1 How our Founding Father’s designed algorithms for assigning fair representation for the new United States of America How Will the Next Election Be Decided? The Jefferson and Hamilton Debate over Methods of Apportionment Prepared by Fred Annexstein University of Cincinnati CC Some rights reserved Supporting Computational Issues Fairness Division Political Arithmetic Algorithmic Method Paradox Undecidability Core Quantitative Issue Fairness Computation

2 The US Constitution “Representatives and direct taxes shall be apportioned among the several states which may be included within this union, according to their respective numbers, which shall be determined by adding to the whole number of free persons, including those bound to service for a term of years, and excluding Indians not taxed, three fifths of all other Persons. The actual Enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent term of ten years, in such manner as they shall by law direct. The number of Representatives shall not exceed one for every thirty thousand, but each state shall have at least one Representative; and until such enumeration shall be made, the state of New Hampshire shall be entitled to chuse three, Massachusetts eight, Rhode Island and Providence Plantations one, Connecticut five, New York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolin a five, South Carolina five, and Georgia three.” Article 1, Section 2 of the Constitution of the United States (1787).

3 The Problem of Representation Apportionment The problem of actual assignment of the whole number of seats in the United States Congress among necessarily fractional numbers of relative state populations is known as the problem of Apportionment. Notes on US representation: –Not to exceed 1 in 30,000 of population –Women and children are to be counted in population (no voting rights) –Slaves counted as 3/5 white person (no voting rights) –At least one per state –Congress given three years to come up with a practical solution. Two plans were submitted right away. One by Alexander Hamilton, the other by Thomas Jefferson. After heated deliberations, Congress applied method of Hamilton. Jefferson: the bill “seems to have avoided establishing [the procedure] into a rule…lest it might not suit on another occasion…[We need to ] reduce the apportionment always to an arithmetical operation, about which no two men can ever possibly differ.”

4 History of Apportionment I Jefferson convinced Washington to block Hamilton’s plan by exercising the veto power, the first ever by a President. After the veto, Congress adopted Jefferson's method, but with a different number of seats. Senator Daniel Webster’s plan was adopted in 1842, only to be replaced by Hamilton's method in was a very confusing year! First the House size was chosen to be 283 so that Hamilton’s and Webster’s methods would agree. After much political infighting, 9 more seats were added and the final apportionment did not agree with either method. In 1876 Rutherford B. Hayes became President based on the botched apportionment of The electoral college vote was 185 for Hayes and 184 for Tilden.

5 Electoral College Map 2007 A hundred years later, mathematicians M. L. Balinski and H. P. Young showed that Tilden would have won if correct apportionment as required by law had been used. In a celebrated theorem, Balinski and Young (1960s) showed that any fixed method of apportionment could not always be “fair”.

6 Apportionment: Hamilton's Method The following is the apportionment method suggested by Alexander Hamilton was was applied by Congress in Three simple steps to find the apportionment. Hamilton’s Method Step 1. Calculate each state’s standard quota. = Population / # Seats StatePopulationStep1 Quota A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total12,500,

7 Apportionment: Hamilton's Method Hamilton’s Method Step 2. First give to each state its lower quota. StatePopulationStep1 Quota Step 2 Lower Quota A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total12,500,

8 Apportionment: Hamilton's Method Step 3. Give the surplus seats to the state with the largest fractional parts until there are no more surplus seats. StatePopulationStep1 Quota Step 2 Lower Quota Fractional parts Step 3 Surplus Hamilton apportionment A 1,646, First 33 B 6,936, Last139 C 154, D 2,091, Second 42 E 685, F 988, Third 20 Total12,500, Hamilton’s Method Satisfies “The Quota Rule”: No state should be apportioned a number of seats smaller than its lower quota or larger than its upper quota. (When a state is apportioned a number smaller than its lower quota, we call it a lower-quota violation; when a state is apportioned a number larger than its upper quota, we call it an upper-quota violation.)

9 Hamilton’s Method is apparently “fair”. However, it suffers from a fatal “flaw” known as the Alabama paradox. In 1880 it was discovered that Alabama would get 8 seats with a House size of 299 but only 7 with a House size of 300. In general the term Alabama paradox refers to any apportionment scenario where increasing the total number of items would decrease one of the shares. The Alabama paradox StatePopulationStep 1Step 2Step 3Apportionment Bama Tecos 9, Ilnos10, Total20, StatePopulationStep 1Step 2Step 3Apportionment Bama Tecos 9, Ilnos10, Total20, Note: Bama gets screwed since other’s share grows faster

10 Apportionment: Jefferson's Method Jefferson’s Method Step 1. Find a “suitable” divisor D. [ A suitable or modified divisor is a divisor that produces and apportionment of exactly M seats when the quotas (populations divided by D) are rounded down.

11 Apportionment: Jefferson's Method Jefferson’s Method Step 2. Each state is apportioned its lower quota. StatePopulationStandard Quota (SD = 50,000) Lower QuotaModified Quota (D = 49,500) Jefferson apportionment A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total12,500, Watch Out - Jefferson’s method can produce upper-quota violations! To make matters worse, the upper-quota violations tend to consistently favor the larger states.

12 Apportionment: Adam's Method Adam’s Method Step 1. Find a “suitable” divisor D. [ A suitable or modified divisor is a divisor that produces and apportionment of exactly M seats when the quotas (populations divided by D) are rounded up. Step 2. Every state gets its upper-quota. StatePopulationQuota (D = 50,500) Upper Quota (D = 50,500) Quota (D = 50,700) Adam’s apportionment A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total12,500, Surprised?! - Adam’s method can produce lower-quota violations! We can reasonably conclude that Adam’s method is no better (or worse) than Jefferson’s method– just different. Jefferson favors larger states and Adam’s favors smaller states.

13 Apportionment: Webster's Method Not Again! - Webster’s method can produce both lower and upper-quota violations! But this is surprisingly rare. Many consider Webster’s the best method. Webster’s Method Step 1. Find a “suitable” divisor D. [ Here a suitable divisor means a divisor that produces an apportionment of exactly M seats when the quotas (populations divided by D) are rounded the conventional way. Step 2. Find the apportionment of each state by rounding its quota the conventional way.

14 Apportionment: Webster's Method StatePopulationStandard Quota (D = 50,000) Nearest Integer Quota (D = 50,100) Webster’s apportionment A 1,646, B 6,936, C 154, D 2,091, E 685, F 988, Total12,500, Webster’s method was adopted by the Congress in 1842, and then replaced by Alexander Hamilton's in It was again adopted in 1901 and reconfirmed in Finally, it was replaced by Huntington-Hill's method in 1941.

15 Apportionment: Huntington Method In 1921 Edward V. Huntington, a Harvard mathematician proposed a solution known as the “method of equal proportions.” Huntington’s Test of Fairness: Any fair apportionment should have the property that it is not possible to transfer a representative (from one state to another) and improve their relative difference in (average) district sizes. Huntington shows such apportionments always exist and there is a solution method “of little interest except to the computers in the Bureau of the Census.” Transactions of AMS, 1928 TABLE OF MULTIPLIERS …… …….. N1/SQRT(N*N+1)

16 Apportionment: Huntington Method Using Huntington’s method of calculating seats, determine which states would benefit if the number of House seats were raised 15 seats to say 450. Use the 2000 census data that will be made available. What part of the country, if any, would benefit most? Census 2000 Ranking of Priority Values Source: U.S. Census Bureau Seat#StateState Seat Priority Value NY CO PA TX MO CA MN GA IA FL OH CA NC

17 Apportionment: Huntington Method Calculate say 100 "multipliers” and multiply each with the population total for each of the 50 states. The resulting 5000 numbers are called the state priority values. Simply choose the top 385(=435-50) priority values. First 50 seats are reserved one per state. Each of these top-ranked values is associated with another seat in the “fair” apportionment. Tally the total number of seats for each state to arrive at the final apportionment. Census 2000 Ranking of Priority Values Source: U.S. Census Bureau Seat#StateState Seat Priority Value CA TX CA NY FL CA IL PA TX OH NY CA

18 CONNECTICUT DELAWARE GEORGIA KENTUCKY MAINE MARYLAND MASSACHUSETTS NEW HAMPSHIRE NEW JERSEY NEW YORK NORTH CAROLINA PENNSYLVANIA RHODE ISLAND SOUTH CAROLINA TENNESSEE VERMONT VIRGINIA

19 What is a paradox? A paradox is a (seemingly) true statement or group of statements that leads to a contradiction or a situation which defies logic or intuition. Russell's paradox is part of the foundation of mathematics and computability theory: Let R be "the set of all sets that do not contain themselves as members".The problem arises when it is considered whether R is an element of itself. If R is an element of R, then according to the definition R is not an element of R. If R is not an element of R, then R has to be an element of R, again by its very definition. The statements "R is an element of R" and "R is not an element of R" cannot both be true. Thus the question is “undecidable”. In computability theory the Halting problem is a problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. He showed that the Halting Problem is “undecidable”.