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9. Brand new state, Brand new state,

gonna treat you great[1]:

The New State Paradox

The question:

In 1907, Oklahoma was the 46th state to join the Union. The House of Representatives at the time consisted of 386 seats. After having suffered through the Alabama Paradox in 1880 and agonized over the Population Paradox in 1901 (see the chapters on these), the congressmen thought that this time they knew exactly what needed to be done to keep everyone happy. Since the population of the first 45 states stood at 74.5 million, Oklahoma’s population of about one million corresponded to five seats in Congress. So they decided to simply add five seats to the House. Obviously, Oklahoma would receive them, nobody would get hurt and everybody would be happy.


The paradox:

Or so they thought. The five seats were added and to nobody’s surprise, when the new total of 391 seats was re-allocated, Oklahoma got them. But something strange happened along the way. New York lost a seat which Maine picked up! It was infuriating. For once everybody had done everything right, and then this happened. The situation was called the New State Paradox. Let us see how it came about.

According to the allocation method then in force, each state first of all received the integer part of the raw number of seats. Then the remaining seats were allocated to the states with the largest leftover fractions. Before, the additon of Oklahoma, there was one remaining seat and New York’s leftover fraction of 0.606 beat Maine’s 0.595. So, New York got the additional seat.

Once Oklahoma joined the Union, the situation was reversed. Maine’s new leftover fraction of 0.594 beat New York’s 0.589. The second and third digits after the decimal point determined the lucky winner.


          Population                          Population

          before incl.       Seats           after incl.       Seats

          of Oklahoma     raw    rounded      of Oklahoma     raw  rounded                               

New York   7,264,183    37.606*   38          7,264,183      37.589   37

Maine        694,466     3.595     3            694,466       3.594*   4

Oklahoma        --         --      --         1,000,000       5.175    5

Total     74,562,608              386        75,562,608                391


  * rounded up



In the chapters on the Alabama Paradox and the Population Paradox I recounted two of the three problems that may arise when non-integer number of seats must be allocated to the states. Once the Alabama Paradox had been identified, the problem was solved by the Congress’s decision henceforth to hold the number of representatives fixed. (That did not stop Congress from increasing the size of the House once more after the 1910-census to 435, a number where it stands until this day.)

So Alabama no longer was a threat and the Population Paradox of 1901 never actually happened. It was only a numerical exercise by number crunchers who wanted to show what could have occurred, had a re-allocation of seats taken place then.

But the addition of Oklahoma did happen. After addition of the five seats – which went, as they should, to Oklahoma – New York lost a seat to Maine. New York had done nothing untoward, its population had not changed. Nevertheless, through no fault of its own, the state was punished with the loss of a seat. And Maine, through no good deed of its own, gained one. New York had to grit its teeth and wait for three years, until the next census of 1910, at which time the state was allocated 43 of the 435 congresspeople. (Maine remained at 4 and Oklahoma got 8.)



Even though the populations of New York and Maine did not change, and Oklahoma was awarded exactly the five additional seats that it deserved, apportionment of the remaining seats was severely affected. By adding Oklahoma’s population to the nation’s total, the fractional seats of all states diminished somewhat. To its misfortune, New York, being the largest state, lost more of its fraction (0.606 to 0.589) than did Maine, the smaller state (0.595 to 0.594). This is how Maine’s remainder managed to inch past New York’s.

As I pointed out in the chapter on the Population Paradox, remaining fractions are random; their sizes do not reflect the sizes of the states. Hence, with calculations based on the leftover fractions, it is purely by luck or by misfortune that a state gains or loses the additional seat. Therefore, one cannot claim that New York was unduly punished when it lost its seat to Maine because, with the same auhority, one could maintain that New York had been improperly rewarded in 1900 when it obtained the additional seat.


Technical Supplement:

In 1983, Michel Balinski and Peyton Young proved mathematically that divisor methods – that search for a divisor which, after rounding the results up or down, gives just in the desired number of seats – avoid not only the Alabama Paradox and the Population Paradox but also the New State Paradox. Unfortunately, they also proved that the divisor method is not quite fair. (See the chapter on the Population Paradox.)

When New Mexico and Arizona joined the Union in 1912 (1 seat) each), the divisor method was in force and the New State Paradox did not rear its head. It did not either in 1959, when Alaska (1 seat) and Hawaii (2 seats) joined. By then, the ‘method of equal proportions’ – don’t ask -- was employed to assign seats…..

Well, if you must know, the method of equal proportions “assigns seats [in the traditional manner], except it rounds fractional remainders of the quotient of the state population divided by the ratio differently. With this method, an additional seat is assigned if the fraction exceeds the difference obtained by subtracting the integer part of the quotient from the geometric mean of this integer and the next consecutive integer.” (I warned you, not to ask.[2])



[1] Lyrics from the musical Oklahoma!

[2] https://www.census.gov/population/apportionment/about/history.html

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