## The Geometry of Influence: Weighted Voting and Hyper-ellipsoids |

**William S. Zwicker**

Union College

September 27, 2010

4:15 pm

Bailey Hall 207

Refreshments will be served in Bailey 204 at 4:00

Towns in Nassau County (Long Island) vary in population, andvoting weightsfor Nassau’s Board of Supervisors used to be assigned proportionately. If (for example) town A had twice the population of B, then A’s supervisor got twice as many votes as B’s. Any such rule is calledweighted voting. Now, suppose we describe some votingWin completely different terms. Can we test whether it’s possible to also expressWvia weights?

One geometric test is well known. When coalitions of voters are represented as vertices of an n-dimensional hypercube, rule Wis weighted if and only if we can slice the hypercube by a hyperplane, separatingwinning coalitionsfromlosing coalitions. We discuss a new result: ruleWis 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 voter’s weight by herBanzhaf voting power, a measure of influence.But wait a moment . . . wouldn’t a voter’s influence be the same as her voting weight? No, and for this reason in 1967 New York’s highest court threw out Nassau’s voting rule.

*Joint work with Nicolas Houy.

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