## Huygens’ Theorem and the Theory of Voting |

**William Zwicker**

Union College

February 13, 2006

4:30 pm

Bailey Hall 201

Pastries and drinks will be served at 4 pm in Bailey 204

Here are three general methods, each of which can be used to create a variety of specific voting systems:

Scoring rules:Each voter assigns points to each candidate. The winner is the candidate with the most points.

Mean proximity voting:Each voter is identified with a point in space. Each candidate is identified with a point in space. The mean position $g$ of the voters is calculated. The candidate closest to $g$ wins.

Distance-to-unanimity voting:The outcome of any election is clear when the vote is unanimous. When a vote is not unanimous, the winner corresponds to whichever unanimous choice is “closest” to the actual vote.

In what ways are these three general types of voting different? We use a theorem of Huygens to prove that the three classes are exactly the same.

Christiaan Huygens (1629-1695) was a Dutch scientist, mathematician, and astronomer known for important work on optics, telescopes, and pendulum motion, and for his close association with many important researchers of his time: Descartes, Pell, Wallace, Newton, Hooke, Mersenne, and others. One of his theorems concerns sums of squared distances to the mean location (or center of gravity) of a set $S$ of points:

These distances satisfy ${d_1}^2 +{d_2}^2 +{d_3}^2=3f^2 +({e_1}^2+{e_2}^2+{e_3}^2)$, if $g$ is the mean location of $s_1, s_2$, and $s_3$.

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Union College Math Department Home PageComments to: math@union.edu Created automatically on: Sat Apr 21 13:29:03 EDT 2018 |