I'm reading Tim Roughgarden's Twenty Lectures on Algorithmic Game Theory. Lecture 5 is on revenue-maximizing auctions. It claims that in the single-parameter qusilinear environment, for the extremely simple example with only one item to sell and one potential buyer:
The space of direct-revelation DSIC auctions is small: they are precisely the posted prices, meaning take-it-or-leave-it offers.
Question 1: how this is concluded? Is it by Myerson's Lemma?
Assuming the above is correct, in the lecture's video, Tim Roughgarden also mentioned randomization over posted prices.
Question 2: how to do randomization over posted prices? Is the randomized posted prices mechanism is still DSIC?