## 2018 REU Project – Referendum Election Simulation

In referendum elections, voters are often required vote simultaneously on multiple questions or proposals, some of which may be related to each other. The separability problem occurs when a voter’s preference for the outcome of one or more proposals in the election depends on the possible outcomes of other proposals. For instance, suppose a voter wants Proposal A to pass, but only if Proposal B does not pass. This voter’s preferences would be considered nonseparable. When election day comes and both proposals are on the same ballot, the voter will have a difficult time deciding how to vote, particularly on Proposal A.

In order to better understand the separability problem and its potential solutions, recent research has focused on the multidimensional preferences associated with voters in referendum elections. Projects in the 2013-2015 REUs focused on using graph theory to model and generate voter preferences.

The main idea is to view voter preferences in an n-question election as Hamiltonian paths within graphs of order 2n. We can then consider which graphs give rise to preferences with interesting properties, such as separability. Moreover, by choosing the right graph and labeling it in a creative way, we can construct preferences with specific forms of interdependence that cannot be constructed easily by other means.

In the 2018 REU, we will use the graph theoretic models developed in previous years to generate diverse electorates and, via computer simulation, investigate the impact of various forms of separability on the desirability of referendum election outcomes.

This project will involve designing experiments and developing computer programs to carry out these experiments via simulated elections. In particular, we will generate electorates with varying forms and degrees of interdependence, run simulations involving these electorates, and then examine the data to make conclusions about how interdependence impacts the ability referendum elections to accurately represent the will of the people. If we see general patterns in our data, we may try to prove why these patterns hold. So while the main thrust of the project will be experimental, there may be a theoretical component as well.

Although this problem uses graph theoretic models, it is not necessary for applicants to have an extensive background in graph theory. Some exposure to the terminology associated with graphs, and familiarity with some of the more common families of graphs, would be helpful. However, the most important qualification for this project is moderate to extensive computer programming experience and/or experience in data analysis.

For a basic survey of the separability problem in referendum elections, read this paper which appeared in the October 2011 issue of Mathematics Magazine.

In order to better understand the separability problem and its potential solutions, recent research has focused on the multidimensional preferences associated with voters in referendum elections. Projects in the 2013-2015 REUs focused on using graph theory to model and generate voter preferences.

The main idea is to view voter preferences in an n-question election as Hamiltonian paths within graphs of order 2n. We can then consider which graphs give rise to preferences with interesting properties, such as separability. Moreover, by choosing the right graph and labeling it in a creative way, we can construct preferences with specific forms of interdependence that cannot be constructed easily by other means.

In the 2018 REU, we will use the graph theoretic models developed in previous years to generate diverse electorates and, via computer simulation, investigate the impact of various forms of separability on the desirability of referendum election outcomes.

**What will this research involve?**This project will involve designing experiments and developing computer programs to carry out these experiments via simulated elections. In particular, we will generate electorates with varying forms and degrees of interdependence, run simulations involving these electorates, and then examine the data to make conclusions about how interdependence impacts the ability referendum elections to accurately represent the will of the people. If we see general patterns in our data, we may try to prove why these patterns hold. So while the main thrust of the project will be experimental, there may be a theoretical component as well.

**Desirable experiences for applicants**Although this problem uses graph theoretic models, it is not necessary for applicants to have an extensive background in graph theory. Some exposure to the terminology associated with graphs, and familiarity with some of the more common families of graphs, would be helpful. However, the most important qualification for this project is moderate to extensive computer programming experience and/or experience in data analysis.

**Further reading**For a basic survey of the separability problem in referendum elections, read this paper which appeared in the October 2011 issue of Mathematics Magazine.