## Fractional Voting Power

I read an interesting article on the paradoxes involved in allocating seats for the Congress. The problem arises because of two rules: one congressperson has one vote, and the number of congresspeople per state should be proportional to the population of said state.

These two rules contradict each other, because it is unrealistic to expect to be able to equally divide the populations of different states. Therefore, two different congresspeople from two different states may represent different sizes of population.

Let me explain how seats are divided by using as an example a country with three states: New Nevada (NN), Massecticut (MC) and Califivenia (C5). Suppose the total number of congresspeople is ten. Also suppose the population distribution is such that the states should have the following number of congresspeople: NN — 3.33, MC — 3.34 and C5 — 3.33. As you know states generally do not send a third of a congressperson, so the situation is resolved using the Hamilton method. First, each state gets an integer portion of the seats. In my example, each state gets three seats. Next, if there are seats left they are allocated to states with the largest remainders. In my example, the remainders are 0.33, 0.34 and 0.33. As Massecticut has the largest reminder it gets the last seat.

This is not fair, because now each NN seat represents a larger population portion than each MC seat. Not only is this not fair, but it can also create some strange situations. Suppose there have been population changes for the next redistricting: NN — 3.0, MC — 3.4 and C5 — 3.6. In this case, NN and MC each get 3 seats, while C5 gets the extra seat for a total of 4. Even though MC tried very hard and succeeded in raising their portion of the population, they still lost a seat.

Is there any fair way to allocate seats? George Szpiro in his article suggests adding fractional congresspersons to the House of Representatives. So one state might have three representatives, but one of those has only a quarter of a vote. Thus, the state’s voting power becomes 2 1/4.

We can take this idea further. We can use the Hamilton method to decide the number of representatives per state, but give each congressperson a fractional voting power, so the voting power of each state exactly matches the population. This way we lose one of the rules that each congressperson has the same vote. But representation will be exact. In my first example, NN got three seats, when they really needed 3.33. So each congressperson from New Nevada will have 1.11 votes. On the other hand MC got four seats, when they needed 3.34. So each MC representative gets 0.835 votes.

Continuing with this idea, we do not need congresspeople from the same state to have the same power. We can give proportional voting power to a congressperson depending on the population in his/her district.

Or we can go all the way with this idea and lose the districts altogether, so that every congressperson’s voting power will be exactly proportionate to the number of citizens who voted for him/her. This way the voting power will reflect the popularity — rather than the size of the district — of each congressperson.

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1. #### Jonathan:

Or we can conclude that the states themselves are anachronistic. Do we really have a union of sovereign states?

Just divide the total population by 435, and set up districts with that number of people. I guess we could try to avoid crossing the old state lines more than necessary.

And the Senate. You should write about the senate, too. The problem is easier for people to follow. Idaho, Wyoming, Montana, and the Dakotas have 10 senators. About 5 million people. Brooklyn, Queens and Staten Island have 5.2 million, and they elect no senators on their own (they get 2 as part of 19.5 million voters in New York State.)

2. #### Anonymous Rex:

I don’t know if you’re aware, but the House is not apportioned using Hamilton’s method. The method used is called the Huntington–Hill method. I assume people don’t usually mention it because it uses geometric means and hence “is complicated”.

3. #### Christ Schlacta:

I also like the idea of sending 1/4 of a congress person to washington as my vote.

4. #### Philip Petrov:

That works perfect… if the “voters” are really thinking people. They are not. End of the democracy story. Sorry 🙂

5. #### Felipe Pait:

I’m ok with my representative but I don’t think many scientists would argue against the proposal of quartering one of my senators. Maybe feeding the sharks would be too much…

6. #### possiblywrong:

I like your last suggestion as well, better than Szpiro’s idea of “mostly integral plus one fractional” representatives. (I made the same suggestion just a few months ago here after reading his book, Numbers Rule.)

Regarding your comment that “this way we lose one of the rules that each congressperson has the same vote,” it is worth noting that this is not one of the rules. All that is required is that “Representatives shall be apportioned among the several States according to their respective numbers.” See a 2008 paper by Toplak (a Temple law professor) here for some interesting details.

7. #### JBL:

Actually it seems to me that the biggest problem with this suggestion is how to assign particular voters to particular representatives — would you want to be in the district that got a representative who only got 1/4 of a vote? (Well, if your region is 80% Republican and you are a Democrat, then perhaps you would.) It seems to me that proportional representation and/or multi-member districts solve this problem more easily.