##### Department of Mathematics,

University of California San Diego

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### Math 196/296 - Student Colloquium

## John Hall

#### UCSD

## Arrow's impossibility theorem and the geometry of voting

##### Abstract:

Kenneth Arrow's Impossibility Theorem essentially states that in the presence of three or more candidates there is no way to hold a fair election. That this statement is true in practice should not be a surprise to anyone familiar with our current electoral system. It is a little more surprising that it holds even in the abstract world of mathematics. \vskip .1in \noindent In this talk we shall define social welfare functions, discuss a reasonable set of fairness criteria, and sketch a proof of Arrow's theorem. Along the way we shall touch on topics in combinatorics, geometry, logic, and set theory. To end on a positive note, we shall show that fair voting methods do exist when the number of voters is infinite. \vskip .1in \noindent Prerequisites: A small amount of basic set theory and linear algebra will be assumed, but all important terms will be defined as we go. \vskip .1in \noindent Refreshments will be provided!

Host:

### December 1, 2005

### 11:00 AM

### AP&M 2402

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