Player 1 is not constrained to choose numbers randomly from a distribution. Player 1's choices are considered to be arbitrary. Player 2's probability of winning can then be calculated from Player 1's choices and Player 2's privately chosen density function.
Consider a paradox: Player 1 picks a random number X, writes it on a slip of paper, and 2X on another slip, and puts them randomly in front of Player 2 as "left" and "right". Player 2 wins amount of money written on a slip he chose.
Let's say he is about to pick left one (but didn't see it yet). Let's say left has number Y. The right one has either Y/2 or 2Y, with 50/50 probability. Which means right one is more profitable to pick, because it has 1.25Y on average!