  Every positive integer may be written uniquely as a base-$\beta$
decomposition--that is a legal sum of powers of $\beta$--where $\beta$ is the
dominating root of a non-increasing positive linear recurrence sequence. Guided
by earlier work on a two-player game which produces the Zeckendorf
Decomposition of an integer (see [Bai+19]), we define a broad class of
two-player games played on an infinite tuple of non-negative integers which
decompose a positive integer into its base-$\beta$ expansion. We call this game
the Generalized Bergman Game. We prove that the longest possible Generalized
Bergman game on an initial state $S$ with $n$ summands terminates in
$\Theta(n^2)$ time, and we also prove that the shortest possible Generalized
Bergman game on an initial state terminates between $\Omega(n)$ and $O(n^2)$
time. We also show a linear bound on the maximum length of the tuple used
throughout the game.
