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

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.

• You can consult the way MWG draws the education signaling game on p.451. Commented Mar 7, 2019 at 4:32

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}