I was reading a paper by Platz and Østerdal (2014) titled "The curse of the first-in-first-out queue discipline". I was wondering about a particular section of it, particularly the part about last-in-first-out mechanisms. $R(t)$ is defined as the cumulative arrival distribution, and they try and show:
Let $R(t)$ be a Nash Equilibrium for the FIFO (first-in-first-out) queue discipline. Then $R(t)$ is Lipschitz continuous.
This is the second Lemma on page 2.
I am not quite understanding why we are trying to show this property to lead into the third Lemma. What is the importance of Lipschitz continuity here? A basic intuitive answer should be okay, but some math would help me I think. Are they trying to ultimately show that $R(t)$ is a unique Nash equilibrium for FIFO?