District | Tories | Whigs | Total |

I | 51 | 49 | 100 |

II | 51 | 49 | 100 |

III | 33 | 67 | 100 |

Total | 135 | 165 | 300 |

**1.0 Introduction**

This post, amazingly enough, is on current events. Stephanopoulos and McGhee have developed a formula, the efficiency gap, that measures the partisanship of the lines drawn for legislative districts. In this post, I present a numerical illustration of this formula and connect it to current events. I conclude with some questions.

**2.0 Numerical Example**

Consider a population of 300 voters divided between two parties. The Whigs are in the majority, with 55% of the electorate. Suppose the government has a three-member council, with each member elected from a district. And each district contains 100 voters.

**2.1 Drawing Districts**

The Tories, despite being the minority party have drawn the districts. The votes in the last election are as in Table 1. The Tories are in the minority of the population, but hold two out of three council seats.

The Tories, in this example, cannot win all seats. In the seats they lose, they want to pack as many Whigs as possible. So where the Whigs win, they win overdominatingly. Many of the Whig votes in that single district are wasted on running up a victory more than necessary. On the other hand, the Tories try to draw their winning districts to win as narrowly as possible. The Whig votes in the districts in which the Whigs lose are said to be cracked.

This is an extreme example, sensitive to small variations in the districts in which the Tories win. They would probably want safer majorities in those districts.

As far as I can see, the drawing of odd-shaped district lines is not necessary for gerrymandering. Consider a city surrounded by suburbs and a rural area. Suppose, that downtown tends to vote differently than the suburbs and rural areas. One could imagine district lines drawn outward from the central city. Depending on relative populations, that might distribute the urban voters such that they predominate in all districts. On the other hand, one might create a few compact districts in the center to pack many urban voters, with the ones remaining in cropped pizza slices having their votes cracked.

**2.2 Wasted Votes**

Define a vote to be wasted if either it is for a losing candidate in your district or it is for a winning candidate, but it exceeds the number needed for a majority in that district. The number of wasted votes for each party in the numerical example is:

- The Tories have 33 wasted votes.
- The Whigs have 49 + 49 + (67 - 51) = 114 wasted votes.

The efficiency gap is a single number that combines the number of wasted votes in both parties. An invariance property arises here. As I have defined it, the number of wasted votes, summed across parties, in each district is 49. Forty nine is one less than half the number of votes in a district. This is no accident.

**2.3 Arithmetic**

In calculating the efficiency gap, one takes the absolute value of the difference between the parties in the number of wasted votes. In the example, this number is | 33 - 114 | = 81.

The efficiency gap is the ratio of this positive difference to the number of voters. So the efficiency gap in the example is 81/300 = 27%.

**3.0 Contemporary Relevance in the United States**

The United States Supreme Court has decided, in a number of cases over the last decades, that gerrymandering might be something they can rule on. Partisan redistricting is not purely a political issue that they do not want to get involved in. Apparently, however, they have never found a clear example.

But what is gerrymandering? Can they define some sort of rule that lower courts can use? How would politicians drawing up district lines know whether or not their decisions will withstand challenges in court? Apparently, Justice Kennedy, among others expressed a hankering for some such rule in his decision in League of United Latin American Citizens (LULAC) vs. Perry (2006).

Gill vs. Whitford is a current case on the Supreme Court docket. And the efficiency gap, which is relatively new mathematics, may be discussed in the pleadings, at least, in this case.

So the creation of the mathematical formula illustrated above might affect the law in the United States. If so, it will impact how districts are drawn and what some consider fair. It is interesting that I can now raise the issue of the performativity of mathematics in a non-historical context, while the mathematics is, perhaps, performing.

**4.0 Questions**

I am working on reading two of the three references below. (Articles in law reviews seem to be consistently lengthy.) I have some questions and comments.

Berstein and Duchin (2017) seems to raise some severe objections. Suppose the election in a district with 100 voters is decided either 75 to 25 or 76 to 24. The way I have defined it, the difference in wasted votes in this district is (24 - 25) or (25 - 24). That is, this district contributes one vote to the difference in wasted votes. So the definition of the efficiency gap privileges races that are won with 75% of the vote.

Consider a case in which one party has support from 75 percent of the voters. Suppose the districts are drawn such that each district casts 75% of their votes for that party. So this party wins 100% of the seats and the efficiency gap is minimized. Do we want to say this is not an example of gerrymandering?

Is the efficiency gap related to power indices somehow or other? How should the efficiency gap be calculated if more than two parties are contesting an election? Mayhaps, one should calculate the efficiency gap for each pair of parties. This loses the simplicity of a single number. Also, sometimes clever Republican strategists might try to help themselves by helping the Green Party, at the expense of the Democratic Party. How does this measure compare and contrast with other measures? As I understand it, a measure of partisan swing, for example, relies on counterfactuals, while the efficiency gap is not counterfactual.

**References**

- Mira Bernstein and Moon Duchin (2017). A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap.
- Geometry of Redistricting, Tufts University, to be held 7-11 August 2017
- Nicholas Stephanopoulos and Eric McGhee (2015). Partisan Gerrymandering and the Efficiency Gap,
*University of Chicago Law Review*.

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