In this paper we introduce the class of Fibonacci games of order k. Given a positive integer k, a constant sum homogeneous weighted ma-jority game is a Fibonacci game of order k (shortly a k-Fibonacci game) if there is one to one correspondence between two sequences. The first one is the, ordered from the least, sequence of all (but the top for k>1) type weights and the minimal winning quota of the game; the other is an initial string, of coherent length, of the k-Fibonacci sequence. For any given number n>5 of non-dummy players in the game, we show that Fibonacci games are in one to one correspondence with the integers k which are perfect divisors of n-1.
