# When is a half not a half?

Poblano Nate notes an important distinction between cutting the Florida delegation in half vs. giving each delegate half a vote.

The distinction is in the way that the delegates are divided up in individual congressional districts. Take for example a district that Clinton won 70-30, and that originally had 4 delegates. If you do the multiplication, you get 2.8 fractional delegates for Clinton and 1.2 for Obama, which rounds up to a 3-1 delegate take for Clinton.

But now suppose that this district only has 2 delegates because Florida’s delegation has been cut in half. With her 70 percent of the vote, Clinton wins 1.4 fractional delegates, and Obama 0.6. However, Clinton’s number now rounds down to 1 delegate, whereas Obama’s rounds up to 1 delegate.

Of course, if the Democratic Party used D’Hondt like most proportional-representation systems, 70-30 would still give 3-1 in a 4-seat district, but 2-0 in a 2-seat district. ((Hare quotas and largest remainders, on the other hand, would go 3-1 and 1-1. That’s as decent an illustration as any as to why Hare quotas and LR are rarely used in self-contained districts.))

Anyway, Nate has much more to say…
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## 0 thoughts on “When is a half not a half?”

1. Tom Round says:

Australia uses “round up all and only remainders over half the quotient” (and consequently, a variable number of seats) for the House of Reps. As a result, it is possible for – eg, 10 + 10 to equal 19, or 21 (eg, if two States with 9.6 and 9.7 quotients, or 10.4 and 10.3 quotients, of population each were combined).

I should’ve called my own blog “Rounding Error”. Too late now.

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3. Reluctant as I am to round on Tom, the Commonwealth Electoral Act rounds to the closest integer. In 2004 this cut Northern Territory representation from 2 to 1. See the House of Representatives (Northern Territory Representation) Bill 2004 digest for what happens when a territory falls short of a second MHR by 295 people.

And yeah, someone should make it their mission in life to persuade the US Democrats to adopt a universal rule for delegate allocation, and preferably one that involves a rational allocation like d’Hondt with balance seats in the event the statewide allocation does not reflect the popular vote in that state.

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5. Tom Round says:

From Attorney-General (NSW); Ex rel McKellar v Commonwealth (1977) 139 CLR 527, per Stephen J (High Court of Australia):

[Â¶16] The pre-1964 method of dealing with remainders necessarily maintained with greater accuracy the principle of proportionality. I have so far referred in general terms to proportionality and to its importance in ensuring a just share of representation in the lower House as between one State and another, having regard to their respective populations. The quota system provided by Section 24 of the Constitution requires that a particular ratio of representatives to population should be observed, that ratio being one representative for every quotaâ€™s worth of population in a State. This is the proportionality of which Section 24 speaks, the proportion which a Stateâ€™s population bears to its representation. It results in each State being accorded its just share of representation, subject always to a minimum representation of five, because the same ratio or quota is to be applied in the case of each State. (at p 555)

[Â¶17] However, because representatives must be whole men or women, proportionality creates a problem of fractions, to be resolved by some rule of approximation of which Section 24(ii) provides an example. The rule which it provides and which, until 1964, Section 10 of the Act also adopted, provides, in effect, that the nearest whole number shall prevail. By doing so it ensures that no violence is done to the principle of proportionality. In contrast, the present rule of approximation, introduced by the amendment made to Section 10 in 1964, under which any fraction earns further representation, does not preserve to the same degree the principle of proportionality. (at p 555)

[Â¶18] The problem posed by the impossibility of fractional representation in a legislature has long been recognized as intractable. Daniel Webster, in his report of the United States Senate on the apportionment of 1832 (Foster on the Constitution (1896), vol 1, 430-46, at p 434), observed that â€œof representation, there can be nothing less than one representative.â€ Faced with this fact and the imperfect reflections of proportionality to which it gave rise, the report concluded that nevertheless â€œCongress is not absolved from all rule, merely because the rule of perfect justice cannot be applied. In such a case, approximation becomes a rule;â€¦ The nearest approximation to exact truth or exact rightâ€¦ prevailsâ€¦ not as a matter of discretion, but as an intelligible and definite ruleâ€ (p no 435). This is a rule to which Section 10 originally conformed but which, in its present form, it ignores. (at p 556)

[Â¶19] This may be seen from the following illustration. Assume a ratio of one representative to 10,000 inhabitants and imagine a multi-storied building served by a staircase having 10,000 steps between each floor, up which staircase winds a queue of the Stateâ€™s population, one to each step. The attaining of each floor entitles the State to one more representative. As the Stateâ€™s population increases, the head of the queue proceeds up the staircase from floor to floor, its representation increasing by one representative for each floor attained. If population numbers jumped 10,000 at a time there would be no problem, but in fact the queue lengthens or shortens, step by step, not floor by floor, and only by some rule of approximation can entitlement to representation be determined at any particular state of the queue. That rule should be one that best conforms to the constitutional requirement that representation be proportionate to numbers of the people. If the queue extends, say, up past floors one and two, the Stateâ€™s entitlement will then be at least two representatives and if it reaches only a few steps above the second floor an entitlement to two representatives will ensure close approximation to the ratio of 1: 10,000. However, once the queue lengthens so that its head is closer to the third than to the second floor, a closer approximation to the ratio will result from an entitlement to three, rather than two, representatives. (at p 556)

[Â¶20] The rule of the nearest whole number will provide these results. Proportionality will be well maintained, consistently with the need for approximation, by adopting, as a Stateâ€™s entitlement, that floor number, that is to say that whole number of representatives, to which the head of the queue is closest. This is precisely the way in which the pre-1964 rule of approximation operated. (at p 556)

[Â¶21] The post-1964 rule departs from this. Under it, as soon as the first step above the second floor is attained, representing a population of 20,001, the State is, as it were, deemed to have extended its queue up to the third floor, thereby entitling it to a third member although in fact its population has barely risen above the second floor. (at p 557)

[Â¶22] Since it is with approximation that these rules are dealing, their application will always be subject to that degree of departure from the maintenance of true proportionality inherent in any rule of approximation.

However if the mid-point in the quota or ratio (with a quota of 10,000, or a ratio of 1: 10,000, the mid-point is 5,000) represents the point of true proportionality, where the head of the notional queue has just reached the landing of a particular floor, the scope for disproportionality is confined to a minimum since it will never exceed half the quota or ratio, in the example, 5,000; add 5,001 people to the queue and the floor above becomes the nearest to the head of the queue; the State becomes entitled to one more representative and the departure from true proportionality drops to 4,999. In the case of the post-1964 rule the point of true proportionality is not at mid-point but right at one end of the range. Accordingly it provides scope for far greater departure from true proportionality. In its case as soon as the head of the queue rises one step above the second floor an entitlement to a third representative arises, although the third floor is still 9,999 steps above. The scope for lack of proportionality is thus virtually double that under the pre-1964 rule. (at p 557)

[Â¶23] This feature of the post-1964 rule of approximation remains a defect despite the fact that, whichever rule of approximation is applied, the maximum extent of relative inequality of representation which may occur as between any two States when fractions remain after dividing the quota into a Stateâ€™s population, will always be the quota minus one. As already observed, it is proportionality in the sense of the maintenance of a specific quota or ratio of inhabitants of a State per representative that Section 24 requires; it is only consequential upon this requirement, and because the same quota or ratio is applied in all States, that equity with respect to representation in the House of Representatives, is achieved as between States. (at p 557).

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6. Tom Round says:

When Australia phased out one- and two-cent coins in 1991 â€“ leaving five-cent pieces as the smallest physical unit of currency â€“ most supermarkets voluntarily adopted the Federal Governmentâ€™s recommended standard that, in every cash register transaction, amounts of money ending in 1, 2, 8 or 9 cents would be rounded off to the nearest amount ending in zero, while amounts ending in 3, 4, 6 or 7 would be rounded off to the nearest amount ending in 5.

Then some enterprising sort made the news when he started going through check-outs buying one or two grapes at a time â€“ worth only 1 or 2 cents each in total â€“ and insisting that, because this rounded down to zero cents, he was entitled (under the store policy as advertised at every checkout) to get them for free.

I believe this argument even actually succeeded once or twice, until the stores concerned amended their policies to fix a five-cent minimum for every transaction!

(Weirdly, I recently re-read an old physics textbook from the 1980s which seemed to state – in so far as I could cipher the authors’ turgid writing style – that a remainder of exactly 0.5 should be rounded off to the nearest even number, rather than consistently up or down – ie, 7.5 should become 8 but 6.5 should become 6 (6.6 or 7.4 would still become 7). Do I understand this correctly and, if so, what is the rationale?)

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