A Fibonacci approach to weighted majority games. Journal of Game Theory

In this paper we intend to investigate the relationship between game theory and Fibonacci numbers. We call Fibonacci games the subset of constant sum homogeneous weighted majority games whose increasing sequence of all type weights and of the minimal winning quota is a string of consecutive Fibonacci numbers. Exploiting key properties of the Fibonacci sequence, we obtain closed form results able to provide a simple and insightful classification of such games. In detail: we show that the numerousness of Fibonacci games with t types is [(t+1)/2]; we describe unequivocally a Fibonacci game on the basis of its profile as a function of t and of a proper index z=1,…,[(t+1)/2];; we provide rules concerning the behaviour of the total number n(t,z) of non-dummy players in a Fibonacci game. It turns out that there are two kinds of Fibonacci games, associated respectively with z=1 (Fibonacci-Isbell games) and z>1.