Winning Strategy for the Multiplayer and Multialliance Zeckendorf Games

TL;DR

This paper investigates the strategic dynamics of Zeckendorf games with multiple players and alliances, establishing conditions under which no player or alliance has a guaranteed winning strategy for large numbers.

Contribution

It extends the Zeckendorf game to multiple players and alliances, analyzing winning strategies and showing that with three or more players, no guaranteed winning strategy exists for large n.

Findings

No player has a winning strategy for n ≥ 5 with ≥3 players.

In two-alliance games, some situations prevent any alliance from guaranteed victory.

Certain alliance structures are proven to always have winning strategies.

Abstract

Edouard Zeckendorf proved that every positive integer $n$ can be uniquely written \cite{Ze} as the sum of non-adjacent Fibonacci numbers, known as the Zeckendorf decomposition. Based on Zeckendorf's decomposition, we have the Zeckendorf game for multiple players. We show that when the Zeckendorf game has at least $3$ players, none of the players have a winning strategy for $n\geq 5$. Then we extend the multi-player game to the multi-alliance game, finding some interesting situations in which no alliance has a winning strategy. This includes the two-alliance game, and some cases in which one alliance always has a winning strategy. %We examine what alliances, or combinations of players, can win, and what size they have to be in order to do so. We also find necessary structural constraints on what alliances our method of proof can show to be winning. Furthermore, we find some alliance structures which must have winning strategies. %We also extend the Generalized Zeckendorf game from $2$-players to multiple players. We find that when the game has $3$ players, player $2$ never has a winning strategy for any significantly large $n$. We also find that when the game has at least $4$ players, no player has a winning strategy for any significantly large $n$.