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2.3a Collective Action Game

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Collective Action Game

In strategic form theory, a collective action game is one where certain strategy profiles promote the collective interest of the players using spontaneous or coercive collaboration.

Conversely, a zero sum game is one whose strategies do not have a collective action differential, and the sum of the payoffs of each strategy profile amounts to 0.

Examples

Stag Hunt

The stag hunt game has a clear collective action component. Consider the following game matrix:

\begin{document}
 
\begin{tikzpicture}
   % Draw the vertical lines
   \draw (0, 0) -- (0, -3);
   \draw (1.5, 0) -- (1.5, -3);
   \draw (3, 0) -- (3, -3);
 
   % Draw the horizontal lines
   \draw (0, 0) -- (3, 0);
   \draw (0, -1.5) -- (3, -1.5);
   \draw (0, -3) -- (3, -3);
 
   % Add labels for players
   \node at (1.5, 1) {Survivor 2};
   \node[rotate=90] at (-1, -1.5) {Survivor 1};
 
   % Add labels for Player 2's choices
   \node at (0.75, 0.25) {S};
   \node at (2.25, 0.25) {H};
 
   % Add labels for Player 1's choices
   \node at (-0.25, -0.75) {S};
   \node at (-0.25, -2.25) {H};
 
   % Add payoffs for each cell
   \node at (0.75, -0.75) {(3, 3)};
   \node at (2.25, -0.75) {(0, 1)};
 
   \node at (0.75, -2.25) {(1, 0)};
   \node at (2.25, -2.25) {(1, 1)};
\end{tikzpicture}
 
\end{document}

Here, only when both survivors decide to catch a stag do they successfully do so, each receiving a utility of 3.

The stag hunt game has a clear collective action component in . The players are much better off choosing to hunt a stag together than going their separate ways or hunting a hare together.

Rock, Paper, Scissors

Rock, paper, scissors is an example of a game that is zero sum. Consider the following game matrix:

\begin{document}
 
\begin{tikzpicture}
   % Draw the vertical lines
   \draw (0, 0) -- (0, -4.5);
   \draw (1.5, 0) -- (1.5, -4.5);
   \draw (3, 0) -- (3, -4.5);
   \draw (4.5, 0) -- (4.5, -4.5);
 
   % Draw the horizontal lines
   \draw (0, 0) -- (4.5, 0);
   \draw (0, -1.5) -- (4.5, -1.5);
   \draw (0, -3) -- (4.5, -3);
   \draw (0, -4.5) -- (4.5, -4.5);
 
   % Add labels for players
   \node at (2.25, 1) {Player 2};
   \node[rotate=90] at (-1, -2.25) {Player 1};
 
   % Add labels for Player 2's choices
   \node at (0.75, 0.25) {Rock};
   \node at (2.25, 0.25) {Paper};
   \node at (3.75, 0.25) {Scissors};
 
   % Add labels for Player 1's choices
   \node[rotate=90] at (-0.25, -0.75) {Rock};
   \node[rotate=90] at (-0.25, -2.25) {Paper};
   \node[rotate=90] at (-0.25, -3.75) {Scissors};
 
   % Add payoffs for each cell
   \node at (0.75, -0.75) {(0, 0)};
   \node at (2.25, -0.75) {(-1, 1)};
   \node at (3.75, -0.75) {(1, -1)};
 
   \node at (0.75, -2.25) {(1, -1)};
   \node at (2.25, -2.25) {(0, 0)};
   \node at (3.75, -2.25) {(-1, 1)};
 
   \node at (0.75, -3.75) {(-1, 1)};
   \node at (2.25, -3.75) {(1, -1)};
   \node at (3.75, -3.75) {(0, 0)};
\end{tikzpicture}
 
\end{document}

There is clearly no collective action component in this game, since when one player wins, the other loses an equivalent amount of utility.

Graph View

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