Is there really a Nash equilibrium in this example?

Question

I was watching this video on Coursera and worked out the example before the solution was presented. The example begins at 4:20

The presenter says that the Nash equilibrium is achieved with 2,2 at the bottom right but I don't see it. My solution shows no Nash equilibrium.

• I have two 1, 3* with the * asterisk representing the preferred value of player, P2 in the 3rd (Right) column.
• I have two 3*, 1 on the bottom row (Down) for player, P2 because if P1 picks Down, the best outcome is achieved with 3 when P2 picks Left or Middle.

What am I missing here? All my other values agree with the example except for the 3 row and 3rd column.

Answer 1

The Nash equilibrium is indeed (down, right).

Note that your chart has helpfully underlined the max value among all possible strategies each player can play, conditional on what the other player plays. The steps on the graph are useful as well: If no numbers are underlined in a cell, that strategy set is strictly dominated by something else. If one number is underlined, one player will want to unilaterally deviate.

Assume the players play (down, right):

1. P1 will not want to play up or center as their return will be 1 in those cases
2. P2 will not want to play left or middle as their return will be 1 in those cases

You can try this for any other cell with underlined values and see one player will be better off deviating. For example, in (up, middle) P2 would prefer to deviate to (up, left), then P1 would deviate to (center, left), P2 to (center, middle), P1 to (middle, up), etc.

FYI: the equilibrium should be referred to by its strategy rather than its values.

Answer 2

You seem to be mistaken in what payoff goes to which player.

By convention, the first value goes to the Row player (here P1), and the second value goes to the Column player (here P2).