## Quota System

A generalization of simple majority voting in which a list of quotas specifies, according to the number of votes, how many votes an alternative needs to win (Taylor 1995). The quota system declares a tie unless for some , there are exactly tie votes in the profile and one of the alternatives has at least votes, in which case the alternative is the choice.

Let be the number of quota systems for voters and the number of quota systems for which , so

where is the Floor Function. This produces the sequence of Central Binomial Coefficients 1, 2, 3, 6, 10, 20, 35, 70, 126, ... (Sloane's A001405). It may be defined recursively by and

where is a Catalan Number (Young et al. 1995). The function satisfies

for (Young et al. 1995). satisfies the Quota Rule.

References

Sloane, N. J. A. Sequence A001405/M0769 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Taylor, A. Mathematics and Politics: Strategy, Voting, Power, and Proof. New York: Springer-Verlag, 1995.

Young, S. C.; Taylor, A. D.; and Zwicker, W. S. Counting Quota Systems: A Combinatorial Question from Social Choice Theory.'' Math. Mag. 68, 331-342, 1995.