How can I represent players with different information given continuous choice ranges?

Question

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.

Answer 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}