Where's the Nash Equilibrium here? $ \begin{pmatrix} (2,-2) & (2,-2)\\ (1,-1) & (3, -3) \\ \end{pmatrix} $

Question

I just opened a book on Game Theory, so I'm totally new to this. My book says that the only Nash Equilibrium in the example below is (2, -2) -first row, first column-, and I really don't see why... First of all, why couldn't the (2, -2) in the first row second column also be a Nash Equilibrium? And second, Isn't (3, -3) greater than (2, -2), and therefore, is a Nash Equilibrium? Thanks in advance $$ \begin{pmatrix} (2,-2) & (2,-2)\\ (1,-1) & (3, -3) \\ \end{pmatrix} $$

Answer 1

We can identify a Nash equilibrium by reasoning from each player's perspective. First, reason from Row's perspective: 1. If Column plays Left, I should play Up, because 2 is bigger than 1. 2. If Column plays Right, I should play Down, because 3 is bigger than 2.

Column's perspective: 1. If Row plays Up, I can play either Left or Right -- I'm indifferent as to which. 2. If Row play Down, I should play Left, because -1 is bigger than -3. So, I should play Left, because no matter what Row does, it's my best outcome.

Now, in game theory we usually assume that each player is perfectly rational, that each knows the other is perfectly rational, and that each knows that the other knows this, etc...

So, Row knows that Column will play Left, by examining the game from her perspective. And, as we saw from Row's own reasoning, when Column plays Left, Row should play Up.

So, Row plays Up, and Column plays Left, and that is the Nash Equilibrium.

Up/Right is not a Nash Equilibrium, because Row does better by playing Down if Column plays Right ( 3 > 2 ). Also, Column will not play Right because her dominant strategy is Left. A dominant strategy is one that yield payoffs at least as good as the other strategy, no matter what the other player does.

Answer 2

You can think of a Nash Equilibrium has a pair of actions in which no player has any incentive to change their action, even if they knew what the other player would pick. In your example, we have the following: the first $(2, -2)$ is not a Pure Nash Equilibrium, as $P_2$ would want to change his action to the one that gives him $-1$ utility instead. On the other hand, the $(2, -2)$ on the top right corner is a Pure Nash Equilibrium since neither $P_1$ nor $P_2$ would want to change their action knowing what the other player chose. In the case of $P_1$, switching actions would also yield a utility of $2$, so $P_1$ has no incentive to change their action since they are already receiving a utility of $2$. Similarly, if $P_2$ was to change their action, they would instead receive $-3$ utility, which is less than the current $-2$ they are receiving. So neither player would change their current action, and so, this is a pure Nash Equilibrium.