Does this simple continuous game have Nash equilibria?

Question

Two players simultaneously pick a number from $[0, 1]$. The payoff of the first player (equal to the loss of the second) is the distance between those numbers.

1. Does there exist a pure-strategy Nash equilibrium?
2. Does there exist a mixed-strategy Nash equilibrium? How many?

I found value of the game $v=\frac12$, pure optimal strategy for the second player $y=\frac12$ and the mixed optimal strategy for the first player $x = \{0, 1\}$ with probabilities $p =(\frac12, \frac12)$.

I am a bit stuck with finding other mixed strategies, could you help me, please?

Answer 1

1. There is no pure-strategy Nash equilibrium: Assume player 1 choose $a\in[0,1]$ and player 2 choose $b\in[0,1]$, show that either player 1 or player 2 can do better.

2. There is a mixed-strategy Nash equilibrium which is the one you suggested, but I cannot think of any other equilibrium with mixed strategies.