*Friday, July 28, 2:00 p.m. - 5:00 p.m., Continental Ballroom A*

Democracy is fraught with different meanings that mathematics can help to make more precise. This session will include talks on the properties of voting systems that best reflect the will of the people in electing a single winner (e.g., for mayor or president), or best represent different factions in electing multiple winners (e.g., to a committee or council). Among other topics discussed will be different ways of apportioning representatives to states, or seats in a legislature to political parties; methodologies for drawing district lines to avoid gerrymandering; and the avoidance of different social-choice paradoxes.

**Organizer:**

**Steven Brams**, *New York University*

#### Political Hypotheses and Mathematical Conclusions

*2:00 p.m. - 2:20 p.m.*

**Paul H. Edelman**, *Vanderbilt University*

When modeling or analyzing democratic processes, mathematicians may find themselves in unfamiliar territory: political philosophy. How we proceed mathematically may depend heavily on our conception of representative democracy and theory of government. I will give a number of illustrations to show how contestable political principles lead to differing mathematical analyses. Our mathematical conclusions are inherently governed by our political hypotheses.

#### Multiwinner Approval Voting: An Apportionment Approach

*2:30 p.m. - 2:50 p.m.*

**D. Marc Kilgour**, *Wilfrid Laurier University*

Approval voting is extended to the election of multiple winners—roughly proportional to their approval in the electorate—who may be either individual candidates, elected to a committee, or members of a political party, who fill one or more seats in a legislature. The sequential version of the divisor apportionment methods of Jefferson and Webster iteratively depreciate the approval votes of voters who have one or more of their approved candidates elected. The nonsequential versions of these methods, which are computationally complex but feasible to use in many elections, tend to elect more representative and diverse bodies than the sequential methods. Whereas the Webster method better satisfies representativeness and diversity than the Jefferson method, the latter, whose vote thresholds for winning seats duplicate those of cumulative voting in 2-party elections, seems fairer.

#### Voting and the Symmetric Group

*3:00 p.m. - 3:20 p.m.*

**Michael Orrison**, *Harvey Mudd College*

Suppose you are voting in an election that requires you to submit a complete ranking of the candidates, from your most preferred candidate all the way down to your least preferred candidate. If you enjoy thinking about abstract algebra, then you might be tempted to view your ranking as a permutation in the symmetric group on the set of candidates. In this talk, I will explain why doing so is worth your while, and how it can quickly lead to new insights and powerful techniques for wrestling with ideas in voting theory.

#### Consistent Criteria, Problematic Outcomes, and the Hypercube

*3:30 p.m. - 3:50 p.m.*

**Tommy Ratliff**, *Wheaton College*

Not all voting consists of selecting a winner from a set of candidates. For example, consider a tenure committee where the criterion is that a successful candidate must be excellent in both teaching and research. There are simple examples that lead to what is known as the discursive dilemma: The committee reaches one conclusion using the majority vote based on the recommendations of each member applying the criterion individually but obtains a different conclusion by first using the majority vote on each category and then applying the criterion.

What happens if there are more categories or if the criterion is more complicated than a simple boolean AND? We can use the geometry of the hypercube and some graph theory to characterize all logical statements that lead to discursive-type dilemmas.

#### Ready for Redistricting 2020

*4:00 p.m. - 4:20 p.m.*

**Karen Saxe**, *Macalester College* and *AMS*

Every ten years the seats of the US House of Representatives are reapportioned to the states and then each state commences to redraw its congressional district lines. In this talk we will give an overview of how the states do this and what changes (legal and procedural) have taken place since the last time we did this. We will highlight how mathematics is used to aid in the redistricting process and help detect when gerrymandering has taken place.

#### Orthogonal Decomposition and the Mathematics of Voting

*4:30 p.m. - 4:50 p.m.*

**William S. Zwicker**, *Union College*

Suppose several teachers are assessing the level of preparation of their common students, with the goal of splitting them into one group ready to tackle more abstract and challenging mathematical concepts, and a second group needing more review. An election is held, in which a ballot recommends a particular split, and the outcome is a collective decision on how to group. This seems quite different from an election in which a ballot is a ranking of candidates for President, and the outcome selects a winning candidate, but in both cases we are aggregating several binary relations of a specified type into a single binary relation (of a possibly different type).

It turns out that there are "universal" rules for aggregating binary relations, which generate a surprising diversity of well-known aggregation rules as special cases. Differences between universal aggregators F and G can arise when an orthogonal decomposition separates ballot information into two components, with F using both and G discarding one of them. We'll discuss two decompositions, related to the two types of elections mentioned above, and to a single voting rule proposed by John Kemeny.