Donald G. Saari, Institute for Mathematical Behavioral Sciences, University of California, Irvine (mailto:[email protected])
Michael A. Jones (Guest Lecturer), Department of Mathematical Sciences, Montclair State University ([email protected])

Elections and voting

We have done this all the time starting from that kindergarten class when our teacher asked for a "show of hands." Sounds simple, but is it? We know from too many examples, and from Arrow's Theorem, which asserts that "no election rule is fair," that this can be a complicated topic. During this week, we will introduce "the mathematics of voting" to see how the muscle power of mathematics can resolve many complex issues in elections. The results are surprising and discouraging.

Topics covered include showing how to understand, and create, all possible voting paradoxes that could occur with any of the standard methods, a discussion of how symmetry groups help us understand decision procedures, commentary on the mathematics of strategic and manipulative behavior (which will show you how to "win" during your next departmental discussion), and a discussion showing that the famous Arrow's Theorem does not mean what we have believed for the last half century. Time permitting, other discussions will show how all of this material extends to topics such as "power indices" (a way to measure the political power of a group), economics, and even statistics.

All of the material can be incorporated in undergraduate courses

While the research frontier of this topic involves complicated material, it turns out that it is possible to package even recently found conclusions in a manner that can be presented in undergraduate courses. In one of the sessions, the speaker will show how even simple algebra and combinatorics can be used to construct surprising paradoxes. He also will indicate how to tie in this material with orbits of symmetry groups from algebra. The material on strategic behavior includes both notions from symmetry groups and basic geometry involving gradients.

All of this leads to material that will be surprising, entertaining, and highly educational for undergraduates.

Prerequisites

Anyone in a math or computer science department will be able to handle this material. For instance, much of the material should be assessable for those comfortable with calculus. We will be using some material from a first course in abstract algebra, but will be reminding the audience of the basic concepts.

References

Most of the material will come from the following:

Saari, D. G., Chaotic Elections! A Mathematician Looks at Voting, American Math Society, Providence, 2001.

Saari, D. G., Decisions and Elections: Explaining the Unexpected, Cambridge University Press, New York, 2001

Saari, D. G., Explaining all three-alternative voting outcomes, Jour Econ. Theory 87 (1999), 313- 355.

Saari, D. G., Disturbing aspects of voting theory, Economic theory 22 (Oct. 2003), 529-556. 2