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Some distinguished types of voters, as vetoers, passers or nulls, as well assome others, play a significant role in voting systems because they are eitherthe most powerful or the least powerful voters in the game independently of themeasure used to evaluate power. In this paper we are concerned with the designof voting systems with at least one type of these extreme voters and with fewtypes of equivalent voters, with this purpose in mind we enumerate thesespecial classes of games and find out that its number always follows aFibonacci sequence with smooth polynomial variations. As a consequence we findseveral families of games with the same asymptotic exponential behaviorexcepting of a multiplicative factor which is the golden number or its square.From a more general point of view, our studies are related with the design ofvoting structures with a predetermined importance ranking.

The golden number and Fibonacci sequences in the design of voting structures