Pure Nash equililibrium

Question

I’m really confused. I have the following game (zero sum).

1 2. 4 1. (2x2 matrix)

Is there a pure Nash equililibrium in this game or not and if there is can you explain Thanks

Answer 1

It's unclear from your question what, exactly, the matrix looks like, but I'm assuming that you're reading across, left to right, top to bottom, and that the numbers provided are all player 1's. As it's a zero-sum game, we get the following matrix:

     A L I C E
B (1,-1) | (2,-2)
O ---------------
B (4,-4) | (1,-1)

Why would Alice play this game, by the way?

Since you ask specifically for a pure Nash equilibrium, we only need to consider what single option each player can take to maximize their payout, regardless of the other player's actions.

We start with Bob. If he takes the top option, the most he can get is $2$, but if he takes the bottom option, the most he can get is $4$. As either way, the worst he can get is $1$, he takes the bottom option.

Alice, too, wants to maximize her payout, or, in this case, minimize her loss. If she takes the left option, the best she can hope for is $-1$, but she risks a payout of $-4$. The much safer option is to take the right option, where, while she can't do any better than $-1$, the worst she can do is only $-2$.

Knowing that Alice will pick the right option, though, Bob sees that if he switches to the top option, he'll increase his payoff from $1$ to $2$.

Knowing that Bob will pick the top option, though, Alice will switch from right to left, to decrease her loss from $-2$ to $-1$.

Knowing that Alice will pick the left option, Bob will switch from top back to bottom, increasing his payoff from $1$ to $4$.

Knowing that Bob will pick the bottom option, Alice will switch from left to right, decreasing her loss from $-4$ to $-1$.

You see? It's cyclic. No matter what anyone picks, one player can benefit from switching. So there is no pure Nash equilibrium.

^{If I completely misunderstood what you meant for the matrix, please edit your OP to clarify and ping me here, so that I can update this appropriately.}

Answer 2

there is no pure equillibrium. A pure nash equillibrium would exist, by definition, if there were strategies that each player could choose so that neither of them wished to change there strategies, knowing the other player would not change there strategy. If you go through all 4 strategy combinations you will see that this is never the case.