chapter 10: the manipulability of voting systems

Chapter 10: The Manipulability of Voting Systems

October 3, 2013


Last Time

1 An Introduction to Manipulability

2 Majority Rule and Condorcet’s Method

3 The Manipulability of Other Voting systems for Three or MoreCandidates


This time

Hare system and other voting systems.

The Gibbard-Satterhwaite Theorem.

The Chair’s Paradox


Manipulability

If only one voter changes their ballot it is called a unilateralchange.

Manipulability

A voting is said to be manipulable if there exist two sequences ofpreference list ballot and a voter (call the voter Jane) such that

1 Neither election results in a tie.

2 The only ballot change is by Jane.

3 Jane prefers - assuming that her ballot in the first electionrepresents her true preferences- the outcome of the secondelection to that of the first election.


May’s Theorem for Manipulability

May’s Theorem for Manipulability

Among all two-candidate voting systems that never result in a tie,majority rule is the rule is the only one that treats all votersequally, treats both candidates equally, and is non manipulable.


Nonmanipulable Voting systems

Condorcet’s Method

Borda Count with three Candidates

Plurality


The Nonmanipulability of the Borda Count with ExactlyThree Candidates

Suppose you prefer A to B, but B is the winner. There are 3possible cases:

Your sincere ballot is A over B over C.

Your sincere ballot is C over A over B.

Your sincere ballot is A over C over B.


Manipulable Voting systems

Borda Count with Exactly Four or More Candidates

Sequential Pairwise Voting


Sequential Pairwise Voting

Agenda Manipulation

Agenda manipulation refers to the ability to control who wins anelection with sequential pairwise voting by a choice of the agenda.


Agenda Manipulation

Agenda: A,C ,B,DRank Adam Beth Jane

1st A C B2nd B A D3rd D B C4th C D A

How can we make A win?


Manipulability of Hare System

Rank Adam Beth Carol Dan Jane

1st B C C D A2nd A B B B B3rd C A A C C4th D D D A D

How can Jane manipulate the vote?


Manipulability of Plurality Runoff Rule


1st A C C B A2nd B A A C B3rd C B B A C

How can Jane manipulate the vote?


Plurality Voting

Group Manipulability

A voting system is group-manipulable if there are elections inwhich a group of voters can change their ballots so that the newwinner is preferred to the old winner by everyone in the group,assuming that the original ballots represent the true preferences ofeach voter in the group.

The Group Manipulability of Plurality Voting

Plurality voting cannot be manipulated by a single individual.However, it is group manipulable.


Question

Is there a voting system with three or more candidates thatsatisfies the following conditions

Elections (with an odd number of voters) never result in ties.

It satisfies the Pareto condition.

It is nonmanipulable.

It is not a Dictatorship.


The Gibbard-Satterthwaite Theorem

The Gibbard-Satterthwaite Theorem

With three or more candidates and any number of voters, theredoes not exist - and there will never exist- a voting system thatalways produces a winner, never has ties, satisfies the Paretocondition, is nonmanipulable, and is not a dictatorship.


A Weak Version

A Weak Version fo the GS Theorem

Any voting system for three candidates that agrees withCondorcet’s method when-ever there is a Condorcet Winner - andthat additionally produces a unique winner when confronted by theballots in the Condorcet voting paradox- is manipulable.


Proof of Weak GS Theorem

Assume Condorcet voting Paradox, and the resulting winner is C .Rank Adam Beth Jane

1st B C A2nd C A B3rd A B C

Rank Adam Beth Jane

1st B C B2nd C A A3rd A B C

The winner is the second election is B, so the system ismanipulable.


Proof of Weak GS Theorem cont

Assume Condorcet voting Paradox, and the resulting winner is B.Rank Adam Beth Jane

1st B A C2nd C B A3rd A C B

Rank Adam Beth Jane

1st B A A2nd C B C3rd A C B

The winner is the second election is A, so the system ismanipulable.


Proof of Weak GS Theorem cont

Assume Condorcet voting Paradox, and the resulting winner is A.Rank Adam Beth Jane

1st A C B2nd B A C3rd C B A

Rank Adam Beth Jane

1st A C C2nd B A B3rd C B A

The winner is the second election is C , so the system ismanipulable.


Majority Rule with tie-breaking power

Consider the following ballot for deciding what snack should beserved. Assume the vote is decided with majority rule where thechair has tie-breaking power

Rank Chair Students Teachers

1st Carrots Candy Cake2nd Candy Cake Carrots3rd Cake Carrots Candy




With three voters and three candidates, the voter with tie-breakingpower can, if all three voters act rationally in their ownself-interest, end up with his or her least preferred candidate as theelection winner.


Problem

Find an agenda for the ballot below in which

1 B is the winner using sequential pairwise voting

2 C is the winner using sequential pairwise voting

3 D is the winner using sequential pairwise voting

Rank Adam Beth Jane



Problem Answer

Find an agenda for the ballot below in which

1 B is the winner using sequential pairwise votingAnswer: A,C ,D,B

2 C is the winner using sequential pairwise votingAnswer: D,B,A,C

3 D is the winner using sequential pairwise votingAnswer: B,A,C ,D

Rank Adam Beth Jane



Problem

Use the following election to illustrate the manipulability of theBorda count with three voters and four candidates:

Rank Adam Beth Jane

1st A B B2nd B A A3rd C C C4th D D D

Show that the Borda count is manipulable if there are fourcandidates and five voters.


Problem Answer

Change Adam’s vote.Rank Adam Beth Jane

1st A B B2nd C A A3rd D C C4th B D D



Problem Answer cont.



1st A B B A B2nd B A A C D3rd C C C D C4th D D D B A

Borda Count of A: 10, B: 11, C : 6, D: 3Change the ballot


1st A B B A B2nd C A A C D3rd D C C D C4th B D D B A

Borda Count of A: 10, B: 9, C : 7, D: 4


Problem

Consider the voting rule in which an alternative is among thewinners if it receives at least one first-place vote. In one sentence,explain why this voting system is not manipulable.Answer:It is not maniplable since the only way for the election to have aunique winner is for all voters to have the same first-place votethus no one has motivation to change their vote.


Problem

Consider the voting system for two candidates and three voters inwhich the candidate recieving an odd number of first-place voteswins. Is the system manipulable?Answer:Yes it is.


1st A A A B B2nd B B B A A

A is the winner.Rank Adam Beth Carol Dan Jane

1st A A A B A2nd B B B A B

However, after Jane changes her vote, B is the winner.


Next Time

Quiz exercises: 9-21 on page 372-373


chapter 10: the manipulability of voting systems

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