How to visually present a simultaneous game with continuous strategies

Question

I've been asked to model a game where two players have an infinite choice of strategies and move simultaneously.

The question asks to present the game in game table or extensive form, and to justify that choice. I understand that a game table wouldn't work, given the infinite strategies.

I understand the diagram below to be a game with continuous strategies and sequential moves. Is there any way to adapt it to show simultaneous moves?

Answer 1

This thread has some game trees. In case you want put them on your documents, you can do like this (using LaTeX with istgame package):

\documentclass{standalone}

\usepackage{istgame}

\begin{document}

\begin{istgame}[font=\scriptsize]
\setistgrowdirection{east}
\cntmdistance{20mm}{20mm}
\cntmAistb{q_1=0}[at end,below]{q_1=1,000}[at end,above]
\istrootcntmA(0){1}
  \istbA[draw=none]
  \endist
\cntmAistb{q_2=0}[at end,below]{q_2=1,000}[at end,above]
\istrootcntmA(1)(0-1)<[xshift=3pt]90>{2}
  \istbA[draw=none]{}{\pi_1,\pi_2}
  \endist
\end{istgame}

\end{document}

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

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

\begin{istgame}[scale=1.5,font=\scriptsize]
\cntmApreset{.9}
\cntmdistance{15mm}{20mm}
\istrootcntmA(0)[null node]{1}
  \istbA(.977)[->,thick]{q_1}[r]
  \istbm
  \endist
\cntmAInfosetO[thick,dashed](0)[-11](2em)
\istrootcntmA(1)(0-1)[null node]<180>{2}
  \istbm
  \istbA(.977)[->,thick]{q_2}[r]
    {\vpay{\pi_1(q_1,q_2)\\\pi_2(q_1,q_2)}}
  \endist
\end{istgame}

\end{document}

\documentclass{standalone}

\usepackage{amsmath}
\usepackage{istgame}

\begin{document}

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

\begin{istgame}
\xtdistance{15mm}{40mm}
\cntmdistance{15mm}{30mm}
\cntmApreset[draw=none]{1}[red!50,opacity=.2]
\istroot(0){Challenger}
  \istb{In}[al]  \istb{Out}[ar]  \endist
\cntmAistb[draw=none]
\istrootcntmA(1)(0-1)<135>{Incumbent}
  \istbA{q_I}[left]  \endist
\cntmAInfosetO(1)
\cntmAistb[draw=none]
\istrootcntmA(2)(0-2)<45>{Incumbent}
  \istbA*{q_I}[left]
    {\vpay{\pi_C(0,q_I)\\\pi_I(0,q_I)}}
  \endist
\cntmApreset[draw=none]{1}[blue!50,opacity=.2]
\cntmAistb[draw=none]
\istrootcntmA(3)(1-1)<180>{Challenger}
  \istbA*{q_C}[midway,left]
    {\vpay{\pi_C(q_C,q_I)\\ \pi_I(q_C,q_I)}}
  \endist
\end{istgame}

\end{document}

Answer 2

You can indicate on the diagram that all the nodes (there is a continuum of them) after player $1$'s action lie in the same information set. This means that player $2$ cannot distinguish which node they are on following player $1$'s action, making their problem strategically equivalent to one where both players move simultaneously (similarly for player $1$).

The convention for indicating that two nodes lie in the same information set is to connect them by a dotted/dashed line, as in the Wikipedia article linked above.

Alternatively, notice that a table is just a convenient representation for a map from action pairs into payoffs. Is there some other way of writing down this mapping, other than via explicitly writing down all the pairs of actions and their corresponding payoffs explicitly in a table?

Answer 3

In an extensive form game, a continuous action space is typically drawn using an arc connecting two branches representing the upper and lower bounds of the action space. One can additionally draw a circle around the arc to indicate that the nodes after those actions all belong to the same information set. The following are a couple of examples:

_{Related: How to draw game trees with continuous action space in LaTeX.}