How many Pure Strategy NE does this game have?

Question

   L      R
U (3,3) (1,5)
D (5,1) (0,0)

The game is as above. Nash Equilibrium that I got for this game is (1,5) and (5,1).

According to me, this game has no Pure Strategy. because Player 1 plays U and D depending on what Player 2 plays. Likewise, with Player 2.

So, Pure Strategy NE in this game should be 0 but the correct answer is 2. How?

Answer 1

This is an anti-coordination game

• If Player 1 always plays D and Player 2 plays L, then player 1 is not going to change ($5$ is best outcome overall) and player 2 is not going to change unless player 1 changes, so this is a Nash equilibrium

• If Player 2 always plays R and Player 1 plays U, then player 2 is not going to change ($5$ is best outcome overall) and player 1 is not going to change unless player 2 changes, so this is a Nash equilibrium