What is the monotone hazard rate assumption used for? I was reading this article on advance purchase discounts and pricing in which the author uses the monotone hazard rate assumption. Why does the author use it?

The article is Advance-purchase discounts as a price discrimination device (PDF) by Nocke Peitz and Rosar

Below you can find a screenshot of the part in question.

Thank you very much!
The motivation for this assumption is to have a unique solution. In this article, it is necessary to prove the 1st lemma, i.e. the quasi-concavity of the profit function.  You find a similar motivation in Tirole’s book (Industrial Organization, p. 156).