Mathematics helps develop definitions to compare different means to make something better or more fair. But there are inherent limitations, expressed as mathematical impossibility theorems...

Impossible?

Joe Malkevitch
York College (CUNY)

Introduction

When you got up from sleeping last night, what were the possibilities that you could achieve in the new day? This is far from the issue of if humans have free will, or whether their choices are constrained by events, genetics or economics that limit their freedom. Humans are not genies who in mythology have the power to make a person's wishes come true. What is possible for you and what is not? Here, I am interested in the limits that mathematics places on what we can know using mathematics as a tool. What are the consequences of mathematical impossibilities for the way that democracies (or autocracies, for that matter) can be run for the benefit of their citizens? I'll say more on the fairness issues after setting the stage for insights into the limits of mathematical insights.

Impossibility in mathematics

People debate whether mathematics is discovered or invented. The first framework is that the facts of mathematics, often described as theorems, are out there and await some clever person (or AI system?) to find them. That mathematics is an invented framework suggests that until a person (or perhaps a computer?) uses their reasoning skills and creativity to invent mathematical ideas and concepts, in the same way that humans have invented devices such as the cell phone, washing machine or electric automobile, those ideas do not exist. Lurking in the background of this debate is the question of what can be achieved and what cannot in understanding the "world of mathematics." After some initial looks at the limits of mathematics, I want to discuss the limits that mathematics places on the ways individuals or groups of individuals can implement being fair!

The remarkable fact often known as the Pythagorean Theorem has been thought about by many cultures. In modern notation, the Pythagorean Theorem states that for a right angle triangle, in what is commonly called the Euclidean Plane, where $a$ and $b$ are the lengths of the sides that meet at a right angle, and $c$ is the length of the side of the triangle opposite the right angle that

$$a^2 + b^2 = c^2.$$

It is worth noting how many ideas and conventions are needed to write down this theorem. The result was stated in English rather than (for example) Bengali and the symbols $+$ and $=$ are used in writing the theorem down. We need to know what the Euclidean Plane is, what a triangle in the Euclidean plane is, what is meant by the length of the side of a triangle, etc. We also need to know what an angle is, what a right angle is, that $7^2$ means $7 \times 7$ and thus has the value 49 expressed in decimal place notation arithmetic, etc.

Part of the reason the Pythagorean Theorem is such a rich fact is that it can also be interpreted as a statement about areas rather than about lengths. Thus, for a right triangle the areas of the squares on the sides $a$ and $b$ of the triangle add up to the area of the square on the third side $c$, often called the hypotenuse of the triangle, using language derived from discussions of ancient Greek mathematics.

Pythagorean Theorem diagram showing squares along each side of a right triangle

The Pythagorean Theorem illustrated. (Diagram courtesy of Wikipedia)


The richness of the equation $a^2 + b^2 = c^2$ as a source of ideas and questions, and the fact that it was in essence found by many cultures who did not have much contact with each other, has been used to argue that the equation and its ramifications were mathematics waiting to be discovered rather than invented.

Once mathematicians get started with a nifty idea, they continue to mine this territory for interesting patterns. For example:

  1. What positive integer triples $(a, b, c)$ satisfy $a^2 + b^2 = c^2$?
  2. Are there triples $(a, b, c)$ that form an arithmetic progression and satisfy $a^2 + b^2 = c^2$?
  3. Are there pairs $(a, b)$ and choices for $c$ where exactly 7 choices for the pair $(a, b)$ satisfying $a^2 + b^2 = c^2$ are possible?
  4. What is the analog of the Pythagorean theorem in geometries other than the Euclidean plane (e.g., Euclidean 3-dimensional geometry, or what has come to be known as hyperbolic geometry, also known as the Bolyai-Lobachevsky plane)?
  5. Invent your own question to place here!

In developing ideas about the concept of numbers to count and to measure it was noticed that something seemed to be special about the length of the hypotenuse of a right triangle whose other sides (its legs) are 1 and 1. Equation $a^2 + b^2 = c^2$ implies that this length is a number whose square is 2. We usually denote this number $\sqrt{2}$ and it is referred to as the square root of 2.

Can this number be written in the form $a/b$ where $a$ and $b$ are positive integers? This collection of numbers is now known as the rational numbers. So the question we raise is whether there are numbers which are not rational and how to describe such numbers. It can be shown that it is impossible to represent the hypotenuse of some particular Euclidean triangles as rational numbers. (Some right triangles can have rational length hypotenuses: for example, triangles with legs of lengths 3 and 5 or 5 and 12, which have, in fact, integer length.)

Early on it was shown that humans could demonstrate the impossibility of writing the square root of 2 as a rational number. It is possible to find rational numbers, where the representation $a/b$ gives a very good approximation to the square root of 2. For example 41/29 or 99/70 might be used as an approximation.

Compass and straight edge construction impossibilities

Euclid's important book on geometry, the Elements, includes discussions of what lengths of segments, as well as other constructions can be accomplished using a straight edge (a ruler without length markings) and compass (allowing one to draw a circle with a particular center and radius). Three famous problems emerged out of these discussions:

  1. Trisecting any angle (Given an arbitrary angle, can one obtain an angle whose measure is $\frac{1}{3}$ the angle of the measure we started with?)
  2. Duplicating the cube (Finding the edge length of a cube whose volume was exactly double the volume of a cube with side length 1 (and hence, volume 1.)
    Note: To this day, despite many proofs that one cannot trisect arbitrary angles with a straight edge and compass, many people claim that they can accomplish this feat. These individuals have not been convinced that this task is impossible!
  3. Squaring the circle (Find a square whose area is equal to the area of a given circle.)

Eventually, it was proven that it is impossible to carry out these constructions with straight edge and compass. Of course, one has to specify what constitutes a proof. Intuitively, a proof is a collection of logical steps based on known facts (theorems) or axioms (statements whose validity is assumed true) which lead to the desired statement (in a finite number of steps). The understanding of what constitutes a rigorous proof has changed with time. Today, computer systems are being used to "make sure" that certain theorems are really facts. The reason for this recent thread of mathematical work has been because the proofs humans have in some cases provided for mathematical conjectures have become so long and complicated that it is unclear if they can be checked for certain by other mathematicians! Some important mathematical facts have required proofs that go on for hundreds of pages. Making sure the proofs do not suffer from subtle errors is required. Sometimes theorems generally accepted by the mathematical community are shown not to apply as fully as thought, and sometimes new proofs have had to be provided because an existing proof was shown to have a mistake, often because some case that might a priori occur was not considered in the proof.

In high school you probably learned a formula which, given the coefficients of a quadratic equation, enabled you to write down an algebraic expression that gives the roots of that quadratic equation. Over time mathematicians developed similar "closed form" formulas for polynomial equations with integer coefficients of degree 3 or 4 (the cubic and quartic equations). However, it came as a surprise when Évariste Galois (1811-1832), Niels Heinrich Abel (1802-1829) and Paolo Ruffini (1765-1822) developed tools that showed that for the analogous polynomial equations of degree 5 or higher it was not possible to find such a formula! Similarly, if you have studied some calculus you know that when functions can be expressed in terms of polynomials, logarithms, exponential functions and trigonometric functions, it is usually not that difficult to find the derivative of such a function. But for relatively simple examples (e.g., $\sqrt{\sin(x)}$) it is not possible to find the indefinite integral of such a function without using tools specially invented to describe the solutions of such integrals, (e.g., elliptic functions).

Condensing the issues considerably, a pioneer in trying to understand what mathematics was knowable from formal systems where one had a collection of undefined terms and axioms involved was David Hilbert (1862-1943).

Photo of the mathematician David Hilbert

Photo of David Hilbert (Courtesy of Wikipedia)



Reacting to programs developed by Hilbert (1862-1943), the mathematician Kurt Gödel (1906-1978) totally surprised the mathematical community by showing that there are legal statements that in a formal mathematical system cannot be proven or disproved. Such statements are sometimes called undecidable.



Photo of the mathematicianKurt Godel

Photo of Kurt Gödel (Courtesy of Wikipedia)

This result of Gödel is often labeled his Impossibility Theorem. Others after Gödel, notably Julia Robinson (1919-1985) (who was a President of the American Mathematical Society) and Yuri Matseyovich showed that long standing questions that interested the mathematics community were undecidable.

Photo of the mathematician Julia Robinson

Photo of Julia Robinson (Courtesy of Wikipedia)

Photo of Yuri Matseyovich (Courtesy of Wikipedia)

After many years of the study of impossibility and undecidability issues, examples of statements which are undecidable have been found in many subareas of mathematics ranging from geometry to group theory (algebra) and to combinatorics. In a general way, however, these impossibility questions belong to the domain of mathematics known as logic.

Impossibility in computer science

As a scholarly discipline, computer science is much newer than mathematics. Being able to study for a degree in computer science at a college first became possible relatively recently. It appears that Purdue University was the first American university to form a department of computer science in 1962. Some computer scientists are primarily concerned with making the hardware that makes it possible to solve problems using that hardware. Other computer scientists are interested in designing languages for computers to solve problems using particular hardware, as well as those scholars who are concerned with the algorithms that can be used to solve problems on a computer. A pioneer of computers in both the sense of what they could do and making hardware to actually do the work was Alan Turing (1912-1954). Today there is a vast array of complexity classes which take questions and lump them together in terms of how hard it is to solve these problems using algorithms.

Photo of Alan Turing

Photo of Alan Turing (Courtesy of Wikipedia)

Democracies

In a representative democracy, the goal is to "translate" the wants and desires of the people being governed using a legislature or parliament to carry out the will of the people. When democracies work well, while majority opinion is implemented, the interests and rights of minorities are maintained. The system used by the United States, having a strong President to execute the laws passed by the Congress and a bicameral legislature, where the President is elected directly by the people and the members of Congress are elected from districts in the individual states as legislators in the House of Representatives and two senators are elected for each state, is unusual. The system common in most European democracies has a single parliament (legislature) where the representatives are selected based on the percentage of the vote that competing parties get. The parties represent the range of views of the populace. The aim is that for a parliament with $h$ seats (house size, a positive integer), if party A gets 17% of the vote it would get about 17% of the seats. The Dutch parliament has 150 seats so in this example we might calculate $.17(150) = 23.5$ seats and conclude party A "deserves" 23.5 representatives. So should party A get 23 seats, 24, or perhaps 25 seats and would this be considered a fair apportionment of seats to party A? In recent years many European democracies have been using a hybrid system to elect legislators, where a combination of percentage vote for parties and electing representatives from districts is used.

To get the major ideas across here I will confine myself to discussing the election of, say, a mayor as the chief executive officer of a small city where any eligible voter can vote for the Mayor. In principle there may be many candidates running for Mayor, where perhaps each candidate is linked to an affiliation to some party, though for this discussion the issue of party is not relevant. To model what is going on we will assume there are $n$ eligible voters who actually voted, and $c$ candidates (here $c$ is a positive integer at least 2).

In an election, each voter uses a ballot, which is a way to express voter views about the candidates. Based on the ballots cast, a prespecified decision method, which I will refer to as an election method, is used to "translate" the views of the individual voters into a SINGLE choice (no ties) for who becomes mayor. This may sound simple but when trying to design an "ideal" system there is little agreement! To see the complications, consider the issue of what kind of ballot to use. In practice, and in theory, here are some examples of the kinds of ballots used in America in the past or proposed for use. Of course, the nature of the election method will have to be tailored to the kind of ballot that is used. For a specific type of ballot there are usually many choices as to what election method to use based on the ballots.

Ballot types

  1. (Plurality ballot) Vote for exactly one of the $c$ candidates.
  2. Rank all the candidates without ties, from most preferred to least preferred.
  3. There are many ways to represent such a ballot, but let me offer two. The first representation uses a diagram with the most preferred candidates towards the top:

    Preference ballot showing an arrow pointing up with B at the top, C in the middle, and A at the bottom

    This ballot means that of the three candidates this voter preferred B to C and to A, and C to A. Note that "how much," or with what intensity B is preferred to B, or C preferred to A, is not information available to those who must count the ballots in this system. Another way to represent this ballot, using the symbol > to mean preferred (for numbers this is the symbol used for greater than), is:

    $$B > C > A.$$

    Ballots that allow one to express an order of preference but not intensity are sometime called ordinal ballots, to contrast them with cardinal ballots where some system to express intensity of preference is offered. Note that some words of explanation are necessary for most people to understand how to read the meaning of the ballot. Perhaps without words a person will understand that the first notation system ranks candidate B above C and A. Some people might find the second notation less clear because they are not familiar with the greater than symbol.

  4. Rank all the candidates with ties, from most preferred to least preferred.
  5. Here is a way to represent a ballot for 5 candidates in this framework.

    Preference ballot with ties. Again the arrow points up, but this time D and E are placed on the same level.

    or

    $$A > D = E > C > B.$$

    The equal sign is used to indicate ties in preference.

    Note that while D and E are liked equally the ballot does not allow the voter to express that E is preferred to C by a small amount but that the voter likes C a lot more than B.

  6. Rank some subset of the candidates with or without ties allowed.
  7. This kind of ballot allows a voter to purposely truncate (omit) the collection of candidates the voter chooses to rank. In the example below, where A, B, and C are the choices seeking office, a voter has ranked only A and C, and B does not appear on the ballot.

    Preference ballot with truncation. The arrow points up but only the top choice of A and the second choice of C are listed.


    Why might a voter not list all of the candidates on the ballot? It might be because the voter has no information about some of the candidates and hence does not feel comfortable listing some candidates. However, another reason may have to do with the fact that the voter has knowledge about the election method which will be used to decide the election. The voter might predict, while sincerely preferring B to C, that listing B might result in B being elected rather than the voter's preferred choice A. Putting it baldly, the voter "lies" about what the voter truly feels and produces an insincere ballot. This approach to voting by one or more voters is often called strategic voting. Strategic voting might be more widely practiced if there are polls which show how other voters in the electorate are leaning. This additional information might convince some voter or group of voters to vote using something other than one's sincere preferences. A remarkable impossibility theorem shows that strategic voting is advantageous!

  8. The voter indicates which candidates are "approved."
  9. What is meant by approving a candidate? The Board of Elections presumably indicates instructions for the meaning of this term and based on the ballots, what decision method will be used to find a winner. One way to think of giving a candidate an approval vote is that one is willing to have this person serve if elected. To the extent that an approval ballot allows a voter to express the amount that the voter likes the candidates it is very "crude" in that it treats all of the approved candidates equally. It is similar to a preference ballot with truncation where all of the candidates that are not omitted are tied at the same level of intensity.

  10. For each candidate the voter indicates a vote of yes or no for that candidate.
  11. Again, the voter might be allowed not to provide a yes/no vote for some subset of the candidates. If the voter is required to vote yes or no on every candidate, this system is identical to approval voting.

How might one design choice and voting systems where the ballot allows for voters to indicate the intensity of their feeling about the candidate? One way of thinking of such a ballot is that each voter gives a "grade" to each choice (candidate) in much the same way that teachers give grades to students in school. What are some of the possible grading or intensity of preference scales? Here are some examples which perhaps you have used in the past or are knowledgeable about.

  • A, B, C, D, F
  • A+, A, A-, B+, B, B-, C+, C, C-, D+, D, D-, F
  • 0 to 100 (100 high)
  • 0 to 100 (100 low)
  • 1 to 99 (99 high)
  • 1, 2, 3, 4, 5 (5 high)
  • 0,1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (10 high)
  • Very poor, poor, neutral, good, very good

Note: There are many variants of verbal scales that solicit voters' (choosers') feelings about choices.

Cardinal ballots can be thought of as offering each voter the opportunity to express an intensity of opinion by giving a numerical grade or value (or using a verbal description) to the candidates. If on the 0 to 100 scale (higher scores are better) Heather gives Mary a score of 90, John a score of 84 and Susan a score of 45, then one can conclude that Mary is preferred to John is preferred to Susan. Mary is above John by 6 points and John is above Susan by 39 points. If Constance gives Mary, John and Susan the score 84, we can conclude that Constance is indifferent to the three choices, but can one conclude that Heather likes Mary twice as much as she likes Susan? Do equal differences in assigned scores mean something? Can voters meaningfully assign a 45 rather than a 44 to a particular candidate?

Most of us are used to the fact that temperatures can be reported by using either the Fahrenheit scale or the Centigrade scale. What makes these temperature scales different from the scales listed for "grading" candidates is that there is a fixed way to get from the number on one scale to the other—water at sea level boils at 100 Centigrade, 212 Fahrenheit. If one wants to change from Centigrade to Fahrenheit then $F = 9/5(C) +32$ is the standard way to do this. Few people agree on how to convert the grade of B+ on an mathematics writing project to a score on the scale 0-100. Scholars intensely debate the tradeoffs between using ordinal ballots and cardinal ballots as well as the pros and cons of different kinds of these ballots.

Impossibility in achieving "fairness"

Perhaps after this discussion you have a new respect for the complexity of making a democracy work. A community of scholars, ranging from those who were trained as philosophers, mathematicians, computer scientists, economists, political scientists, etc. have contributed further insights into the complications of running a democracy. Let me begin with the mathematical economist Kenneth Arrow (1921-2017).

Photo of Kenneth Arrow

Photo of Kenneth Arrow (Courtesy of Wikipedia)

While Arrow was a tremendously important figure in election theory and mathematical economics, his perhaps best-known work stems from his book Social Choice and Individual Values (1951, second edition 1970). Some of this work is already present in his doctoral dissertation in economics written (1951) under the direction of Harold Hotelling, (1895-1973) a mathematical statistician at Columbia University.

Arrow proved that it is IMPOSSIBLE to design an election method that meets a small list of fairness rules. His initial result had a small technical error, and over the years the fairness rules on Arrow's list have been reformulated and expanded in various ways, but the essence of what he showed, because it is a theorem rather than a theory, has to be lived with. The original framework of Arrow was an election where rather than choosing a single winner, what was required was a ranking for society (allowing ties) based on the input ballots of the individuals making up the society. Here are some samples of the kind of fairness rules that Arrow looked at. Typically, what is involved is consistency in what should happen in two closely related sets of ballots, the result in two similar elections.

  1. There is no dictator. The choice for society does not always coincide with the choice of a particular voter - the dictator.
  2. There is no imposed choice. The ranking (winner) depends on the votes of the voters rather than being selected on the basis of some expert or wise person (oracle).
  3. Monotonicity. Getting more votes should not harm a candidate.
  4. Any set of ballots should be assigned a "winner." Those counting the votes based on ballots cannot reject some election results as being too "weird" to be acted upon.
  5. Independence of irrelevant alternative. The relative position of two candidates in the society ranking should depend only on the rank of these two candidates and not the presence of other choices.

Intuitively, one can state that Arrow's Impossibility Theorem means that for elections (or economic decision making) where there are 3 or more candidates (choices) using ordinal ranked ballots with ties allowed (system 2 described above), there are no election decision methods which are "perfect" in the sense that they obey a short list of fairness conditions.

In addition to Arrow's Theorem there is another in some ways more dramatic impossibility theorem. This result is known as the Gibbard-Satterthwaite Theorem named for the philosopher Alan Gibbard and the economist (who teaches at a business school) Mark Satterthwaite. What they independently showed is that in very general systems of voting where there are 3 or more choices (candidates) to decide among, the only election decision method that avoids the incentive for voters to use strategic voting (submit ballots that do not represent what they truly believe, that is lie about their preferences) is dictatorship.

There are by now many different impossibility theorems of this kind, where the properties that the election decision method obeys differ in the differing results but the theorems have in common that they show that if certain fairness rules are important in your view, there is no method that obeys all of the desirable conditions. Part of the reason that attempts to reform or improve election methods fail is that reformers typically differ on what fairness conditions are the essential ones. Another complexity is that some appealing election decision methods are computationally hard. For elections with many voters and candidates no computer can decide the winner/ranking in a reasonable amount of time. Experts also differ on whether the voters will accept a voting method to replace plurality if the description of the system is very complex to explain.

To help give you a sense of these matters using a specific example, let's consider the election below where 55 people have ranked 5 candidates, where there are no ties or truncations.

Collection of ballots


Take a moment to decide who you think should win this election.

Now verify that these five election decision methods choose a different winner!

  1. Plurality (Candidate with most first place votes wins.)
  2. Run-off (A run-off between the two persons with highest number of first places votes is the winner). The voters do not have to go to the polls to vote again because the ballots code the information about their preferences about all the choices. Of course, if the voters went to the polls a second time their preferences might change but typically many fewer people vote when there is a physical run-off election needed at a later date than the original vote.
  3. Sequential run-off (If no candidate has a majority eliminate the candidate with the lowest number of first place votes, transferring votes for the eliminated candidates to the next highest ranked person, until a single winner emerges.)
  4. Borda Count (Candidates are assigned points from a ballot based on the number of candidates below the particular candidate and the candidate with the most points wins. Using the ballots above B gets $18(0) + 12(4) + 10(3) + 9(1) + 4(3) +2(1)$ points.)
  5. Condorcet (If there is a candidate who beats every other candidate in a two-way race that candidate wins. There are ballots for which no candidate meets the criterion.)

The example above shows that for some elections there are many appealing methods which do not agree as to who should win!

While the general public widely underestimates the role that mathematics plays in the technologies (e.g. cell phone, streaming video) that it enjoys so much, it is also not commonly understood that mathematics pays so much attention to questions about fairness and equity. Unfortunately, it is also true that though mathematics and computer science have shown some of the issues which make it hard to design fair systems for the benefit of society, the public has not taken advantage of designing ways to DO BETTER than what we currently do even if these better systems have faults. Thus, most scholars believe that using election systems that are based on ordinal preference ballots is superior to voting using a plurality ballot, but very few elections are conducted using ordinal preference ballots. Since impossibility theorems show that no perfect system is available, reformers constantly argue about what method to move to, and when a change is made, sometimes an election where the results seem "unintuitive" causes the reform to be undone or discourages reforms in other places.

There are impossibility theorems that affect how fully fair a system of assigning parties a fair share of seats can be in a legislature with $h$ seats. This theorem is known as the Balinski-Young Theorem, after the mathematicians Michel Balinski (1933-2019) and H. Peyton Young. Here is an intuitive statement of this theorem.

(Balinski-Young) There exists no method to assign the $h$ seats in a parliament based on the percentage of the votes for each party that is:

  1. Population monotone (Having a larger percentage vote for a party does not decrease the number of seats the party gets.)
  2. Obeys quota (The number of seats that a party gets is its fraction of the vote times the house size, say $s$, rounded (if not precisely a positive integer) down to the integer below $s$ or rounded up to the integer above $s$.)

The reason for the term population monotone above is related to a version of the apportionment problem for a legislature that arises in the United States. Each state has a certain population. The number of seats assigned to each of the states (currently 50) every 10 years based on census data is required by the US Constitution to be proportional to the population of the state. (A complication in the US version of the problem is that each state is required to get at least one seat in the House of Representatives.) A method used in the past allowed a state to lose representation (based on the same population data for the states) when the total house size $h$ increased. This phenomenon came to be known as the Alabama Paradox, named for the state it affected.

Algorithmic fairness

The rapid growth of the availability and power of computers has spurred the development of artificial intelligence methods. Increasingly, decisions about who should get bail after being arrested for a crime, who should get a loan for a house they want to buy, or who should get access to affordable housing are being carried out by computer programs. Some of these programs are "trained" to make decisions based on data about the success of decisions made in the past by humans. However, it has become apparent that some of the systems trained on data that was biased to begin with exhibit the same biases that originally occurred when the decisions were made by biased humans (see the July 2020 Feature Column for one example). Researchers are beginning to explore axioms describing fairness in this context. Extrapolating from voting theory, one might expect to obtain impossibility theorems showing that no matter how an algorithm is trained, there are axioms one might wish to hold that cannot be met simultaneously.

In conclusion, in democratic societies we pursue ways to make these systems work better or more fairly. Mathematics helps develop definitions to compare different means to make something better or more fair. But there are inherent limitations, expressed as mathematical impossibility theorems, that show the limits of how much it is possible, no matter how well meaning a society may be, to be totally fair.


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