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For a hypothetical world, how can I build a political system in which:
Having the maps drawn by an independent 3rd party, or an algorithm, is not a solution as people tend to self-gerrymander.
Gerrymandering is only an issue because a 50.001% majority for a precinct and an 80% majority are considered equivariant. We also consider 2 politicians as having equal vote on bills regardless of district size. In America; Montana has 994k people per congressman, Rhode island 1st district has 526k people. Those politicians have equal voices in congress despite one having twice the people as the other.
We also have an issue where the "losing" party's members have no voice. Democrats in that Montana district (who make up at least 40% of the population) may not feel represented as there is only a single congressman, who is Republican.
So - how do we fix this:
Now:
Would also suggest:
Downside:
Note: This answer was written when the question was on Worldbuilding SE, the site for building consistent fictional worlds.
The German System (simplified):
Germany uses a form of Mixed-member proportional representation:
A number of seats are allocated to individuals in first-past-the-fencepost districts. At the same time, voters cast a second vote for proportional party preference. Then the size of the legislature is adjusted so that the combined delegations have the right proportions.
Example: There are 100 districts. Party A gets 80 of them, party B gets 19, one goes to an independent. Overall, A gets 60% of the proportional vote, B gets 35%, party C gets 5%. The legislature consists of 80 direct candidates from A, 20 direct candidates and the top 26 list candidates from B, and the top 6 list candidates from C, and the independent.
Drawbacks: In theory, the size of the legislature could become infinite. And the outcome depends on complex calculations.
A prescribed allocation mechanism:
Simply declare that the boundaries of election districts must run north-south. When there are shifts in demographics the lines are shifted, but they are never bent into a pretzel.
Drawbacks: Inconvenitently shaped districts.
Mediaeval Iceland style:
Throw out the geographical restrictions entirely. Anyone who gets the backing of a certain number of people becomes a representative entitled to speak and vote at the meetings. Where those people live doesn't matter. Who the representative is doesn't matter. There are no fixed-time elections, any person can move their backing to someone else at any time with some notice period.
Voting power of representatives is proportional to their number of backers. Number of actual representatives in the government may vary. If desired the lower limit may be removed and individuals may come vote in person if they wish. The practicality of individual voting depends on population size.
Note that this does not require representatives to be from the local geographic area of most of their backers, but if local representation is actually important to the people, then they will choose their representatives accordingly, and if local representation is not the people's primary concern then it is difficult to argue why they should be forced to choose what they consider to be an inferior candidate just because that person lives nearby. Unlike a fixed-district system they aren't disenfranchising anyone by their choice.
This also mitigates the other main problem of fixed-size districts which is that the majority faction gets the candidate they want and everyone else has no voice to speak of. In any fixed-district that is closely divided as much as 50%-1 of the population may be getting utterly shafted at any given election cycle.
The disadvantage is that it's more bookwork to keep track of who has how many votes, leaving more opportunity for fraud. My personal suggestion for reducing the incidence of fraud would be to set the penalty somewhere between "burnt at the stake" and "flayed alive", but that is likely influenced by my personal distaste for most politicians.
Vote is not the only way to reach democracy. In fact, some argues that it prevent real democracy, and lead to oligarchy, because by selecting representetives, you delegate power, because vote is sensible to corruption, because those that take part of an election want power, and many other arguments.
That's why Ancient Athens, one of the first democracy, that's been seen as an example by modern democracys, used another system : Demarchy
The kleroterion, a sorting machine, was used to choose randomly citizens that will speak in the Agora.
Law of large number makes it very representative : if half the population is poor, half of the agora will be poor. If a third of the Agora lived in a given neighborhood, a third of the Agora will be from that neighborhood.
This have a caveat that you need large enough numbers to have the most representative representents. But if you have enough of them, both of your bullet points will be respected.
If this isn't good enough for you, go even further, and adopt direct democracy. Grant your citizen the capacity of doing popular referendum, and build a constitution around it, so most of the power come directly from the citizens and not some representatives. You can't have a better representent for yourself than yourself itself.
The most interesting approach to this I've heard of was during the Alternative Vote referendum in the UK, where "oh but local representatives" was a major opposition argument. The proposal was essentially to divide the region into constituencies, and conduct a single-winner (ie put-a-cross-against-one-candidate) ballot across the whole region, and tabulate the results. (The ballot is for individuals representing parties; independent candidates are fairly screwed in this system, unless they are grouped into a 'None of the Above Party'.) The electoral commission then adds up the total number of non-spoilt ballots and divide by the desired number of seats in the chamber; this is your parties' 'target threshold'.
Each party looks at the scores they achieved in each constituency. They can now announce a winner by picking one or more geographically contiguous constituencies which, taken together, provided enough votes for the party to take them over the threshold. They pick one of the individuals who were on the ballots in those constituencies, and that person is elected the member for that party, for those constituencies. At the same time, other parties are using their votes in the same constituencies to appoint their own members.
The result of this is a chamber which is essentially proportionate, with an 'overlay' of constituency mappings. Minor parties might have a couple of candidates whose 'territory' spans dozens or hundreds of constituencies, whereas the major parties will have more members with much smaller territories. Almost every voter should be able to identify an individual member of the party they voted for who 'represents them'.
I think that this system is much less susceptible to gerrymandering than a normal winner-takes-all system, although probably not immune. I haven't seen it formally defined anywhere, though, so I don't know if any proper analysis of it has been done.
In much of Europe it is proportional representation. Gerrymandering is not usually an issue.
For example, here we have X parties for any election (frequently around 20), which decide to field candidates in any given voting districts.
After the election, all the parties that got at least 5% of total votes across the country, are assigned seats in parliament proportional to their results.
It usually is around 5-7 parties that get there, and then they can decide on coalitions to have parliamentary majority.
People are represented by someone local to them.
That depends on your definition of "local". As soon as you want to make it more local, you are nudged into direction of direct democracy with people representing themselves only. As long as somebody else is representing you, you will get non-local representation at some point.
The number of representatives elected representing a movement in total remains proportional to the popular vote?
Seems a potentially significant problem in first-past-the-gate system only.
No system can perfectly solve this problem mathematically, fundamentally it is an issue of aggregation and discretisation. When you aggregate a voter pool, you necessarily lose information about the individual opinions of those voters and imperfectly capture those in their representation (by party/candidate). The level you discretise to will determine how much information is lost, and at some point you must discretise because a law/policy must be enacted or not; you can't have 42.15% of a Death Penalty!
However, there have been various moderately successful attempts at "Mixed-member proportional representation" systems. For example in Scotland where regional Members of Scottish Parliament (MSPs) are elected by FPTP and then List candidates are selected based on the votes that went to unsuccessful FPTP candidates.
Of the 129 MSPs, 73 are elected to represent first past the post constituencies and are known as "Constituency MSPs".... The remaining 56 MSPs, called "List MSPs", are elected by an additional members system, which seeks to make the overall results more proportional, countering any distortions in the constituency results
https://en.wikipedia.org/wiki/Scottish_Parliament#Members,_constituencies_and_voting_systems
It is also worth noting that this system has it's issues too, including theorectical discussion as to whether splitting a party into separate FPTP/PR parties could increase the number of seats from the same voting pool by gaining more list candidates, without losing FPTP candidates.
Add each representative to the system
If in a district 20% vote on Alice, 30% on Bob and 50% for Charlie, all 3 are added to the system. When voting on issues, each get their respective piece of the vote. All votes of all districts are then summed, meaning each n % is added for the total votes for an issue. Everyone is represented by their percentage of vote.
This way it doesn't matter how a district is drawn, as in your examples it'll always add to the same amount of votes.
A lower bound is recommended to prevent 0,2% of vote representatives, so there won't be 100 representatives in a district. Each of the voters on a person below the threshold should vote again so their vote isn't lost.
I actually built a system like this: Full description, including examples and notes. Is it any good? Don't know, but it's exactly what you asked for. The condensed version follows:
Benefits:
How it works (voting):
The first key difference is that when representatives vote on legislation, their vote is no longer just worth one vote. Instead, they have a base voting power equal to the number of votes they received in the most recent election. If rep A got 1234 votes in that election, their "yes" vote on a piece of legislation is really 1234 votes.
The second key difference is that the losers in an election will pass their votes to their party. The party will then distribute those votes to the members of the party that made it to the House.
(Optional but recommended.) There are also a few at large seats reserved for parties that were unable to seat any district-based reps.
How it works (elections):
Each election is individual, and can not change the results of any other election. Voters vote for a candidate who had a party affiliation. Once all votes are counted, the candidate with the lowest vote total is eliminated, and their votes distributed to other candidates as desired. Repeat until there is only one candidate left.
Example ballot:
Note that this is basically a First-Past-The-Post ballot. The only difference is that the party marking is actually important, and the bit at the bottom that shows how votes will be passed between parties.
This is a simple mathematical question. It has already been solved, in an objective/empirical manner. It's called the "shortest split line" algorithm.
https://en.wikipedia.org/wiki/Gerrymandering#Shortest_splitline_algorithm
It is provably, mathematically correct. Gerrymandering is eliminated with this method. It can't be gamed or cheated. For all non-trivial real-world populations, it just works. And it was possible to implement this with 1960s computing, if not earlier.
I don't know if someone has already mentioned what New Zealand does, called Mixed Member Proportional?
https://elections.nz/democracy-in-nz/what-is-new-zealands-system-of-government/what-is-mmp/
I think it covers exactly what you're asking, each voter gets TWO votes, one for a local member and one for the party they want to represent them. Local members, if elected, get a seat, and the remaining seats are filled by the parties proportionally.
Hope this covers your question; have a read of their election coverage, too, they just had an election this weekend gone.
Biproportional Apportionment is immune to gerrymandering, and ensures local representation.
Briefly, it proceeds in two steps:
As we can see in step 1, district boundaries have no effect on the number of seats a party is assigned, making gerrymandering impossible. Also, each region is represented by candidates from that region. However, regions are not necessarily represented by the majority candidate of that region. For instance, if you have 3 districts, with the following total votes:
| District | Party A | Party B |
|---|---|---|
| A | 60% | 40% |
| B | 55% | 45% |
| C | 52% | 48% |
District C would be represented by the candidate from party B, since party B deserves a seat due to having over 33% of the vote.
Abolish district boundaries. Instead of districts, the legislature just has "seats". A candidate runs for Seat N. Voters are randomly allocated to each seat for every election. Every representative represents every part of the whole.
Yes, this could mean that "your" representative lives on the other side of the state/nation/planet. But there's a good chance that that person also got votes from some of your neighbors. Likewise the representative that lives just down the street probably got votes from other of your neighbors.
Since this is World Building*, and because you chose the utopia tag, I'm going to assume the world can handle some near-future sci-fi. Start by developing a benevolent super artificial general intelligence, then have it conduct polling and monitoring (utopian monitoring, not big brother monitoring) of all citizens. Then give it the power to select potential representatives based on who will represent the people the best, and who has been determined to create the most positive outcome for the entire population. The people still get some choice, but ultimately the utopian AGI is guiding the outcome.
*Note: This was originally posted on the world building area with the utopia tag.
Geographic constraints are always in conflict with proportionate representation
The requirement for local representatives is in conflict with proportionate representation. It also leads to a two-party system where neither party truly represents any group of people or ideals and instead both parties are mainly fighting for popularity. (See the United States)
To achieve proportionate representation you could instead drop all geographic constraints and have voters vote for the party they are most aligned with instead of an individual representative. Then the seats in the legislative body can be distributed to the parties proportionately based on the votes. The parties can then determine which representatives they will send to the legislature, possibly allowing registered party members to vote on this.
This of course does not satisfy your "local representatives" constraint, but I would argue that a party delegate 1000mi away is more representative than a local majority delegate 5mi away.
As long as the government can change the law without the people's consent — it's impossible
Gerrymandering exists in any form and in any government because the government can adjust district boundaries without the consent of the people. Not that the consent of the people would significantly help or is even the issue. My point is, as long as the incumbents have the authority to change those districts, the people are proverbially hosed.
1. Redistricting will always be necessary, so it must be permitted
Today you may have a thriving residential neighborhood housing 1,000 voters. Tomorrow that same geographic location may have a shopping mall, a junk yard, and low-value commercial property housing 100 voting stalwarts who think nothing should ever change. On the flip side, today you may have 500 acres of farm land housing 2 voters and a lot of cows, but tomorrow there's a thriving master-planned community with 1,000 voters. There will always be reasons to redistrict, so the process of redistricting must exist.
2. How you redistrict isn't as important as controlling the process of redistricting
Once we admit we need redistricting, the next problem is ensuring that, once we develop a fair and equitable solution (if such a thing exists, both tend to be in the eye of the beholder), we need to make sure it's whomping hard for Representative Lastforever to simply change those boundaries to ensure his continuing reign of terror public service.
Using The United States as an example, we could create a Constitutional Amendment that identifies exactly how redistricting occurs and that's the law. Period. End of Story. Representative Lastforever loses the race if the demographics after constitutional redistricting don't favor him. However...
Part of the problem with the U.S. Constitution (and it's there for a very good reason) is that all powers of authority not explicitly identified as belonging to the Federal government automatically belong to the States. And that's one reason why gerrymandering exists in the U.S. Because redistricting is controlled at the state level and it's a lot easier to control 50 state constitutions than it is 1 federal constitution.
An astute observer might say, "yeah, but you could achieve the same effect by enshrining the process of redistricting in state constitutions." That's 100% true, kinda. I believe most state constitutions in the U.S. are changed by majority vote of the population (otherwise known as "popular vote") which means the old big-city vs rural-town problem exists. The Federal constitution requires 67% of Congress to propose a new Amendment and 75% of state legislatures to ratify it. In practice, that's honking hard to do. It has only happened 27 times in 244 years (and 10 of them happened right on top of ratifying the Constitution in the first place).
So, we remove from local government (and especially from incumbent legislators) the ability to change how redistricting occurs.
3. All that leaves is coming up with a "fair and equitable" solution
The problem with what I just proposed is that it's basically considered better to tolerate gerrymandering than to encode in so rigid a manner a solution that (perish the thought) needs to change. The problem with law, politics, philosophy, and people, is that they all change. The U.S. today is fundamentally a two-party governing system. We've tried to bring more parties into it — but it's basically a two-party system. If, by an act of pure magic, more parties suddenly come into being, that would change how redistricting should, would, or could occur — except that we just entombed it in a legal device that's specifically designed to be whomping hard to change.
That same astute observer might point out that the problem here is that people's political leanings change — and the point of gerrymandering is to take advantage of that fact. So long as you're trying to provide a "fair and equitable" solution on the basis of the changing opinions of the population, you'll always lose. To quote Whopper: "An interesting game... it appears the only way to win is to not play at all." In other words — the only way to be completely immune to gerrymandering to to never district anything.
Conclusion: it remains impossible
Can you come up with a system that's "fair and equitable?" Ignoring the simple truth that "fair and equitable" is always interpreted in the eye of the beholder, the answer is "sure, for today." Once you come up with one, there are ways to make it basically impossible to gerrymander.
But the price you pay for that convenience today might incur a high price in the future.
Which is why the fight to discontinue the use of the Electoral College has been so long, and will continue indefinitely. It's very hard to predict the future, and sometimes the devil you know is better than the devil you don't.
It has a whole bunch of problems of its own, especially if your population distribution changes dramatically, but it's worked for centuries in a variety of political systems (notably still including the US Senate), and is completely impossible to gerrymander after the initial setup.
In each district (which can be gerrymandered to your heart's content), following an election the sealed ballots are collected and one is drawn at random to select the representative. The probability that a candidate wins is proportional to the percentage of votes cast for that candidate.
This has several potential advantages
Possible disadvantages
Randomize then contigu-ize
EDIT: Now that this question has been moved from Worldbuilding to Politics, this answer may not be appropriate.
This is deliberately overcomplicated and based on assumptions that are not met in the real world, but could be fun in a whimsical hypothetical world. I think it would work but I could be wrong. There would definitely be cases where it failed, even when a better outcome is possible. It is designed to give random composition of districts and try "pretty hard" to make them "contiguous-ish."
Suppose that everyone had an anonymous "account" in some central system, and this account kept track of where they lived and how they had voted in the past. Then you could start by throwing out the geographic constraints and randomly assigning everyone to a district.
Once you've done that, you could use some kind of algorithm to identify pairs of people that are similar to each other (based on their voting record), far away from the rest of the people in their district, but close to the people in the opposite district. Those pairs of people would be switched. You can generalize this from pairs of people to pairs of sets of people.
You would repeat this a bunch of times. Each step would maintain random-ish party composition and move closer to contiguousness. But you would probably not end up with "full contiguousness."
Would that work?
Having the maps drawn by an independent 3rd party, or an algorithm, is not a solution as people tend to self-gerrymander.
I'm not sure what you're referring to by 'self-gerrymander'. Gerry-mandering is trying to force as many of the other party's votes into one district as possible, leaving a majority for the other party in as many other districts as possible.
But it isn't that black and white either. With an equitable system, both parties would be drawn more to the middle to garner as many votes as possible. In your diagram I think the 'proportionate outcome' where you have 3 solid districts for one party and 2 solid districts for the other party would be the worst outcome. At that point all representatives are basically party appointments and there is no real choice at the ballot box.
I think an impartial algorithm is precisely the way to fix it.
One possible system that I haven’t seen mentioned yet is the Weimar Germany system. Overall, this system gives proportional representation, i.e. parties are represented by a number of members in parliament that corresponds to their vote share. However, Weimar implemented a neat idea (in my opinion) that also ensures local representation: regionalised lists.
In very basic terms, Germany was split into 35 electoral districts along political subdivisions. (Ideally this might be along state lines only but considering Weimar Germany had a huge state called Prussia alongside very tiny ones like Anhalt or Lippe-Detmold the larger states were split into multiple districts while several smaller states or smaller states and provinces of Prussia were combined into single districts. The important point is to have non-volatile lines to prevent gerrymandering.) These districts were grouped into district collectives while maintaining some regional indentity within these district collectives (e.g. the district of East Prussia – what is now northeastern Poland and the Kaliningrad Oblast – was its own district collective as it was an exclave; the mostly Bavarian-speaking parts of Bavaria were combined as a district collective, the Franconian-speaking parts of Bavaria were a separate district collective, etc.).
Parties would submit lists in each electoral district. After tallying the votes, a party would gain one member from a district for 60,000 votes. If a party had more than 30,000 remaining votes in a single district, this party’s votes in all districts of the district collectives would be added; for each sum of 60,000 remaining votes the party would gain an additional seat which went to the list of the district that gained the most votes. Finally, the remaining unallocated votes of district collectives were transferred to a national level; here again 60,000 votes would give an additional seat. If ultimately at least 30,000 unallocated national votes remained, the party gained one more additional seat. (There were additional rules which do not matter here.)
Obviously, this system can be broken down more; in modern Germany, there are 2 or 4 subdivisions depending on the state (not counting the city states) you are in: federal states, (provinces in some states such as Bavaria,) counties (Landkreis) or county-free cities (kreisfreie Stadt), and finally communities (Gemeinde) or sometimes city boroughs (Stadtbezirk, Stadtteil) in county-free cities. One could easily use the county level as the lowest allocation level (carving out larger towns from within counties if necessary) and then simply move upwards along existing political subdivisions. This would ensure that each area has a local-ish representative for most elected parties; the more votes a party received in an area the closer its next candidate would be.
One can easily toy around with the list submission requirements. Maybe a single person is enough per district if a party does not expect to gain more than one member (or if it is mathematically impossible to do so). Maybe lists are submitted at a higher level but candidates broken down to individual lower levels according to their place of residence, where applicable. Maybe a national list is used for all remaining votes elevated to the national level instead of using the local overhang (Weimar did this).
Furthermore, one does not have to use a fixed number of votes to gain a seat. For example, one could also use one threehundredth of the valid votes total if one desires a 300-member legislature. One could also require a party to at least gain a certain threshold of national votes before they are considered at all. And so on.
Remove the individual politician and assign localism after the election on party specific boundaries.
Elect the whole group of legislators in a single PR election. Assign parties a number of representatives proportional to their vote share with a minimum percentage capped by seat allocation. (Not sure on math here it's not just divide by seats 50% required in 2 seat state is obviously wrong).
After the election each party publishes its own localisation map showing geographic responsibility for each elected members. Can be pre-published estimates if locally this is something voters want to see.
Parties may choose to have their own multimember districts if they feel that allows better allocation of party resources.
Disclaimer: This was a Worldbuilding answer, more suitable as a starting point for creating an interesting and plausible political system for a fictional society. Not a "we should implement this now in the real world" answer!
OK, so the self-gerrymandering that you mention in your question is not actually gerrymandering. Gerrymandering results in a disproportionate representation compared to what you would expect looking at the overall population.
Consider 100 people, divided into 10 districts, and half support the purple (P) party, half support the grey (G) party. We expect to end up with 5 P representatives and 5 G representatives. We can create some example districts... The first example is self-segregated, the second and third are gerymandered (and the third has unevenly sized districts):
PPPPPPPPPP PPPPPPPPPP PPPPPPPPPP PPPPPPPPPP PPPPPPPPPP GGGGGGGGGG GGGGGGGGGG GGGGGGGGGG GGGGGGGGGG GGGGGGGGGG
GGGGGGGGGG GGGGGGGGPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP GGGGPPPPPP
GGGGGGGGGG GGGGGPPPPPP GGGGGPPPPPP GGGGGPPPPPP GGGGGPPPPPP GGGGPPPPPP GGGGPPPPP GGGGPPPPP GGGGPPPPP GGGGPPPPP
In the first (self-segregated) case, you actually get "perfect" representation--everyone is represented by the person they voted for. In the second and third case, only just over half of people are represented by someone they voted for (the P voters, plus the G voters who live in a district that is allowed a G representative). Specifically, in the second case you end up with 8 P representatives and in the third case you end up with 9 P representatives!
So here is my frame challenge: Why is it important that someone is represented by someone who is local to them? If your world has even modern technology, you may have a lot more in common with someone halfway around the world than with a person who lives down the street. On Stackoverflow we vote for moderators not based on where we live, but based on which sites we belong to. Etc.
From Robert Heinlein's The Moon is a Harsh Mistress:
Suppose instead of election a man were qualified for office by petition signed by four thousand citizens. He would then represent those four thousand affirmatively, with no disgruntled minority, for what would have been a minority in a territorial constituency would all be free to start other petitions or join in them. All would then be represented by men of their choice. Or a man with eight thousand supporters might have two votes in this body. Difficulties, objections, practical points to be worked out — many of them! But you could work them out. . . and thereby avoid the chronic sickness of representative government, the disgruntled minority which feels — correctly! — that it has been disenfranchised
The same character also has a few other ideas for how representatives could be chosen, and it's easy to come up with variations and then see how it could work in your world.
Allow me to pull out one of my pet theories: electoral quorums.
First, Let's understand the problem. In the Founders' era, communities were small, well-defined, stable, and relatively homogenous. Districts were meant to reflect meaningful real-world differences between these communities, and it was assumed that the community would choose one of its own members: someone who was both familiar with the needs of the community itself and motivated (through close association) to represent them fairly and well. I imagine this worked reasonably well through the first few election cycles: perhaps up and through the Civil War. But as the industrial era set in, with its dramatic increase in immigrant labor, the rapid expansion of territories in the West, and the ever-increasing mobility of populations, the idea that an electoral district had any relation to real-world communities began to evaporate.
These days, electoral districts are little more than abstract divisions of the population. They may have a rough geographic cohesion, but they do not (generally) depict cohesive, real-world communities. As such, most of the nineteenth century assumptions about the relationship of a district to its representative are defunct. Representatives may not know or associate with the constituents they ostensibly serve; they may not even be long-term residents of those districts (except in the minimal, formal sense that they own a residence there). As a rule, Representatives have closer ties to the party that runs their campaigns, to the caucus they join in Washington, and to corporate interests that have financial investments in the district they represent, than they have with the actual citizens living there.
This distancing of the representative from the electorate is what leads down the road to gerrymandering. Pragmatically, a party has no reason to worry about representing the interests of the citizens in any particular district, because the party assumes that such abstract divisions do not represent actual communities, and such an abstract grouping will never act as a cohesive group in any notable or problematic way. Instead, the party is interested in maximizing the number of representatives it can extract from these abstract groupings. That naturally evolves into gerrymandering: redefining these abstract districts so that as many districts as possible have a clear minority of opposing party voters.
The ideal gerrymander (which is rarely if ever achievable in practice) would be to lump all of the opposition voters possible into one district — where they will always win one representative in a landslide victory — and to distribute all the other voters equally across all the remaining districts, so that the opposition becomes a minority in those districts, and loses. Thus for a state with six districts and a roughly 50/50 split between voters for party A and party B, an ideal gerrymander for party B would create one district that was 100% voters for party A, and then equally distribute the remaining voters across the other districts. District 1 would be a 100% landslide for party A, guaranteeing one 'A' representative in Congress; Districts 2 through 6 would would be 60/40 majorities for party B, guaranteeing five 'B' representatives in Congress (net +4 to party 'B'). Obviously few gerrymanders reach that perfect ideal, but it's easy enough to milk out one or two or three extra seats for one party or the other, regardless of the actual preferences of constituents.
Now, lots of different people have wrestled with various mathematical, geometric, geographic, or demographic solutions to this puzzle, many of which have some merits. But the weakness of all of these approaches is that they do not address the central problem: the problem of the representative's political detachment from his constituents, in which constituents are abstracted away by the political system. Our goal is to somehow re-bind the representative to the constituents, so that the interests or parties, caucuses, and corporate investments do not so spectacularly outweigh the interests of ordinary citizens.
The solution as I see it, is to institute electoral quorums: rules which state that an election is not valid unless a minimum given percentage of the eligible voting population — the quorum — casts a vote. If the quorum is not met, no one wins that district: the challenger does not go to congress, the incumbent leaves office, no one can be appointed by the governor... The seat stays empty until a new election is held which meets the quorum requirement, no matter how many elections that might take. It is the constituents' opportunity to say: "We don't like what we are seeing, and we're taking our ball and going home until we get what we want." For instance, consider what might happen in the ideal gerrymander I discussed above under quorum rules. In District 1, people are happy, and vote in the party 'A' candidate with gusto, giving party A one representative. In Districts 2 through 6, though, party 'A' voters are angry about the obvious gerrymander, and decide to abstain from the ballot; if the quorum rule is set high enough, all of these elections become invalid, and no one goes to Congress from either party. (net +1 to party 'A'). Alienating voter blocks by trying to manipulate or ignore them becomes decidedly bad business for political parties — and worse business for candidates — so gerrymandering and a number of other political ills will fly right out the window.
Quorums of this type are extremely common; they are explicitly written into Robert's Rules of Order, which is the foundation of decision-making rules in Congress, institutional settings, and corporate boardrooms. In fact, public elections are the only large decision-making system I can think of that does not institute some form of a quorum system to protect against political shenanigans. Adding them to our election process would re-empower the citizenry and force many public representatives to adhere closely to the interests of their constituents, both to ensure their own election and to prevent the opposition from gaining effective power because of vacancies where quorums are not met. It would obviate gerrymandering as a strategy, and act as a check-and-and balance on a representative's ability to stray from what their public wants.
Single Member Districts With Bonus Seats
Start with some variant on the single member district system. Have voters cast a single vote for the candidates running in their single member district. This will give you a baseline number of seats in the legislature for each party.
But, also track, how many voters are cast for candidates of each party collectively. Award bonus seats to a party that receives fewer single member district seats than it does a percentage of the vote, to reconcile the total proportion of seats in the body to the percentage of the vote favoring members of each party.
Perhaps a threshold requirement of one single member seat is required to get bonus seats so that a party has to have majority support somewhere, to exclude extremists parties that aren't very popular in any one place.
Example
For example, suppose that you have 100 districts. Party A wins 45 seats, Party B wins 40 seats, Party C wins 10 seats and Party D gets 5 seats.
But, Party A got 40% of the vote, Party B got 40% of the vote, Party C got 15% of the vote and Party D got 5% of the vote.
If you give 5 bonus seats to Party B, 7 bonus seats to Party C, and 1 bonus seat to Party D, for a total of 113 seats, you now have:
Party A 39.8% of the 113 seats with 45 single member seats Party B 39.8% of the 113 seats with 40 single member seats and 5 bonus seats Party C 15.0% of the 113 seats with 10 single member seats and 7 bonus seats Party D 5.3% of the 113 seats with 5 single member seats and 1 bonus seat.
Overall there are 100 single member seats (88.495% of the total) and 13 bonus seats that represent voters who were unrepresented because of how district lines were drawn and how voters are distributed, which are selected from party lists.
Analysis
The numbers in the example aren't random. Big national parties with a big share of the vote tend to receive close to their share in single member districts with a moderate bias in favor of one or the other (like Parties A and B). Dispersed political support from smaller percentages of the population leads to much great underrepresentation in single member districts (like Party C). But some small parties do well in single member districts because they are geographically defined (e.g. Party D might be a nationalist party seeking autonomy for some geographical district).
It has the virtue of being predominantly territorial, while adjusted to the minimum extent necessary to avoid gerrymandering bias, and also reduces the incentive to gerrymander in the first place, since it doesn't lead to political advantage due to bonus seats.
It also had the virtue of being very easy for a typical voter to understand their role in.
New Zealand once had a system like this before it switched to a mixed member proportional representation system in 1996 that captures the same concept in a somewhat different way.
