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Quota System


A generalization of simple majority voting in which a list of quotas {q_0,...,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=|_n/2_|)^nQ(n,r)=(n+1; |_n/2_|+1),
(1)

where |_x_| is the floor function. This produces the sequence of central binomial coefficients 1, 2, 3, 6, 10, 20, 35, 70, 126, ... (OEIS A001405). It may be defined recursively by Q(0)=1 and

 Q(n+1)={2Q(n)   for n even; 2Q(n)-C_((n+1)/2)   for n odd,
(2)

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

 Q(n,r)=(n+1; r+1)-(n+1; r+2)
(3)

for r>n/2-1 (Young et al. 1995). Q(n,r) satisfies the quota rule.


See also

Binomial Coefficient, Central Binomial Coefficient

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References

Sloane, N. J. A. Sequence A001405/M0769 in "The On-Line Encyclopedia of Integer Sequences."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.

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Quota System

Cite this as:

Weisstein, Eric W. "Quota System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuotaSystem.html

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