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

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

I found value of the game v=1/2, pure optimal strategy for the second player y=1/2 and the mixed optimal strategy for the first player x={0, 1} with probabilities p=(1/2, 1/2).

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

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  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.

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