Two countries are trying to form a cartel. To keep the price of gold high, they agree to limit their production to 4 tons and 1 ton, respectively. For each country, cheating is equivalent to producing 1 more ton per day. Depending on your decisions, total production would be 5, 6, or 7 tons per day, with corresponding profit margins of $ 16, $ 12, and $ 8,000 per ton. Describe this situation as a non-zero sum game, writing the game's matrices.
My attempt
\begin{array}{c|lcr} I/II & \text{1} & \text{2} \\ \hline 1 & (4,1) & (4,2) \\ 2 & (5,1) & (5,2) \\ \end{array}
Is correct? Should I use the profit figures?
Thanks in advance for any comments.
A rational firm would likely want to maximize its profit regardless of the number of units sold, so you should use the profit margin times the number of units sold. This gives the game, with units in thousands of dollars,
\begin{array}{c|lcr} I/II & \text{1} & \text{2} \\ \hline 1 & (64,16) & (48,24) \\ 2 & (60,12) & (40,16) \\ \end{array}
This game only has a single pure Nash equilibrium.