1. The Mathematics of Voting: Apportionment and Power
TCM Conference Jan. 27., 2017
Dan Teague
NCSSM

2. Apportionment and Power
The Apportionment problem focuses on th ability to fairly distribute representatives. Fairness is the goal. The Power problem focuses on how that fairness in distribution translates into a distribution of power, that is, the ability to pass laws.

3. 10-year Census
Given the populations of each state estimated by the decennial census, and the total population of the US, fairly apportion the 435 representatives in the US House of Representatives to the 50 states.

5. A Simple Example: 20 Delegates

6. How would you share the 20 delegates?

7. A Simple Example: 21 Delegates

8. Jefferson Apportionment

9. Ratios are Very Badly Behaved

10. We Need a Model (and a Metric) for Fairness
Consider two states,
20,000 p/d
X with 100,000 people and 5 representatives
15,000 p/d
Y with 60,000 people and 4 representatives.
Is this fair apportionment?
Unfair to X by 500 p/d

14. More Than Two Districts?

20. Relative Unfairness

24. Reapportionment of 2000 Utah vs North Carolina

26. Utah vs North Carolina Reapportionment of 2000
857

28. Power
Example 1: A company has 1,000 shares all owned by one of three people.
Person A owns 499 shares,
person B owns 498 shares, and
person C owns the remaining 3 shares.
A proposal requires a majority vote (501 shares) for approval. What is the distribution of power among the 3 shareholders?

29. Example 2: A commission is formed with 4 members, A, B, C, and D
Example 2: A commission is formed with 4 members, A, B, C, and D. Person A has 5 votes, person B has 4 votes, person C has 3 votes, and person D has only 1 vote. At least 7 votes are required to pass a particular proposal. We can describe this situation using the notation [7; 5, 4, 3, 1]. How is the power distributed among the members?

30. [ 8; 6, 3, 2, 2] What would a Dictator look like? [ 8; 8, 3, 3, 1]
How about someone with Veto power?
[ 8; 6, 3, 2, 2]

36. Power Polynomial

40. Voting Power in the Electoral College
NICHOLAS R. MILLER  Department of Political Science  University of Maryland Baltimore County (UMBC) 

41. The Mathematics of Voting: Apportionment and Power
TCM Conference Jan. 27., 2017
Dan Teague
NCSSM

42. userpages.umbc.edu/~nmiller/POLI309/VOTING%20POWER.ppt