Quota System

A generalization of simple majority voting in which a list of quotas $\{q_0, \ldots, q_n\}$ 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 $k$, there are exactly $k$ tie votes in the profile and one of the alternatives has at least $q_k$ votes, in which case the alternative is the choice.

Let $Q(n)$ be the number of quota systems for $n$ voters and $Q(n,r)$ the number of quota systems for which $q_0=r+1$, so

$$Q(n)=\sum_{r=\left\lfloor{n/2}\right\rfloor }^n Q(n,r)={n+1\choose \left\lfloor{n\over 2}\right\rfloor +1},$$

where $\left\lfloor{x}\right\rfloor$ 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 $Q(0)=1$ and

$$Q(n+1)=\cases{ 2Q(n) & for $n$\ even\cr 2Q(n)-C_{(n+1)/2} & for $n$\ odd,\cr}$$

where $C_k$ is a Catalan Number (Young et al. 1995). The function $Q(n,r)$ satisfies

$$Q(n,r)={n+1\choose r+1}-{n+1\choose r+2}$$

for $r>n/2-1$ (Young et al. 1995). $Q(n,r)$ satisfies the Quota Rule.

See also Binomial Coefficient, Central Binomial Coefficient

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.