  Zeckendorf proved that every positive integer $n$ can be written uniquely as
the sum of non-adjacent Fibonacci numbers; a similar result holds for other
positive linear recurrence sequences. These legal decompositions can be used to
construct a game that starts with a fixed integer $n$, and players take turns
using moves relating to a given recurrence relation. The game eventually
terminates in a unique legal decomposition, and the player who makes the final
move wins.
  For the Fibonacci game, Player $2$ has the winning strategy for all $n>2$. We
give a non-constructive proof that for the two-player $(c, k)$-nacci game, for
all $k$ and sufficiently large $n$, Player $1$ has a winning strategy when $c$
is even and Player $2$ has a winning strategy when $c$ is odd. Interestingly,
the player with the winning strategy can make a mistake as early as the $c + 1$
turn, in which case the other player gains the winning strategy. Furthermore,
we proved that for the $(c, k)$-nacci game with players $p \ge c + 2$, no
player has a winning strategy for any $n \ge 3c^2 + 6c + 3$. We find a stricter
lower boundary, $n \ge 7$, in the case of the three-player $(1, 2)$-nacci game.
Then we extend the result from the multiplayer game to multialliance games,
showing which alliance has a winning strategy or when no winning strategy
exists for some special cases of multialliance games.
