### Instruction: For how many positive integer values of $x$ is $x^{-1}>x$?### Response: This problem is asking about the relationship between a positive integer $x$ and its reciprocal, $x^{-1}$.
I know that the reciprocal of a positive integer is always between $0$ and $1$, so it is smaller than the original integer.
Therefore, the inequality $x^{-1}>x$ can never be true for any positive integer $x$.
That means the answer is zero.