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

Some distinguished types of voters, as vetoes, passers or nulls, as well as some others, play a significant role in voting systems because they are either the most powerful or the least powerful voters in the game independently of the measure used to evaluate power. In this paper we are concerned with the design of voting systems with at least one type of these extreme voters and with few types of equivalent voters. With this purpose in mind we enumerate these special classes of games and find out that its number always follows a Fibonacci sequence with smooth polynomial variations. As a consequence we find several families of games with the same asymptotic exponential behavior except for 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 of voting structures with a predetermined importance ranking.

► We study voting structures with some distinguished types of voters, e.g. vetoers. ► Those special types are either the most powerful or the least powerful voters. ► Exact enumeration results for complete simple games. ► Enumeration formulas are either polynomials or Fibonacci numbers plus a polynomial. ► This research is related with the determination of importance rankings.