Let’s take a class vote. How many of you are registered

to vote?

How groups can best arrive at decisions (goals).

find an out come that “reflects the will of the people.”

What do you think is the best way for millions of people to make one decision?

Preference List Ballot (def)

A ballot consisting of a rank ordering of candidates.

Usually in the form of a vertical list with our most

preferred candidate on top.

No ties allowed.

Examples of Preference List Ballot

Australia uses it too!

The Number of Voters Assumption (rule)

For the sake of avoiding excessive annoyances in the theory…

Throughout the chapter, we will assume that the number of

voters is odd. In reality, numbers of voters are often so

high that ties are unlikely anyways.

Majority RuleThe most obvious of voting methods…

3 desirable properties:

Do we take these simple traits for granted??

The first desirability fails in the system called a Dictatorship, where all ballots except the dictators are ignored…The second desirability fails in imposed rule, where a certain candidate wins regardless of who votes for whom …The third desirability fails in minority rule, where fewest votes win…

May’s Theorem (Kenneth May, 1952)

Who would beat whom if two candidates faced an election 2 at

a time?

Rank Number of Voters (3)

first A B C

second B C A

third C A B

A would defeat B (2:1)

B would defeat C (2:1)

Condorcet’s Method (procedure)

Extending majority rule to three or more candidates…

Example- Condorcet’s Method

Rank 6 5 3 1

first GB AG RN PB

second AG RN AG GB

third PB GB GB AG

fourth RN PB PB RN

Number of voters (15) Amount of voters that voted in the same order.

Who is the winner using Condorcet’s Method?

AG

too good to be true. There is one tragic flaw…

Condorcet’s Voting Paradox

Remember this slide???

Rank Number of Voters (3)

first A B C

second B C A

third C A B

A would defeat B (2:1)

B would defeat C (2:1)

C would defeat A (2:1)

Explain why it’s impossible to have two winners using Condorcet’s method with an

odd amount of voters

By definition of the method, a person wins by beating all others in head to head elections.

Since amount of voters is odd, no one head-to-head election will have a tie.

How can B beat A if A has already been determined to beat all other candidates??

Plurality Voting

Can you see a potential problem with this method???

1980 Senate Race in NY

22% 23% 15% 29% 7% 4%

D D H H J J

H J D J H D

J H J D D H

D- Alfonse D’Amato- ConservativeH- Elizabeth Holtzman- LiberalJ- Jacob Javitz- Liberal

Is there a Condorcet winner?

Who won using Plurality Voting?

Yes, Elizabeth Holtzman

Yes, Alfonse D’Amato

Condorcet Winner Criterion

Manipulability

Can you think of an example of this? It is a problem for Plurality voting, but not the Condorcet

Method

Which is your favorite to win the tournament?

Rank Methods (procedure)

Condorcet’s contemporary, Jean-Charles de Borda (1733-1799)

Borda Count (procedure)

How much is the last place vote worth??

Borda Count Failure

Independence of Irrelevant Alternatives (IIA)

if it is impossible for a candidate B to move from nonwinner status to winner status

unless at least one voter reverses the order in which he or she had B and the winning

candidate ranked.

Independence of Irrelevant Alternatives

Show that the Borda Count Method does not

satisfy IIA.Rank Number of Voters (5)

first A A A C C

second B B B B B

third C C C A A

Rank Number of Voters (5)

first A A A B B

second B B B C C

third C C C A A

Sequential Pairwise Voting (procedure)

Start with an agenda and pair the first two candidates in a one-on-one

contest. The winner moves on to confront the third candidate in the list,

one-on-one. Losers are deleted. The candidate remaining at the end wins.

Example: Sequential Pairwise Voting

Rank NUMBER OF VOTERS (3)

first A C B

second B A D

third D B C

fourth C D A

Who Wins?

Anything wrong

with that?

Pareto Condition (failed by SPV method)

The Hare System (procedure)

Repeatedly delete candidates that are “least preferred” in the sense of being at the top of the

fewest ballots. Number of Voters (13)

Rank 5 4 3 1first A C B B

second B B C A

third C A A C

B and C are both eliminated in the

first round

Hare SystemNumber of Voters (13)

Rank 5 4 3 1first A C B A

second B B C B

third C A A C

Suppose the voter in the last

column moved A up in his or her

vote. The only

change made is

favorable to A.

Hare System

Number of Voters (13)

Rank 5 4 3 1first A C B A

second B B C B

third C A A C

Reapply the Hare system and see that only B is eliminated in the first round.

Number of Voters (13)

Rank 5 4 3 1

first A C C A

second C A A CC wins!

Wait…what do you mean C won?!?!

A won the original election and the only change was favorable to A!

There is a failure in Monotonicity with the

Hare System

Plurality Runoff Method (procedure)

A runoff method (new election using the same ballots), where the two candidates with the most first place votes are pitted

head-to head. This method is not monotone.

Number of Voters (13)

Rank 4 4 3 2

first A B C D

second B A D C

third C C A A

fourth D D B B

A and B tie with four first

place votes each

Plurality RunoffNumber of Voters (13)

Rank 4 4 3 2

first A B A A

second B A B B

A wins. Who would have won with the Hare system??

Number of Voters (13)

Rank 4 4 3 2

first A B C C

second B A A A

third C C B B

In the first round only

D would have been

eliminated.

Practice Problems

pg. 408 1-7 (skills check)

also, exercises 5-10

With so many different ways to count votes, and with so many different winners, how do

we tell who the true winner is???

Arrow’s Impossibility Theorem

Approval Voting