MGF Chapter 13

Modified Divisor

If the standard divisor is slightly changed, the numbers are forced to workout. IF THE SUM IS TOO HIGH, USE A LARGER DIVISOR. IF THE SUM IS TOO LOW, USE A SMALLER DIVISOR.

in the Borda count method, however many choices you have determines how many ___ each place gets

POINTS -> if there are four choices, 1 place is worth 4 points, 2nd place is worth 3, 3rd is worth 2, and 4th place is worth 1 point

Plurality Method

The winner is the candidate who receives the most first place votes.

Standard Divisor

Total Population / Number of Seats (allocated items)

pairwise comparison method example

Voters are asked to rank four brands of soup: A,B, C, and D. The votes are summarized in the following preference table. # of votes. 34. 30. 6. 2 1st choice A. B. C. D 2nd choice B. C. D. B 3rd choice C. D. B. C 4th choice D. A. A. A - There are four choices so we need to use our formula to see how many comparisons need to be made. c= n(n-1)/2 = 4(3)/2 = 6 comparisons - find comparisons by going in alphabetical order A: B -> A=34 B=30 +6 +2, B wins* A: C -> A=34 C=6 +30 +2, C wins A: D -> A=34 D=2 +30 +6, D wins B: C -> B=30 +34 +2 C=6, B wins* B: D -> B=30+34 D=2 +6, B wins* C: D -> C=6 +34 + 30 D=2, C wins all the pairs should add up to the total number of votes! B WINS BY THE PAIRWISE METHOD.

Borda Count Method example

Voters are asked to rank four brands of soup: A,B,C and D. The votes are summarized in the following preference table. PTS. # of votes 34. 30. 6. 2 4 1st choice. A. B. C. D 3 2nd choice B. C. D. B 2. 3rd choice C. D B. C 1 4th choice. D. A. A. A a: 34(4) + 30(1) + 6(1) + 2(1) = 174 b: 34 (3) + 30 (4) + 6(2) + 2(3) = 240 c: 34 (2) + 30 (3) + 6(4) + 2(2) = 186 d: 34(1) + 30 (2) + 6(3) + 2(4) = 120 B wins.

plurality method example

Voters are asked to rank four brands of soup: A,B,C, and D. The votes are summarized in the following preference table: number of voters: 34. 30. 6. 2 first choice. A B. C. D second choice B. C. D. B third choice C D. B. C fourth choice D. A. A. A a HAS the most first place votes, so it wins by the plurality method.

satisfy the quota rule

a group's apportionment should be either its upper or its lower quota. An apportionment method that guarantees that this will always occur will satisfy this rule.

Modified Quota

a quota that corresponds with a modified divisor; a group's population divided by the standard divisor

Plurality with Elimination Method

a voting method that chooses the candidate with a majority of the votes and when there isn't one it eliminates the candidate(s) with the least votes and transfers those votes to the next highest candidate on those ballots, continuing this way until there is a majority.

pairwise comparison method

compare all pairs to one another and the winner of each pair gets a point. Whichever candidate has the most points win.

apportionment problem

determines a method for rounding standard quotas into whole numbers so that the sum of the numbers is the total number of allocated items

Irrelevant Alternatives Criterion

if a candidate wins a straw vote (1st election) and in a 2nd election (real election) one or more candidates are removed, the winner of the first election should still win.

Monotonicity Criterion

if a candidate wins the straw poll (first election), and then gains additional support without losing any of the original support in the second election, then the candidate should win the second election

head-to-head comparison

if there is one candidate who is favored by the voters when compared to each of the other candidates , then that candidate should win the election.

Modified Lower Quota

modified quotients that are rounded down

modified upper quota

modified quotients that are rounded up

Arrow's Impossibility Theorem

none of the four voting methods discussed always satisfies each of the fairness critera

Plurality with Elimination Method example

number of votes 50. 65 60 1st choice A. B. C 2nd choice B. C. A 3rd choice C. A. B - find the total number of votes and determine how many are needed to find the majority -> 50+60+65 = 175; 175/2 = 87.5 = 88 is the majority -if no one has the majority, drop the choices with the fewest first place votes. Give their votes to the next choice. - Repeat this process until one candidate has the majority. -> a would get dropped, the second choice is b, so b would have 115 votes and win the majority.

Standard Quota

population of a specific group/standard divisor

straw vote

preliminary vote that doesn't count like testing the waters

preference table

shows how often each particular outcome occurred

quota rule

the apportionment for every group under consideration in Hamilton's method should be the lower or upper quota

modified rounded quota

the modified quotients that are rounded

the process for evaluating the head to head criterion is similar to what method?

the pairwise comparison method

lower quota

the standard quota rounded DOWN to the nearest whole number

upper quota

the standard quota rounded UP to the nearest whole number

borda count

voters list candidates in order of their preference and candidates are assigned points for ranking. Whoever has the most points wins.

preference ballots

Ballots in which voters are asked to rank all the candidates in order of preference

how to find the number of comparisons that need to be made (pairwise method)

C = n(n-1)/2 where n = number of candidates c = comparisons

Majority Criterion

If a candidate receives more than 50% of the votes, they should win.

how to find the minimum requirement to obtain the majority (even numbers)

(total number of votes/2) + 1 = majority

how to find the minimum requirement to obtain the majority (odd numbers)

(total number of votes/2) = mixed number = round up that number to get a whole number = majority

how to find the upper quota

- regardless of the decimal, add 1 to the whole number

how to find the lower quota

- round down to the whole number, regardless of how high or low the decimal

majority criterion can be three things:

1) satisfied - a candidate has the majority and wins 2) violated - a has the majority and loses 3) not apply - no one has the majority

Head to Head comparison can be three things

1) satisfied - candidate a beats all other candidates in an H2H comparison and wins 2) violated - candidate a beats all others and loses the election 3) doesn't apply - no candidates beats all others H2H

Hamilton's Method

1. Calculate each groups standard quota 2. Round each standard quota DOWN TO THE NEAREST WHOLE NUMBER, thereby finding the lower quota. Initially, give to each group its lower quota. 3. Give the surplus items, one at a time, to the groups with the largest decimal parts in their standard quotas until there are no more surplus items.

Jefferson's Method

1. Find the modified divisor (d) such that when each group's modified quota (group's population divided by d) is rounded down to the nearest whole number, the sum of whole numbers for all the groups is the number of items to be apportioned. The modified quotients are rounded down. 2. apportion to each group its modified lower quota

Adam's method

1. find a modified divisor (d) such that when each group's modified quota (a group's population divided by d) is rounded UP TO THE NEAREST WHOLE NUMBER, the sum of whole numbers for all groups is the number of items to be apportioned. The modified quotients that are rounded up are called modified upper quotas. 2. apportion to each group its modified upper quota

Webster's method

1. find a modified divisor (d) such that when each group's modified quota is ROUNDED TO THE NEAREST WHOLE NUMBER, the sum of whole numbers for all groups is the number of items to be apportioned. The modified quotients that are rounded are called modified rounded quotas. 2. Apportion to each group its modified rounded quota