# Do any countries use a mathematical formula to divide districts, so that gerrymandering is avoided?

The US is famous for being heavily influenced by gerrymandering, to the point where some of the districts look absolutely ridiculous to the plain eye. The obvious solution is to use some sort of a deterministic mathematical formula for dividing a territory into equal parts, so that no side gets an advantage.

But have any countries actually implemented such a policy? I know gerrymandering is easy to avoid altogether by using proportional voting, however I'm curious about systems that do have voting districts.

• This doesn't avoid gerrymandering. It shifts it to the selection of the deterministic mathematical formula. Because some side will always get some advantage under some circumstance. So the question will be who can better manipulate creation of the formula to match their needs. Commented Jun 30, 2017 at 1:40
• "being heavily influenced by gerrymandering" FiveThirtyEight called, they said your sources are way wrong. "Gerrymandering and other partisan efforts at redistricting do play a role, but it is mostly around the margin. A study by John Sides and Eric McGhee found that redistricting after the 2010 Census, which was controlled by Republicans in many key states, produced a net swing of only about seven House seats toward Republicans.. Fake news is so fake. Commented Jun 30, 2017 at 3:32
• @Brythan, Re "some side will always get some advantage under some circumstance": that always doesn't match those somes -- it amounts to "some side sometimes gets some advantage". The goal of such algorithms being to minimize such occasional advantages impartially -- Governor Gerry's goal was the opposite -- maximizing his party's advantage. Unless someone has a proof that all such algorithmic maximums and minimums are necessarily equal or virtually equal, then better districting algorithms remain a worthy goal.
– agc
Commented Jun 30, 2017 at 4:00
• @user4012 - two points to that - 1) swinging seven seats is a pretty huge deal. 2) your "seven seats" hand-waving seems to assume that there was never any gerrymandering in place before, vs having been done, increasingly, over many, many years. That seven seats represents an increase in the effect of gerrymandering, not the total amount. Commented Jun 30, 2017 at 14:44
• @user4012 One effect of bi-partisan redistricting is gerrymandering to create safe seats. That way both sides know that they don't have to spend resources fighting for those seats at election time, and can then hand the candidacy to those seats to whatever party hack they owe a favour to. This doesn't show up as a party imbalance in the number of seats won, but rather a lack of real competition within each seat. Commented May 22, 2018 at 12:51

## 6 Answers

This is a difficult question to answer because any answer will be heavily dependent on one's definition of bias. To attempt to answer, I'll be working off of the Merriam-Webster definition of gerrymandering:

to divide (a territorial unit) into election districts to give one political party an electoral majority in a large number of districts while concentrating the voting strength of the opposition in as few districts as possible

I believe that Norwegian system of reapportionment avoids both prongs of this definition of gerrymandering. Wikipedia outlines the Norwegian process:

Out of the 169 seats in the Storting, 150 are apportioned among the 19 Counties of Norway with deliberate bias in favor of rural areas. The number of seats for a county is decided using a formula in which a county receives 1 point for every inhabitant and 1.8 points for every square kilometer of land area. However, the bias is reduced by the 19 compensation seats, which are given to parties that are underrepresented. Thus the system does not have a great effect on the partisan composition of the Storting, but does result in more MPs coming from rural counties. Electoral researcher Bernt Aardal calculated that if the 2009 parliamentary election had been conducted without this bias, the Labour Party and Progress Party would both have lost a seat, while the Red Party and Liberal Party would each have gained one, reducing the majority of the Red-Green Coalition from 3 seats to 1.

Specifically:

1. Does the Norwegian system divide election districts to give one political party an electoral majority?

• According to a well-credentialed political scientist, removing geographic weights in the system would have been inconsequential in its outcome. Check out his research (which highlight several other, perhaps even more proportionately designed, electoral systems) (Aardal, B. (2011). The Norwegian Electoral System and its Political Consequences. World Political Science, 7(1), pp. -. Retrieved 30 Jun. 2017)
• The proportionate design of the Norwegian system gives less prominent party organizations more power than the US system does.
• The geographic districts remain largely the same over time
2. Does the Norwegian system concentrate the voting strength of the opposition (in this case, we'll assume that's the minority party(s)) in as few districts as possible?

• Since the geographic weights are counterbalanced with compensation seats, no geography would be disproportionately favored (as outlined in the Aardal study).
• The majority party cannot punish the minority party by changing or modifying district geographies once in power

In conclusion, I believe that the Norwegian system avoids both 1 and 2, uses a mathematical formula, and therefore satisfies the requirements of this question.

I'm also sure that several other countries have similar systems and I'm sure the study I've linked provides further insight into how they work/operate.

• This isn't responsive to this question. This question: does any country use a mathematical formula for dividing into equal parts. This answer: Norway doesn't try to divide into equal parts and instead uses two formulas to compensate for that unequal division. The very existence of compensation seats reinforces my point that there is no way to fairly determine geographic districts. Commented Jun 30, 2017 at 14:55
• The question literally reads " divide districts, so that gerrymandering is avoided". This answer is responsive to that question OP mentions an "obvious solution" being equal parts but doesn't ask about countries that use equal apportionment. Commented Jun 30, 2017 at 15:09
• The title is a summary, not the question. The question says "dividing a territory into equal parts" and "But have any countries actually implemented such a policy?" Commented Jun 30, 2017 at 15:19
• "The obvious solution is to use some sort of a deterministic mathematical formula for dividing a territory into equal parts, so that no side gets an advantage." "But have any countries actually implemented such a policy? I know gerrymandering is easy to avoid altogether by using proportional voting, however I'm curious about systems that do have voting districts." Agree to disagree @Brythan -- just because OP suggests an answer to his own question doesn't mean it's the only answer. Commented Jun 30, 2017 at 15:31
• (-1) Norway uses PR, gerrymandering is a lot less relevant. The OP might have been a bit imprecise in his choice of words but the intent behind the question seems clear to me and this answer does not address it. Commented Jul 1, 2017 at 9:24

While this does not truly address OP's question about other countries, there are states in the USA which, while not purely from mathematical formula, use objective, consistent criteria for drawing their lines.

They way they do this is, by law, instead of the "winning" party getting to drive the process, the process is always done by a non-political, non-partisan body (very, very different from "bi-partisan").

Iowa is the most well-known example of this -

Boston Globe: Iowa redistricting takes the partisanship out of mapmaking

• And the Iowa non-partisan commission drew three Republican districts and one Democratic district. Meanwhile, it would have been trivial to have drawn two Democratic districts and two Republican districts, matching the partisan lean of the state. In California, Democrats should outnumber Republicans, but there are more Republican voters than are seen in the congressional results. Allegedly "non-partisan" commissions are heavily vulnerable to partisan manipulation or simple luck. And, as you note, this doesn't answer the question. Commented Jun 30, 2017 at 15:24
• @Brythan - the question is asking for a foreign non-biased example, based on the assumption that all the USA methods are biased, so it is addressing what they wanted, it just doesn't accept the flawed assumption. Your assumption is also flawed. A "district" is a local sub-division of the state. It is very much geographically based. This idea that the number of districts has to match the total partisan breakdown of the state is an artificial one. Iowa having a three to one breakdown does not represent a flaw. Also, the fact that three voted that way this time does not make it always so. Commented Jun 30, 2017 at 15:32
• If we took a poll of how voters self-identify and tried to artificially create a result that would always match that, that would be a form of gerrymandering, itself. There would be no point to having candidates or elections, no point in having politicians discuss issues if you are going to artificially impose a partisan-centered result like that. Artificial "equality" is just as artificial as an skewed partisan result. Commented Jun 30, 2017 at 15:34
• @Brythan Actually, it wouldn't have been "trivial" to create 2 Democratic districts. If, in an attempt to get 2 Democratic districts, you took 28,000 Democratic votes from the only Democratic district and transplanted them into the district the Republicans won most narrowly, the result would instead have been 4 Republican districts, at least for this election. But this was an election where Trump won Iowa by over 9 points. In 2018, one of those Republican districts could easily swing Democratic, if the winds blow the other way.
– D M
Commented Jun 30, 2017 at 15:58
• "Non-partisan" is not the same as "objective". By definition, "objective" criteria would be criteria that produce the same answer every time, no matter who is doing the deciding. Commented Apr 17, 2019 at 13:35

Short Answer

Single member district election systems are inherently biased outside certain rare distributions of voters that usually aren't present.

Maximal bias can be prevented with historic voting blind formulas for drawing districts, but minimal bias consistent with single member districts (i.e. gerrymandery bias free results) can't be achieved without considering historic voting practices.

Single member district election systems trade systemic regime stability enhancements that it provides, for a less accurate reflection of the public will than proportional representation systems, and whether that is worth it is a value choice.

Long Answer

The Inherent Biases Of Single Member District Only Systems

A system of exclusively electing legislators to a parliament or multi-member legislature from single member districts of approximately equal population is inherently biased.

It is biased against homogeneously dispersed minority factions, even if they are substantial. It is biased in favor of factions that have majorities in geographic concentrated areas (and especially in favor of factions that have majorities in geographic concentrated areas that co-exist with minorities in another faction or factions in the same area).

This bias relative to proportional representation is potentially present in almost all case except those where almost everyone in any given location favors just one dominant political party and the population of those pockets of support for a political party are quite large relative to the population of a typical legislative seat.

When this condition does not hold, a match between a pure proportional representation outcome and the actual allocation of elected officials by party is extremely difficult to secure unless that regions where this doesn't hold almost exactly balance each other out and you have a two party system.

A mathematical formula for drawing boundaries is generally insufficient to prevent this bias from emerging.

There exist maps that minimize the bias that arises from single member district systems relative to proportional representation systems that have only the bias completely inherent in a single member district system. Arguably, when you talk about a map not being gerrymandered, in a context in which a single member district system is a foundational assumption, this is what one means.

But, it is not possible, in general, to minimize that bias merely from knowing the geographic distribution of people on a map. Without knowing their historical partisan preferences, no formula consistent or almost always minimizes the map's bias relative to a proportional representation system.

You can use a "voting history blind" formula to prevent a maximal bias relative to a proportional representation benchmark, but you can't minimize it.

Drawing perfect districts is even harder when accurately representing the relative power of the political parties competing is not the only goal.

For example, the districts that maximize that goal are not the same as the districts that maximize ethnic diversity in the legislative body for which the elections are held are both are legitimate considerations.

These conflicts between competing goals are not nearly so stark in proportional representation systems.

The Case For And Against Single Member Districts Only

There are still arguments in favor of single member districts.

1. Not all factions present an equal threat to the stability of a state. A faction holding a majority support in a geographically contiguous area is a secession and insurgency threat, even if that localized majority is a small share of the nation's total population. So, it is important that such regions perceive that they are well represented in the overall national political process. In contrast, a faction that is a much larger share of the nation's total population, but is a minority everywhere rarely presents a secession and insurgency threat, so it is less important from a national stability perspective to give that dispersed large majority a full political voice relative to its numbers.

2. A single member district, plurality vote system is also very simple to understand and administer. You count votes in each district which is independent of every other district. The person who gets the most votes wins.

Furthermore, in the vast majority of those districts, the outcome won't be remotely close. You have close votes that change control of the country and really matter only when the competing legislative coalitions are very close to 50-50 (which admittedly a two party system naturally gravitates to over time) and in which the swing districts are very close to 50-50. But, in if that happens, a disputed election boils down to just a very simple counting process in just a handful of close races, at a time when the country is almost equally divided between two major parties or coalitions. Both the confined nature of the bona fide disputes and the simplicity strongly disfavor outcomes where a credible election contest is possible, and particularly at fragile moments for a nation's survival, clarity of succession can be more valuable than accuracy (especially when almost exactly half of the country favors each side).

1. Closely related to this point is that when the nation is not almost equally divided 50-50 and one party instead has an exaggerated edge, the winning party will tend to have a legislative majority that is much safer than their electoral majority. This system over rewards winners and over punishes losers. This, in turn, makes it easier for the winning party or coalition to govern after the election in a stable way. Razor thing legislative majorities like the one that the U.S. has now in Congress are rare.

2. Also related to that point is that single member district systems strongly favor the development of a two party political system. Two party political systems are much more prone to having clean majorities for one party or another after an election than systems with three or more parties. A single member district system forces politicians to form their coalitions before the election rather than after it to get elected. So, post-election delays in determining which party is in control that have been common in Belgium, Israel, and historically, in Italy, and which are currently an issue in Germany, rarely arise, avoiding another form of potential instability and uncertainty.

3. And, between elections, a single member district system provides a very direct and clear avenue for a citizen to complain about the government from someone who is more likely than not to be sympathetic to them, without regard to who is currently in power. This sense of being heard, by a particular person who is responsible for them also can reduce the sense of futility that can lead to insurrection and government instability.

In a single member district system you are essentially gaining a system that favors the stability of the regime in the short run, over a system that more accurately reflects the wishes of the population as a whole. Whether the tradeoff is worth it or not is ultimately a judgment call that doesn't have a right or wrong answer. In theory, modern technology and civil order reduce the risks of instability that single member district systems minimize to a tolerable level.

But the fact that the U.S. has experienced widely believed factually false election disputes even in 2020, and the fact that geographically compact majorities tried to leave the Union in 1861 along geographic lines that still largely match modern political division in the u.S., both suggest that the concerns about instability from a system that isn't as simple can't be lightly disregarded.

Persistent, long term bias of the system towards one party and against the other, which most single member district systems are inherently prone to give rise to, can also eat away at public support for the political system and cynicism in the long run, especially if the ends sought in politics are zero sum and high stakes (like control of the U.S. Supreme Court in the U.S. political system).

In South Australia there is a "Fairness Rule", which in practice requires the boundaries to be redrawn after each election so that the party which won the two party preferred vote would have won the election. This does tend to favour the major parties somewhat, but does prevent excessive gerrymandering.

The obvious solution [to gerrymandering] is to use some sort of mathematical formula for dividing a territory into equal parts, so that no side gets an advantage.

A 'deterministic' mathematical formula is not a panancea. It will have spurious precision if the underlyimg assumptions are not examined and shown to be consisyent wirh the aims that the modeller is trying to achieve, here fairness.

Take for example in the formula you suggest and let us see what happens when one district is mostly rural with a low population and the other side is mostly metropolitan with several large cities, or just one very large city with a very high population? Then a party has to win far fewer votes in the former to elect a representative than in the latter and this seems manifestly unfair, seeing that the latter has many more people

Take for example London, with 8 million people whereas Devon has less than a million. A better solution is to go by demography and have districts with roughly the same number of people. For example the constituency districts of Devon have roughly 80,000 people.

An even more important reason that mathematics is not a universal panancea is that electoral process needs to be secure from manipulation and no mathematical formula, deterministic or otherwise, can protect from such manipulation. This requires an electoral supervisory body that is free of political pressure.

For example, in the 2019 Israeli elections, over 1200 Israeli activists with bodycams were sent to Arab-Israeli polling booths which, as was widely noted, intimidated that electoral demographic. This isn’t going to be fixed by a ‘deterministic mathematical formula.’

• I think you're making an assumption that London wouldn't be subdivided by the mathematical formula into multiple seats. The point is to get the formula to draw the lines for where those subdivisions are, rather than people; the simplest way would be to draw the shortest line that cuts the number of people on each side in half, and then repeating this process until you have seats of the desired size. Commented Oct 19, 2021 at 2:09
• @nick01200: The mathematical formula mentioned by the OP only refers to 'territory'. He says nothing about demography. Moreover, the formula you're suggesting is too mathematical and takes no account of the politics of the situation. This is why independent redistricting commissions that know an area are a much better idea. Commented Oct 19, 2021 at 2:46
• @nick012000: Since you are so keen on mathematics, the circle has many such lines - in dact, infinite - and it's possible to design other shapes that have many more than one. Commented Oct 19, 2021 at 4:54
• @nick012000 many counties in the US are square, so there will be at least two equally short lines that cut the county's population in half. Also, your algorithm will need to be modified for states whose congressional delegations are not a power of two. For example, if there are to be three districts, what do you do after drawing the first line that divides the population into two halves? Maybe you have to change the first step: divide the population into two parts, one of which is twice as large as the other. In a square there are four such lines, and the choice will be politicized. Commented Oct 19, 2021 at 9:51
• @nick012000 another factor is that where political sentiment is evenly spread geographically, blindly balanced districts will result in congressional delegations that overrepresent the majority. For example, suppose there are 40 Needle party voters for every 60 Thread party voters. If every district reflects that proportion then there will be no Needle party representatives whatsoever. Gerrymandering can bring district-based systems closer in effect to proportional representation as well as farther from it. Commented Oct 19, 2021 at 10:24

I don't know if such a policy has ever been used. But it's worth noting that such policies do exist. For example, one could implement a law that says that anyone can submit a proposed set of districts after a census, and the proposed set of districts with the lowest average distance between a pair of voters in the same district wins.

For another possibility, one could define the "diameter" of a district to be the greatest crow-flies distance between any two points in the district, and say that the winning set of districts is the one in which the average diameter of all the districts is lowest.