How to graph a rational reaction set? In my game theory class, I am asked to graph the rational reaction set for the follow matrix:
$\begin{bmatrix}
(3,-1) &(-1,-4) \\ 
(-1,2) & (1,2) 
\end{bmatrix}$
I am not sure how to graph the rational reaction set, and the book doesn't say how to do so either. Can someone explain how to graph the rational reaction set?
 A: I've never heard the term rational reaction set, but I am guessing it means best response (set). In normal form games, this is typically done by underlying the response. The custom is that the row player's payoff is listed first.
So let us underline the row player's best responses to any strategy that the column player can play:
$\begin{bmatrix}
(\underline{3},-1) &(-1,-4) \\ 
(-1,2) & (\underline{1},2) 
\end{bmatrix}$
Clearly, if column 1 is played, then the best response of the row player is to play row 1. Similarly, if column 2 is played, then row 2 is the best response. We do the same for the column player:
$\begin{bmatrix}
(\underline{3},\underline{-1}) &(-1,-4) \\ 
(-1,\underline{2}) & (\underline{1},\underline{2}) 
\end{bmatrix}$
If row 2 is played, then the column player may rationally play 1 or 2, hence the best response is not unique. The underlined payoffs are now the best response sets of the players.
By definition, if in a cell both payoffs are underlined, i.e., the strategies are best responses to each other, then this is a Nash equilibrium. The game has 2 pure Nash equilibria.
