# How to find pure strategy NE if you have a n X n matrix (n players) [closed]

Consider the down below

which I have trouble with solving. I am not used to find NE for n players, but rather for a simple $$2 x 2$$ matrix or $$3 x 3$$, but how does one find NE when you have N players? Can you use some symmetric argument or anything similar? To be honest, I have no idea.

I thought to make a 2 x 2 matrix and then argue that $$(help, don't help)$$ and $$(don't help, help)$$ in this particular game is a pure strategy NE, but could you then make an argument that this holds true for n players? Or what am I supposed to do? Thanks in advance for any help. Here is a picture of the $$2 x 2$$ matrix

Think about the incentives of player $$i$$: If he knew that no one else helps, he'd want to help. If he knew that at least one other player helps, he'd rather not help. A single player helping would make everyone happy, but no one wants to be that single player, because it's costly to help. This is the classical problem of finding a volunteer, so that's why this game is called the volunteer's dilemma.
For finding a pure-strategy Nash equilibrium just think about the 3 classes of possible outcomes: (1) Nobody helps, (2) a single player $$i$$ helps, (3) two or more players help. For each case, think about whether some player would like to unilaterally deviate and switch his choice to the opposite one. If there is no such player, you have a Nash equilibrium. That shouldn't be too difficult.
As to the representation of the game: You can indeed generalize your 2x2 bi-matrix game to $$n$$ players, but not via an $$n$$x$$n$$ matrix. (So the title of your question is misleading.) Remember that the number of rows and columns in the matrix corresponds to the number of pure strategies a player has. This number stays the same, it's always 2 pure strategies (help or don't help). Rather, the number of players is higher. Therefore you would need a 2x2x2x ... x2 $$n$$-matrix (i.e. an $$n$$-dimensional matrix where each cell has $$n$$ payoff entries). But you cannot write down this thing on a 2-dimensional sheet of paper, so you just use the payoff function as defined in the 3 bullet points in the question.