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US Senate Allocation

lpetrich

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I'm posting about this here because it is an interesting mathematical conundrum.

Constitution for the United States - We the People specifies the term lengths of its elected officials, and various laws and traditions have filled in the gaps.
  • House of Representatives: 2 years
  • Senate: 6 years
  • President: 4 years

House elections occur on even-numbered years, years with numbers divisible by 2. So on every even-numbered year, the whole House is elected, with the members' terms lasting until the next even-numbered year.

Presidential elections occur on years with numbers divisible by 4. This gives the right term length and it makes a President elected for every two House elections.

Senators could be elected all at once, every 6 years, but the creators of the Constitution decided to stagger the elections of the Senators, making some of the Senators elected each two years, when the Reps are elected. This results in three classes of Senators: Class I, elected on years 6n+2, Class II, elected on years 6n+4, and Class III, elected on years 6n. The three classes are to have numbers of Senators as close as possible to each other, differing by at most 1.

Each state admitted to the Union gets its two states randomly assigned to the three classes, to within that class-size constraint. Let's show how the assignment works with a simple example:
Code:
    1   2   3
A   A   A
B   AB  A   B
C   AB  AC  BC
In order of Senator assignment, each triplet of states thus has some permutation of all the possible class assignments: 12, 13, 23.

It is not known to me what order of Senator assignment the first 13 states had, but Rhode Island (the 13th in) and the rest of the states all fit this pattern.


But if the Senate has an all-at-once election, then the House, the Senate, and the Presidency have elections in a 12-year cycle:
Code:
H H H H H H
S     S
P   P   P
 

lpetrich

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I checked on how well the states fit this pattern, and I found that all those after the first 13 states did so. For the first 13, the date of ratification did not give a good fit. But alphabetical order did.
Code:
#   State           Admission Date      Class 
1   Connecticut     January 9, 1788     13    
2   Delaware        December 7, 1787    12    
3   Georgia         January 2, 1788     23    
4   Maryland        April 28, 1788      13    
5   Massachusetts   February 6, 1788    12    
6   New Hampshire   June 21, 1788       23    
7   New Jersey      December 18, 1787   12    
8   New York        July 26, 1788       13    
9   North Carolina  November 21, 1789   23    
10  Pennsylvania    December 12, 1787   13    
11  Rhode Island    May 29, 1790        12    
12  South Carolina  May 23, 1788        23    
13  Virginia        June 25, 1788       12    
14  Vermont         March 4, 1791       13    
15  Kentucky        June 1, 1792        23    
16  Tennessee       June 1, 1796        12    
17  Ohio            March 1, 1803       13    
18  Louisiana       April 30, 1812      23    
19  Indiana         December 11, 1816   13    
20  Mississippi     December 10, 1817   12    
21  Illinois        December 3, 1818    23    
22  Alabama         December 14, 1819   23    
23  Maine           March 15, 1820      12    
24  Missouri        August 10, 1821     13    
25  Arkansas        June 15, 1836       23    
26  Michigan        January 26, 1837    12    
27  Florida         March 3, 1845       13    
28  Texas           December 29, 1845   12    
29  Iowa            December 28, 1846   23    
30  Wisconsin       May 29, 1848        13    
31  California      September 9, 1850   13    
32  Minnesota       May 11, 1858        12    
33  Oregon          February 14, 1859   23    
34  Kansas          January 29, 1861    23    
35  West Virginia   June 20, 1863       12    
36  Nevada          October 31, 1864    13    
37  Nebraska        March 1, 1867       12    
38  Colorado        August 1, 1876      23    
39  North Dakota    November 2, 1889    13    
40  South Dakota    November 2, 1889    23    
41  Montana         November 8, 1889    12    
42  Washington      November 11, 1889   13    
43  Idaho           July 3, 1890        23    
44  Wyoming         July 10, 1890       12    
45  Utah            January 4, 1896     13    
46  Oklahoma        November 16, 1907   23    
47  New Mexico      January 6, 1912     12    
48  Arizona         February 14, 1912   13    
49  Alaska          January 3, 1959     23    
50  Hawaii          August 21, 1959     13
 

lpetrich

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Adding {1,2} for a possible 51st state, I find these triplets of states:
Code:
13 12 23
13 12 23
12 13 23
13 12 23
12 13 23
12 13 23
13 12 23
23 12 13
23 12 13
12 23 13
13 12 23
23 12 13
12 23 13
23 12 13
23 12 13
23 12 13
23 13 12
The triplets have frequencies
  • 12 13 23 - 3
  • 12 23 13 - 2
  • 13 12 23 - 5
  • 13 23 12 - 0
  • 23 12 13 - 6
  • 23 13 12 - 1
I did a chi-squared test with the null hypothesis being equal probability, and I found a p-value of 0.091 - a probability of 1/11 that a random selection would produce a chi-squared value greater than what i found: 9.47.

I also notice some "runs" in the data with the same triplet repeated a few times.
 

lpetrich

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I have extended this work to US states and territories. I have used Ballotpedia and Wikipedia as my sources for almost all of the data that I have collected.

My list includes all 50 states, the US itself, and 5 US territories: Puerto Rico, US Virgin Islands, Guam, Northern Marianas, and American Samoa (PR, VI, GU, MP, AS). I have omitted Washington DC because its government is essentially a city government.

The governors of Vermont and New Hampshire (VT, NH) have 2-year terms, while the President and governors of all the rest have 4-year terms.

The next issue is how their terms fit into our calendar, because (year number) modulo (term length) must be a constant. Here are the remainders after dividing the year number by term length:
  • 0: 14 (11): AS, DE, IN, MO, MT, NC, ND, NH, PR, US, UT, VT, WA, WV
  • 1: 2 (2): NJ, VA
  • 2: 37 (34): AK, AL, AR, AZ, CA, CO, CT, FL, GA, GU, HI, IA, ID, IL, KS, MA, MD, ME, MI, MN, MP, NE, NM, NV, NY, OH, OK, OR, PA, RI, SC, SD, TN, TX, VI, WI, WY
  • 3: 3 (3): KY, LA, MS
Number of US states is in ().

So most state governors are elected in midterms (years 4n+2), and most of the rest in President years (years 4n). VT's and NH's governors are elected in both sorts of years.
 

lpetrich

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I'll now turn to legislatures.

Of these entities, only three have single-chamber legislatures: NE, VI, and GU. Nebraska's is called the Senate because its House was abolished in 1934. The other two are simply called the Legislature.

All the rest have two chambers, a lower house and an upper house. The lower house goes under various names, most often the House of Representatives, while the upper house is always called the Senate. The alternate names for the lower house are:
  • Assembly: CA, NV, WI
  • General Assembly: NJ
  • House of Delegates: MD, VA, WV

Lower houses' terms are almost always 2 years, with the exceptions having 4 years: AL, LA, MD, MS, ND, NE, PR

Some upper houses also have 2-year terms: AZ, CT, GA, ID, MA, ME, MP, NC, NH, NY, RI, SD, VT
All the others have 4-year terms, or mixtures of 2-year and 4-year terms, with the exception of the US Senate, with its 6-year term.

Chambers with more than 2 years per term may have all their elections at once, or their elections staggered, distributed over 2-year intervals.
 

lpetrich

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Here are the one-chamber entities:
  • Governor: 4n+2, Senate: 4n + (0,2) (staggered) -- NE
  • Governor: 4n+2, Legislature: 2n -- GU, VI
The <number>n+<number> is election years. The coefficient of the n is the length of the term. Thus, 4n+2 means 4-year terms and midterm elections.

Here are the entities with all terms equal:
  • 2n -- NH, VT
  • 4n -- PR
  • 4n+2 -- AL, MD
  • 4n+3 -- LA, MS
  • Governor: 4n, House, Senate: 4n+(0,2) -- ND
This covers all entities where the lower house has term length 4 years. All the others have term length 2 years.

For the upper house (Senate) with term length 2:
  • 2n, 2n, 2n -- NH, VT
  • 4n, 2n, 2n -- NC
  • 4n+2, 2n, 2n -- AZ, CT, GA, ID, MA, ME, MP, NY, RI ,SD
Numbers are for the Governor, House, and Senate

For the upper house (Senate) with term length 4:
  • 4n, 2n, 4n+(0,2) -- AS, IN, MO, MT, UT, WA, WV
  • 4n+1, 2n+1, 4n+3 -- VA
  • 4n+2, 2n, 4n -- KS, NM, SC
  • 4n+2, 2n, 4n+2 -- MI
  • 4n+2, 2n, 4n+(0,2) -- AK, CA, CO, IA, NV, OH, OK, OR, PA, TN, WI, WY
  • 4n+3, 2n, 4n+(0,2) -- KY

For the upper house (Senate) with terms having sequences 2 years - 4 years - 4 years
  • 4n, 2n, (2-4-4) -- DE
  • 4n+1, 2n+1, (2,4,4) -- NJ
  • 4n+2, 2n, (2,4,4) -- AR, FL, HI, IL, MN, TX

Here are the ways of doing 2-4-4:
  • 10n+(0,2,6) -- MN
  • 10n+(1,3,7) -- NJ
  • At year 10n+2, all Senators get randomly assigned 2-4-4, 4-2-4, and 4-4-2 for their next terms -- IL
  • At year 10n+2, all Senators run, and they get randomly assigned 2-4-4 and 4-4-2 for their next terms -- AR, DE, HI, TX
  • At year 10n+2, all Senators run, and they get assigned 2-4-4 and 4-4-2 for their next terms to give them 4n for odd district number and 4n+2 for even district number -- FL

This is done because of the redistricting that happens after each US Census, something done at year 10n.

Finally, the national government:
  • President: 4n
  • House: 2n
  • Senate: 6n+(0,2,4)
 
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