  We study a geometrically constrained coalescence model derived from spin
systems. Given two probability distributions $\mathbb{P}_R$ and $\mathbb{P}_B$
on the positive reals with finite means, colour the real line alternately with
red and blue intervals so that the lengths of the red intervals have
distribution $\mathbb{P}_R$, the lengths of the blue intervals have
distribution $\mathbb{P}_B$, and distinct intervals have independent lengths.
Now, iteratively update this colouring of the line by coalescing intervals:
change the colour of any interval that is surrounded by longer intervals so
that these three consecutive intervals subsequently form a single monochromatic
interval. We say that a colour (either red or blue) wins if every point of the
line is eventually of that colour. Holroyd, in 2011, asked the following
question: under what natural conditions on the initial distributions is one of
the colours almost surely guaranteed to win? It turns out that the answer to
this question can be quite counter-intuitive due to the non-monotone dynamics
of the model. In this paper, we investigate various notions of advantage one of
the colours might initially possess, and in the course of doing so, we
determine which of the two colours emerges victorious for various nontrivial
pairs of distributions.
