  We propose a classification of all one-dimensional discrete statistical
models with maximum likelihood degree one based on their rational
parametrization. We show how all such models can be constructed from members of
a smaller class of 'fundamental models' using a finite number of simple
operations. We introduce 'chipsplitting games', a class of combinatorial games
on a grid which we use to represent fundamental models. This combinatorial
perspective enables us to show that there are only finitely many fundamental
models in the probability simplex $\Delta_n$ for $n\leq 4$.
