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Social Choice Theory.
Social Choice Theory.
Understanding Social Choice Theory.
Understanding Social Choice Theory.
INTRODUCTION
Social choice theory is an economic theory that considers whether a society can be ordered in a way that reflects individual preferences. The theory was developed by economist Kenneth Arrow and published in his book Social Choice and Individual Values in 1951.
Understanding Social Choice Theory
Frenchman Nicolas de Condorcet laid the groundwork for social choice theory in a 1785 essay. The essay included the jury theorem. In the theorem, each member of a jury has an equal and independent chance of making the correct judgment on whether a defendant is guilty.
Condorcet showed that the majority of jurors are more likely to be correct than each individual juror, thereby making the case for collective decision-making. Condorcet's paradox builds upon his previous theorem and proposes that majority preferences can be irrational. Thus, Condorcet showed that while collective decision-making is preferable to individual decisions, there are still problems associated with it.
In the 20th century, Arrow broadened the theory of social choice beyond the investigation of the properties of majority rule. Arrow's generalization of the theory of social choice asks whether it is possible to find a rule that aggregates individual preferences, judgments, votes, and decisions in a way that satisfies minimal criteria for what should be considered a good rule.
Arrow's social choice theory considers all sorts of individual choices, not just political choices, and all sorts of possible rules for reaching collective decisions beyond a simple majority voting rule.
Arrow's Five Conditions
Ordering society in a way that reflects these many and varied individual preferences is difficult. Arrow specified five conditions that a society's choices must meet in order to fully reflect the choices of its individuals. They are:
Ø Universality: The decision rule must yield a complete ranking of all preferences and do so consistently under identical conditions.
Ø Responsiveness: An increase in an individual preference for an alternative must also either increase or at least not change, but never decrease, the overall social preference for that alternative.
Ø Independence of Irrelevant Alternatives: The inclusion or exclusion of certain alternatives must not alter the rank order of other alternatives with respect to one another.
Ø Non-imposition: The set of aggregated social preferences must be a product of one or more combinations of individual preferences.
Ø Non-dictatorship: The rule must actually reflect the preferences of multiple parties, and not simply of a single individual.
Using these conditions, Arrow developed his Impossibility Theorem. Arrow's Impossibility Theorem states that it is impossible to order society in a way that reflects individual preferences without violating one of the five conditions. Therefore, selecting a social choice rule will always involve sacrificing or compromising among Arrow's five axiomatic conditions.
Special Considerations
Another notable contributor to social choice theory is Jean Charles de Bourda, a contemporary of Condorcet, who developed an alternative voting system known as Borda Count. Other contributors to the theory include Charles Dodgson (better known as Lewis Carroll) and Indian economist Amartya Sen.
Example of Social Choice Theory
To consider a political example, under a dictatorship, decisions about social choices and the ordering of society are made by a single individual. Meanwhile, in an open democratic society, each individual has an opinion about how society should best be ordered. Both of these systems violate Arrow's Impossibility Theorem and are thus flawed methods for reaching social decisions that reflect the preferences of society.
A dictatorship obviously violates the non-dictatorship condition. A majoritarian democracy, on the other hand, violates the conditions of independence of irrelevant alternatives. This is because in majority voting, cycling (an endless loop of alternatives with no preferred solution) of preferences is possible, which makes the order and selection of alternatives presented a deciding factor in which alternative will be preferred.
For example, consider three voters voting for three alternatives:
Ø Voter 1 prefers option A over option B and option B over option C
Ø Voter 2 prefers option B over option C and option C over option A
Ø Voter 3 prefers option C over option A and option A over option B
All voters prefer A over B, B over C, and C over A, and a majority of voters will always vote against each of the possible options. Only if one of the options is excluded can a majority vote reach a decision in this situation, which means that the social rank order is dependent on the presence (or rather absence) of an irrelevant alternative.
In practice what this means is that in democracy, the outcome of majority voting may often be a function of the allowable alternatives that voters are permitted to consider and not a reflection of the voters' true preferences.
Fairness Criterion.
Fairness Criterion.
FAIRNESS CRITERIA METHOD
The Fairness Criteria are things that should always be true according to common sense but aren't always true.
Majority Criterion
· If there is a choice that has a majority of the first-place votes in an election, then that choice should be the winner of the election.
· If someone gets the majority of the votes, then they should win.
Condorcet Criterion
· If there is a choice that in a head-to-head comparison is preferred by the voters over every other choice, then that choice should be the winner of the election.
· If Bob is preferred over Jon in a one-on-one comparison and Bob is preferred over Stacey and Bob is preferred over Shelley, then Bob should win the election.
Monotonicity Criterion
· If choice X is a winner of an election and, in reelection, the only changes in the ballots are changes that only favor X, then X should remain a winner of the election.
· Bob wins an election. For some reason, there is reelection. Some people change their minds and rank Bob higher in their preference ballots. Bob should still win the election.
Independence of Irrelevant Alternatives Criterion
· If a candidate or alternative X is a winner of an election and one (or more) of the other candidates or alternatives is removed and the ballots recounted, then X should still be a winner of the election.
· Bob wins an election. Jon decides to remove himself from the election after the ballots are counted. Shelley calls for a recount because Jon is no longer a candidate. Since Bob won initially, he should still win.
Methods of Voting.
Methods of Voting.
MAJORITY RULE
WHAT IS MAJORITY RULE?
Five friends hanging out on a Friday night are trying to decide where they should go for dessert. Some of them want ice cream while others want frozen yogurt. When they put it to a vote, three votes for ice cream and two for frozen yogurt so the five friends head to the ice cream shop. Ice cream wins by the power of majority rule.
Majority rule is a decision-making system. In a choice or vote between two or more options, the option that wins over 50% of the vote wins. Although we use this strategy in daily life as in the ice cream/frozen yogurt dilemma, more often we think of majority rule in government. When a new law is voted on or a president is being elected, we often use majority rule. The will of the majority of people is respected and controls the outcome almost all the time.
Majority rule is similar but slightly different from a plurality system. When a plurality system is used, the winning candidate only needs to win more votes than the other candidates. In a majority rule system, a candidate needs to win over 50% of the overall vote.
ADVANTAGES OF MAJORITY RULE
One of the biggest advantages of majority rule is that decision-making is quick and clear. There is no need for compromise or negotiation after the decision has been made. The final decision is respected as the will of the majority of the people.
2. Also, many believe that majority rule remains one of the best and fairest ways to have a democratic election because it follows May's Theorem. May's Theorem states that a group decision must be decisive; in other words, there is one clear winner. It should protect the anonymity of each voter. It treats all of the options equally or with neutrality. And finally, it assures what is called positive responsiveness. It holds in essence that if there is a tie, and if one voter changes his preference while all other votes remain the same, whatever option that voter changes to will win. It is important to know that May's Theorem only holds true if there is an odd number of voters.
DISADVANTAGES OF MAJORITY RULE
One of the biggest disadvantages of majority rule is the chance of tyranny of the majority in politics. When there is a tyranny of the majority, there is no incentive for the majority to cooperate or compromise with the minority. Most political systems place regulations on the majority by constitutional or statutory protections. These protections attempt to guarantee protection for the minority against tyranny. Nevertheless, they have not always been effective.
We can look at the example of segregation in the American South. For decades, the white political majority in the South upheld segregation in law and policy. The black minority was ignored and oppressed. It wasn't until the Supreme Court decision in Brown vs. Board of Education that things started to change. The Supreme Court found that separating black and white children in unequal schools violated minority rights. This decision forced integration although the majority still favored segregation.
PLURALITY METHOD
The voting method we’re most familiar with in the United States is the plurality method.
In this method, the choice with the most first-preference votes is declared the winner. Ties are possible and would have to be settled through some sort of run-off vote.
This method is sometimes mistakenly called the majority method, or “majority rules”, but it is not necessary for a choice to have gained a majority of votes to win. A majority is over 50%; it is possible for a winner to have a plurality without having a majority.
Step-by-step procedure:
1. Count the number of first-choice votes. A candidate with a majority of second-choice votes is declared the winner.
2. If no candidate has a majority of first-choice votes, the candidate with the lowest number of first-choice votes is eliminated from the schedule. These votes are allocated to other candidates based on second-choice preferences.
3. Steps 1 and 2 can be repeated until one candidate attains an absolute majority.
Example:
In our election from above, we had the preference table:
For the plurality method, we only care about the first choice options. Totaling them up:
Anaheim: 1+3=41+3=4 first-choice votes
Orlando: 3 first-choice votes
Hawaii: 3 first-choice votes
Anaheim is the winner using the plurality voting method.
Notice that Anaheim won with 4 out of 10 votes, 40% of the votes, which is a plurality of the votes, but not a majority.
CONDORCET METHOD
One of the people who arguably founded the modern field of voting theory was Marie Jean Antoine Nicolas de Caritat, better known by his title of Marquis de Condorcet. He was a mathematician and political philosopher, whose more political work eventually brought his life to an untimely end in 1794.
Within the study of the mathematics of voting systems, he is best known for his discussion of pairwise majority votes — voting between two choices at a time. Pairwise votes are very often seen in committees. A candidate who wins every pairwise vote is called a Condorcet winner; a candidate who loses every pairwise vote is called a Condorcet loser.
It seems perfectly reasonable that Condorcet winners should generally place first in an election, while Condorcet losers should generally not place first in an election. Respectively, those are the Condorcet winner criterion and the Condorcet loser criterion. Alarmingly, a simple plurality vote violates both criteria. In fact, it turns out that very few voting systems fulfill both criteria. Those that do are known as Condorcet methods.
Step-by-step Procedure:
To find the Condorcet winner from a voter-preferences table:
1. List all head-to-head matchups possible among the candidates.
2. With each matchup, run through all columns of the table, awarding all the voters in that column to the preferred candidate from the matchup.
3. If one candidate has beaten all other candidates in matchups, they are the Condorcet winner.
Comparison between Samuel Adams and Guinness (the number of supporters in the first row represents the number of voters who prefer Samuel Adams to Guinness, and vice-versa for the second row):
The winner for this pair is Samuel Adams
If there is a candidate who 'wins' EVERY comparison with all other candidates, then this candidate is the winner. If there is no such candidate, then there is no Condorcet winner.
Note: you can define a "winning" candidate as that candidate having a number of preferential votes which is greater than or equal to the number of preferential votes of all other candidates when the candidates are compared pairwise. There isn't always a Condorcet winner. If no candidate satisfies this condition for winning, then there is no Condorcet winner.
BORDA COUNT METHOD
The Borda count method is a point-based election system in which voters number their preferred choices in order. This type of election method was developed independently in many different locations and time periods throughout history. There is evidence it was in use as early as the thirteenth century and possibly even earlier. It was named after a French mathematician and engineer named Jean Charles de Borda.
The Borda count method is often used as a way to find a workable option or compromise for a group of people. For example, if an organization wants to have a national meeting, they need to pick a city to host. They might narrow down the choices to the five cities with the largest number of members. Then, they can let everyone vote. Some people may want to have the voting locally. Others might want the opportunity to travel. Some people may use this as an excuse to visit friends or family in one of the cities while they are in town. Everyone brings their own reasoning to the table and ranks the order they would prefer to have the meeting.
Step-by-step procedure:
1. Each voter ranks each option with a number listing one for their top choice, two for their second, and so forth.
2. The votes are collected and tallied. There should be 25 tallies - each city most likely had at least one person rank them in each of the positions. This means for city A, there should be a tally of how many times it was ranked 1st, 2nd, 3rd, 4th, and 5th. This will repeat for each city.
3. Next, the number of tallies is multiplied by the score for that ranking. For each first place tally, they get five points. This continues until for each fifth place tally they get one point.
4. Each city's score is totaled.
5. The city with the highest score should host the meeting.
Example:
Find the Borda Count winner of the election whose preference table is below.
Notice that we have added a column to the left of the preference schedule indicating how many Borda points are awarded to each ranking. Since there are 4 candidates, a first-place ranking is worth 4 points.
To find the score of Candidate A, we look at the four times A appears on the preference table. The first column says that 6 people ranked A first, giving A 6(4)=246(4)=24 points. Everyone else ranked A last, giving A 1 point each. This adds up to 5+4+2=115+4+2=11 points. Therefore, A has a total of 24+11=3524+11=35 points.
We can calculate the Borda points of the other candidates similarly, as follows:
B: 6(3)+5(3)+4(2)+2(4)=496(3)+5(3)+4(2)+2(4)=49
C: 6(1)+5(2)+4(4)+2(3)=386(1)+5(2)+4(4)+2(3)=38
D: 6(2)+5(4)+4(3)+2(2)=486(2)+5(4)+4(3)+2(2)=48
Thus, Candidate B wins this election when the Borda Count is used.
SEQUENTIAL PAIRWISE VOTING
It is an electoral system that extends the concept of approval voting to a multiple winner election. It is a simplified version of proportional approval voting. Proposed by Danish statistician Thorvald N. Thiele in the early 1900s, it was used (with adaptations for party lists) in Sweden for a short period from 1909-1921, and was replaced by a cruder "party-list" style system as it was easier to calculate.
Sequential pairwise Voting is to elect multiple winners equitably by selecting a candidate in each round and then reweighing the approvals for the subsequent rounds.
Each ballot is assigned a value equal to the reciprocal of one more than the number of candidates approved on that ballot who have been designated as elected. Each ballot is counted at its current value as a vote for all continuing candidates approved on that ballot. The candidate with the most votes in the round is elected. The process continues until the number of elected candidates is equal to the number of seats to be filled.
Step-by-step procedure:
Fix an ordering (also called an agenda of the candidates (chosen however you please, ex A,D,B,C,F,E)
Have the first two compete in a head-to-head (majority rules) race, the winner of this race will then face the 3rd candidate on the list in a head-to-head race, the winner of that race will face the next candidate... continue until the last candidate in the ordering is in a head-to-head race with the winner of the previous head-to-head and the winner of that race is declared the winner of the general election.
The societal preference order then starts with the winner (say C) with everyone else tied, i.e. C>A=B=D=E=F.
Example:
Given any agenda, could a Condorcet winner ever lose in the sequential pairwise voting system. So, have we found a voting system that satisfies CWC? Does sequential pairwise voting satisfy CLC?
Recall the race for the Graduate Student Representative from the previous section.
Ø From previous activities we learnt that using the plurality method produced the societal preference order P>S>J>L, and Borda count produced S>J>P>L. Now, who would win this election under sequential pairwise voting with agenda P,S,J,L?
Find an agenda under which Peter would win. Do the same for Sergio and for Liz.
Ø Suppose that all the voters switched J and L on their preference orders. Use the original and new preference schedule, with agendas S,L,P,J, to show that sequential pairwise voting is not neutral.
HARE SYSTEM
The Hare system is a system of proportional representation that aims to achieve party representation in the closest proportion to actual voting strength by transferring votes beyond those needed to elect a candidate from that candidate to the next indicated choice.
The Hare System can be used in the single transferable vote (STV-Hare) system and the largest remainder method (LR-Hare) and other quota rule compatible methods of party-list proportional representation. Both versions are named after the political scientist Thomas Hare, but the largest remainder method in which it is used is also sometimes called the Hare–Niemeyer method (after Horst Niemeyer) or the Hamilton method (after Alexander Hamilton).
Step-by-step procedure:
1. Check Majority Rule
2. If no one owns by majority rule, eliminate the candidate with the fewest number of votes.
3. If we don’t see the winner eliminated, we eliminate the candidate with the fewest number of votes again, until we found the winner.
4. Find the winner using the Hare Method.
Example:
APPROVAL VOTING
Approval voting is a method of voting that allows an elector to cast a vote for as many of the candidates per office as the elector chooses. The winner of each office is the candidate who receives the most votes or, for elections in which multiple candidates fill open seats, the winners are those candidates, in a number equal to the number of seats being filled, attaining the greatest number of votes.
Approval voting allows voters to select all the candidates whom they consider to be reasonable choices. With Approval Voting, the ballot asks you to mark all choices that you find acceptable. The results are tallied, and the option with the most approval is the winner.
Additionally, Approval Voting is susceptible to strategic insincere voting, in which a voter does not vote their true preference to try to increase the chances of their choice winning.
Step-by-step procedure:
1. Determine the winner
2. Evaluate the fairness of an election using the Approval Voting method
3. Understand the voting process
4. Identify the winner
Example:
A group of friends is trying to decide upon a movie to watch. Three choices are provided, and each person is asked to mark with an “X” which movies they are willing to watch. The results are:
Totaling the results, we find:
Titanic received 5 approvals
Scream received 6 approvals
The Matrix received 7 approvals.
In this vote, The Matrix would be the winner.
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