2.3b Pareto Efficiency

Pareto Efficiency

An outcome is Pareto efficient if it is beneficial to the collective good such that no other outcome makes any player better off without also making some player worse off.

Relatedly, an outcome $A$ is a Pareto improvement over outcome $B$ if $A$ makes every player better off, and no player worse off.

Properties

The collective good in a strategic form game is a subset of the Pareto efficient outcomes.

All finite games have at least one Pareto efficient outcome.

A Pareto efficient outcome has no Pareto improvements.

Example

Stag Hunt, Revisited

The stag hunt game has a clear outcome which is in the collective interest of the players. 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}

Then, the outcome of $(S,S)$ is the only Pareto efficiency in this game. This is because any other choice will negatively influence at least one of the players in terms of utility.

Example