The Geometry of Influence: Weighted Voting and Hyper-ellipsoids
William S. Zwicker
September 27, 2010
Bailey Hall 207
Refreshments will be served in Bailey 204 at 4:00
Towns in Nassau County (Long Island) vary in population, and voting weights for Nassauís Board of Supervisors used to be assigned proportionately. If (for example) town A had twice the population of B, then Aís supervisor got twice as many votes as Bís. Any such rule is called weighted voting . Now, suppose we describe some voting W in completely different terms. Can we test whether itís possible to also express W via weights?
One geometric test is well known. When coalitions of voters are represented as vertices of an n-dimensional hypercube, rule W is weighted if and only if we can slice the hypercube by a hyperplane, separating winning coalitions from losing coalitions. We discuss a new result: rule W is weighted if and only if we can separate winning from losing coalitions via a hyper-ellipsoid centered at the mean winning coalition. To convert voting weights to the proportions of this ellipsoid, one multiplies each voterís weight by her Banzhaf voting power, a measure of influence.
But wait a moment . . . wouldnít a voterís influence be the same as her voting weight? No, and for this reason in 1967 New Yorkís highest court threw out Nassauís voting rule.
*Joint work with Nicolas Houy.
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