# How to get rid of gerrymandering: the math is surprising

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I be mad for items, because they are the fancy!
Gerrymandering is bad, right? Okay, so let’s get rid of gerrymandering.

How do we do it?

Let’s re-examine the nation of Katechon, which we have mentioned before. (Let’s assume everyone has been magically healed.)

This nation has exactly 1,000,000 adults who can vote.

It is divided up into 10 districts, each of which sends one representative to the national legislature, which consists of 10 people.

The nation has 2 parties, the blues and the violets. During most elections, 600,000 adults vote for the blues, and 400,000 vote for the violets.

However, through a quirk of history, the violets were briefly able to draw the district lines, and they drew them most unfairly:

4 districts hold 400,000 blues

6 districts hold 33,333 blues and 66,666 violets.

So the violets win 6 districts and therefore they gain control over the national legislature, even though they only make up 40% of the voters.

That’s rotten!

What would be more fair?

Well, the public is divided 60% blues and 40% violets, so surely it would be fair to divide up each district that way?

10 districts each hold 60,000 blues, and 40,000 violets

Congratulations, you’ve now created a totalitarian one-party dictatorship! The blues hold 60% of every district, so they win every district, so 100% of the national legislature is in the control of the blues, even though they only hold 60% of the vote.

Hmm, okay, so what would be fair? Perhaps we want to give each party a fighting chance in each election? Isn’t that fair?

2 districts each hold 100,000 blues

8 districts hold 50,000 blues and a 50,000 violets

So now the violets have a 50% chance of winning 8 districts. The blues are granted two districts automatically, but that is fair, since they are more popular than the violets. Each party has some chance of winning.

However, in this model, if 1 blue voter in each district changes their mind, then the violets end up with 80% of the legislature. We’ve engineered a system that is very unstable! If Katechon is like the nation of Hungary, where a 2/3rds majority can amend the constitution, then Katechon is now vulnerable to an authoritarian takeover, much like what happened in Hungary after 2010. And that’s because 8 people out of a 1,000,000 changed their vote.

6 districts hold 100,000 blues

4 districts hold 100,000 violets

Perfect! Now the national legislature will be 60% blues and 40% violets, just like the people themselves.

But wait, what if someone changes their mind? What if the blue politicians, knowing they have an unbeatable advantage, start to become corrupt, arrogant, out of touch, immoral?

The problem with the system that we just built is that it is very rigid. In the blue districts, if 10% of the public turns against the blues, the vote in that district becomes:

90,000 blue

10,000 violet

If 20% changes their mind:

80,000 blue

20,000 violet

The blues would need to lose more than 50% of their vote, in two of their districts, before they would lose control of the legislature. The system is rigid and will tend to keep the blues in power long after they’ve become unpopular. If it was unfair when the violets took control of the government, with just 40% of the vote, it is also unfair when the blues do it. And we have now created a system where the blues can remain in power, even when the majority of the public is voting against them.

In fact, to push the blues out of all 6 of their districts would require 300,006 people to shift their vote, and that’s 30.00006% of all voters. Assuming the discontent, against the blues, builds up equally in all districts, it would take a landslide to finally end the rule of the blues. They would be down to just 29.00094% public support before they finally lost power.

In short, nothing but a shocking landslide could push the blues out of power, and till that moment comes, the blue politicians might grow rather corrupt, knowing how entrenched their power is.

What would help?

Above all else, we could make the districts smaller. But again, if the discontent is evenly spread among the districts, it takes a landslide to push the blues out of power: 300,000 people.

What if we make the unrealistic assumption that the discontent piles up in one district, until everyone in that district changes their mind, and then the discontent starts to build up in the next district, and then the next, and then next?

In other words, what is the minimum number of people who need to change their mind to get the unpopular blues out of power?

We start with 10 districts of 100,000 people each. 6 districts with 600,000 blues. To change 1 and a half districts:

150,001 people need to change their mind

What about 100 districts of 10,000 people each? 60 districts with 600,000 blues. To change 10 and a half districts:

105,001

What about 1,000 districts of 1,000 people each? 600 districts with 600,000 blues. To change 100 and a half districts:

100,501

What about 10,000 districts of 100 people each? 6,000 districts with 600,000 blues. To change 1000 districts and a half:

100,051

What about 1,000,000 districts of 1 person each? 600,000 districts with 600,000 blues. To change 100,001 districts:

100,001

So the numbers look like this, as the districts shrink in size:

150,001

105,001

100,501

100,051

100,001

In summary, we can say, as the districts get smaller, then the size of the shift needed (to punish the previous majority party) describes the asymptote of the tangent, converging towards the reality of what the public actually wants.

You may not be happy with the conclusion: Gerrymandering is universal in any political system that has geographic units that demand representation. No matter how you draw the lines on the map, those lines on the map will have an impact that creates some distance between the will of the voters and the actual outcome. Some people want to believe that there must be some way of drawing the lines on a map so that the result is intuitive, natural, and fair — but they are wrong. The math proves it. In the USA the 1964 Voting Rights Act tried (in part, indirectly) to limit the use of gerrymandering for results that were blatantly unfair — but this task is hopeless. As soon as you draw geographic districts and grant each block of land its own representation in a higher level assembly, you need to surrender any hope of results that are intuitive, natural, or fair.

Some countries, such as Israel, have purely party based parliamentary systems with no geographic representation. We wrote about this previously:

Some nations have fine-grained representation of land but course-grained representation of ideas

Some nations have fine-grained representation of ideas but course-grained representation of land

You cannot have both, unless you’re willing to have a legislature that has something like 5,000 people in it, which is not practical. (For instance, the USA has 50 states, so if we wanted 100 representatives from each state, the legislature would have 5,000 people in it — clearly not realistic.)

Keep in mind, if we get rid of all geographic units, we still haven’t reached perfection. In 1953, Kenneth Arrow published his Impossibility Theorem about voting. He later won a Nobel Prize in Economics (doesn’t exist, blah, blah, blah) for this work among others. He showed that rank voting of any kind fails to aggregate voter preferences. He later expressed some optimism about “approval voting” where voters get an infinite number of votes (or rather, enough votes to vote for every candidate). Among the many problems to be solved, we need a voting system that is fine-grained enough that small random variations don’t lead to a landslide in the unpopular direction, swamping the main trend.

This is the crucial point: there are many problems with our current systems of voting. Many of these problems have been mathematically proven and well-documented for much of a century. If we want to make democracy stronger, then we are eventually going to have to get serious about fixing these problems. Some of these problems are tightly bundled with other problems so that several problems will have to be addressed simultaneously.

But let’s focus on gerrymandering for a moment. If we want to attack it, then we have to get rid of geographic units. Whether at the local level, or the national level, everyone voting in a given polity should be free to vote for the entire slate of candidates who have been nominated for that level of government.

For instance, the city council. Suppose there are 100 candidates. Suppose we implement approval voting, so the voters have infinite number of votes (or rather, in this case, 100 votes). Suppose the top 9 vote getters will be declared the winner, and then constitute the new city council.

According to Arrow’s informal speculation about approval voting, such a system should do a better job of representing the will of the majority (certainly compared to one vote=one candidate systems).

However, we now run the risk of establishing a tyranny of the majority, so it might be necessary to ensure some minority representation. Perhaps every voter gets 5 votes and the top 9 vote getters are elected — that ensures some minority representation. Here the logic is the opposite of the Parliamentary systems, which typically enforce a minimum cutoff for representation in the legislature (5% in Germany, 3% in Finland, effectively 0% in Israel). By contrast “5 votes, top 9 winners” describes a maximum of minority representation. As with all of these cutoffs, the decision is arbitrary and has to be made based on what is seen as good for the nation. Germany enforces a high cut off because it has traumatic memories of the 1920s and 1930s and so it hopes to keep small, extremist parties from getting a foothold in the legislature (because once inside, the small, extremist parties can gain some legitimacy from the mere fact of being in the legislature).

We previously looked at this idea as it might apply to staggered elections to the legislature, with several candidates elected each year, for very long terms. We also explored the idea of monthly voting, combined with long terms, so that elections to the national legislature might be extremely staggered, and thus insulated from any particular panic or rage or other extreme public passion.

But let’s go back to the issue of gerrymandering. Here we can say, as a matter of basic math, gerrymandering will exist in any voting system that has geographic representation. You can reduce the influence of gerrymandering by making geographic districts small, but the only way to fully eliminate the influence of gerrymandering is to eliminate all geographic representation. The whole citizenry of a polity (of whatever size, and at any level, city or province or national) needs to be free to vote for any candidate who is running for leadership of that polity.

If we get rid of gerrymandering, what do we lose?

Your town assembly can still take care of the concerns of your town.

Your regional assembly can still take care of the concerns of your region.

Your National Assembly can still take care of the concerns of your nation.

We can get rid of gerrymandering and still have different levels of government, each devoted to a particular level of localism. You can still have an elected person who is responsible for your local concerns. But there is no reason to have one level of localism interfere with the affairs of a higher level of localism.

UPDATE:

Conversation on Hacker News.

Someone on Hacker News wrote:

An unstated assumption here is a winner-takes-all election system

In terms of the math, increasing the number of representatives from each district is the same as making the districts smaller. As we said above:

“As the districts get smaller, then the size of the shift needed (to punish the previous majority party) describes the asymptote of the tangent, converging towards the reality of what the public actually wants.”

And the same applies to increasing the number of representatives from each district: as the number increases, the shift needed (to punish the previous majority party) describes the asymptote of the tangent, converging towards the reality of what the public actually wants. But there will remain some distortion, however small. You eventually get into infinitesimals that have no practical impact, but it is still there. If you have districts that only have 1,000 people each, and each district elects 1,000 representatives (everyone in the district is elected) then the distortion disappears, but if you have districts with 1,000 people, and they only elect 999 representatives, the distortion reappears, however small that might be. At some point the distortion is so small that it does not matter, but reaching that level would take a very large assembly.

In the comments, someone asked why I dismiss the possibility of a very large assembly. It is impossible for us to repeat every idea in every essay, but we suggested a very large intermediate assembly a few days ago.

UPDATE 2022-05-12

Some people, who wrote comments in response to this essay, were focused on the intentional dishonesty of gerrymandering, and they wrote things like this:

“You can easily get rid of gerrymandering simply by proscribing an algorithmic process for districting.”

There were several people who said this, here and on Reddit and on Hacker News. I believe they are defining gerrymandering as a deliberate and intentional act. By contrast, I was using the word “gerrymandering” to refer to the fact that any geographic boundary will establish a gap between the will of the voters and the final result of voting. So even if an algorithmic process is followed, and the damage done is accidental and unintentional, I am still referring to that as “gerrymandering.” Indeed, the whole point of my essay is that if one group does its best to be fair when drawing lines on the map, and another group is evil and immoral and corrupt, and does its worst when drawing lines on a map, the damage done by both groups will be about the same, over the long-term.