Please consider the following exercise:
Suppose that we are considering a game between two players with the following game bimatrix $$ (A,B) = \begin{pmatrix} (2,1) & (0,0)\\ (0,0) & (1,2) \end{pmatrix} $$ Suppose that the players cannot cooperate. The feasible set of non-cooperative solutions is given by $$ \{(u,v) = (p^TAq, p^TBq): p\in X^*, q \in Y^*\} $$ Graph the set of non-cooperative solutions.
My approach: I know that the set of feasible solutions lie inside the convex hull of the points $(1,2), (2,1), (0,0)$ as these are the values corresponding to the four possible combinations of pure strategies ($p = 1$ with $q = 1$ or $q = 0$ and $p = 0$ with $q = 1$ or $q = 0$). In order to determine the solution within this convex set, I tried to write out the definition of $(u,v)$. Suppose that we have $p = (p_1, p_2)$ and $ q = (q_1,q_2)$. Then $$ (u,v) = (2p_1q_1 + p_2q_2, p_1q_1 + 2p_2q_2) $$ I think that this is correct, but unfortunately this is a function that depends on four variables (and cannot be plotted in three dimensions).
The solution: Apparently the graph should look like this:
This graph comes with the following note: the curve joining $(1,2)$ and $(2,1)$ has parametric form $(x,y) = (1 - 2a + 3a^2, 2 - 4a + 3a^2)$.
My question: How do you arrive at the form of $(x,y)$ as given in the solution? How do you solve this exercise? I feel like I always struggle when visualising functions, or when trying to come up with alternative expressions for sets. I have no idea how I would have been able to arrive at $(x,y) = (1 - 2a + 3a^2, 2 - 4a + 3a^2)$. Is there a general way of approaching these kinds of problems, or do you just have to spend a lot of time on them to get a feel for it?