Representativeness is said to be a desirable feature of government. If candidate A gets only 20% of the votes, and candidate B gets 80%, surely candidate B should win. That intuitively makes sense. However, there are profound issues with the concept of representativeness. David Deutsch explores these issues in chapter 13 of his book The Beginning of Infinity.

Deutsch explains that the quota is the exact number of seats a state should be assigned under strict proportionality. For example, if Congress has 100 seats (I'm simplifying here), and your state has 10% of the country's population, then the state should receive exactly 10 seats in Congress. To stay within the quota means that under a given apportionment rule, the number of seats assigned to the state never deviates more than one whole seat from the quota. Since the quota is rarely a whole number, it has to be rounded somehow. The rounding method is called the apportionment rule. With the above numbers, there is no need to round, but if, say, Congress instead had 105 seats, then your state's quota would be 10.5. The apportionment rule then has to either round that down to 10 or round it up to 11, all while trying to remain as representative as possible. The rule may take any information into account, but if it is capable of deviating from a state's quota by more than 1, it is said to "violate quota."

Consider the following example of a tricky election outcome, which example is thanks to Deutsch. Imagine a country of 4 states with a congress of 10 seats total, with the first state having just under 85% of the population, and the remaining states having just over 5 percent each. Recall that a representative apportionment rule must never deviate more than 1 seat from the quota. That means that the first state must receive either 8 or 9 seats. This could be achieved through an apportionment rule that says to simply round each state's quota to the nearest integer. It would assign 8, 1, 1, and 1 seats, bringing the total number of seats to 11---which, as Deutsch says, doesn't seem to be a big problem at first, until you realize that 85% of 11 is 9.35, not 8. So 85% of the population would be underrepresented in Congress. That apportionment rule violates quota.

As Deutsch points out, the problems resulting from apportionment rules always seem to merely require a quick fix---which is why he aptly describes them as "why don't they just..." suggestions. Why don't they just round to the nearest integer? Why don't they just add another seat to Congress? Etc. Depending on the apportionment rule, such ad-hoc fixes may temporarily work, but they will always fail in the long run.

Apportionment rules, despite being devised to improve representativeness, paradoxically lead to less representative and "unfair" outcomes. One such paradox is the population paradox. Deutsch explains that the population paradox is when, according to a given apportionment rule, a state whose population has grown since the last election loses seats to a state whose population has shrunk. It's a paradox because, although the given apportionment rule is designed to be as representative as possible, it has resulted in an outcome that's less representative. A state whose population has grown should---intuitively, at least---be awarded more seats.

There are other paradoxes, too. One is the Alabama paradox, which is when an increase in the number of seats leads to a decrease in seats for one state. There is also the New-states paradox, which is when the addition of a new state leads to a reduction of seats for existing states. Why don't they just increase the number of seats accordingly? Because that can trigger the Alabama paradox.

According to the Wikipedia article explaining the above paradoxes, an apportionment rule may be free of the Alabama paradox. It may even be free of the population paradox. But Balinski and Young proved that whenever a rule stays within the quota, it cannot be free from the population paradox. In other words, to avoid the population paradox, a rule must violate quota.

Deutsch then explains that the purpose of elections is not to make government as representative as possible, and therefore such paradoxes are not as problematic as they seem. Instead, their purpose is to implement Popper's criterion of democracy by making it possible to remove bad leaders and policies without bloodshed. However, I want to linger on the concept of representativeness and apportionment paradoxes as they relate to ideas of so-called "social justice."

Social-justice warriors (SJWs) have long been calling for "diversity" and "representativeness" in the workplace. For example, when a company's workforce consists almost exclusively of white people, it is said to lack diversity and be unrepresentative of the general populace. Of course, those of us who are individualists and don't believe in group grievances already know that there is only one valid hiring strategy---qualification---and that it is no company's job to play the role of social do-gooder. But let's take SJWs' argument seriously and see if it leads to any contradictions.

I believe Balinski & Young's theorem can help us uncover the same paradoxes in this context. It reaches beyond elections. To see how, simply replace "state" with, say, "race," "Congress" with a company, and "seats" with, say, "number of employees."

In other words, instead of a state being allocated its "fair" share of seats in Congress, think of a group of people of the same race being allocated their "fair" share of jobs in a company. Applying our previous example to this thought experiment, instead of four states there would be four races, with one (say, white people) making up just under 85% of the population, and the remaining making up just over 5% each. Imagine a company with 10 employees total, whose CEO would be looking to (or, let's be honest, forced to) "make changes" to his workforce to make it more racially representative. The CEO can't hire 8.5 white people, so he'd need to round somehow. Under the same rule, he'd round down to 8 white people, and 1 of each remaining race---but his workforce is only 10 people strong, not 11! And if the CEO did hire an additional person, what race should that person be? And how would the company account for the vast majority of the population now being underrepresented in the workforce? Etc.

A company looking to employ an apportionment rule to have a workforce that's as representative as possible will eventually violate quota or be subject to the above paradoxes. That is still true if it is only parts of the company's workforce that are required to be "representative," such as the board of directors. The population paradox will result in one group of employees that belong to a race that has been growing in number in the general populace losing jobs to another race that has been shrinking. The Alabama paradox will lead to an employee of a certain race being fired when the company hires additional people---when the workforce grows in total number. The new-state paradox means that when a new category of race is added to the grievance catalog, existing hires of a certain race may get fired. And so on.

According to the Wikipedia article, such paradoxes occur whenever there are three or more alternatives. There are more than three recognized racial categories, so the above paradoxes would invariably happen to companies trying to stay within the quota. The same holds for other categories SJWs are concerned with, such as sexual orientation and disability. They would have had an easier time convincing people that the two genders could be represented perfectly in companies---but now that they have convinced themselves that there are dozens of genders, "representative" hiring practices based on gender will be subject to the same paradoxes as well.

In the case of hiring, it gets more complicated than in the case of elections, because here we are told to optimize across multiple dimensions of representativeness: not just along race, but also along gender, sexual orientation, etc. I believe that this can only lead to additional paradoxes and undesirable consequences, such as a black man having to be fired to hire a white woman and vice versa.

Presumably, it is likewise impossible to make all teams and organizations within a company and the company as a whole perfectly representative, as intra-company transfers of employees would have to happen between teams and organizations. Such transfers would, I conjecture, result in similar paradoxes. (If anyone knows of a theorem that proves these two conjectures, please let me know.)

The population paradox is a particularly damning one for SJWs. Surely they wouldn't want black people to be fired from companies when the population of black people grows? But that's exactly what companies would need to do in the name of representativeness that SJWs so desire. Keep in mind that ad-hoc fixes of the sort "when in doubt, just keep the minority hire" are likewise apportionment rules---and there can be no apportionment rules that stay within the quota without resulting in the population paradox. There is no way out of this mathematical reality.

At least, for now. Our knowledge of mathematics does not have a monopoly on the truth that people think it has. It contains mistakes. Balinski & Young's theorem may later be found to be wrong. Should that happen, that does not mean that laws forcing companies to have "representative" workforces or boards of directors would be a good thing. They still wouldn't be because the only sensible hiring rule is based on qualifications. SJWs would first need to come up with a good, non-refuted moral explanation for doing such a thing. Since their answers usually involve force, I doubt they could produce one.

An SJW might try to wiggle his way out of this conundrum by pointing out that apportionment paradoxes typically don't occur every election. Likewise, an apportionment rule is said to violate quota if it is capable of deviating from the quota by more than one---but that need not happen every election either. While that is true, companies don't hire every four years. They hire all the time. With hundreds of thousands of companies in the US alone, said paradoxes would result in what SJWs consider injustices every day.

In reality, calls for "diversity"---in whose service the idea of "representativeness" stands---are hidden calls for every company to be the same. In a flowery, perfectly "socially just" (but mathematically impossible) world, every company would need to have the same proportions of their employees belonging to certain groups. Every company would look the same in terms of demographics. That is not what happens in a free society with free hiring practices. In an open society, you'll see different kinds of companies with various degrees of representativeness, most of them averaging out anyway.

A workforce need not be representative of the general populace. As with democracies, representativeness is not and cannot be its purpose. A workforce need only be qualified to solve the problems the company hires them to solve. Indeed, hiring managers should be free to discriminate based on any characteristics they like---but of course, they would be foolish to take anything other than qualification into account. Any company that discriminates only based on merit has a larger pool of qualified candidates to choose from and therefore has a better chance to outperform bigoted competitors. So there's no need to make bigotry illegal. The free market has built-in mechanisms to promote tolerance and discourage irrational discrimination. Laws and regulations determining hiring and firing based on "representativeness" not only increase discrimination---because now, hiring managers are forced to look at characteristics other than qualification---but also lead to paradoxes that achieve the opposite of what the lawmakers had in mind when they drafted those laws.

To those who think representation in the workforce---or anywhere---is important, surely being mathematically proven wrong will change their minds? But it won't, because nothing can. So do you see now how irrational SJWs are?