Hill's method An Unsuccessful Attempt to Improve on Webster/Sainte-Lague: by Mike Ossipoff
I've kept saying that Webster/Sainte-Lague puts each party's seats as close as possible to what is called for by a common ratio between seats & votes. It minimizes the difference between each party's seats and the number of seats called for by that common ratio.

Likewise, in one article I showed that, starting with a Webster allocation, transferring a seat from 1 party to another would always increase the difference between the seats per vote ratios of those 2 parties. Again, we're talking about the difference between those ratios.

But someone could say: Isn't the ratio between those parties' seats per vote ratios the more important measure of how much they differ? The factor , or ratio by which they differ from each other, rather than their difference? Similarly, when we divide a party's votes by the common divisor, to determine how many seats it should have, and we want to put that party's seats as close as possible to that number, which isn't a whole number, is not the factor or ratio by which that party's seats differs from what the common divisor would give it the more important thing? Shouldn't we be minimizing that instead of the difference?

Yes, that is more appealing, I would say. The trouble is that it results in a biased seat allocation method, a method that consistently favors the smaller parties. This disqualifies it as a PR method.

So then we can't go by factor, and we have to go by difference. Roundng off to the nearest whole number minimizes the difference between the party's whole number of seats, and the number of seats that that common divisor would give it.

The seat allocation that would result if we went by the factor by which 2 numbers differ, rather than their difference, has a name: "Hill's method".

Hill's method is currently in use to apportion the U.S. House jof Representatives. In 1941 Congress enacted Hill's method to apportion the House, replacing Webster's method, which had been law since around 1900, and which had actually been in use for much longer than that.

Hill's method differs from Webster's method in how it rounds to the nearest whole number. Webster does it in the way that we would expect, and Hill does it so as to minimize the factor or ratio by which that whole number differs from the original number to be rounded.

Not only is Hill's method biased toward small parties (or small states when used for apportionment), but it's also considerably more complicated than other allocation rules.

But there's one possible way that Hill's method could be part of a U.S. proposal: In a country which apportions seats in the national legislature to subunits of the country (states, provinces, etc.) a PR proposal could dodge the issue of allocation rules by just saying: "The seats in each state (or province, etc.) shall be allocated to parties according to those parties' votes by the same allocation rule that apportions seats to those states (or provinces, etc.) according to their populations. In the U.S., this would amount to proposing Hill. One could argue that, biased or not, if Hill's method is officially regarded as the unbiased, proportional allocation, then who could argue with it for PR?

But bias is much more tolerable and likely to be accepted in apportionment than in PR, so the wisdom of such a proposal in the U.S. is debatable. For instance, Congress is already heavily weighted in favor of small states, by giving each state 2 senators & at least 1 member of the House of Representatives. The additional bias caused by Hill is minor. But a PR proposal biased toward small parties could generate much anger & opposition.

The allocation methods and their merits is the subject of a book entitled Fair Representation, by Balinski & Young (Yale University Press).