I have the following 2 player payoff bi-variate matrix
$$ \begin{matrix} 0,1 & 1,0 & 0,0 & 1,3 \\ 1,1 & 2,3 & 2,2 & 2,3 \\ 0,3 & 0,0 & 1,3 & 3,3 \\ 1,0 & 0,0 & 0,0 & 1,4 \\ \end{matrix} $$
I want to mark all equilibria in this matrix, and so I marked it as such:
$$ \begin{matrix} 0,1 & 1^*,0 & 0,0 & 1^*,3^* \\ 1,1 & 2^*,3^* & 2^*,2 & 2^*,3^* \\ 0,3^* & 0,0 & 1,3^* & 3^*,3^* \\ 1^*,0 & 0,0 & 0,0 & 1^*,4^* \\ \end{matrix} $$ And so I claim that the equilibria are at points (1,3)$R_1C_4$, (2,3)$R_2C_2$, (2,3)$R_2C_4$, (3,3)$R_3C_4$, and finally (1,4)$R_4C_4$
However, upon review of my answer, my notes claim that the following are the markings of the equilibria: $$ \begin{matrix} 0,1 & 1,0 & 0,0 & 1,3^* \\ 1^*,1 & 2^*,3^* & 2^*,2 & 2,3^* \\ 0,3^* & 0,0 & 1,3^* & 3^*,3^* \\ 1^*,0 & 0,0 & 0,0 & 1,4^* \\ \end{matrix} $$ And so the equilibria are at points (2,3)$R_2C_2$ and (3,3)$R_3C_4$. Why?
As a side note, the notes also claim that the Pareto optimal payoffs are (3,3) and (1,4)
Any help as to why the results are the way they are is much appreciated!
Customarily, the first payoff refers to the Row player and the second payoff to the Column player. Thus, when you search for the best replies of the Row player, you fix a column and look which row yields the highest payoff among the first element of each payoff pair. Similarly for the Column player. This gives you the second construction. (You searched across columns for the Row player and across rows for the Column player.)