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 $$\varPsi (t)$$ Ψ ( t ) of such games as a function of t, number of non-dummy players’ types. It turns out that the growth rate of $$\varPsi (t)$$ Ψ ( t ) is exponential. The main result of our paper is that, for two consecutive t values of the same parity, the ratio $$\varPsi (t+2)/\varPsi (t)$$ Ψ ( t + 2 ) / Ψ ( t ) converges toward the golden ratio $${\varPhi }$$ Φ .