Mathematics in the Social Sciences

Module 9 Mathematics in the Social Sciences Introduction

• Mathematical theory and tools are valuable in social sciences like economics, demography, psychology, and political science.

• Computational methods and technology such as geospatial information systems are used to study and simulate social behavior and systems.

• Examples of social issues mathematics can help examine:

  • How the number of seats for a region is determined in the Philippine Congress.

  • Who controls power among shareholders in a company.

  • How the US President is elected.

  • Whether voting systems are fair.

• Module will discuss how mathematics can provide more informed and scientific answers to these questions.

• The mathematics involves elementary concepts, but answers can be complicated.

Learning Outcomes

• Apply four different methods of voting to determine the winner of a ranked election.

• Discuss the concept of principles of fairness in voting and define at least 5 of these principles.

• Create a hypothetical preference schedule that produces different winners using different voting methods.

• Reflect on the implications of Arrow’s Impossibility Theorem.

• Identify winning and losing coalitions and critical players in a weighted voting system.

• Compute the Banzhaf power distribution given a weighted voting system.

• Clarify distinctions between dictators, dummies, and players with veto power.

• Reflect on the power relations in the Philippine Lower House and Senate in terms of coalitions and bloc voting.

• Differentiate between quota and divisor methods.

• Apply the quota methods of Hamilton and Lowndes given an apportionment problem.

• Apply the divisor methods of Jefferson, Adams and Webster given an apportionment problem.

1.0 Voting Theory and the Mathematics of Social Choice

• Democracy relies on the right of citizens to participate in elections.

• Elections involve more than just counting votes; experts dedicate their careers to this field.

• Key questions in the mathematics of voting:

  • Is there a best (fairest) way to conduct an election and transform individual preferences into a single societal preference (winner)?

  • How can mathematics design, analyze, and compare different election methods?

  • How can mathematics define "fair"?

1.1 The Mathematics of Voting

• Voting theory is part of Social Choice theory.

• Social Choice theory consolidates varied and conflicting choices into a single group (or society) choice that reflects individual desires as much as possible.

• Kenneth Arrow's 1952 paper, “Social Choice and Individual Values,” examined fundamental concepts of this theory.

• Arrow proved an important result addressing the question: “Is there an ideal voting system?”

• The mathematical concept underlying voting systems is order.

• Order Theory studies ways objects can be ordered.

Voting Systems

• A voting system is a method for a group to select a winner from several candidates.

• Choosing is easy with two alternatives or one person choosing.

• With several people choosing from three or more alternatives, the process is more complex.

• Example: In an election with 8 candidates, three got most of the votes; Tito won with 36.99% using the plurality method.

• With the winner obtaining only about 37% of the votes, it is uncertain whether the system reflected the voters' will.

• Scenarios:

  • Supporters of Vic and Joey had Tito as their second choice. (Tito might be the best candidate).

  • Supporters of Vic and Joey liked Tito the least. (Tito might not be the best representative).

  • One of the 5 others may be perfectly satisfactory to all voters. (That one person may even be the second choice of all, and perhaps should have been the winner).

Ranked Voting Systems

• Ranked (or preferential) voting systems ask voters to rank candidates.

• Analysis is simplified by assuming voters rank all candidates in order of preference, with no ties allowed.

Transitivity

• Voting systems should satisfy certain conditions or features.

• Individual preferences are assumed to be transitive: if a voter prefers X to Y and Y to Z, they prefer X to Z.

• Transitivity means relative preferences are not altered by eliminating candidates.

• “Fairness” principles are also desirable conditions that make elections “fair” from the voters' perspective.

• In designing a voting system, these desirable “fairness conditions or principles” are imposed.

• Math studies don't deal with cheating or manipulation.

Example 1: Non-transitivity can cause problems!

• Leni chooses from Alex (A), Billy (B), and Carl (C).

• She ranks them by intelligence, looks, and income:

• In pairs, Leni prefers A to B, B to C, and C to A, which is non-transitive and problematic.

• Which suitor to choose?

• Resolve by considering other attributes or weighting attributes differently.

• Translating to a larger population:

  • An election with 3 candidates (A, B, C) asks voters to rank them, producing three distinct rankings:

• In pairs, 2/3 prefer A to B, 2/3 prefer B to C, and 2/3 prefer A to C!

• No candidate stands out.

1.2 Voting Methods

• “Voting method” means the mathematical process, algorithm, or manner of counting and consolidating individual votes to choose a winner.

• It does not refer to voting machines, polling precincts, etc.

• Plurality voting (pataasan) is the most popular, where the candidate with the most votes (most 1st-place votes) wins.

• Example 2: Plurality isn’t always the best method

• In a barangay election, 100 residents elected their leader from 5 candidates (R, H, C, O, S).

• Voters ranked the candidates, and the results were consolidated:

• Using plurality, R wins with 49 first-place votes, but is the last choice of a majority (51).

• H almost won (one vote behind) and is the second choice of the rest.

• H might be a more acceptable winner.

• Plurality method failed to choose H.

• Candidate H's performance can be evaluated through one-to-one comparisons with other candidates.

  • H got 51 votes versus 49 for R.

  • H had 97 votes versus 3 from C.

  • H is preferred to both O and S by all voters.

• Plurality method failed to satisfy a basic principle of fairness called the Condorcet criterion.

Condorcet Criterion

• If a candidate wins in a one-to-one comparison with any other alternative, that candidate should win the election.

• Due to Marquis de Condorcet (1743-1794).

• Several voting methods and fairness criteria will be examined using a fixed election's results.

• Four methods:

  1. Plurality Method

  2. Borda Count Method

  3. Method of Pairwise Comparisons

  4. Plurality with Elimination Method

• Plurality is simplest; the others are also popular.

Borda Count Method

• Weighted voting method.

• Each place on a ballot is assigned points.

• N candidates: N points for first, N-1 for second, down to 1 point for last place.

• Tally points, and the candidate with the highest total wins.

• Named after Jean-Charles de Borda (1733-1799).

Method of Pairwise Comparisons

• Every candidate is matched one-on-one with every other candidate.

• Each one-to-one pairing is a pairwise comparison.

• When pairing candidates (X or Y), each vote is assigned to either X or Y by the preference order.

  • (X gets votes of all voters ranking X higher than Y).

• The winner of each head-to-head match-up gets 1 point.

• The election winner is the candidate with the most points.

Plurality with Elimination Method

• Sophisticated version of the plurality method, carried out in rounds.

• Eliminate the candidate with the fewest first-place votes.

• Remove the candidate from the preference schedule along with all assigned votes.

• Repeat until a candidate with a majority of 1st place votes emerges.

Example 3: The Math Lovers Club Election

• Four candidates for President: Alice (A), Ben (B), Cris (C), and Dina (D).

• 37 members rank the candidates:

1. Plurality Method: A has the most 1st-place votes (14) and wins.

2. Borda Count 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 winner is Ben (B); Alice places last!

1. Method of Pairwise Comparison

• Compare A and B. A is preferred by 14 over B, and B is preferred by 23 over A. B wins and gets 1 point.

• Compare all other pairs:

  • A vs C (14 to 23) → C gets 1 pt

  • A vs D (14 to 23) → D gets 1 pt

  • B vs C (18 to 19) → C gets 1 pt

  • B vs D (28 to 9) → B gets 1 pt

  • C vs D (25 to 12) → C gets 1 pt

• Candidate Cris (C) has the most points (3) and is the winner.

1. Plurality with Elimination Method

• Round 1: B has the least 1st-place votes; eliminate B.

• Combine the identical columns for Round 2.

• C has the least 1st place votes (11); eliminate C, combine identical 2nd and 3rd columns for Round 3.

• A has fewer 1st place votes than D, eliminate A, leaving candidate Dave (D) as the winner.

Summary of Winners

• Four methods yield four different winners!

Activity 1

Find the winners of an election with 55 voters whose preference schedule is given below using the four different voting methods.

1.3 Other Fairness Principles

• Condorcet Criterion: If there is a candidate who wins in a one-to-one comparison with any other alternative, that candidate should be the winner.

1. Majority Criterion: If a candidate is the first choice of the majority (50%+1) of voters, that candidate should be the winner.

2. Monotonicity Criterion: If a candidate X is the winner, and in a re-election, all voters change their preferences favorably to X, then X should still win.

3. Independence of Irrelevant Alternatives Criterion: If a candidate X is the winner, and one or more candidates are removed and votes recounted, then X should still be the winner.

4. Unanimity: If every individual prefers a certain option to another, then so must the resulting societal choice.

5. Non-dictatorship: The outcome should not simply follow the preference order of a single individual while ignoring all others.

Problems with the Different Voting Methods

• Plurality method may violate Condorcet criterion.

• All four voting methods have weaknesses: election outcomes may violate fairness principles.

• Example 4. The Board of Trustees of a university is choosing its Chair. There are 11 board members and 4 candidates. The Board will use the Borda Count Method to choose its Chair. The results of the election are given below:

| No. of Voters | 6 | 2 | 3 |
| ----------- | - | - | - |
| 1st place: 4 pts | A | B | C |
| 2nd place: 3 pts | B | C | D |
| 3rd place: 2 pts | C | D | B |
| 4th place: 1 pt | D | A | A |

• Computing the total weighted points for each candidate, we find that B has 32 points, C has 30 points, A has 29 points, and D has 19 points. By the Borda Count method, candidate B wins.

• With this outcome, the Majority Criterion is violated. Candidate A has 6 out of 11 1st-place votes, a majority, but the Borda method failed to produce A as a winner.

• Example 5. A student club election is held with 3 candidates A, B, C. The winner is to be decided by the plurality with elimination method. The preference schedule is given below:

| No. of votes | 7 | 8 | 10 | 4 |
| ----------- | - | - | -- | - |
| 1st choice | A | B | C | A |
| 2nd choice | B | C | A | C |
| 3rd choice | C | A | B | B |

• Eliminating B (who has the least number of 1st place votes), we obtain the reduced table

• With this round, we see that candidate C wins.

• However, the Election Committee declared the election null and void due to some irregularity and asked the voters to cast their votes again. A second election is held and the 4 voters (in the last column of Round 1) decide to change their ranking. They switch their 1st and 2nd choices (between A and C), expecting to be on the winner’s side after the new election. No other voters change their choices. Since C won the first election, and the new votes only increased C’s votes, we expect C to win again. Let’s determine the winner of the second election.

| No. of votes | 7 | 8 | 10 | 4 |
| ----------- | - | - | -- | - |
| 1st choice | A | B | C | C |
| 2nd choice | B | C | A | A |
| 3rd choice | C | A | B | B |

• A is eliminated after Round 1.

• B wins in the new election!

• This outcome violates the Monotonicity Criterion.

• Example 6. The preference schedule for a certain election is shown below. The winner is to be determined using the method of pairwise comparison.

| No. of votes | 5 | 3 | 5 | 3 | 2 | 4 |
| ----------- | - | - | - | - | - | - |
| 1st place | A | A | C | D | D | B |
| 2nd place | B | D | E | C | C | E |
| 3rd place | C | B | D | B | B | A |
| 4th place | D | C | A | E | A | C |
| 5th place | E | E | B | A | E | D |

• Comparing all pairs, A gets 3 points. B, C, and D have 2 points and E has 1 point. Candidate A wins.

• The Comelec, for some reason, wanted to have a recount, without new voting. Before they started their recount, losing candidates B,C, and D conceded and requested that their votes be omitted (essentially dropping out from the race). This means that the new preference schedule is now —

| No. of votes | 5 | 3 | 5 | 3 | 2 | 4 |
| ----------- | - | - | - | - | - | - |
| 1st choice | A | A | | | D | |
| 2nd choice | | D | E | C | C | E |
| 3rd place | C | B | D | B | B | A |
| 4th place | D | C | A | E | A | C |
| 5th place | E | E | B | A | E | D |

• And this simplifies to —

• The winner is now E!

• Originally, the winner was A, but when some candidates dropped out and no re-vote was made, the winner became E. This violates the Independence of Irrelevant Alternatives fairness criterion.

• Four different methods produced four different winners.

• Instances of elections using certain voting methods violated fairness criteria.

• There is no ideal voting method!

1.4 Arrow’s Impossibility Theorem

• Arrow’s Impossibility Theorem (1952)

• It is impossible to design a voting system that would simultaneously obey in all voting instances all of the following fairness conditions: monotonicity, independence of irrelevant alternatives, unanimity, and non-dictatorship.

• The theorem states that there is no consistent method by which a democratic society can make a choice that is always “fair” when that choice must be made from among three or more alternatives.

• The concept of “fair” is taken in the sense that a fairness principle or condition is not violated.

• Kenneth Arrow (1921-2017) examined these ideas for his Ph.D. dissertation “Social Choice and Individual Values”.

• He published his results in a 1952 essay “A Difficulty in the Concept of Social Welfare”.

• Arrow was awarded the Nobel Prize in Economics in 1972.

• Arrow’s theorem applies to ranked or preferential voting systems.

• The mathematical proof uses concepts of order theory.

• It proves that no voting method can satisfy all reasonable fairness criteria at the same time.

• Are there alterntive methods? There are other voting methods that are also used, even in some legislatures. An example is approval voting.

• In this system, voters are not asked to rank the candidates in order of preference. Given a set of candidates, voters can give their approval to as many (or as few) of their choices. Voting experts consider this strong system, but more applicable to smaller voting populations, like some legislatures or organizations.

• This method work for those voters who cannot or who prefer not to make distinct rankings among the candidates.

According to advocates of the method, approval voting —

• is easy to understand and simple to implement.

• gives voters flexible options and increases voter turnout.

• helps elect the strongest candidates.

• unaffected by the number of candidates.

• will reduce negative campaigning.

Also, in the approval method minority candidates are not greatly disadvantaged.

• Four voting methods discussed:

  • Plurality method

  • Borda count method

  • Method of pairwise comparison

  • Plurality with elimination method

• Several fairness principles introduced:

  • Condorcet criterion

  • Majority principle

  • Monotonicity

  • Independence of irrelevant alternatives

  • Unanimity

  • Non-dictatorship

• Arrow’s “Impossibility Theorem” demonstrates that no voting method will always satisfy all fairness criteria in all voting instances.

Activity 2

• What are advantages of ranked voting systems, as compared to just voting for a single winner without specifying ranks?

• In your opinion, which fairness principles would you least like to be violated?

• Does Arrow’s Impossibility Theorem render elections useless?

2.0 Weighted Voting Systems and the Measurement of Power

• In many elections, the principle of one-person, one-vote is followed.

• In some voting situations, individuals or groups are not always equal, and some voters are given more say than others.

• Weighed voting systems are systems where voting rights are not equally divided among all voters.

• Examples:

  • Corporate shareholder’s meetings, where each shareholder has as many votes as the number of shares owned;

  • Some legislatures, where bloc voting is followed because of strict party loyalties. Here the parties are the players and the sizes of the blocs are the weights of the votes; and

  • Committee voting, where the chair has tie-breaking or veto power.

• What has more say in weighted voting systems or who has more power?

• How is power measured?

• Is there a mathematical treatment of power?

• Understanding how power is distributed will be useful in many ways.

2.1 Weighted Voting Systems

• Analyze voting situations with two alternatives (yes/no votes), assuming no abstentions.

• A weighted voting system has three main ingredients:

  • Players

  • Weights

  • Quota

  • Players: voters, denoted as $P<em>1, P</em>2, …, P_N$ (N players).

  • Weights: number of votes each player controls, denoted as $w<em>1, w</em>2, …, w_N$.

  • Quota: minimum number of votes needed to pass a motion, usually a strict majority (q > 50%, usually 50%+1), but may require more (up to 100% for unanimous).

  • The quota must satisfy: $\frac{1}{2} (w<em>1+ w</em>2 +…+ w<em>N) ≤ q ≤ w</em>1+ w<em>2 + …+ w</em>N$

• Notation: $[q: w<em>1, w</em>2 , … , w_N]$ indicates a weighted system with quota q and N players, with weights listed after the quota in decreasing order.

• Example 1: [13: 8, 6, 4, 3, 2, 1]

  • This weighted voting system has 6 players: P1, P2, P3, P4, P5, P6, with weights 8, 6, 4, 3, 2, 1, respectively. The quota or number of votes to win is 13. There are 24 total number of votes.

• Example 2: [21: 9, 8, 2, 1]

  • Since the total number of votes is only 20, no motion could ever pass because the quota is too high. This is not a valid weighted system.

• Example 3: [15: 10, 9, 8, 7]

  • The quota is less than half the total number of votes. If P1 and P2 voted yes and P3 and P4 voted no, both groups win. So this is not a legal voting system.

• Example 4: [20: 7, 7, 7, 7, 7]

  • Three players need to vote yes to meet the quota. In this case, all players have equal power and this is essentially just a one-person, one-vote system. It can equivalently be described by the system [3: 1, 1, 1, 1, 1].

• Example 5: [6: 7, 3, 1, 1]

  • The first player can pass any motion alone. Such a player is called a dictator – a player whose weight is bigger than or equal to the quota. In systems with dictators, all the other players have no power. A player whose vote has no outcome in the election is called a dummy. In this system, the last 3 players are all dummies.

• Example 6: [8: 7, 3, 1, 1]

  • Player 1 is not a dictator but he can single-handedly prevent any group of players from passing a motion. Player 1 is said to have veto power.

• Example 7: [51: 48, 47, 5]

  • Does player P3 really have very little power compared to P1 and P2? Observe that P1 cannot pass a motion without the help of P2 or P3. Similarly, P2 (and P3) need votes of at least one other player. At least two players are needed to pass a motion. In this case, we see that P3 has just as much power as the two other players!

  • This example shows that in a weighted voting system, a player with the most number of votes does not necessarily hold the most power.

• Example 8: [51: 26, 26, 26, 22]

  • Suppose 4 partners hold shares of 26%, 26%, 26% and 22% respectively. The weights are almost uniform. Can we say that all 4 players more or less have the same power? Not necessarily.

  • The last shareholder is really a dummy. P4 has no power – the weight of P4 ‘s vote (22%), even if close to the others, is not enough to make a losing coalition win, or a winning coalition lose. In this example, the first three partners share equal power. The last has none.

• Example 9: [15: 5, 4, 3, 2, 1]

  • All votes are needed to meet the quota (unanimous vote). Thus, each vote counts as much as the other. The system is equivalent to [5: 1,1,1,1,1] and is essentially a one-person, one-vote system.

• We should thus regard the “one-person, one-vote” principle to mean that all players have an equal say in the outcome of the election, instead of just having an equal number of votes.

2.2 Banzhaf Index and the Mathematical Measurement of Power

• A player’s power cannot be measured simply by the number of votes.

• Example: X has four votes, Y has two votes; X doesn’t necessarily have twice the power of Y.

• If two players have the same number of votes, expect them to have the same power.

• Power measurements:

  • Banzhaf index: power measured by winning. The player whose vote influences the outcome the most has the most power. (The Shapely-Shubik index is another system).

• Example 7 (Revisited): [51: 48, 47, 5]

• Winning combinations:

  • P1 and P2 (95 votes)

  • P1 and P3 (53 votes)

  • P2 and P3 (52 votes)

  • P1, P2 and P3 (100 votes)

• Coalition: A set of players who join forces to vote together.

• Weight of coalition: total number of votes in a coalition.

• Winning coalitions: coalitions with enough votes to win; otherwise, losing coalitions.

• Example 7:

• Coalition G is the grand coalition.

• Winning coalitions for [51: 48,47,5]:

• In coalitions D, E, and F, both players are needed to win.

• In G, no single player is essential.

• Critical player (pivotal player): a player whose desertion turns a winning coalition into a losing one., or a voter who can cause the measure to fail by changing his/her vote from yes to no.

Banzhaf Power Index

• Power is proportional to the number of times the player is critical.

• Previous example: each player is critical twice, so all have equal power. Each holds 1/3 power; Banzhaf power index is 1/3.

• (1/3, 1/3, 1/3) is Banzhaf power distribution for the three players.

Computing the Banzhaf power index:

1. List all coalitions. (N players: $2^N – 1$ coalitions).

2. Determine which are winning coalitions.

3. In each winning coalition, determine which players are critical.

4. Count the number of times player P is critical and call this number $C_P$.

5. Count the total number of times all players are critical and call this number T.

The Banzhaf power index of player P is given by the ratio $\frac{C_P}{T}$, that is,

Power Index of Player P = \frac{\text{# Critical instances of player P}}{\text{# Critical instances of all players}}

• Example 10: Banzhaf distribution for [4: 3, 2,1]

• 1. 7 coalitions: {A}, {B}, {C}, {A,B}, {A,C}, {B,C}, {A,B,C}

• 2-3

i. Based on module 9, in what ways is math present in the Social Sciences?

Mathematical theory and tools are valuable in social sciences like economics, demography, psychology, and political science, helping to examine social issues and provide informed answers.

ii. What mathematical concepts are involved in the election process?

The mathematical concepts involved in the election process include order theory, which underlies voting systems. Voting methods such as plurality, Borda count, pairwise comparisons, and plurality with elimination use mathematical processes to consolidate individual votes and choose a winner. Fairness principles such as the Condorcet criterion, majority criterion, monotonicity criterion, and independence of irrelevant alternatives are also relevant.

iii. Of the four different voting methods presented, which of these is currently used in the Philippines? If you were asked to suggest a different voting method, which would you choose? Why?

The plurality method is used in the Philippines, where the candidate with the most votes wins. If asked to suggest a different voting method, the approval voting might be valuable. In this system, voters can give their approval to as many (or as few) of their choices.

iv. Aside from the ones mentioned in the modules, what are the other applications of mathematics in Social Sciences?

Other applications of mathematics in social sciences could include network analysis in sociology, statistical modeling in market research, and game theory in political negotiations.

v. Can you recall and share a personal experience wherein you recognize the importance of mathematics in Social Sciences?

The importance of mathematics in social sciences can be recognized in situations such as understanding election results where different voting methods yield different outcomes, or in analyzing power distribution in weighted voting systems.

vi. Without mathematics, what is life in Social Sciences?

Without mathematics, social sciences would lack the tools for precise analysis, potentially leading to less informed and less accurate conclusions.