the value of <span class="math-container">$w$</span>, but the Congressmen are only aware that <span class="math-container">$w\sim U[0,1]$</span>. The benefits to the congressmen and the experts are given by <span class="math-container">$-x^2$</span> and <span class="math-container">$-(x - 0.5)^2$</span>, respectively.</p>
<p>So, the sequence of the game is the following: the nature player &quot;chooses&quot; <span class="math-container">$w$</span>, then the experts choose an <span class="math-container">$a_1$</span> to recommend, and finally the congressmen choose <span class="math-container">$a_2$</span>, the acutal policy.</p>
</blockquote>
<p>My problem is the following: I can't figure out a way to represent the fact that the experts know the value of the continuous <span class="math-container">$w$</span>, but the congressmen don't.</p>

CC BY-SA 4.0
2019-03-07T04:32:25.473Z
You can consult the way MWG draws the education signaling game on p.451.
Herr K.
<game-theory>
How can I represent players with different information given continuous choice ranges?
Yair Deb
# How can I represent players with different information given continuous choice ranges?

(Asked by: username_0 on 2019-03-06)

I must represent in extended (tree) form the following Game Theory problem:

> A committee of experts must recommend a policy $a\_1\in \mathbb R$ to Congress. Congress will then choose a policy $a\_2\in \mathbb R$. The result of this policy is uncertain, determined by $x = a\_2 + w$, where $w$ is random variable uniformly distributed between $[0,1]$, chosen by the "nature" player. The experts know the value of $w$, but the Congressmen are only aware that $w\sim U[0,1]$. The benefits to the congressmen and the experts are given by $-x^2$ and $-(x - 0.5)^2$, respectively.
>
> So, the sequence of the game is the following: the nature player "chooses" $w$, then the experts choose an $a\_1$ to recommend, and finally the congressmen choose $a\_2$, the acutal policy.

My problem is the following: I can't figure out a way to represent the fact that the experts know the value of the continuous $w$, but the congressmen don't.

**Tags:** game-theory
> **Comments:**
> **username_1**: You can consult the way MWG draws the education signaling game on p.451.  

---

## 1 answers

### Answer by username_2 on 2019-12-01. Score: 1

One way to represent the game tree (assuming that the congressman observes the expert's choice):
The game tree is drawn using LaTeX with istgame package.

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}[scale=2,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntm(0)[chance node]{Nature}
  \istb{w}[r]  \istb{w'}[r]         \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntm(1)(0-1)<-170>{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\istrootcntm(2)(0-2)<-10>{Expert}
  \istb{a_1}[r]  \istb{a_1'}[r]     \endist
\cntmdistance{15mm}{20mm}
\istrootcntm(3)(1-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(4)(1-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntm(5)(2-1)
  \istb*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntm(6)(2-2)
  \istb*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfoset(3)!.55!(5){Congressman}
\xtCInfoset[dashed](4)!.45!(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}

\end{document}

If you want an arc type continuum of actions:

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

\NewDocumentCommand\vpay{m}
{ \begin{matrix} #1 \end{matrix} }

\begin{istgame}[scale=2]%,font=\scriptsize]
\cntmdistance{20mm}{70mm}{50mm}
\istrootcntmA(0)[chance node]{Nature}
  \istbA(.865){w}[r] \istbA(.865){w'}[r]     \endist
\cntmdistance{20mm}{40mm}{25mm}
\istrootcntmA(1)(0-1)<-170>{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\istrootcntmA(2)(0-2)<-10>{Expert}
  \istbA(.915){a_1}[r] \istbA(.915){a_1'}[r] \endist
\cntmdistance{15mm}{20mm}
\istrootcntmA(3)(1-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(4)(1-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\istrootcntmA(5)(2-1)
  \istbA*{a_2}[r]{\vpay{-(x-0.5)^2\\-x^2}}   \endist
\istrootcntmA(6)(2-2)
  \istbA*{a_2'}[r]{\vpay{-(x-0.5)^2\\-x^2}}  \endist
\xtCInfosetO(3)(5){Congressman}
\xtCInfosetO[dashed](4)(6){Congressman}
\xtCommentTo[draw=none]($(0)!.5!(2)$)(1.5,0)
  {$w\sim U[0,1]$}[r]
\xtCommentTo[draw=none](3-1)(-1,-1)
  {where $x=a_2+w$ and similrarly for other cases}[r]
\end{istgame}

\end{document}


---

