  Zeckendorf proved that any positive integer has a unique decomposition as a
sum of non-consecutive Fibonacci numbers, indexed by $F_1 = 1, F_2 = 2, F_{n+1}
= F_n + F_{n-1}$. Motivated by this result, Baird, Epstein, Flint, and Miller
defined the two-player Zeckendorf game, where two players take turns acting on
a multiset of Fibonacci numbers that always sums to $N$. The game terminates
when no possible moves remain, and the final player to perform a move wins.
Notably, studied the setting of random games: the game proceeds by choosing an
available move uniformly at random, and they conjecture that as the input $N
\to \infty$, the distribution of random game lengths converges to a Gaussian.
  We prove that certain sums of move counts is constant, and find a lower bound
on the number of shortest games on input $N$ involving the Catalan numbers. The
works Baird et al. and Cuzensa et al. determined how to achieve a shortest and
longest possible Zeckendorf game on a given input $N$, respectively: we
establish that for any input $N$, the range of possible game lengths
constitutes an interval of natural numbers: every game length between the
shortest and longest game lengths can be achieved.
  We further the study of probabilistic aspects of random Zeckendorf games. We
study two probability measures on the space of all Zeckendorf games on input
$N$: the uniform measure, and the measure induced by choosing moves uniformly
at random at any given position. Under both measures that in the limit $N \to
\infty$, both players win with probability $1/2$. We also find natural
partitions of the collection of all Zeckendorf games of a fixed input $N$, on
which we observe weak convergence to a Gaussian in the limit $N \to \infty$. We
conclude the work with many open problems.
