Daniel J. Velleman's American Mathematical Monthly, volume 117, number 1, january PDF

By Daniel J. Velleman

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X t y t t y t x t y ↓ x t y t t x Figure 4. Swapping endpoints to eliminate a containment. p. q, either the support of q is unchanged by this transformation, or x leaves the support and y enters the support; in either case, a R (q) = a S (q). So a(R) = a(S). Supposing that S is (k, m)-agreeable, we claim that R is (k, m)-agreeable. Let M ⊆ R V S = V R be any set of m voters. p. p. There is some tedium in considering all possible cases, but the argument stands on the following S R facts: a{x,y} ( p) = a{x,y} ( p) for all p (used in case 2), A Sy ⊆ A yR (used in case 1), and S R A y ⊆ A x (used in case 3).

Experimentation with a few societies suggests (k − 1)/m as a lower bound on the agreement proportion of a (k, m)-agreeable circular society. This turns out to be correct; the difficulty lies in proving it, and the rest of the paper is devoted to the following theorem. 2. (a) If S is a (k, m)-agreeable circular society, then a(S) k−1 > ; |S| m (2) |S| + 1. equivalently, a(S) ≥ k−1 m (b) For every N ≥ m ≥ k ≥ 1, there is a (k, m)-agreeable circular society S of size N with a(S) = k−1 N + 1. m We prove the sharpness condition (b) in Section 2, and establish the lower bound (a) in Section 4.

Assume that there is a finite set V of voters, and each voter v has an approval set Av of platforms. 28 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 We define a society S to be a triple (X, V, A) consisting of a spectrum X , a set of voters V , and a collection A of approval sets for all the voters. Of particular interest to us will be the case of a linear society, in which X is a closed subset of R and approval sets in A are of the form X ∩ I where I is either empty or a closed bounded interval in R.

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American Mathematical Monthly, volume 117, number 1, january 2010 by Daniel J. Velleman


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