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The Logic and Fairness/Unfairness of Gerrymandering
Rick Garlikov

Gerrymandering legislative districts is the structuring of populations into districts in a way which is mathematically more likely to get one's party elected in more districts, even if there may be a majority of voters of the other party in total in the area covered by all the districts as a whole.  And it is considered to be an unfair way of influencing or 'rigging' elections.  While contriving districts to be populated in ways whose election outcomes are likely predetermined can be unfair, it can also be fair in some cases to the extent of at least reducing the amount of unfairness which would otherwise occur.  However, I will argue that if this is done mathematically alone by some sort of merely geographic grouping of districts, it will never be able to be fair in some important ways no matter how it is structured.

To start with, suppose that a large section of a state is 60% Republican and 40% Democratic.  A Democratic legislature in the state at the time district lines are drawn or redrawn could make one or a few of the districts contain most of the Republicans, who then handily win those districts, leaving a much larger number of districts with majorities of Democratic voters.  And if the state legislature passes legislation based on the votes of the majority of district representatives, that would give the minority party Democrats far more power than their proportion of voters in the state would justify or be fair.

Just to show how the math of this works, suppose we were trying to divide 500 marbles into 25 boxes of 20 marbles each.  And suppose 300 marbles are red (representing Republicans if you like, or just being marbles) and 200 are blue (representing Democrats if you like, or just being marbles).  I will show that it is possible and not terribly difficult to divide the marbles/voters in a way that can allow the red ones to control everything, the blue ones to control everything, or for there to be an equal number of red and blue districts/boxes.  Hence, how districts are structured can change or control the power structure and balance of power immensely.

Dividing These Voters Into Districts to Give Red Total Dominance of Everything
If you divide the marbles proportionally into the boxes, each box will have 12 red ones and 8 blue ones.  All 25 boxes will be red dominant.  If these are legislative districts, all the districts would be majority Republican, and, all else being equal, would always elect the Republican candidate or vote for the Republican position on any issue.  (Of course, not all else is equal; some Republicans may vote the Democratic side of an issue or vote for a particular Democratic candidate in preference to the particular Republican candidate.  But just for the sake of illustration assume that districts will vote for the policy or person that their majority party wants, with any independents splitting their votes in a way that does not affect the outcome.)  So the boxes would look like this:

 Box 1 Box 2 Box 3 Box 4 Box 5 Box 6 Box 7 Box 8 Box 9 Box 10 Box 11 Box 12 Box 13 Box 14 Box 15 Box 16 Box 17 Box 18 Box 19 Box 20 Box 21 Box 22 Box 23 Box 24 Box 25

If votes were taken by simple majority rule along strict party/color lines for each district then every vote would be won by the red marbles/Republicans, and 60% of the marbles/voters would determine 100% of the policies and programs, effectively allowing 40% of the marbles voters no say in any policies, and thus effectively disenfranchising them, even though they get to vote; it is simply that their vote cannot determine anything.

Dividing These Same Voters Into Districts to Give Blue Total Dominance of Everything
But if we were to put 138 of the red marbles all into 5 boxes of only red marbles and into 2 boxes of all but 1 red marbles (see boxes 3 and 5 below), and divide the other 162 red marbles evenly into the remaining 18 boxes, those 18 boxes will have 9 red marbles and 11 blue ones, allowing the blue marbles to outnumber them.  So if these boxes were voting districts, the blue districts would outnumber the red ones 18 to 7, even though the red marbles within the districts outnumbered the blue marbles 300 to 200.  This essentially then disenfranchises 60% of the marbles/voters, even though they get to vote, since, as before with the blue marbles, now the votes of the red marbles and their voters cannot determine any laws, policies, or programs. Thus, under either system, a large percentage of the voters, possibly even a majority, have no meaningful say in the outcome of any vote.

 Box 1 Box 2 Box 3 Box 4 Box 5 Box 6 Box 7 Box 8 Box 9 Box 10 Box 11 Box 12 Box 13 Box 14 Box 15 Box 16 Box 17 Box 18 Box 19 Box 20 Box 21 Box 22 Box 23 Box 24 Box 25

Thus, how voters are distributed among districts matters, and can predetermine the outcome.  Or it can at least result in different outcomes from the same individual voters voting in the same ways.  So what then would be fair?  Is there a fair way to do this or is every way of doing it going to be unfair?

I believe that no mathematical way alone of doing this will be able to be fair in one sense of fairness, because no matter what mathematical model is used or the districts drawn, they all determine and predetermine the outcome of every election or vote on any issue as long as people vote pretty much along party lines (which is the way politicians tend to campaign because it is perhaps easiest for them), and as long as the demographics of the party preferences are known when the lines are drawn.  (Again I am assuming that the votes of independents split in such a way that doesn't affect the outcome and that sufficient blue or red voters 'cross over' to vote for the other side.)   There is a sense of fairness in which it is always unfair to predetermine the outcome of what is supposed to be a contest if that predetermination is based on something not germane to the spirit or significance of the contest or of what is being contested.  In this case it is how the district lines are drawn that strongly influence or totally predetermine the results, and that seems irrelevant to what should determine election results.  This is perhaps particularly egregious when it is done by legislators intentionally to help their party win in future elections, but it would occur no matter how the lines are drawn, even if by some mathematical algorithm chosen at random or for its simplicity, neutrality, and conformity with common sense that groups together adjacent neighborhoods into districts with equal numbers of residents in each instead of drawing odd, irregularly shaped districts that look contorted or like some sort of Rorschach diagrams on a map of the districts.

There are ample examples in sports of creating divisions and groupings of athletes in order to make the competition be fair among them because otherwise the outcomes would be generally, though not necessarily always, predetermined by characteristics that are not germane to what the competition is supposed to demonstrate.  For example people compete within 'divisions', such as boxers in different weight classes, because all else being equal skill-wise a lightweight has little chance against a heavyweight boxer.  So to make the contests (i.e., in this case, boxing matches) fair and about who has the greater skill at the sport, there have to be weight divisions.  Similarly in golf with regard to age, since after a certain age, it is difficult and highly unlikely to win a tournament open to all, though it is possible.  Now, it might be fair for a person in a lower weight division to seek the title in a higher weight division (without adding weight), but it would not be fair for someone in the higher weight division to seek the title in a lower weight division without losing weight to meet the limits of the division.  Similarly in golf, it is fair for a 'senior' to play in a standard tournament, but not for a young player to play in a seniors' tournament.  Or in some sports it would not be sporting or fair not to separate the men's league or part of the tournament from the women's because the outcome would most likely be determined ahead of time by the different attributes in general of men and women, where men tend to have, say more upper body strength and women tend to have better balance and flexibility.  So if men had to compete against women in gymnastics on a balance beam, that would put men at a serious disadvantage just as it would generally put women at a disadvantage to compete against men on rings in gymnastics or in the hammer throw in track or in weightlifting.   This is not to say that no woman could beat a man at a typical 'men's sport' or that no man can beat a woman at a typical 'women's sport', just as it is not to say that no lightweight can beat any heavyweight.  We are talking about likely probabilities.  And as before with regard to aspiring upward in boxing, it seems to me that in general the principle about competing outside one's normal division/class is that it would be fair to have someone in any normally disadvantaged bracket voluntarily challenge for the title in the advantaged bracket, but not vice versa.  E.g., it seems to me it would be fair if a top woman tennis player or golfer wanted to play a major male (side of a) tournament.   But it would not be fair for a male golfer or tennis player to play for the title on the women's side or tournament.

The tennis match between Billie Jean King and Bobby Riggs was fair because the outcome was not predetermined, given that though for tennis he was a member of the advantaged male class, she was a member of the advantaged age class.  Plus he had boasted that he could beat any woman basically because he did not think much of women's tennis in terms of quality of play.  She did not have to accept the challenge or try to prove him wrong.  So even if he was correct, she would have been the underdog, and thus the most likely disadvantaged, so her willingness to play him was fair in that way too.  For more about this and other male/female tennis matches, see https://en.wikipedia.org/wiki/Battle_of_the_Sexes_(tennis).  Most notably for trying to make the outcome not essentially predetermined were the rules for the Jimmy Connors/Martina Navratilova match.  He was 40 and she was 35 at the time so relative age was not so much a factor as natural strength, speed, etc.  To make that not be a factor as much as tennis ability itself (whatever that means, but probably at least shot-making skill and strategy), he only was allowed one serve each point, while she had the normal two serves, and she was able to hit into the doubles court on his side while he had to hit into the singles court on her side. By the time of that match women's tennis was strongly accepted and so instead of being considered a battle of the sexes in the way the Riggs/King match (and the earlier Bobby Riggs/Margaret Court match) were, it was called the battle of champions, though Navratilova is said to have called it in her typical dry, wry, pointed manner just a "battle of egos."

Now, the concept of fairness is difficult to articulate though people tend to have intuitions about it.  E.g., if two people run a race where we genuinely don't know which one will win, that seems to be fair even if it turns out that one of them is much, much faster than the other, as we can see once they race and the faster one leaves the slower one in the dust.  If that were to happen, any subsequent race between them without the slower one somehow being able to improve through training, experience, conditioning, growth, etc. or without the faster one losing speed through age or inactivity, etc. would be unfair in some sense because we already know who is faster and who will win (barring a fall or muscle pull, cramp, or some such by the faster runner).  However, if what we want to see is not which one is faster, but which one would win if the slower one is given a head start or if they run a vastly different distance race (e.g., a marathon instead of a 100 meter sprint), that would be a fair contest in itself, though in the case of the head start, that is essentially handicapping the contest in the way, for example golf does when it has contests where golfers' handicaps count.  You are not then trying to see who is better at the skill, but who can win if something is done to make their relatives skills and abilities not be so decisive or not alone be the determining factor.  It would also be unfair in a different way to base life and death on who gets to have food and who doesn't by who wins a race between someone obviously much faster than someone else.  While in nature, speed might have survival value, in the modern world, it should not.

Now, it seems to me that if one candidate or one side of a referendum issue is clearly favored by a vast majority of voters, and everyone knows that going into the voting, that does not make it unfair for the more desired outcome to win; and it would make it unfair to have a system that let the far less desired outcome win, particularly if it were designed knowingly for that purpose.   Hence, gerrymandering would be wrong if the particular way it is done is one of the only relatively few ways it could have the inferior or less desired policy or candidate win, or if it were somehow easier for the inferior or less desired policy or candidate to win by exploiting it in some way that itself is not right.  We do have a sense of cheating even when the rules of a contest do not yet specifically prohibit a  particular manner of competing.  When those forms of competing are first used, the consensus tends to be that as soon as the rules can be changed to prohibit them, they should be.  (A classic example of that was the "four-corners offense" in basketball, used by Dean Smith at the University of North Carolina, which essentially turns basketball into a game of 'keep-away' and was different from just patient strategy to try to take only higher percentage shots, because the goal was to run out the clock while ahead rather than to actively try to score by taking the best shot one could get.  The NCAA instituted a shot-clock to prohibit the four-corners offense.)  So some forms of competing violate the spirit of the contest even when not violating the letter of the rules.  One humorous example of all this is the story, whether true or not, about the blind golfer (there are blind golfers, who just need help with knowing distances and directions to aim) who challenges a top pro to a golf match for a large sum of money.  The pro thinks this will be easy pickings and says to just name the time and place.  So the blind golfer says "midnight on a moonless, cloudy night" with no artificial lighting or night-vision goggles, etc.  It would be fair for a blind golfer to challenge a sighted one, but not vice versa.  However, it would not then be fair to the sighted golfer to play the match in the pitch dark.  As in gerrymandering, you cannot fairly 'stack the deck' against an opponent so that you tailor the rules to let you most likely win when you would not otherwise be known to be likely to win.

And another humorous, but pointed example of all this was in an episode of the original TV series Maverick, with James Garner playing Bret Maverick, a likeable gambler in the old West.  This was an episode where a banker swindles Maverick out of a lot of money by some loophole in a contract.  Maverick gets all his money back and more by swindling the banker back.  They go back and forth in this manner, with people watching to see which one will finally triumph.  It comes down to a poker game between hands, when the banker pulls out a fresh deck of cards and offers to bet Maverick all the money on his being able to cut the ace of spades on the first try when cutting the cards.  Maverick gets to shuffle the cards, of course, first.  So Maverick takes the bet and shuffles the cards and sets the deck on the table for the banker to cut the cards.  At that point, the banker pulls out a big knife, jams its point down through the deck and picks up all the cards at once impaled on the knife, saying "There, I 'cut' the ace of spades on the first try."  All the guys at the table bust out laughing because the banker has clearly tricked Maverick and gotten the better of him in this bet.  But Maverick calmly holds up his left arm and pulls the ace of spades out of the sleeve, and, showing it to everyone, says "No, you didn't."  All the guys double up in laughter except the banker, who knows he was had at his own game.  All of this would be unfair, of course, except that in the show, the contest had become not about who could most fairly be successful at whatever they were doing, but about who could cheat the other successfully in the most clever way.  Maverick won that because he cheated better at the other guy's game and because he had outsmarted the banker's cheating and gone him one better in response to it during the same competition.  It was like having the world's greatest comeback to the world's greatest putdown.  And a great comeback always trumps a great putdown because it uses it, builds on it, and improves on it by turning it to one's own advantage.

And rules, or their interpretation, themselves can sometimes violate the spirit of a competition, as in a sport's rules not allowing instant replay evidence when it shows a referee or umpire clearly made a wrong call that is particularly crucial.  Sports has long recognized that people can disagree about whether an umpire's or referee's call about something was correct or not, and so they put 'judgment calls' solely in the hands of the responsible or designated official, making his/her call the official call in the sense of final call for the record.  However, some of what used to be a mere matter of instant perception which there was no way to prove was wrong, can now, with instant replay from the right position, be clearly shown in some cases to have been mistaken, taking it out of the realm of 'judgment calls'.  What that means is there can be conflicts between rules that prescribe what the official should call to be correct, and what the official does call.  For the specific rules and details of an especially unfair case of the interpretation of the existing rules in favor of an umpire's mistaken judgment over clear evidence of reality, in this case in Major League Baseball, see "Fairness as Moral and Conceptual Relevance".

Dividing These Same Voters into Districts to Make Red and Blue Equal, But Thus Ignoring the Overall Red Majority
In the same way that seemingly neutral rules can still unfairly yield a wrong result, districts drawn without intentional gerrymandering can still be unfair in the sense of favoring one party or candidate over another by accident of geography and demographics, which seems should not matter.  Even if we were to randomly drop marbles into the boxes with no particular pattern, once they are there, the outcome is essentially predetermined just by their placement alone, which seems that it should be irrelevant to electing someone for office or whether a referendum passes or not -- particularly since we know that if some people simply moved from one area to another, the outcome would be different even though everyone votes the same way they did.  But it seems wrong in some way that where voters live within a given body of voters or given government jurisdiction should determine the outcome of elections.  But if mathematical groupings based on geography are what determine how votes are counted, there is no escaping this kind of unfairness, no matter who wins or loses, with the possible exception of a district map that would yield a relatively equal number of red/blue voting districts, as in the following table, though this still is unfair then to the majority because it arbitrarily takes away their majority.

 Box 1 Even (10/10) Box 2 Red Majority (14/6) Box 3 Red Majority (18/2) Box 4 Red Majority (18/2) Box 5 Red Majority (18/2) Box 6 Red Majority (18/2) Box 7 Red Majority (18/2) Box 8 Red Majority (18/2) Box 9 Red Majority (18/2) Box 10 Red Majority (18/2) Box 11 Red Majority (18/2) Box 12 Red Majority (18/2) Box 13 Red Majority (18/2) Box 14 Blue Majority (7/13) Box 15 Blue Majority (7/13) Box 16 Blue Majority (7/13) Box 17 Blue Majority (7/13) Box 18 Blue Majority (7/13) Box 19 Blue Majority (7/13) Box 20 Blue Majority (6/14) Box 21 Blue Majority (6/14) Box 22 Blue Majority (6/14) Box 23 Blue Majority (6/14) Box 24 Blue Majority (6/14) Box 25 Blue Majority (6/14)

Now, if we are talking about an election, what these different tables mean about campaigning is that Red and Blue will both basically ignore the people in Table 1 and Table 2 because Red will win no matter what in Table 1 and Blue will win no matter what in Table 2, if neither party or candidate just totally self-destructs.  Table 3 requires both parties to try to meet the perceived interests or wishes of as many Blues AND Reds as they can, which would be good for both Reds and Blues, but it does so at the cost of fairness to at least 100 Reds whose votes for their perceived needs get ignored in Boxes 2 - 13.

I think that if one just tries to devise districts purely mathematically in these ways, no way of doing it will be fair.  I think that what is needed is some form or variation of voting schemes devised or advocated by Lani Guinier which tries to make candidates and proposals have to attract the votes of a fair number of voters of any numerical minority (whether by race, religion, gender, party, etc.) along with having a majority of the votes in the district.  This would force candidates and government officials trying to pass issue votes to develop proposals and plans to try to meet the perceived needs and interests of minorities and not just ignore them.

For example, just considering Red and Blue here for numerical minority and majority, the ratio given is 3:2 in favor of Red.  So one way to honor that majority (as Table 1 does, but which Table 2 totally ignores and Table 3 unfairly removes) without disenfranchising Blues (as Table 1 does) or giving them more control or power than they deserve (as Tables 2 and 3 do, particularly Table 2), would be to have a system whereby a candidate or referendum issue has to win a 'super-majority' based on, for example the number of votes equal to 1) 50% of the district votes PLUS 2) some reasonable percentage of the minority voters based on their percentage in the district.  For example, in each of the boxes 8 - 25 of Table 2, Red comprises 45% of the district, and Blue comprises 55% of the district, so to win the district the candidate would need at least 10 votes plus a percentage of the 9 Red votes as well.  I haven't worked out all the math to see what a fair percentage of the minority vote in each case would be, but say that it should be half the percentage of their percentage of the district, which, in these boxes we are talking about  would be 22.5% of 9, or 2 (rounded off).  Thus, the winner of any of those districts will need 12 of the 20 votes in the district.  I don't know whether that is high enough to be fair to minorities or not, and that would require specific arguments to try to judge; but if it needs to be higher, the fraction of the percentage of the minority needed could be raised accordingly.  The operating principle should be one that encourages meeting the (perceived) needs of minorities, not one that is strictly numerically based in a way that leads to gaming the system or winning legalistically or through some legal loophole.  The math should serve the moral principle, not replace it with a flawed numerical legal rule that is difficult to amend even though clearly unfair.

I don't know what the option should be for districts that can not achieve the super-majority necessary, but it should be something that gives both groups in the district representation but penalizes them for not being able to come up with policies or a candidate that can win the required super-majority by serving both groups fairly.

For Red to win in boxes 1 - 7, it need not get any Blue votes since there is no (or insufficient) Blue minority in those districts.  The votes of the 'orphan' Blues in boxes 3 and 5 won't matter, but since the Red candidates or policies need to appeal to Blue voters in boxes 14 - 25, that should serve as some sort of protection for any isolated blue voters outside those districts, along with whatever protections based on "rights" can be put into the constitution of the constituency (whether the state constitution or the country's).  Or it could be that if there are any minority voters in a district, at least one minority vote must accompany the majority vote to qualify the majority vote as a winning one.  I repeat, I don't have the details worked out here, but the overall point is to make the math meet the principle of fair minority (and majority) representation and influence.  And if the math needs to be flexible to do that, it should be able to be.

 This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking.  But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do.  I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account.