1. Political Redistricting By Saad Padela

2. The American Political System Legislative bicameralism  Number of seats in lower house is proportional to population Single-member districts First-past-the-post (or plurality) voting “One man, one vote”

5. The Case for Redistricting New Census data every 10 years # of Representatives = α * Population  0 < α < 1 # of Representatives = # of Districts Population rises => More seats Districts must be redrawn

6. Gerrymandering

8. Types of Gerrymandering Partisan  Democrats vs. Republicans Bipartisan  Incumbents vs. Challengers Racial and ethnic  Majority vs. Minority groups “Benign”  In favor of minority groups

9. Gerrymandering Strategies Different election objectives  To win a single district  To win a majority of many districts Partisan  Own votes Win districts by the smallest margin possible Minimize wasted votes in losing districts  Opponent's votes Fragment them into different districts Concentrate them into a single district

10. Gerrymandering Strategies Bipartisan  Maximize number of “safe” districts Racial and ethnic  Fragment supporters of minority candidates “Benign”  Maximize chances of minority representation by concentrating them into single districts

11. A Linear Programming Formulation? Easy to see Small scholarly literature  Those who are involved in it like to keep their work secret

12. Detection of Gerrymandering A rich literature Hess, S.W. 1965. “Nonpartisan Political Redistricting by Computer.” Operations Research, 13 (6), 998-1006.

13. Good Districts are... Equally populous Contiguous Compact

15. Equal population Easy to write as a constraint

16. Contiguity Highly intuitive Sometimes tedious to code

17. Compactness Ambiguous Difficult to measure Niemi et al. 1990. “Measuring Compactness and the Role of a Compactness Standard in a Test for Partisan and Racial Gerrymandering.” The Journal of Politics, 52 (4), 1155-1181.  “A Typology of Compactness Measures” (Table 1)‏ Dispersion Perimeter Population

18. A Typology of Compactness Measures: Dispersion District Area Compared with Area of Compact Figure  Dis7 = ratio of the district area to the area of the minimum circumscribing circle  Dis8 = ratio of the district area to the area of the minimum circumscribing regular hexagon  Dis9 = ratio of the district area to the area of the minimum convex figure that completely contains the district  Dis10 = ratio of the district area to the area of the circle with diameter equal to the district's longest axis

19. A Typology of Compactness Measures: Dispersion District Area Compared with Area of Compact Figure  Dis7 = ratio of the district area to the area of the minimum circumscribing circle  Dis8 = ratio of the district area to the area of the minimum circumscribing regular hexagon  Dis9 = ratio of the district area to the area of the minimum convex figure that completely contains the district  Dis10 = ratio of the district area to the area of the circle with diameter equal to the district's longest axis

20. A Typology of Compactness Measures: Dispersion Moment-of-inertia  Dis11 = the variance of the distances from all points in the district to the district's areal center for gravity, adjusted to range from 0 to 1  Dis12 = average distance from the district's areal center to the point on the district perimeter reached by a set of equally spaced radial lines

21. A Typology of Compactness Measures: Perimeter Perimeter-only  Per1 = sum of the district perimeters Perimeter-Area Comparisons  Per2 = ratio of the district area to the area of a circle with the same perimeter  Per4 = ratio of the perimeter of the district to the perimeter of a circle with an equal area  Per5 = perimeter of a district as a percentage of the minimum perimeter enclosing that area

22. A Typology of Compactness Measures: Population District Population Compared with Population of Compact Figure  Pop1 = ratio of the district population to the population of the minimum convex figure that completely contains the district  Pop2 = ratio of the district population to the population in the minimum circumscribing circle Moment-of-inertia  Pop3 = population moment of inertia, normalized from 0 to 1

23. Warehouse Location model Hess, S.W. 1965. “Nonpartisan Political Redistricting by Computer.” Operations Research, 13 (6), 998-1006. Garfinkel, R.S. And G.L. Nemhauser. 1970. “Optimal Political Districting By Implicit Enumeration techniques.” Management Science, 16 (8). Hojati, Mehran. 1996. “Optimal Political Districting.” Computers and Operations Research, 23 (12), 1147-1161. All these formulations have class NP

24. Heuristic Methods Hess, S.W. 1965. Garfinkel, R.S. And G.L. Nemhauser. 1970. Hojati, Mehran. 1996. Bozkaya, B., Erkut, E., and G. Laporte. 2003. “A tabu search heuristic and adaptive memory procedure for political districting.” European Journal of Operational Research, 144, 12-26.

25. Statistical physics? Chou, C. and S.P. Li. 2006. “Taming the Gerrymander – Statistical physics approach to Political Districting Problem.”

26. Criticisms of Compactness Altman, Micah. 1998. “Modeling the effect of mandatory district compactness on partisan gerrymanders.” Political Geography, 17 (8), 989-1012.  Nonlinear effects – “electoral manipulation is much more severely constrained by high compactness than by moderate compactness”  Context-dependent, and purely relative  Asymmetrical effects on different political groups  Compactness can also disadvantage geographically concentrated minorities

27. More Sophisticated Measures Niemi, R. and J. Deegan. 1978. “A Theory of Political Districting.” American Political Science Review, 72 (4), 1304-1323.  Neutrality v% of the popular vote results in s% of the seats  Range of Responsiveness % range of the total popular vote over which seats change from one party to the other  Constant Swing Ratio rate at which a party gains seats per increment in votes  Competitiveness % of districts in which the “normal” vote is close to 50%

28. Balinksi, Michel. 2008. “Fair Majority Voting (or How to Eliminate Gerrymandering).” The American Mathematical Monthly, 115 (2), 97- 114.