chalkdust

A mathematical view of

voting systems

Philip Halling

Alexander Bolton

S

 aer I began my undergraduate degree, Barack Obama won the 56th United States presidential election. The next day, my pure maths tutor asked me if I had followed it closely, saying that elections really interested him. I was surprised to hear this: surely the only maths involved was adding up the number of votes? In reality, voting systems hold considerable interest for mathematicians, and there are several mathematical results and theorems concerning electoral processes. The main thing that I like about the language of mathematics is that it allows us to make extremely precise statements without ambiguity and, as you’ll see, we can make precise mathematical statements about voting systems—with some surprising results.

Transitive preferences and voting systems We define an electorate E to be a vector of voters (v1 , . . . , vn ). Each voter vi has a ranking (with ties allowed) of the vector of candidates C = (c1 , . . . , cm ). We write x ≻i y to mean that voter vi prefers x to y. The ranking is transitive, meaning that if x ≻i y and y ≻i z then we must have x ≻i z. An interesting property of the votes is that if a majority of voters prefer x to y, and a majority prefer y to z, it does not necessarily follow that a majority prefer x to z. A simple example with three voters is x ≻1 y ≻1 z, z ≻2 x ≻2 y and y ≻3 z ≻3 x. I’ll use the notation “≻”, without the subscript i, to refer to the preferences of a group of voters. 11

spring 2016

Chalkdust, Issue 03

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