Each of the apportionment methods has at least one weakness. Some potentially violate the quota rule and some are subject to one of the three paradoxes.

 The quota rule says that each state should be given either its upper quota of seats or its lower quota of seats.

Example (PageIndex{1}): Quota Rule Violation

A small college has three departments. Department A has 98 faculty, Department B has 689 faculty, and Department C has 212 faculty. The college has a faculty senate with 100 representatives. Use Jefferson’s method with a modified divisor of d = 9.83 to apportion the 100 representatives among the departments.

 State A B C Total Population 98 689 212 999 Standard Quota 9.810 68.969 21.221 100.000 d = 9.83 9.969 70.092 21.567 quota 9.000 70.000 21.000 100

District B has a standard quota of 68.969 so it should get either its lower quota, 68, or its upper quota, 69, seats. Using this method, District B received 70 seats, one more than its upper quota. This is a Quota Rule violation.

 The population paradox occurs when a state’s population increases but its allocated number of seats decreases.

A mom decides to split 11 candy bars among three children based on the number of minutes they spend on chores this week. Abby spends 54 minutes, Bobby spends 243 minutes and Charley spends 703 minutes. Near the end of the week, Mom reminds the children of the deal and they each do some extra work. Abby does an extra two minutes, Bobby an extra 12 minutes and Charley an extra 86 minutes. Use Hamilton’s method to apportion the candy bars both before and after the extra work.

 State Abby Bobby Charley Total Population 54 243 703 1,000 Standard Quota 0.594 2.673 7.734 11.000 Lower Quota 0 2 7 9 Apportionment 0 3 8 11

With the extra work:
Abby now has 54 + 2 = 56 minutes
Bobby has 243 + 12 = 255
Charley has 703 + 86 = 789 minutes

 State Abby Bobby Charley Total Population 56 255 789 1,100 Standard Quota 0.560 2.550 7.890 11.000 Lower Quota 0 2 7 9 Apportionment 1 2 8 11

Abby’s time only increased by 3.7% while Bobby’s time increased by 4.9%. However, Abby gained a candy bar while Bobby lost one. This is an example of the Population Paradox.

 The new states paradox occurs when a new state is added along with additional seats and existing states lose seats.

A small city is made up of three districts and governed by a committee with 100 members. District A has a population of 5310, District B has a population of 1330, and District C has a population of 3308. The city annexes a small area, District D with a population of 500. At the same time the number of committee members is increased by five. Use Hamilton’s method to find the apportionment before and after the annexation.

 State A B C Total Population 5,310 1,330 3,308 9,948 Standard Quota 53.378 13.370 33.253 100.000 Lower Quota 53 13 33 99 Apportionment 54 13 33 100

 State A B C D Total Population 5,310 1,330 3,308 500 10,448 Standard Quota 53.364 13.366 33.245 5.025 105.000 Lower Quota 53 13 33 5 104 Apportionment 53 14 33 5 105

District D has a population of 500 so it should get five seats. When District D is added with its five seats, District A loses a seat and District B gains a seat. This is an example of the New-States Paradox.

In 1980, Michael Balinski (State University of New York at Stony Brook) and H. Peyton Young (Johns Hopkins University) proved that all apportionment methods either violate the quota rule or suffer from one of the paradoxes. This means that it is impossible to find the “perfect” apportionment method. The methods and their potential flaws are listed in the following table.

 Paradoxes Method Quota Rule Alabama Population New-States Hamilton No violations Yes Yes Yes Jefferson Upper-quota violations No No No Adams Lower-quota violations No No No Webster Lower- and upper-quota violations No No No Huntington-Hill Lower- and upper-quota violations No No No

An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense.

To apportion is to divide into parts according to some rule, the rule typically being one of proportion. Certain quantities, like milk, can be divided in any proportion whatsoever others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between our desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values. This results, at times, in unintuitive observations, or paradoxes.

Several paradoxes related to apportionment, also called fair division, have been identified. In some cases, simple adjustments to an apportionment methodology can resolve observed paradoxes. Others, such as those relating to the United States House of Representatives, call into question notions that mathematics alone can provide a single, fair resolution.

## Population

• Hamilton
• State
• Insert Below
• % Representation
• Numbers
• Integer Part
• Fractional Part
• Members Manually

The question now becomes, are these seats all apportioned fairly? To find out we need to know the “Average Constituency” of each state.” The Average Constituency measures the fairness of an apportionment (Pirnot, n.d. pg. 534).” To find the Average Constituency one would take the population of a state and divide it by the assigned seats, and the compare them to determine fairness. Giving an example from the calculations above, one can see that state 1 has a population of 15475 and state 2 has a population of 35644. State 1 has 3 assigned seats and state 2 has 7 (Pirnot, n.d.). 15457/3 = 5158

## Impossibility result

In 1982 two mathematicians, Michel Balinski and Peyton Young, proved that any method of apportionment will result in paradoxes whenever there are three or more parties (or states, regions, etc.). [2] [3] More precisely, their theorem states that there is no apportionment system that has the following properties (as the example we take the division of seats between parties in a system of proportional representation):

• It follows the quota rule: Each of the parties gets one of the two numbers closest to its fair share of seats (if the party's fair share is 7.34 seats, it gets either 7 or 8).
• It does not have the Alabama paradox: If the total number of seats is increased, no party's number of seats decreases.
• It does not have the population paradox: If party A gets more votes and party B gets fewer votes, no seat will be transferred from A to B.

Methods may have a subset of these properties, but can't have all of them:

• A method may follow quota and be free of the Alabama paradox. Balinski and Young constructed a method that does so, although it is not in common political use. [4]
• A method may be free of both the Alabama paradox and the population paradox. These methods are divisor methods, [5] and Huntington-Hill, the method currently used to apportion House of Representatives seats, is one of them. By the impossibility result, these methods will necessarily fail to always follow quota.
• No method may always follow quota and be free of the population paradox. [5]

The history of the way that the House of Representatives in the United States has been apportioned is fascinating. It involves many colorful and powerful figures in United States history. The United States is very unusual in having a legislative branch with two independent parts, the Senate and the House of Representatives. This somewhat unique situation came about because the people responsible for creating the United States needed a way to compromise concerning their views about the new country they were forming. On the one hand, there were people from relatively unpopulated states where there were many landowners and there were people from relatively highly populated states with large urban populations. By having one legislative branch in which each former colony would have equal numbers of representatives (two senators for each state) and one legislative branch based on population, a compromise was reached. However, from the very start, people from different states were anxious to protect their own interests in the House of Representatives. The more seats one had, the safer it would be for one's point of view. It is useful to remember that during the early days of the United States no one foresaw the effect of having two dominant political parties at nearly all times of the country's history. The Constitution, in fact, makes no mention of political parties.

Among the very first causes of friction between the people who created the new country was how to apportion the House of Representatives. Right from the beginning there were various proposals. The reason for the difference was the observation that different methods resulted in different numbers of seats for the different former colonies. Using the principle more is better, the protagonists involved typically came down in support of that method which did the best for their personal interests. Furthermore, it was also true that one could defend different choices of how to apportion with reasonable sounding arguments. After the 1790 census, rival methods emerged for how to apportion the House of Representatives, using what today have come to be called Hamilton's method and Jefferson's method, which will be described below. George Washington, faced with a bill which gave rise to an apportionment that he was unable to support, vetoed the bill that Congress presented. This was the first time in the history of the new country that the veto power of the President was used and only one of two times that Washington used the veto power.

Jefferson's method was used for the 1790 apportionment (h=105) and continued to be used until 1840, when a method suggested by Daniel Webster was adopted. During this period the debate centered around the statistical fact that Jefferson's method was systematically giving large states more than their share, that is, the method is biased in favor of large population states. In 1850 Vinton's method, in essence Hamilton's method, became law and this method remained on the books until the turn of the 20th century. Yet for complicated reasons filled with political wrangling, ad hoc approaches to apportionment were used. In 1901 Webster's method was used, reacting in part to the realization that Hamilton's method was subject to allowing a state to lose seats when the House of Representatives increased in size, the so-called Alabama paradox, and strange behavior when a new state was added to the union (known as the new state paradox). In 1911 Webster's method was used with special provision for what to do if a new state entered the Union. Astonishingly, in 1920, despite the new census, no new apportionment of the House of Representatives occurred, in essence, because Congress could not agree on a way to carry out an apportionment which could be enacted into law! For the 1930 census Webster's method was used.

Of particular interest to mathematicians is the work of Edward Vermilye Huntington (1874-1952). Huntington was associated with Harvard University for much of his career, having been appointed there as an instructor in 1901 and having retired in 1941. In addition to his work on apportionment he is known for his work in axiomatic geometry. (Huntington served as President of MAA, vice-President of AMS, and President of AAAS, a remarkable accomplishment!) During the years of the First World War, when Huntington did work for the military in the area of statistics, or shortly thereafter, he learned of the apportionment ideas of Joseph A. Hill. He revised ideas of Hill to obtain a rigorous apportionment method, which he referred to as the Method of Equal Proportions. Huntington's method was a member of the same family of methods as those of Jefferson, Webster, President John Quincy Adams and James Dean (not the actor), who was a Professor of Mathematics at the University of Vermont. Although Dean's method and that of Adams have never been used in America, they have been part of the debate and of court cases that deal with the best method to use.

Extensive debate among experts about which of the many methods that might be used for apportionment simmered in the background as Congress was unable to decide on an apportionment during the first part of the 20th century. The Speaker of the House, Nicholas Longworth, suggested that the National Academy of Sciences make an objective study of the problem. A committee of mathematicians consisting of G.A. Bliss (1876-1951), E.W. Brown (1866-1938), L. P. Eisenhart (1876-1965) and Raymond Pearl (1879-1940) was formed to investigate the situation.

Their report indicated support for Huntington's method. Rather later an even more prestigious group of mathematicians provided a report to the President of the National Academy of Sciences on apportionment methods. The report was signed by Harold Marston Morse (1892-1977), John Von Neumann (1903-1957), and Luther Eisenhart (who had been involved with the 1929 report also). The report again supported the Huntington-Hill method.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.

In 1982, two mathematicians, Michel Balinski and H. Peyton Young, published the very important book, Fair Representation: Meeting the Ideal of One Man, One Vote, in which they reported in detail on the history of the apportionment problem and described work of their own on the mathematics of the apportionment problem that had appeared in a variety of research papers. This work built on the earlier work of Huntington but carried the mathematical theory of apportionment much further. In particular, they followed in the footsteps of Kenneth Arrow's work in understanding fairness in voting and elections by looking in detail at fairness issues growing out of apportionment problems. Specifically, they noted the tension between different views of the essential fairness questions. These fairness questions take the form of stating various axioms or rules that an apportionment method should obey. Many of these issues are quite technical but an intuitive overview follows. There are now many variants of similar sounding axioms which differ in their details.

Here are some fairness issues that might be raised: Is an apportionment method house monotone (i.e. avoids giving fewer seats to a state in a larger house)? Does an apportionment method obey quota? Is an apportionment method biased (in the sense that when used to decide many apportionment problems, it tends to be unfair to small or large states in a systematic way)? Is an apportionment method population monotone? (For example, in comparing the results of applying the same apportionment method to two consecutive censuses, could a state whose population went down get more seats than it did previously, while at the same time a state whose population went up lose seats?) Does an apportionment avoid the new states paradox? They also examined the consequences of a state splitting into two states to get more seats. (This is an important issue for the AP in the European context.)

Balinski and Young showed that these fairness conditions do not mix well. Informally their results (some of which were known to earlier researchers) can be stated:

** If a method is well behaved with regard to changes in population, then it must be a divisor method (rounding rule method). Balinski and Young reject the use of an ingenious method they developed referred to as the quota method. This method, though it obeys quota and is house monotone, does not avoid the population paradox.:

** No divisor methods guarantee giving each state its lower or upper quota. (In fact, no method which avoids the population paradox guarantees giving every state its lower or upper quota.)

** Divisor methods are house monotone.

** Divisor methods (rounding rule methods) avoid paradoxical results when new states are added to the apportionment mix.

Balinski and Young also call attention to the issue of bias of an apportionment method which involves the consequences of using this method time after time. If a method tends to give more seats to large states or more seats to small states this might be deemed a strike against it. The difficulty is arriving at either a theoretical or empirical framework for analyzing bias. The issues involved here are a classic example of the difficulties in the interface between theoretical results in mathematics and how they are applied.

Welcome to the
Feature Column!

These web essays are designed for those who have already discovered the joys of mathematics as well as for those who may be uncomfortable with mathematics.

### PRESH TALWALKAR

I run the MindYourDecisions channel on YouTube, which has over 1 million subscribers and 200 million views. I am also the author of The Joy of Game Theory: An Introduction to Strategic Thinking, and several other books which are available on Amazon.

(As you might expect, the links for my books go to their listings on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.)

By way of history, I started the Mind Your Decisions blog back in 2007 to share a bit of math, personal finance, personal thoughts, and game theory. It's been quite a journey! I thank everyone that has shared my work, and I am very grateful for coverage in the press, including the Shorty Awards, The Telegraph, Freakonomics, and many other popular outlets.

I studied Economics and Mathematics at Stanford University.

People often ask how I make the videos. Like many YouTubers I use popular software to prepare my videos. You can search for animation software tutorials on YouTube to learn how to make videos. Be prepared--animation is time consuming and software can be expensive!

Feel free to send me an email [email protected] . I get so many emails that I may not reply, but I save all suggestions for puzzles/video topics.

### MY BOOKS

If you purchase through these links, I may be compensated for purchases made on Amazon. As an Amazon Associate I earn from qualifying purchases. This does not affect the price you pay.

Book ratings are from June 2021.

Mind Your Decisions is a compilation of 5 books:

The Joy of Game Theory shows how you can use math to out-think your competition. (rated 4.2/5 stars on 200 reviews)

40 Paradoxes in Logic, Probability, and Game Theory contains thought-provoking and counter-intuitive results. (rated 4.1/5 stars on 30 reviews)

The Irrationality Illusion: How To Make Smart Decisions And Overcome Bias is a handbook that explains the many ways we are biased about decision-making and offers techniques to make smart decisions. (rated 4/5 stars on 17 reviews)

The Best Mental Math Tricks teaches how you can look like a math genius by solving problems in your head (rated 4.2/5 stars on 57 reviews)

Multiply Numbers By Drawing Lines This book is a reference guide for my video that has over 1 million views on a geometric method to multiply numbers. (rated 4.1/5 stars on 23 reviews)

Mind Your Puzzles is a collection of the three "Math Puzzles" books, volumes 1, 2, and 3. The puzzles topics include the mathematical subjects including geometry, probability, logic, and game theory.

Math Puzzles Volume 1 features classic brain teasers and riddles with complete solutions for problems in counting, geometry, probability, and game theory. Volume 1 is rated 4.4/5 stars on 75 reviews.

Math Puzzles Volume 2 is a sequel book with more great problems. (rated 4.3/5 stars on 21 reviews)

Math Puzzles Volume 3 is the third in the series. (rated 4.3/5 stars on 17 reviews)

### KINDLE UNLIMITED

Teachers and students around the world often email me about the books. Since education can have such a huge impact, I try to make the ebooks available as widely as possible at as low a price as possible.

Currently you can read most of my ebooks through Amazon's "Kindle Unlimited" program. Included in the subscription you will get access to millions of ebooks. You don't need a Kindle device: you can install the Kindle app on any smartphone/tablet/computer/etc. I have compiled links to programs in some countries below. Please check your local Amazon website for availability and program terms.

### MERCHANDISE

Grab a mug, tshirt, and more at the official site for merchandise: Mind Your Decisions at Teespring.

## Biden Is a Prisoner of His Own Paradoxes

Democratic National Convention via AP

Joe Biden and his handlers know that he should be out and about, weighing in daily on the issues of the campaign.

In impromptu interviews, Biden should be offering alternative plans for dealing with the virus, the lockdown, the economic recovery, the violence and the looting, and racial tensions.

Yet Biden’s handlers seem to assume that if he were to leave his basement and fully enter the fray, he could be capable of losing the election in moments of gaffes, lapses, or prolonged silences.

So wisely, Team Biden relied on the fact that the commander in chief is always blamed for bad news — and there has been plenty of bad news worldwide this year.

That reality was reflected in the spring and early summer polls that showed growing discontent with the incumbent Trump as if he were solely responsible for one of the most depressing years in U.S. history.

But news cycles, like polls, are not always static.

What was true in July is not necessarily so in September and especially in November. Volatile years produce volatile voters. Now, many voters think they see a waning of the virus, a need to get their kids back in school, and a glimmer of hope that the economy is recovering.

A large segment of the public is becoming irate at the nightly looting, destruction, and arson that no longer seem to have much to do with the May death of George Floyd while in police custody. Where are the police, the mayors, and the governors to protect the vulnerable, the law-abiding, and the small-business owners?

##### Is Antifa Holding America’s Safety Hostage… to Elect Joe Biden?

Biden knows the mercurial polls now tell him that he must re-emerge and cease being a virtual candidate. Yet he knows that if he does, he risks losing the race. So his surrogates talk of mandatory fact-checking of the debates — or even canceling them entirely.

Hillary Clinton recently said that Biden “should not concede under any circumstances,” apparently even if he loses the November election. If the rules no longer favor Biden, then it seems time to change the rules.

So Biden has become a tragic prisoner of his own paradoxes.

He is an old centrist who forged a Faustian bargain with socialist Bernie Sanders and his hard-core leftist supporters. That alliance was felt necessary to win the Democratic nomination and the general election.

The hard left provided the urban fireworks this summer that seemed to drive down Trump’s poll numbers. Blue-state governors and mayors contextualized the violence as a “summer of love” or “largely peaceful.”

Biden stayed mum — both because the polls suggested he should remain so, and because he could hardly criticize those whose often violent acts were creating a sense of national anarchy under Trump’s presidency and thus undeniably aiding the Biden candidacy.

But as CNN news anchor Don Lemon recently warned his fellow leftists, now the polls are changing. Lemon apparently fears that the public is sick of seeing the urban unrest. Suddenly, many members of the media want Biden to condemn the rioting and violence.

But if Biden did, he might alienate his now-critical left-wing Bernie base. Yet Biden’s continued reluctance to unequivocally fault the rioters and arsonists may be alienating moderate suburban swing voters.

The same paradox surrounds the debates. Should Biden, as promised, debate Trump?

Yes. But would he thereby blow up his candidacy in a moment of incoherence?

No. But would he end up ridiculed in absentia, like Clint Eastwood’s empty chair at the 2012 Republican convention?

Trump never sits still. So should Biden match the president’s frenzied pace and hold town halls, impromptu interviews, tarmac rallies, photo-ops on the campaign trail, and daily unscripted press conferences? But to do so could well confirm to voters that he is frail and confused.

How did Biden become a prisoner of his own paradoxes?

Perhaps he knew that he was not physically or cognitively up to running a real campaign. But he ran all the same.

Perhaps he knew that the violence of antifa and other agitators could eventually hurt more than help him, but for months he kept silent about the violence all the same, given the perceived political damage to Trump.

Perhaps he knew that he had always opposed the wacky agenda of Sanders, but Biden wrongly felt he could pose as a moderate in 2020 yet if elected keep a promise to the socialists of the more radical wing of his party to govern as a leftist.

Perhaps he knows that his new progressive allies would be happy for him to win them a presidency but even happier for him to then disappear as soon as possible.

Paradoxes happen when what seems real is not — and is known not to be real by those who act as if it is.

Victor Davis Hanson is a classicist and historian at the Hoover Institution, Stanford University, and the author of “The Second World Wars: How the First Global Conflict Was Fought and Won,” from Basic Books. You can reach him by e-mailing [email protected] .

The Venice Commission in its Code of Good Practice in Electoral Matters specifies that (single-seat) constituencies should be drawn so that the size difference of a constituency’s size from the average should not exceed a fixed limit while its borders must not cross the borders of administrative regions, such as states or counties.

Assuming that constituencies are of equal size within each of the administrative regions, the problem is equivalent to the apportionment problem, that is, the proportional allocation of voting districts among the administrative regions. We show that the principle of maximum admissible departure is incompatible with common apportionment properties, such as monotonicity and Hare-quota. When multiple apportionments satisfy the smallest maximum admissible departure property we find a unique apportionment by the repeated application of the property. The allotment such that the differences from the average district size are lexicographically minimized can be found using an efficient algorithm. This apportionment rule is a well-defined allocation mechanism compatible with and derived from the recommendation of the Venice Commission. Finally, we compare this apportionment rule with mainstream mechanisms using data from Hungary, Germany and the United States.

## The elimination paradox: apportionment in the Democratic Party

To award delegates in their presidential primary elections, the US Democratic Party uses Hamilton’s method of apportionment after eliminating any candidates (and their votes) that receive less than 15% of the total votes cast. We illustrate how a remaining candidate may have his or her delegate total decline as a result of other candidates being eliminated this leads to a new elimination paradox. We relate that paradox to the new states, no show, and population paradoxes and show that divisor methods are not susceptible to the elimination paradox. We conclude with instances in which the elimination paradox may occur in other contexts, including parliamentary systems.

This is a preview of subscription content, access via your institution.