What is your favorite food?

Preference Schedule

A Preference Schedule is a way to represent the order in which people like (prefer) certain items.

The top of the preference schedule is the item they like the most and the bottom of the preference schedule represents the item they like the least.

The number at the bottom is the total number of people who organized their preferences in that same exact order.

Most A B C D Least E

Total of this chart 10

Ranking Methods

Plurality – The item that receives the most First Place votes. (All items receive the same number of total votes)

If an item receives over half of the total first place votes, then they are the majority winner. The schedule does not have to have a majority winner. The majority winner must be the plurality winner but the opposite is not true.

Borda Count – A method where a point value is assigned to each place an item placed on the preference chart and a total is obtain from all charts. Highest point total wins. If there are n items on the chart, then the first place item receives n points and second receives n-1 points all the way down to last place receiving 1 point. Remember to multiply the points received for the place the item is by the number at the bottom of the chart.

Ranking Methods

Runoff Method - The two choices with the most first place voters are kept and all other choices are eliminated. The number of first place votes are now recalculated using only the two remaining choices.

Example: 25 people audition for a part in a movie and 5 producers are choosing the person for the part. Two people get a callback and the producers must make their choice from those two.

Sequential Runoff – The choice with the fewest first place votes is eliminated from the schedules. The number of first place votes is now recalculated with the remaining choices. This process is repeated until a winner is found.

Example: Think American Idol, if only the same people could vote each week until one winner is named.

Algorithm – the step by step process of how to complete a task.

Example: How do I get to the cafeteria?

Consider the following set of preferences.

A B B CC A D AD D C BB C A D

6 8 4 7A.) Determine the 4-3-2-1 Borda winner.B.) Determine the plurality winner.C.) Determine the runoff winner.D.) Determine the sequential runoff winner.

Borda MethodA B B CC A D AD D C BB C A D

6 8 4 7A = 6(4) + 8(3) + 4(1) + 7(3) = 73

4 points

3 points

2 points

1 point

A B B CC A D AD D C BB C A D

6 8 4 7A = 6(4) + 8(3) + 4(1) + 7(3) = 73 WINNER!B = 6(1) + 8(4) + 4(4) + 7(2) = 68C = 6(3) + 8(1) + 4(2) + 7(4) = 62D = 6(2) + 8(2) + 4(3) + 7(1) = 47

4 points

3 points

2 points

1 point

Plurality

A B B CC A D AD D C BB C A D

6 8 4 7First Place votes:A = 6B= 8 + 4 = 12 WINNERC = 7D = 0

Runoff MethodA B B CC A D AD D C BB C A D

6 8 4 7First Place votes:A = 6B= 8 + 4 = 12C = 7D = 0

A and D are eliminated because B and C have the most first place votes

A B B CC A D AD D C BB C A D

6 8 4 7First Place votes:A = 6B= 8 + 4 = 12C = 7 + 6 = 13+ 6 = 13 WINNERD = 0

Sequential Runoff

A B B CC A D AD D C BB C A D 6 8 4 7

First Place votes:A = 6B= 8 + 4 = 12C = 7D = 0

Eliminate D because it has the least number of first place votes

A B B CC A D AD D C BB C A D

6 8 4 7First Place votes:

A = 6B= 8 + 4 = 12C = 7D = 0

Eliminate A because it now has the least number of first place votes, its votes get transferred to C

A B B CC A D AD D C BB C A D

6 8 4 7First Place votes:

A = 6B= 8 + 4 = 12C = 7 + 6 = 13+ 6 = 13 WINNERD = 0

More Ranking Methods

Condorcet – Compare each choice with every other choice. (Whichever choice is ranked higher on the preference schedule receives the number of votes listed at the bottom of the preference schedule.) The choice that receives the most votes is the winner in that comparison. To be the Condorcet winner a choice must win each comparison (go undefeated).

Example: Five teams are having a round robin tournament (playing every other team) the Condorcet winner would need to have a record of 4 wins and 0 losses.

Condorcet Method

A B B CC A D AD D C BB C A D

6 8 4 7A is the Condorcet WINNER because it won all three

comparisons.

Choose any two and compare them.

A vs B A vs C A vs D

6 8 6 4 6 4

7 4 8 7 8

13 12 14 11 7

21 4

Pairwise voting – Choose any two of the choices and compare them (in same way as Condorcet method). Once a choice loses a comparison it is eliminated from the method. The winner is then compared to another choice and again the winner advances and the loser is eliminated. Repeat this pattern until only one choice remains.

Example: Think of a bracket type tournament (the Super Bowl), the winner of the Super Bowl advanced to the last game of the year and won and is the champion but did not necessarily beat every team in the tournament.

Pairwise MethodA B B CC A D AD D C BB C A D

6 8 4 7

Choose two and compare them, winner advances and the loser goes home.

C 13

D 12

C 11

A 14

A 13

B 12

A WINNER

Note:

If there is a Condorcet winner, then the same option must be the pairwise winner.

AA

A A

B

B

B

BC

C

C C

D

D D

D

8 10 7 5

Find the Condorcet winner.

Find the Pair-wise winner.

Paradox – is a statement that leads to a contradiction. The outcome does not logically make sense.

orIn this case where we did the process correctly

but were unable to find a winner.

Consider the following set of preferences.

C A D BA B C AB C B DD D A C 12 10 13 21

A.) Determine the 4-3-2-1 Borda winner.B.) Determine the runoff winner.C.) Determine the sequential runoff winner. D.) Determine the Condorcet winner.E.) What does the fact that there is a Condorcet winner

tell us about the pairwise winner?

a.) Bb.) Bc.) Bd.) Be.) Since we have a Condorcet winner the pair-wise winner must be the same. SO the pair-wise winner would be B as well.

Arrow’s Conditions

1.) Nondictatorship – The preference of a single individual should not become the group ranking without considering the preference of the others.

2.) Individual Sovereignty – Each individual should be allowed to order the choices in any way and to indicate ties.

3.) Unanimity – If every individual prefers one choice to another, then the group ranking should do the same.

4.) Freedom from Irrelevant Alternative – The winning choice should still win if one of the other choices is removed. (The choice removed is the irrelevant alternative)

5.) Uniqueness of the Group Ranking – The method of producing the group ranking should give the same result whenever it is applied to a given set of preferences. The group ranking should also be transitive.

Arrow’s Examples1.) I like Twix more than Kit Kats, Kit Kats more than Snickers, and Snickers more than

Twix.

2.) Everyone decided that we would take the test on Friday until little miss teachers pet showed up and said she wanted to take it on Monday, so of course I “caved” and moved the test to Monday.

3.) I tell you to put all four of your classes in order from most to least favorite but you are not allowed to put my class last

4.) The reigning Miss American is “stripped” of her crown and the votes are recounted and the second runner-up is the new Miss America, not the first runner-up.

5.) We use pairwise voting and choice A beats choice C for the overall win because C eliminate choice B earlier, even though B would have beaten A if they had been compared.

Approval Voting

All the choices are listed and each person is allowed to vote for as many choices as they want. The person may even vote for all of the choices or none of the choices. After everybody has voted, the total number of votes is tallied and the choice with the most votes wins. If there is a tie, then two choices may win.

Example: What your favorite movie?

Section 3A panel of sportswriters is selecting the best football team in

a league, and the preferences are distributed as follows:

A B CB A BC C A 52 38 10

a.) Determine the winning team using a 3-2-1 Borda count.b.) The 38 who ranked B first and A second decide to lie in

the order to improve the chances of their favorite and so they change their rank of C second and A third. Determine the winner using a 3-2-1 Borda count.

Weighted Voting

Weighted voting – is where some members of a voting body have more votes than others.

Example: Corporate share holders (more shares – more votes)

Coalitions – a collection of voters, small as one voter all the way up to the group including all the voters

The number of votes needed to pass an issue will be set in each problem, if a number is not set use the simple majority.

Winning Coalitions – all the coalitions that have enough votes to pass an issue.

Power Index – the true measure of power a voter has in the group

Finding the power index

1.) List all winning coalitions and their total votes.2.) Select any voter and if once the selected voter is removed

from the winning coalition that coalition no longer has enough votes to pass the issue that selection receives one point on their index. If the coalition still has enough to pass even after your selection has been removed, then the selection does not receive a point for that coalition.

3.) Continue with the same selection until you have tested it in all the winning coalitions.

4.) Select a different voter and following steps 2 and 3, continue until all voters have been selected.

Consider a situation in which A, B, C, D have 5, 4, 2, and 2 votes, respectively, and in which 6 votes are needed to pass an issue.

a.) List all possible coalitions. b.) Mark off the non-winning coalitions.c.) Determine the power index for each voter.

a.) All possible coalitions.

A – 5 AB – 9 ABC – 11 ABCD - 13

B – 4 AC – 7 ABD - 11

C – 2 AD – 7 ACD - 9

D – 2 BC – 6 BCD - 8

BD – 6

CD - 4

A – 5 AB – 9 ABC – 11 ABCD – 13

B – 4 AC – 7 ABD – 11

C – 2 AD – 7 ACD – 9

D – 2 BC – 6 BCD – 8

BD – 6

CD – 4

b.) Mark off all non-winning coalitions.

The Power Index.

A – 4

B – 4

C – 2

D – 2

Weighted Voting and Voting Power

1.) Consider a situation in which A, B, C, D have 4, 2, 2, and 1 votes, respectively, and in which 5 votes are needed to pass an issue.

a.) List all possible winning coalitions.b.) Determine the power index for each voter.c.) What group or groups is this situation the least fair for? And Why?

2.) Consider a situation in which A, B, and C have 10, 5, and 4 votes respectively, and a simple majority is needed to pass an issue.

a.) List all possible winning coalitions.b.) Determine the power index for each voter.c.) What term is used to describe voter A and the what term for voter B

and C?

Section 5

Consider a situation in which A, B, C have 3, 2, and 1 votes, respectively, and in which 4 votes are required to pass an issue.

a.) List all possible winning coalitionsb.) Determine the power index for each voter.