IMPOSSIBILITY AND MANIPULABILITYSection 9.3 and Chapter 10

Review of Conditions and Criteria

There are many conditions and criteria that are used to determine if an election is “fair”

These criteria often refer to voters changing their ballots in some way, and the result of the election changing (or not)

Condorcet Winner Criterion

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

Plurality does not satisfy this criterion, but Condorcet’s method does

Independence of Irrelevant Alternatives A voting system is said to satisfy IIA if it is

impossible for a candidate A to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had A and the winning candidate ranked.

The Borda count does not satisfy IIA, but Condorcet’s method does

The Pareto Condition

If everyone prefers one candidate (say, B) to another candidate (say, D), then this latter candidate (D) should not be among the winners of the election.

Plurality satisfies the Pareto condition, but sequential pairwise voting does not

Monotone

If a candidate is a winner, and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain a winner.

Plurality is monotone, but the Hare system is not

The Search for a Perfect Voting System All of the methods we have discussed are flawed

in some way

Why didn’t I just tell you about the “best” voting system in the first place?

Recall May’s Theorem says that majority rule is the “best” method for deciding the winner of an election with two candidates

Arrow’s Impossibility Theorem

Named after Kenneth Arrow, an American economist

Essentially, the theorem states that there is no perfect voting method

It doesn’t just say that we haven’t thought of a perfect system yet; it says that we can never create one

Arrow’s Impossibility Theorem

With three or more candidates and any number of votes, there does not exist (and will never exist) a voting system that: always produces a winner satisfies the Pareto condition satisfies the independence of irrelevant alternatives

condition is not a dictatorship

Proving the Theorem

Arrow’s Theorem is hard to prove (he earned a Nobel prize in 1972 for his work in this area)

We will prove a weaker version of his theorem

To prove that it is impossible to create a voting system, we will assume that we have created such a system

This assumption will lead to an impossibility

Weak Version of Arrow’s Theorem

With three or more candidates, there does not exist (and will never exist) a voting system that: satisfies the Condorcet winner criterion satisfies the independence of irrelevant alternatives

condition always produces at least one winner

A Hypothetical Voting System

Let’s assume that we have a hypothetical voting system that satisfies the Condorcet winner criterion satisfies the independence of irrelevant alternatives

condition always produces at least one winner

In other words, we’re assuming that we have exactly the kind of voting system that Arrow’s Theorem says should not exist

A Problematic Profile

Consider this voter profile

Since our system “always produces at least one winner,” we might wonder who the winner should be

We will show that A is not the winner

Voters Preference

1 A > B > C

1 B > C > A

1 C > A > B

A New Profile

What about this profile?

Since C is the Condorcet winner, and our hypothetical system satisfies the Condorcet winner criterion, C must be the winner using our hypothetical system also

Voters Preference

1 A > B > C

1 C > B > A

1 C > A > B

Modifying the Profile

Now we’ll modify the second profile to turn it into the first one:

Notice that the only change was for the second voter to change from C > B > A to B > C > A

B is irrelevant to the question of A versus C, so since C was the previous winner and A was a previous non-winner, IIA means that A must continue to be a non-winner

Voters Preference

1 A > B > C

1 B > C > A

1 C > A > B

Voters Preference

1 A > B > C

1 C > B > A

1 C > A > B

The Problematic Profile

So we have just shown that for this profile, A is not a winner

A similar argument shows that B and C are also non-winners

But we assumed that our hypothetical system always finds a winner

Therefore our hypothetical system can’t exist

Voters Preference

1 A > B > C

1 B > C > A

1 C > A > B

Summary of Chapter 9

One “best” way to determine the winner of an election with two candidates: majority rule

Many ways to determine the winner of an election with more than two candidates

All of these methods are “unfair” in some way

We use criteria to be very specific about the ways in which the methods are “unfair”

It is impossible to find a completely fair system

Chapter 10: Manipulability

Sometimes, in order to achieve the election result you prefer, you submit a ballot that misrepresents your actual preferences

This type of strategic voting is called manipulation, and the misrepresented ballot is referred to as an insincere or disingenuous ballot

An Example

Consider this voter profile, with just two voters

If we use the Borda count to determine the winner, B wins with 5 points

Assuming that Voter #1 knew the ballot that Voter #2 was going to submit, could Voter #1 have submitted her ballot so that A wins?

Voter Preference

#1 A > B > C > D

#2 B > C > A > D

A Manipulated Outcome

What if Voter #1 changes her ballot like this:

This ballot is insincere: #1 likes B better than C or D, but she has ranked B last to try to change the result

Using this new ballot, A is now the Borda count winner

Voter #1 prefers this outcome according to her original ballot

Voter Preference

#1 A > D > C > B

#2 B > C > A > D

Manipulability

A voting system is said to be manipulable if there are two sets of ballots and a voter (we’ll call him “Bob”) such that neither election ends in a tie the only difference between the two sets of ballots is

Bob’s ballot Bob prefers (according to his actual preferences as

expressed in the first election) the outcome of the second election to that of the first

Majority Rule

In an election with two candidates (A and B), is majority rule manipulable?

In order to manipulate the election, A would have to be the winner, but your true preference would have to be B > A

The only change you can make to your ballot is to change it to A > B

Since May’s Theorem guarantees that majority rule is monotone, changing your vote from a vote for the loser to a vote for the winner cannot change the outcome

Condorcet’s Method

Condorcet’s Method is also non-manipulable

If you prefer B, but the winner is A, then A beats B head-to-head even with your vote preferring B over A

No matter how you change your ballot, A will still beat B head-to-head

The Perfect System?

Condorcet’s Method has some very nice properties elections never result in ties (assuming the number of voters is

odd) satisfies the Pareto condition non-manipulable not a dictatorship

However, Condorcet’s Method also sometimes doesn’t produce a winner

Is there a “perfect” system that satisfies all of these properties and always gives a winner?

The Gibbard-Satterthwaite Theorem

With three or more candidates and any number of voters, there does not exist (and never will exist) a voting system that always produces a winner, never has ties, satisfies the Pareto condition, is non-manipulable, and is not a dictatorship.