I have learned how to compute the mixed Nash equilibrium (NE) of a game many times, however, whenever I am faced with a matrix, I just cannot recall the exact way/logic of going about computing it.
I think the main problem is that the formulation process is just too long. I need to either pose it as a linear program (but who can remember how to formulate the objectives) or use a graphical method such as the best response curve. But I just keep on forgetting them because I lack the intuition.
For example, I am given the matrix,
I can visually check that (Fink, Fink) or $(2,2)$ is the pure NE. But I cannot remember how exactly to compute the mixed NE.
I have some rules of thumbs, which are unreliable, for example: by inspection I can see that the players probably want to earn a higher payoff than $1$ sometimes. So the players probably use a mixed strategy $(2/3, 1/3)$ of sorts to switch between Quiet and Fink. As I said, totally unreliable.
What is the simplest and most intuitive method that you use to compute the NE for matrix games (and does it generalize to larger matrices)?
This game has no mixed-strategy NE. The only NE is (Fink, Fink) as playing fink is a strictly dominant strategy.
In mixed-strategy NEs players are indifferent between the strategies they play with positive probability. This gives a simple algebraic solution.
In your game let $q$ be the probability of playing Quiet. Note that the game is symmetric so it suffices to look at the indifference condition for one player.
In a mixed strategy NE player 1 is indifferent if $2q=3q+1(1-q)$. Rearranging would give the mixed-strategy NE if there was one. In this case we get $0=1$ and there is no mixed-strategy-NE.
You can also derive mixed-strategy NEs as the intersection of best-response functions.