  Zeckendorf proved that every positive integer can be written uniquely as the
sum of non-adjacent Fibonacci numbers. We further explore a two-player
Zeckendorf game introduced in Baird-Smith, Epstein, Flint, and Miller: Given a
fixed integer $n$ and an initial decomposition of $n = nF_1$, players alternate
using moves related to the recurrence relation $F_{n+1} = F_n + F_{n_1}$, and
the last player to move wins. We improve the upper bound on the number of moves
possible and show that it is of the same order in $n$ as the lower bound; this
is an improvement by a logarithm over previous work. The new upper bound is $3n
- 3Z(n) - IZ(n) + 1$, and the existing lower bound is sharp at $n - Z(n)$
moves, where $Z(n)$ is the number of terms in the Zeckendorf decomposition of
$n$ and $IZ(n)$ is the sum of indices in the same Zeckendorf decomposition of
$n$. We also studied four deterministic variants of the game, where there was a
fixed order on which available move one takes: Combine Largest, Split Largest,
Combine Smallest and Split Smallest. We prove that Combine Largest and Split
Largest realize the lower bound. Split Smallest has the largest number of moves
over all possible games, and is close to the new upper bound. For Combine Split
games, the number of moves grows linearly with $n$.
