Proper strong-Fibonacci games

We define proper strong-Fibonacci (PSF) games as the subset of proper homogeneous weighted majority games which admit a Fibonacci representation. This is a homogeneous, type-preserving representation whose ordered sequence of type weights and winning quota is the initial string of Fibonacci numbers of the one-step delayed Fibonacci sequence.We show that for a PSF game, the Fibonacci representation coincides with the natural representation of the game. A characterization of PSF games is given in terms of their profile. This opens the way up to a straightforward formula which gives the number Ψ(t) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of Ψ(t) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio Ψ(t + 2)/Ψ (t) converges toward the golden ratio Φ.