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8. Virginia versus Maine:

The Population Paradox

The question:

In representative democracies, seats in legislators are usually allocated according to the size of the population of the district. The proportion of the district’s population in the nation’s total is computed and… that’s where the problems start: virtually all such proportions are fractional numbers but the number of representatives must be whole numbers, i.e., integers. Rounding the proportions to the nearest integers usually results in a total that is one or two seats long or short of the desired number of representatives.

As we will see below, a method was devised that allocates only integer numbers to the districts. But the states’ populations keep growing and they usually do so at different rates. Then, when the time comes to re-allocate seats, a state whose population grew more during the previous period than the population of another state always gets a larger part of the pie.



The paradox:


Or at least, sometimes it is wrong.

The method that the US House of Representatives used to allocate integer numbers to the districts, proceeded in two rounds. In the first, an appropriate divisor was sought, such that when the states’ populations were divided by this divisor and all results were rounded down, only a few seats were left over. In the second round, the leftover seats were allocated to the states with the highest leftover fractions. But there was a problem.

According to the census of 1900 -- after allocating the integer parts of the ‘raw’ seats (9 to Virginia, 3 to Maine) -- Virginia’s leftover fraction, 0.599. was larger than Maine’s, 0.595. Hence, according to the method in use, Virginia received the leftover seat.

But the populations grew during the following year, Maine’s by 4,648 and Virginia’s by 19,767.  if a new House had been appointed in 1901, Virginia – whose remaining fraction this time around would have been 0.509 and hence smaller than Maine’s 0.548 – would have lost a seat to Maine, in spite of its larger population growth.


                  1900                          1901

                Population  Seats            Population Seats

                            raw    rounded                raw   rounded

Virginia       1,854,184   9.599*  10        1,873,951   9.509   9

Maine           694,466    3.595    3          699,114   3.548*  4

Total          74,562,608         386       76,069,522         386

* rounded up     

A paradox!



In principle, a state with three times the population of another state, should have three times as many representatives. This is the principle of proportionality. But populations continue to increase and since the size of the House of Representatives is fixed at 435, whenever one state gains a seat, another loses one. If this leads to a violation of the requirement of proportionality, Congress is faced with the dreaded population paradox.

The ‘raw seats’ of both states fell in 1901 because the nation as a whole grew faster than either of the two states – from 74,562,608 to 76,069,522. The fall for Virginia was less (from 9.599 raw seats to 9.509, i.e., 0.94%), than for Maine (1.31%). But this was not the reason for the dis-proportionality. To glean the real cause for the paradox, the fractional part must be scrutinized in isolation.



In 1900, Virginia obtained the additional seat purely by chance. After awarding the integer part of the raw seats, it was the third digit after the decimal point that determined which state was due another representative in the House. A year later, it was the second digit after the decimal point that awarded the additional seat, this time to Maine.

Fractional remainders vary randomly between 0 and 1 and do not reflect the states’ relative sizes. Thus, it is no wonder that they may violate proportionality. Any method that is based in some way on fractional remainders may lead to the population paradox.

A method was required that respects proportionality. In the 1980s, the mathematicians Michel Balinski and Peyton Young stipulated that a good allocation method should be ‘fair’. (See the Technical Supplement.) They found that the so-called divisor method was a possible candidate. In this method an appropriate number is sought – the divisor – such that, when dividing each state’s population by this divisor and then rounding up or down, the appropriate number of seats is allocated. One simply tries one divisor, and if the total number of seats turns out to be too small or too large, a larger or a smaller divisor is chosen, and the process is repeated.

Alas, the divisor method is not always fair. Even worse, Balinski and Young proved that any method of apportionment that is not based on divisors falls victim to the population paradox. The depressing conclusion is that allocation methods are either unfair or are susceptible to the population paradox. What a letdown for representative democracy!


Technical Supplement:

To be ‘fair’ an allocation method must award each state no more and no less than its quota, and it must be unbiased.

By quota, the two mathematicions meant the number of raw seats, rounded down or up by no more than 0.5. If a state’s number of raw seats is, say, 5.8, the fair share is either 5 or 6.

By unbiased is meant a system that does not systematically favor either large states or small states. Over the long run, advantages and disadvantages must average out. It does not come as a great surprise, that rounding at the arithmetic mean, i.e., at 0.5, is historically the least biased method.

So all that remains is to satisfy the quota requirement. At first glance it seems that this should be easy. Of course, rounding up or down leaves the allocation in quota.

Well, it does not. Balinski and Young proved that if there are four or more districts, and the legislature has at least three more seats than there are districts, then “there is no method that avoids the population paradox and always stays within the quota.

Let’s say that 36 seats are to be apportioned among four states. A first atttempt to allocate seats – with the divisor 44,444 and rounding up or down at 0.5 –  results in a total of 37 seats. So we redo the calculations with the divisor 46,000. This time the total works out to the desired 36 seats.

Fair share (Quota)

                      ‘raw’                   Divisor 46000          

    State Population  Seats*   Rounded         Seats Rounded   


    AA      70000     1.58     2             1.52     2      

    BB     112000     2.52     3             2.57     3      

    CC     208000     4.68     5             4.61     5      

    DD    1200000    27.23    27            26.30    26


    Tot.  1600000    36.01    37            35.00    36      


But check this out: State DD received only 26 seats even though the raw amount was 27.23. The ‘fair share,’ would have been either 27 or 28 seats. Hence, surprisingly, simple rounding can violate the quota requirement.

The reason is that rounding the raw seats of a small state entails a greater adjustment than when the same is done for a large state. The quota of a state with 1.5 raw seats spans 66 percent (33 percent when rounded up from 1.5 to 2, and another 33 percent when rounded down). A state whose raw allocation is 41.5, on the other hand, has a quota that spans less than 2.5 percent. Hence, for a large state the quota requirement is more easily violated.

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