Examining different measures used to detect gerrymandering.

Abstract

Gerrymandering is the act of changing political boundaries so as to favor one political party or class in an election. Much effort has been made in recent years to mathematically detect gerrymandering in order to prevent this manipulation. Markov Chain Monte Carlo (MCMC) methods have been used to sample the distribution of legal and reasonable redistrictings of a particular area, which allows comparison of specific districtings with the space of reasonable legal districtings. Depending on the state, districting plans in America are often required to be {esc}(3z{esc}(Bcompact{esc}(3y{esc}(B, though this term is loosely defined by the law. Fortunately, the MCMC method can take into account the compactness of districts, but of course this requires a strict definition of compactness. Unfortunately, different groups have used a multitude of measures of compactness. An open question in this field is what relationship, if any, exists between diffferent measures of compactness. Two of the methods that have been used are the Isoperimetric Score and the Cut Edge Score of a districting plan. Isoperimetric Score looks at the geography of a districting plan, while Cut Edge Score examines the relational structure of a districting plan. Looking at these two seemingly different measures of compactness, I have found that prioritizing Cut Edge Score with MCMC methods may result in more compact districting plans, both in terms of Isoperimetric Score and Cut Edge Score, more easily than when taking into account Isoperimetric Score. Moreover, there is consistently a strong relationship between the two scores. In ad- dition, I looked at the Population Balance Score of districting plans, which measures how close a districting plan is to having an equal population in each district. Look- ing at this score, I have found that in certain cases prioritizing Cut Edge Score in MCMC methods results in better Population Balance Scores than when prioritizing Isoperimetric Score.