Please help me out with the conditions for existence of a pure strategy Nash equilibrium. The game is one of two players with symmetric strategies. After that, please help me out with the conditions needed to show uniqueness of my closed form solution. Even just a link to a resource would be extremely valuable. I looked up resources but a lot of them seem too technical and not applied, and I really wanted to see a proof of existence in practice. Thank you in advance!
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I really wanted to see a proof of existence in practice.
Proofs are usually theoretical, especially non-constructive proofs of existence. I'm sorry you have to take this class, because if you find the proofs in the resources very technical, you might not get a lot out of it.
Having said that, since the game you describe is "small":
The game is one of two players with symmetric strategies.
meaning
$$\begin{matrix} &\#2 \\ \#1 & \begin{array}{c|c|c} &A &B \\ \hline A &a_1,a_1 &a_2,a_3 \\ \hline B &a_3,a_2 &a_4,a_4 \end{array} \end{matrix} $$
they probably want you to look at all possible cases. In case of pure strategies, only the order of the utilities (of the pure strategy profiles) matter, as a player will deviate from a strategy profile if they can get a bigger utility by changing her strategy. Thus you have to look at stuff like 'case $a_1>a_2>a_3>a_4$' etc.
You can figure out all the equilibria in each case, see if 1. one exists 2. it's unique.
E.g.; in case $a_1>a_2>a_3>a_4$:
Player 1 will strictly prefer $(A,A)$ to $(B,A)$ and $(A,B)$ to $(B,B)$, so she will always play $A$. Same goes for player $2$ because of symmetry. Thus $(A,A)$ is a Nash-equilibrium, and it is the only one.
There are a bunch of cases though, so if you can, figure out smart arguments for simplifying them/ignoring some.