chapter 9:social choice: the impossible dreamlhicko2/math181/... · 2013. 9. 18. · chapter...

Chapter 9:Social Choice: The Impossible Dream

September 18, 2013


Last Time

Last time we talked about

Voting systems

Majority Rule

Condorcet’s Method

Plurality

Borda Count

Sequential Pairwise Voting


Condorcet’s Method

Definition

A ballot consisting of such a rank ordering of candidates is called apreference list ballot because it is a statement of the preferencesof the individual who is voting.

Description of Condorcet’s Method

With the voting system known as Condorcet’s method, a candidateis a winner precisely when he or she would, on the basis of theballots cast, defeat every other candidate in a one-on-one contestusing majority rule.


Example

Consider the following set of preference lists:Number of Voters(9)

Rank 3 1 1 1 1 1 1First A A B B C C DSecond D C C C B D BThird C B D A D B CFourth B D A D A A A

Winner: C


Pros and Cons of Condorcet’s Method

Pro:Takes all preferences into account.Con:

Condorcet’s Voting Paradox

With three or more candidates, there are elections in whichCondorcet’s method yields no winners. In particular, the followingballots constitute an election in which Condorcet’s method yieldsno winner.


Plurality Voting

Plurality Voting

Each voter pick their top choice.


Example


Rank 3 1 1 1 1 1 1First A A B B C C DSecond D B C C B D CThird B C D A D B BFourth C D A D A A A

Winner: A


Pros and Cons of Plurality Voting

May’s Theorem

Among all two-candidate voting systems that never result in a tie,majority rule is the only one that treats all voters equally, treatsboth candidates equally, and is monotone.

Condorcet Winner Criterion

A voting system is said to satisfy the Condorcet winnercriterion(CWC) provided that, for every possible sequence ofpreference list ballots, either (1) there is no Condorcet winner or(2) the voting system produces exactly the same winner for thiselection as does Condorcet’s Method.

Manipulability

A voting system is subject to manipulabity if there are elections inwhich it is to a voter’s advantage to submit a ballot thatmisrepresents his or her true preferences.


Borda Count

Rank Method

A rank method of voting assigns points in a nonincreasing mannerto the ordered candidates on each voter’s preference list ballot andthen sums these points to arrive at the group’s final ranking.

Borda Count

The rank method in which there are n candidates with eachfirst-place vote worth n − 1 points, each second-place vote worthn − 2 points, and so on down to each last-place vote worth 0points is known as the Borda Count. The point totals are referredto as a candidate’s Borda score.


Example



Winner: B


Pros and Cons of Borda Count

Pro:Takes all preferences into account.Con:

Independence of Irrelevant Alternatives

A voting system is said to satisfy independence of prevalentalternatives (IIA) if it is impossible for a candidate X to movefrom nonwinner status to winner status unless at least one voterreverses the order in which he or she had X and the winningcandidate ranked.


Sequential Pairwise Voting

Description of Sequential Pairwise Voting

Sequential pairwise voting starts with an agenda and pits thefirst candidate against the second in a one-on-one contest. Thewinner then moves on to confront the third candidate in the list,one on one. Losers are deleted. This process continues throughoutthe entire agenda, and the one remaining at the end wins.


Example



With Agenda A,B,C ,DWinner: DWith Agenda D,C ,B,AWinner: B


Pros and Cons of Borda Count

Pro:Efficient way to determine a winner. Always a unique winner withan odd number of voters.Con:Agenda dependent

Pareto Condition

A voting system is said to satisfy the Pareto condition providedthat in every election in which every voter prefers one candidate Xto another candidate Y , the latter candidate Y is not among thewinners.


Hare System

Description of the Hare System

The Hare system proceeds to arrive at a winner by repeatedlydeleting candidates that are “least preferred“ in the sense of beingat the top of the fewest ballots. If a single candidate remains afterall others have been eliminated, he or she alone is the winner. Iftwo or more candidates remain and all of these remainingcandidates would be eliminated in the next round (because they allhave the same number of first-place votes), then these candidatesare declared to be tied for the win.


Example



Winner: C


Possible Problems with the Hare System

Monotonicity

A voting system for three or more candidates is said to satisfymonotoncity provided that, for every election, if some candidateX is a winner and a new election is held in which the only ballotchange made is for some voter to move this winning candidate Xhigher on his or her ballot (and to make no other changes), then Xwill remain a winner.

Another reason: Chicago lost being the host city to the 2016Summer Olympics because of this system.


Example


Rank 5 4 3 1First A C B BSecond B B C AThird C A A C

Winner: ANumber of Voters(12)

Rank 5 4 3 1First A C B ASecond B B C BThird C A A C

Winner: C


Plurality Runoff Method

Description of Plurality Runoff Method

Plurality runoff is the voting system in which there is a runoff(that is, a new election using the same ballots) between the twocandidates receiving the most first place votes. If there are ties,then the runoff is among either those tied for the most first-placevotes, or the lone candidate with the most first-place votes alongwith those tied for the second-most first-place votes (and pluralityvoting is used).


Example


Rank 3 1 1 1 1 1 1First A A B B B C DSecond D B C C C D CThird B C D A D B BFourth C D A D A A A

Winner: B


Hare System vs Plurality Runoff Method


Rank 4 4 3 2First A B C DSecond B A D CThird C C A AFourth D D B B

Plurality Runoff Method Winner: AHare System Winner: C


Practice Problems

1 Does the Condorcet’s rule satisfy the Pareto Condition?Monotonicity? Why?

2 Does plurality voting satisfy the Pareto Condition?Monotonicity? Why?

3 Does Borda count voting satisfy the Pareto Condition?Monotonicity? Why?

4 Does sequential pairwise voting satisfy the CWC?Monotonicity? Why?

5 Does Hare system voting satisfy the Pareto Condition?

6 Does plurality voting satisfy IIA?


Practice Problems


Rank 2 2 1 1 1First A B A C DSecond D D B B DThird C A D D AFourth B C C A C

Calculate the winner using

1 plurality voting.

2 the Borda count.

3 the Hare system.

4 sequential pairwise voting with the agenda B, D, C , A.


Practice Problems


Rank 2 2 1 1 1First C E C D ASecond E B A E EThird D D D A CFourth A C E C DFifth B A B B B

Calculate the winner using

1 plurality voting.

2 the Borda count.

3 the Hare system.

4 sequential pairwise voting with the agenda A, B, C , D, E .


Next Time

Quiz over 3.5 and 9.1-9.3Practice problems exercises 13-28 on pages 351-353


chapter 9:social choice: the impossible dreamlhicko2/math181/... · 2013. 9. 18. · chapter...

Documents