Voting Part One

Question 1

social choice function

Answer 1

the decision procedure that is used to render the result of an election
always get the same outcome
does not incorporate any value judgements about whether the function is fair or reasonable or appropriate for elections in a democratic system

Question 2

simple majority method

Answer 2

the social choice function that, in a two-candidate election, selects as the winner the can­didate who gets more than half of all the votes cast. If each candidate gets exactly half of the votes, then the result is a tie.

Question 3

super majority method

Answer 3

the social choice function that selects as the winner the candidate who gets a fraction p or more of all the votes. In particular, if there are t voters, a candidate must get at least pt votes to win. If no candidate gets pt votes, then the result is a tie; p greater than 1/2 but less than 1
need greater than pt votes to win the election
The minimum number q of votes that a candidate needs to win is called the quota.

Question 4

status quo method

Answer 4

one candidate as status quo and one as challenger
when there’s a tie, the status quo wins
a way of breaking a tie

Question 5

weighted voting method example

Answer 5

used during election of the president of the US- each state is weighted based upon the population/representatives (# of reps + # of senators); imposes the question of does this give certain states more power

Question 6

reasons why he two-candidate case is simpler than the general case

Answer 6

Easy design of the ballot, it is clear what decision needs to be rendered in the end, and it prevents an anomaly that can occur when there are more than two candidates

Question 7

quota

Answer 7

The minimum number q of votes that a candidate needs to win
given by the smallest whole number greater than or equal to pt. The mathematical notation for this is q = ⌈pt⌉ (read “the ceiling of pt”), where the expression ⌈x⌉ denotes the smallest integer greater than or equal to x
As p increases toward 1, the standard for winning becomes increasingly difficult to meet. In the ultimate case, p = 1, the supermajority method requires unanimity (or consensus) to produce a winner.

Question 7

quota

Answer 7

The minimum number q of votes that a candidate needs to win
given by the smallest whole number greater than or equal to pt. The mathematical notation for this is q = ⌈pt⌉ (read “the ceiling of pt”), where the expression ⌈x⌉ denotes the smallest integer greater than or equal to x
As p increases toward 1, the standard for winning becomes increasingly difficult to meet. In the ultimate case, p = 1, the supermajority method requires unanimity (or consensus) to produce a winner.

Question 8

weighted voting method

Answer 8

Sup­pose there are n voters: 1,2,…,n, and, for each i from 1 to n, voter i is assigned a positive number wi of votes (called the weight of voter i). Let t = w 1 + w2 + ··· +wn be the total number of votes (the sum of the weights). A candidate who gets more than half of all the votes cast (i.e., more than t/2 votes) is the winner. If no candidate gets more than half of the votes, then the result is a tie.

Question 9

hybrid of the supermajority, status quo, and weighted voting methods

Answer 9

Suppose there are n voters with nonnegative weights w1 ,w2 ,…w n, and two candidates, one designated as the status quo and the other designated as the challenger, and a parameter p satis­fying 1/2 ≤ p ≤ 1. Let t = w 1 + w2 +··· +wn be the sum of the weights (the total numbers of votes). If the challenger gets a fraction p or more of all the votes (i.e., at least pt votes), then the challenger is the winner. Otherwise, the status quo candidate is the winner. (The threshold q = pt is another example of a quota.)

Question 10

Bloc voting method

Answer 10

First, the electorate is partitioned into n blocs (every voter is in exactly one bloc), and, for each i from 1 to n, bloc i is assigned a positive number wi of votes. Each bloc conducts a “popular vote” election using the simple majority method (resolving any ties by some method chosen by that bloc). Then the bloc casts all of its votes in the “national” election for the candidate that won its simple majority election. The winner is the candidate receiving the most votes in the national election.
do this everywhere in US but Maine and Nebraska

Question 11

Monarchy method

Answer 11

one of the candidates is a monarch. That candidate wins no matter how anybody votes.

Question 12

Dictatorship method

Answer 12

one of the voters is the dictator. Whoever the dictator prefers is the winner; the same voter always picks
Consider a weighted voting method based on simple ma­jority with 4 voters having 6,2,2,1 votes, respectively. Then the first voter is (in effect) a dictator

Question 13

Parity method

Answer 13

If just one candidate gets an even number of votes, then that candidate wins. If both candidates get an odd number of votes or if both candidates get an even number of votes, then the result is a tie.
The parity method is anonymous and neutral but violates the monotonicity criterion

Question 14

All-ties method

Answer 14

the election is a tie, no mat­ ter how the electorate votes.

Question 15

Criteria

Answer 15

(we may also call them “conditions”, “rules”, or “properties”) that we may impose on our election methods. We want these criteria to formalize our notions of fairness, appropriateness, or reasonableness.

Question 16

Anonymity criterion aka anonymous

Answer 16

Criteria satisfied if the method treats all the voters the same
method is anonymous if the outcome of the method (either “A”, “B”, or “tie”) does not change if the voters exchange ballots among themselves. In other words, if I were to present your ballot and you were to present mine, this twist should not impact the election outcome. Anonymity is the property of a social choice function that allows elections to be conducted via secret ballots. Up to now, we have not assumed ballot secrecy, and one may imagine that voters are expected to write their names on their ballots. In the absence of anonymity, the result of the election can change if the names on the ballots are permuted. In the presence of anonymity, however, the names on the ballots can be ignored, and therefore the ballots themselves need not even ask for the names of voters
“Dictatorships violate anonymity.” : Clearly a dictatorship favors one voter (the dictator!) over the oth­ers, and therefore cannot operate unless it is possible to discern which ballot was contributed by the dictator. Thus a dictatorship does not sat­isfy the anonymity criterion
Weighted voting methods distinguish between voters, making some voters more powerful than others. Weights are associated to different voters so changing these weights would matter who is who and then this ultimately changes the results. Weighted voting therefore also vio­lates anonymity. While this may be acceptable in certain situations, it violates the fundamental principle of “one person, one vote” for demo­cratic elections to public
It is easy to see that the simple majority method and the superma­jority methods do satisfy the anonymity criterion, as do any status quo methods that are based on these methods. This is because it does not matter which voters vote for a certain alternative; all that matters is how many voters vote for a certain alternative.
Proposition used to determine if the criteria is met: A method is anonymous if and only if its outcomes depend only on the tabulated profile

Question 17

neutrality criterion

Answer 17

A method satisfies the neutrality criterion (or is neutral) if it treats both candidates equally.
Status quo methods violate this criteria
To be more precise, a method is neutral if whenever all the voters for A change their votes to B, and all the voters for B change their votes to A, the winner changes from A to B or from B to A (or the outcome remains a tie if it was a tie before the change).
Clearly status quo methods are not neutral (they favor the status quo candidate). Similarly, monarchy is not neutral (in much the same way that dictatorship is not anonymous). In elections for public office, at least, fairness and impartiality almost always require neutrality. Neither candidate should have an advantage built into the decision procedure.

Question 18

monotonicity criterion

Answer 18

A method satisfies the monotonicity criterion (or is monotone) if the following holds: Suppose the votes are cast and the method selects one candidate as the winner. Then suppose that the method is employed again after one or more voters change their votes from the losing candidate to the winning candidate. The candidate who was the winner before the change must remain the winner after the change. In other words, if candidate A wins when facing one profile but then some voters switch their votes from B to A, then A must still win. This criterion requires the consideration of two profiles, the first before some voters change their minds, the second after. The voters who change their minds all do so in the same direction, deciding to vote for A where they had voted for B before. It would be disturbing if a social choice method gave the election to A before the change, but failed to give the election to A after the change. After all, if such a method were used in practice, voters would be afraid to cast their ballots in an honest way, aware that a vote for A could harm the cause of electing A. We therefore impose the monotonicity criterion in most situations. Another way to phrase the monotonicity criterion informally is: “A vote for a candidate can never harm that candidate.”

Question 19

decisiveness criterion

Answer 19

method satisfies the decisiveness criterion (or is decisive) if it always chooses a winner (i.e., it never ends in a tie).
As disappointing as a tie may be, however, we sometimes need to settle for a method that allows for the possibility of a tie. For example, if we want to use the simple majority method, and the number of voters is even, ties will always be a possibility. On the other hand, we might try to impose a criterion that eliminates ties, except when absolutely necessary.

Question 20

near decisiveness criterion (or is nearly decisive)

Answer 20

Decisive will always give you a winner so nearly decisive gives you a decision if every case but a tie
Satisfied if the only situation in which a tie can occur is if both candidates receive exactly the same number of votes.
Near decisiveness is meant to be inclusive in the sense that any method that is decisive must also be nearly decisive. The converse of this state­ ment, however, is false. A nearly decisive method need not be decisive. For example, the simple majority method with an even number of voters is nearly decisive without being decisive. Many methods fail to satisfy near decisiveness. For example, every super-majority method (without status quo) fails to be nearly decisive, because with a nearly 50:50 split in the electorate, neither candidate will have enough votes to win. The worst violator of the near decisiveness criterion is, of course, the all-ties method, which ends in a tie when presented with any profile whatsoever.

Question 21

slate

Answer 21

a set of candidates or alternatives

Question 22

the electorate

Answer 22

a set of voters

Question 23

Plurality method

Answer 23

the social choice function that selects as the winner the candidate who is ranked as the first choice of the most voters. In the event of a tie, the plurality method selects all the candidates who tie for the most first-place votes. You simply tally the first-place votes for each candidate, and the candidate with the most first-place votes wins. Such a candidate is said to have a plurality of the votes, that is, more votes than anyone else. This is how elections are customarily run in most situations. Note that we do not require a majority, because doing so would sometimes result in no winner being selected. To determine the plurality winner of an election, one need know only the first-choice preference of each voter. In other words, this method permits us to ignore all the information in the profile about second, third, and further preferences of the voters. In many elections, voters are given the chance to submit only a vote-for­ one ballot and we do not have any further information about the voters’ preferences. It is still possible to compute the plurality winner in vote for-one elections, because the lost information from the profile about lower-ranking preferences is not used. But most of the other methods we will consider require more complete knowledge of the profile. As a result, most of the methods we will study cannot be implemented in an election conducted with a vote-for-one ballot. Hence the plurality method often becomes the default method in vote-for-one elections.

Question 24

Borda count method

Answer 24

If there are n candidates in the slate, then assign n − 1 points to a candidate for every voter who ranks that candi­ date first, n− 2 points for every voter who ranks that candidate second, n−3 points for every voter who ranks that candidate third, and so forth until you assign 1 point for every voter who ranks that candidate in position n − 1 (i.e., second-to-last). A candidate receives no points for any last-place votes they get. Tally the points, and the winner is the candidate who gets the most points, or, in the event of a tie, the winners are the candidates who tie for the most points.
The Borda count method is attractive because it uses all the rich information that the profile provides. Still, some may be unhappy about a method that allows a candidate to win when that candidate wouldn’t register even a single vote in a vote-for-one election.