lecture on mathematics of voting and apportionment

1-1

Math Circle

Lecture on Mathematics of Voting

and Apportionment

Ernesto Diaz Assistant Professor of Mathematics Department of Natural Sciences and

Mathematics

Excursions in Modern Mathematics, 7e: 1.1 - 2 Copyright © 2010 Pearson Education, Inc.

1 The Mathematics of Voting The Paradoxes of Democracy

•  Vote! In a democracy, the rights and duties of citizenship are captured in that simple one-word mantra.

•  We vote in presidential elections, gubernatorial elections, local elections, school bonds, stadium bonds, American Idol selections, and initiatives large and small.


1 The Mathematics of Voting The Paradoxes of Democracy

•  The paradox is that the more opportunities we have to vote, the less we seem to appreciate and understand the meaning of voting.

•  Why should we vote?

•  Does our vote really count?

•  How does it count?


1 The Mathematics of Voting Voting Theory

•  First half is voting;

•  Second half is counting. •  Arrow’s impossibility theorem:

A method for determining election results that is democratic and always fair is a mathematical impossibility.


1 The Mathematics of Voting

1.1 Preference Ballots and Preference Schedules

1.2 The Plurality Method

1.3 The Borda Count Method


The Math Appreciation Society (MAS) is a student organization dedicated to an unsung but worthy cause, that of fostering the enjoyment and appreciation of mathematics among college students. The Tasmania State University chapter of MAS is holding its annual election for president. There are four candidates running for president: Alisha, Boris, Carmen, and Dave (A, B, C, and D for short). Each of the 37 members of the club votes by means of a ballot indicating his or her first, second, third, and fourth choice. The 37 ballots submitted are shown on the next slide. Once the ballots are in, it’s decision time. Who should be the winner of the election? Why?

Example 1.1 The Math Club Election


Example 1.1 The Math Club Election


•  Candidate with the most first-place votes (called the plurality candidate) wins

•  Don’t need each voter to rank the candidates - need only the voter’s first choice

•  Vast majority of elections for political office in the United States are decided using the plurality method

•  Many drawbacks - other than its utter simplicity, the plurality method has little else going in its favor

Plurality Method


Under plurality:

Example 1.2 The Math Club Election (Plurality)

A gets 14 first-place votes

B gets 4 first-place votes

C gets 11 first-place votes

D gets 8 first-place votes

and the results are clear - A wins (Alisha)


The allure of the plurality method lies in its simplicity (voters have little patience for complicated procedures) and in the fact that plurality is a natural extension of the principle of majority rule:

Majority Candidate

In a democratic election between two candidates, the candidate with a majority (more than half) of the votes should be the winner.


•  Two candidates: a plurality candidate is also a majority candidate - everything works out well

•  Three or more candidates: there is no guarantee that there is going to be a majority candidate

Problem with Majority Candidate


In the Math Club election: Problem with Majority Candidate

•  majority would require at least 19 first-place votes (out of 37).

•  Alisha, with 14 first-place votes, had a plurality (more than any other candidate) but was far from being a majority candidate

•  With many candidates, the percentage of the vote needed to win under plurality can be ridiculously low


One of the most basic expectations in a democratic election is the notion that if there is a majority candidate, then that candidate should be the winner of the election.

Majority Criterion

If candidate X has a majority of the first-place votes,then candidate X should be the winner of the election.

THE MAJORITY CRITERION


The plurality method satisfies the majority criterion-that’s good!

The principal weakness of the plurality method is that it fails to take into consideration a voter’s other preferences beyond first choice and in so doing can lead to some very bad election results. To underscore the point, consider the following example.

The Condorcet Criterion


Tasmania State University has a superb marching band. They are so good that this coming bowl season they have invitations to perform at five different bowl games: the Rose Bowl (R), the Hula Bowl (H), the Fiesta Bowl (F), the Orange Bowl (O), and the Sugar Bowl (S). An election is held among the 100 members of the band to decide in which of the five bowl games they will perform. A preference schedule giving the results of the election is shown.

Example 1.3 The Marching Band Election


•  Under the plurality method, Rose Bowl wins with 49 first-place votes

•  Bad outcome - 51 voters have the Rose Bowl as last choice

•  Hula Bowl has 48 first-place votes and 52 second-place votes

•  Hula Bowl is a far better choice to represent the wishes of the entire band.



A candidate preferred by a majority of the voters over every other candidate when the candidates are compared in head-to-head comparisons is called a Condorcet candidate

Condorcet Criterion

If candidate X is preferred by the voters over each of the other candidates in a head-to-head comparison,then candidate X should be the winner of the election.

THE CONDORCET CRITERION


The idea behind insincere voting (also known as strategic voting) is simple: If we know that the candidate we really want doesn’t have a chance of winning, then rather than “waste our vote” on our favorite candidate we can cast it for a lesser choice who has a better chance of winning the election. In closely contested elections a few insincere voters can completely change the outcome of an election.

Insincere Voting


Example 1.4 The Marching Band Election Gets Manipulated


•  Insincere voting common in real-world elections

Consequences of Insincere Voting

•  2000 and 2004 presidential elections: Close races, Ralph Nader lost many votes - voters did not want to “waste their vote.”

•  Independent and small party candidates never get a fair voice or fair level funding (need 5% of vote to qualify for federal funds)

•  Entrenched two-party system, often gives voters little real choice


•  Each place on a ballot is assigned points

•  With N candidates, 1 point for last place, 2 points for second from last, and so on

•  First-place vote is worth N points

•  Tally points for each candidate separately •  Candidate with highest total is winner

•  Candidate is called the Borda winner

The Borda Count Method


Let’s use the Borda count method to choose the winner of the Math Appreciation Society election first introduced in Example 1.1. Table 1-4 shows the point values under each column based on first place worth 4 points, second place worth 3 points, third place worth 2 points, and fourth place worth 1 point.

Example 1.5 The Math Club Election (Borda Method)


Tally the points:

Example 1.5 The Math Club Election (Borda Method)

A gets: 56 + 10 + 8 + 4 + 1 = 79 points B gets: 42 + 30 + 16 + 16 + 2 = 106 points

C gets: 28 + 40 + 24 + 8 + 4 = 104 points

D gets: 14 + 20 + 32 + 12 + 3 = 81 points

The Borda winner of this election is Boris! (Wasn’t Alisha the winner of this election under the plurality method?)


In contrast to the plurality method, the Borda count method takes into account all the information provided in the voters’ preference ballots, and the Borda winner is the candidate with the best average ranking - the best compromise candidate if you will. On its face, the Borda count method seems like an excellent way to take full consideration of the voter’s preferences, so indeed, what’s wrong with it? The next example illustrates some of the problems with the Borda count method.

What’s wrong with the Borda Method?


The Borda Method violates two basic criteria of fairness:

What’s wrong with the Borda Method?

•  Majority criterion

•  Condorcet criterion

Despite its flaws, experts in voting theory consider the Borda count method one of the best, if not the very best, method for deciding elections with many candidates.


•  individual sports awards (Heisman Trophy winner, NBA Rookie of the Year, NFL MVP, etc.)

•  college football polls •  music industry awards •  hiring of school principals, university

presidents, and corporate executives

Borda count method in Real Life


2 The Mathematics of Power

2.1 An Introduction to Weighted Voting

2.2 The Banzhaf Power Index


In a democracy we take many things for granted, not the least of which is the idea that we are all equal. When it comes to voting rights, the democratic ideal of equality translates into the principle of one person-one vote. But is the principle of one person-one vote always fair? Should one person-one vote apply when the voters are institutions or governments, rather than individuals?

Weighted Voting


What we are talking about here is the exact opposite of the principle of one voter–one vote, a principle best described as one voter–x votes and formally called weighted voting. Weighted voting is not uncommon; we see examples of weighted voting in shareholder votes in corporations, in business partnerships, in legislatures, in the United Nations, and, most infamously, in the way we elect the President of the United States.

Weighted Voting


The Electoral College consists of 51 “voters”(each of the 50 states plus the District of Columbia), each with a weight determined by the size of its Congressional delegation (number of Representatives and Senators). At one end of the spectrum is heavyweight California (with 55 electoral votes); at the other end of the spectrum are lightweights like Wyoming, Montana, North Dakota, and the District of Columbia (with a paltry 3 electoral votes). The other states fall somewhere in between.

Electoral College


The 2000 and 2004 presidential elections brought to the surface, in a very dramatic way, the vagaries and complexities of the Electoral College system, and in particular the pivotal role that a single state (Florida in 2000, Ohio in 2004) can have in the final outcome of a presidential election. In this chapter we will look at the mathematics behind weighted voting, with a particular focus on the question of power.

Electoral College


Thus, a generic weighted voting system with N players can be written as:

Notation

[q: w1, w2,…, wN]

(with w1 ≥ w2 ≥ … ≥ wN)

GENERIC WEIGHTED VOTING SYSTEM WITH N PLAYERS


Four college friends (P1, P2, P3, and P4) decide to go into business together. Three of the four (P1, P2, and P3) invest $10,000 each, and each gets 10 shares in the partnership. The fourth partner (P4) is a little short on cash, so he invests only $9000 and gets 9 shares. As usual, one share equals one vote. The quota is set at 75%, which here means q = 30 out of a total of V = 39 votes. Mathematically (i.e., stripped of all the irrelevant details of the story), this partnership is just the weighted voting system [30: 10, 10, 10, 9].

Example 2.6 Unsuspecting Dummies


Everything seems fine with the partnership until one day P4 wakes up to the realization that with the quota set at q = 30 he is completely out of the decision-making loop: For a motion to pass P1, P2, and P3 all must vote Yes, and at that point it makes no difference how P4 votes. Thus, there is never going to be a time when P4’s votes are going to make a difference in the final outcome of a vote. Surprisingly, P4–with almost as many votes as the other partners–is just a dummy!

Example 2.6 Unsuspecting Dummies


2 The Mathematics of Power

2.1 An Introduction to Weighted Voting

2.2 The Banzhaf Power Index


In weighted voting the player’s weights can be deceiving. Sometimes a player with a few votes can have as much power as a player with many more votes (see Example 2.4); sometimes two players have almost an equal number of votes, and yet one player has a lot of power and the other one has none (see Example 2.6).

Weighted Voting


A coalition is any set of players who might join forces and vote the same way. In principle, we can have a coalition with as few as one player and as many as all players. The coalition consisting of all the players is called the grand coalition. Since coalitions are just sets of players, the most convenient way to describe coalitions mathematically is to use set notation. For example, the coalition consisting of players P1, P2,and P3 can be written as the set {P1, P2, P3} (order does not matter).

Coalitions


The Parliament of Icelandia has 200 members, divided among three political parties: P1, P2, and P3, with 99, 98, and 3 seats in Parliament, respectively. Decisions are made by majority vote, which in this case requires 101 out of the total 200 votes. Let’s assume, furthermore, that in Icelandia members of Parliament always vote along party lines (not voting with your party is very unusual in parliamentary governments).

Example 2.8 The Weirdness of Parliamentary Politics


We can think of the Parliament of Icelandia as the weighted voting system [101: 99, 98, 3].


In this weighted voting system we have four winning coalitions.


In the two-party coalitions, both parties are critical players (without both players the coalition wouldn’t win); in the grand coalition no party is critical–any two parties together have enough votes to win. Each of the three parties is critical the same number of times, and consequently, one could rightfully argue that each of the three parties has the same amount of power (never mind the fact that one party has only 3 votes!).



4 The Mathematics of Apportionment

4.1 Apportionment Problems 4.2 Hamilton’s Method and the Quota Rule

4.3 The Alabama and Other Paradoxes 4.4 Jefferson’s Method 4.5 Adam’s Method

4.6 Webster’s Method


Without a doubt, the most important and heated debate at the Constitutional Convention concerned the makeup of the legislature. The small states wanted all states to have the same number of representatives; the larger states wanted some form of proportional representation. The Constitutional compromise, known as the Connecticut Plan, was a Senate, in which every state has two senators, and a House of Representatives, in which each state has a number of representatives that is a function of its population.

Apportionment


While the Constitution makes clear that seats in the House of Representatives are to be allocated to the states based on their populations (“According to their respective Numbers”), it does not prescribe a specific method for the calculations. Undoubtedly, the Founding Fathers felt that this was a relatively minor detail–a matter of simple arithmetic that could be easily figured out and agreed upon by reasonable people.

Apportionment


Certainly it was not the kind of thing to clutter a Constitution with or spend time arguing over in the heat of the summer. What the Founding Fathers did not realize is that Article I, Section 2, set the Constitution of the United States into a collision course with a mathematical iceberg known today (but certainly not then) as the apportionment problem.

Apportionment


Obviously, the word apportion is the key word in this chapter. There are two critical elements in the dictionary definition of the word: (1) We are dividing and assigning things, and (2) we are doing this on a proportional basis and in a planned, organized fashion. We will start this section with a pair of examples that illustrate the nature of the problem we are dealing with. These examples will raise several questions but not give any answers. Most of the answers will come later.

Apportionment Problems


It is the year 2525, and all the planets in the Utopia galaxy have finally signed a peace treaty. Five of the planets (Alanos, Betta, Conii, Dugos, and Ellisium) decide to join forces and form an Intergalactic Federation. The Federation will be ruled by an Intergalactic Congress consisting of 50 “elders,” and the 50 seats in the Intergalactic Congress are to be apportioned among the planets according to their respective populations.

Example 4.2 The Intergalactic Congress of Utopia


The population data for each of the planets (in billions) are shown in Table 4-2. Based on these population figures, what is the correct apportionment of seats to each planet?

Example 4.2 The Intergalactic Congress of Utopia

We’ll get back to this later . . .


The basic elements of every apportionment problem are as follows:

The “states” This is the term we will use to describe the parties having a stake in the apportionment. Unless they have specific names (Azucar, Bahia, etc.), we will use A1, A2,…, AN, to denote the states.

Apportionment: Basic Concepts and Terminology


The “seats” This term describes the set of M identical, indivisible objects that are being divided among the N states. For convenience, we will assume that there are more seats than there are states, thus ensuring that every state can potentially get a seat. (This assumption does not imply that every state must get a seat!)



The “populations” This is a set of N positive numbers (for simplicity we will assume that they are whole numbers) that are used as the basis for the apportionment of the seats to the states. We will use p1, p2,…, pN, to denote the state’s respective populations and P to denote the total population P = p1 + p2 +…+ pN).



The standard divisor(SD) This is the ratio of population to seats. It gives us a unit of measurement (SD people = 1 seat) for our apportionment calculations.



The standard quotas The standard quota of a state is the exact fractional number of seats that the state would get if fractional seats were allowed. We will use the notation q1, q2,…, qN to denote the standard quotas of the respective states. To find a state’s standard quota, we divide the state’s population by the standard divisor. (In general, the standard quotas can be expressed as fractions or decimals–round them to two or three decimal places.)



Upper and lower quotas The lower quota is the standard quota rounded down and the upper quota is the standard quota rounded up. In the unlikely event that the standard quota is a whole number, the lower and upper quotas are the same. We will use L’s to denote lower quotas and U’s to denote upper quotas. For example, the standard quota q1 = 32.92 has lower quota L1 = 32 and upper quota U1 = 33.



Our main goal in this chapter is to discover a “good” apportionment method–a reliable procedure that (1) will always produce a valid apportionment (exactly M seats are apportioned) and (2) will always produce a “fair” apportionment. In this quest we will discuss several different methods and find out what is good and bad about each one.

Apportionment Methods



4.1 Apportionment Problems

4.2 Hamilton’s Method and the Quota Rule

4.3 The Alabama and Other Paradoxes


Hamilton’s method can be described quite briefly: Every state gets at least its lower quota. As many states as possible get their upper quota, with the one with highest residue (i.e.,fractional part) having first priority, the one with second highest residue second priority, and so on. A little more formally, it goes like this:

Hamilton’s Method


Step 1 Calculate each state’s standard quota.

Step 2 Give to each state its lower quota.

Step 3 Give the surplus seats (one at a time) to the states with the largest residues (fractional parts) until there are no more surplus seats.

HAMILTON’S METHOD


As promised, we are revisiting the Parador Congress apportionment problem. We are now going to find our first real solution to this problem–the solution given by Hamilton’s method (sometimes, for the sake of brevity, we will call it the Hamilton apportionment). Table 4-6 shows all the details and speaks for itself. (Reminder: The standard quotas in the second row of the table were computed in Example 4.3.)

Example 4.4 Parador’s Congress (Hamilton’s Method)


Hamilton’s method (also known as Vinton’s method or the method of largest remainders) was used in the United States only between 1850 and 1900.

Example 4.4 Parador’s Congress (Hamilton’s Method)


Hamilton’s method is still used today to apportion the legislatures of Costa Rica, Namibia, and Sweden. At first glance, Hamilton’s method appears to be quite fair. It could be reasonably argued that Hamilton’s method has a major flaw in the way it relies entirely on the size of the residues without consideration of what those residues represent as a percent of the state’s population. In so doing, Hamilton’s method creates a systematic bias in favor of larger states over smaller ones.

Hamilton’s Method


This is bad–a good apportionment method should be population neutral, meaning that it should not be biased in favor of large states over small ones or vice versa. To be totally fair, Hamilton’s method has two important things going for it: (1) It is very easy to understand, and (2) it satisfies an extremely important requirement for fairness called the quota rule.

Hamilton’s Method


No state should be apportioned a number of seats smaller than its lower quota or larger than its upper quota. (When a state is apportioned a number smaller than its lower quota, we call it a lower-quota violation; when a state is apportioned a number larger than its upper quota, we call it an upper-quota violation.)

QUOTA RULE



4.1 Apportionment Problems

4.2 Hamilton’s Method and the Quota Rule

4.3 The Alabama and Other Paradoxes


The most serious (in fact, the fatal) flaw of Hamilton’s method is commonly known as the Alabama paradox. In essence, the Alabama paradox occurs when an increase in the total number of seats being apportioned, in and of itself, forces a state to lose one of its seats. The best way to understand what this means is to look carefully at the following example.

Alabama Paradox - Hamilton’s Method


The small country of Calavos consists of three states: Bama, Tecos, and Ilnos. With a total population of 20,000 and 200 seats in the House of Representatives, the apportionment of the 200 seats under Hamilton’s method is shown in Table 4-7.

Example 4.5 More Seats Means Fewer Seats


Now imagine that overnight the number of seats is increased to 201, but nothing else changes. Since there is one more seat to give out, the apportionment has to be recomputed.



Here’s the new apportionment (Hamilton’s method) for a House with 201 seats. (Notice that for M = 200, the SD is 100; for M = 201, the SD drops to 99.5.)



Sometime in the early 1900s it was discovered that under Hamilton’s method a state could potentially lose some seats because its population got too big! This phenomenon is known as the population paradox. To be more precise, the population paradox occurs when state A loses a seat to state B even though the population of A grew at a higher rate than the population of B. Too weird to be true, you say? Check out the next example.

The Population Paradox


Part I. The Apportionment of 2525. The first Intergalactic Congress was apportioned using Hamilton’s method, based on the population figures (in billions) shown in the second column of Table 4-10. (This is the apportionment problem discussed in Example 4.2.) There were 50 seats apportioned.

Example 4.6 A Tale of Two Planets


Part I. The Apportionment of 2525.



Part II. The Apportionment of 2535. After 10 years of peace, all was well in the Intergalactic Federation. The Intergalactic Census of 2535 showed only a few changes in the planets’ populations–an 8 billion increase in the population of Conii, and a 1 billion increase in the population of Ellisium. The populations of the other planets remained unchanged from 2525. Nonetheless, a new apportionment was required.



Part II. The Apportionment of 2535. Here are the details of the 2535 apportionment under Hamilton’s method.



In 1907 Oklahoma joined the Union. Prior to Oklahoma becoming a state, there were 386 seats in the House of Representatives. At the time the fair apportionment to Oklahoma was five seats, so the size of the House of Representatives was changed from 386 to 391. The point of adding these five seats was to give Oklahoma its fair share of seats and leave the apportionments of the other states unchanged.

The New-States Paradox


The Metro Garbage Company has a contract to provide garbage collection and recycling services in two districts of Metropolis, Northtown (with 10,450 homes) and the much larger Southtown (89,550 homes). The company runs 100 garbage trucks, which are apportioned under Hamilton’s method according to the number of homes in the district. A quick calculation shows that the standard divisor is SD = 1000 homes, a nice, round number which makes the rest of the calculations (shown in Table 4-12) easy.

Example 4.7 Garbage Time


As a result of the apportionment, 10 garbage trucks are assigned to service Northtown and 90 garbage trucks to service Southtown.



Now imagine that the Metro Garbage Company is bidding to expand its territory by adding the district of Newtown (5250 homes) to its service area. In its bid to the City Council the company promises to buy five additional garbage trucks for the Newtown run so that its service to the other two districts is not affected. But when the new calculations (shown in Table 4-13) are carried out, there is a surprise: One of the garbage trucks assigned to Southtown has to be reassigned to Northtown!



Notice that the standard divisor has gone up a little and is now approximately 1002.38.



There are two key lessons we should take from this section:

Hamilton’s Method

(1) In terms of fairness, Hamilton’s method leaves a lot to be desired; and

(2) the critical flaw in Hamilton’s method is the way it handles the surplus seats. Clearly, there must be a better apportionment method.

lecture on mathematics of voting and apportionment

Documents