which units are you most interested in covering? unit a –management science unit b – growth unit...

Which units are you most interested in covering?

Unit A –Management Science

Unit B – Growth

Unit C – Shape and Form

Unit D – Statistics

Here’s how you voted(remember?)

A A A A A A A A A A B B B B C C C C D D DB B B B B B D D D D D D D D D D D D B C CC C C C C C B B B B C C C C B B B B C B BD D D D D D C C C C A A A A A A A A A A A

Student (Voter) # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

Here’s how you voted

A A A A A A A A A A B B B B C C C C D D DB B B B B B D D D D D D D D D D D D B C CC C C C C C B B B B C C C C B B B B C B BD D D D D D C C C C A A A A A A A A A A A

If we group identical ballots together, it becomes a littlemore manageable:

Here’s how you voted

A A A A A A A A A A B B B B C C C C D D DB B B B B B D D D D D D D D D D D D B C CC C C C C C B B B B C C C C B B B B C B BD D D D D D C C C C A A A A A A A A A A A

If we group identical ballots together, it becomes a littlemore manageable:

6 4 4 4 1 2

Summarizing:

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

How many

Let’s simplify the question a little: Of the 4 choices, which is the most popular?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

How many

Plurality method:Which has the most first place

votes?6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Plurality method:Which has the most first place

votes?6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Since A has 10 first place votes, and none of the others havemore than 4, A is the easy winner by this method.

The Plurality winner is Unit A – Management Science

Borda CountWe give 4 points for each 1st place vote, 3 for a 2nd placevote, 2 for a 3rd, and 1 for a 4th.

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Method #2:

Borda Count

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

A has 10 first place and 11 4th place votes, so its point total is10*4 +11*1 = 51.

Borda Count

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

A has 10 first place and 11 4th place votes, so its point total is10*4 +11*1 = 51.Similarly,B gets 4*4 + 7*3 + 10*2 = 57C gets 4*4 + 2*3 + 11*2 + 4*1 = 48D gets 3*4 + 12*3 + 6*1 = 54

Borda Count

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

A has 10 first place and 11 4th place votes, so its point total is10*4 +11*1 = 51.Similarly,B gets 4*4 + 7*3 + 10*2 = 57C gets 4*4 + 2*3 + 11*2 + 4*1 = 48D gets 3*4 + 12*3 + 6*1 = 54

The Borda winner is Unit B – Growth

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Method #3:

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Keep eliminating losing options and re-casting ballots until acandidate has a majority of the 1st place votes.

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Since D has the fewest 1st place votes (3), it is “voted off theisland”. All ballots are recast, keeping A, B and C in their original order, but crossing D off.

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B CB B CC B C B C B

C A A A A

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C B CB B C B C BC C A A A A

Note that some columns can now be combined (the first 2 arenow the same, as are the 3rd and 5th, and the 4th and 6th)

Plurality with elimination method:

Get a majority, “survivor-style”6 4 4 4 1 2A A B C B CB B C B C BC C A A A A

Note that some columns can now be combined (the first 2 arenow the same, as are the 3rd and 5th, and the 4th and 6th)

Plurality with elimination method:

Get a majority, “survivor-style”10 5 6A B CB C BC A A

Plurality with elimination method:

Get a majority, “survivor-style”10 5 6A B CB C BC A A

Next to be voted off the island is B:

Plurality with elimination method:

Get a majority, “survivor-style”10 5 6A C CC A A

Now, we can combine columns 2 and 3:

Plurality with elimination method:

Get a majority, “survivor-style”10 11A CC A

And we now have a “survivor” - a majority winner, C.

The Plurality with Elimination winner is Unit C –Shape and Form

Head-to-head comparisons

Given a choice of only A or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Method #4:

Head-to-head comparisons

Given a choice of only A or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Method #4:

Head-to-head comparisons

Given a choice of only B or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

Given a choice of only B or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

Given a choice of only C or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

Given a choice of only C or D, which would mostpeople prefer?

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

Normally, we would continue comparing all pairs of choices,(A-B, A-C, and B-C) but in this example that is not really necessary to find the most popular option (why not?)

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

In fact, if we did compare all pairs of options, we would find that:A wins 0 comparisons;B wins 2 comparisons (B beats A, 11-10, and B beats C, 15-6);C wins 1 comparison (C beats A, 11-10); andD wins 3 comparisons, as we’ve already seen

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Head-to-head comparisons

The winner by the method of Head-to-Head Comparisons is Unit D, Statistics

6 4 4 4 1 2A A B C D DB D D D B CC B C B C BD C A A A A

Ok, back to the question…Which is the most popular choice?A, which got the most 1st place votes?

B, which got the most points in the 4-3-2-1 point counting method?

C, which won the “reality show” elimination method?

Or D, which the majority of you preferred head-to-head against each and every other choice?

Which method is best?

To answer that, we need to decide which of the voting methods is the most fair.

But what do we mean by “fair” anyway?

Weren’t all of the voting methods used so far fair?

Which method is best?

The book presents 4 criteria that most people would agree any fair (democratic?) voting method should satisfy:

The 4 fairness criteria:

1) Majority: If one candidate is the favorite of more than half (a majority) of the voters, then that candidate should win. What is fair: Majority rules!

What isn’t fair: 15 out of 21 like A the best, but B beats A in the election

Note: If no candidate receives a majority of the 1st place votes, then this criterion does not apply – any result would be considered fair

The 4 fairness criteria:

2) Condorcet: If one candidate is preferred in head-to-head comparisons with every other candidate, then that candidate should win.

Fair: If one candidate is preferred to each and every other candidate, it is should win.

Not fair: More voters would choose candidate A over each possible alternative, and yet A loses

Note: There may not be a candidate that wins every head-to-head comparison. In that case, this criterion would not apply, and any result could be fair.

The 4 fairness criteria:

3) Monotonicity: If candidate X would win, and a voter then changes his ballot in a way that favors X, then X should still win.

Not fair:

“I snuck a peak at the votes, and Ryan is about to win the election”

“Oh –I wasn’t going to vote for him, but I might as well just change my vote to support him.”

With this ballot changed to support Ryan … Ryan loses?!?

The 4 fairness criteria:

4) Independence-of-Irrelevant-Alternatives: Adding (or removing) losing candidates to the list of options should not change the winner.

Not fair (and rather weird!):

“Would you like blueberry or cherry pie?”

“Cherry, please.”

“Oh – I just noticed, we also have apple pie.”

“Oh great! In that case I’ll have blueberry”

The 4 fairness criteria:

4) Independence-of-Irrelevant-Alternatives: Adding (or removing) losing candidates to the list of options should not change the winner.

Not fair (alternative version):

“Would you like blueberry, cherry, or apple pie?”

“Cherry, please.”

“Oh – I just noticed, we are out of apple today.”

“In that case give me blueberry.”

The 4 fairness criteria:Majority: If one candidate is the favorite of more than half (a majority) of the voters, then that candidate should win.

Condorcet: If one candidate is preferred in head-to-head comparisons with every other candidate, then that candidate should win.

Monotonicity: If candidate X would win, and a voter then changes his ballot in a way that favors X, then X should still win.

Independence-of-Irrelevant-Alternatives: If X would win, and one or more of the losing candidates drops out of the election, then X should still win.

So, back to the question (again)…

Which voting method is the “fair” one?

And, most importantly, which candidate won our class vote? What will we be studying this semester?

?