In the next matrix, why is (55,0) not a Nash Equilibrium? My book says that the next matrix has no Nash Equilibriums. Still, Im a little confused about row 3, column 2.
Reasoning from player 2's perspectivo, he could say "if player 1 chooses row 3, I Will choose column 2, and that is convenient for both of us". So how is that not supposed to be a Nash Equilibrium?
Although I undestand that if I reason from player 1's perspective, he would go like this: "If player 2 chooses column 2, I Will pick row 1"
I kind of understand that I have already said why is (55,0) not a Nash Equilibrium (because being player 1 , i wouldn't pick that option), but not entirely...
$$
        \begin{pmatrix}
        (35,25) & (80,0) \\
        (30,10) & (60, 0)\\
        (45,-10) & (55,0)\\
                 \end{pmatrix}
$$
 A: In normal form games, here's the best way to find pure strategy Nash equilibria. Take the first player, who can choose the row, and underline the payoff that is best for him for each of the possible strategies of the other player (column 1, column 2). That is, we underline best responses to each of the possible strategies of the other player.
You would underline row 3 column 1 and row 1 column 2.
Now you do the same for the second player. Which is the best column for each of the rows player 1 can choose? You would underline row 1 column 1; row 2 column 1 and row 3 column 2.
A pure strategy Nash equilibrium is a cell where both players underlined the payoff. This just means both strategies are a best response to each other, which by definition is a Nash equilibrium. In your game, we can see that the two players never choose the same cell, so there is no strategy combination where the strategies are best responses to each other. Hence, there is no pure strategy Nash equilibrium (but there are mixed strategy equilibria).
Thus, row 3 column 2 is not a Nash equilibrium, because player 1 best responds by switching to row 1, where he gets $80>55$.
A: In short, an NE has to take views of "all" players. Any player deviating from NE "unilaterally" won't get greater payoffs. So, at (55, 0), even though it's a better choice for player 2, it's not an optimal option for player 1. Furthermore, if player 2 chooses column 2 to play, player 1, instead of choosing row 3, can choose row 1 to obtain greater(than 50) payoff 80 for himself.
