Problems with Borda Count

Jean Charles Borda proposed the Borda count method in 1770. It has the advantage that spoiler candidates are much less likely to be a problem. But Nicholas de Condorcet was very critical of the Borda count method.

Condorcet’s Criterion

According to Condorcet, if a candidate would beat all the others in head-to-head match-ups, then they should win the election.

A candidate that would beat every other candidate in a one-on-one run-off is called a Condorcet candidate. Not every election has a Condorcet candidate, but when there is it seems reasonable that they should win.

Borda Count Fails the Condorcet Criterion

In fact, Borda count doesn’t even always elect the majority candidate! Suppose 25 kids vote on the best ice cream flavor.

Exercise. Is there a Condorcet candidate? Why?

Exercise. What flavor does Borda count pick?

Condorcet Methods

It is easy to make a voting method that passes the Condorcet criteria. Just check for a Condorcet candidate by checking every pairwise comparison.

Checking for a Condorcet Candidate

With 5 candidates, there are 10 possible pairwise match-ups. That could take a long time to check. Save time by only checking match-ups until every candidate has lost one (or you find a Condorcet candidate).

Mexican vs. Italian

Mexican wins.

Mexican vs. Thai

Mexican wins.

Mexican vs. Chinese

Chinese wins.

Chinese vs. Indian

Chinese wins.

Chinese vs. Italian

So far, only Chinese hasn’t lost, but that changes if you compare it with Italian.

So there is no Condorcet candidate, since every restaurant loses at least one match-up.

What Makes a Voting Method Fair?

So far, we have seen some features that an ideal voting system would have:

• Condorcet Criterion - A Condorcet candidate should win, if there is one.

• Monotonicity Criterion - Ranking a candidate highly should never cause a candidate to lose.

• No Spoilers Criterion - A third party candidate who won’t win shouldn’t change the outcome.

Arrow’s Impossibility Theorem

Unfortunately, even though we can find methods that satisfy some of the fairness criteria, it is impossible to satisfy them all. This is a consequence of a famous theorem that was proved by the economist Kenneth Arrow in 1951

What is Impossible?

Here is one version of Arrow’s theorem (there are other more complicated versions that say more).

Why is it Impossible?

Consider an election with three front runner candidates (assume any other candidates are ranked lower by all of the voters and so don’t matter) and just 9 voters.

Exercise. Show that there is no Condorcet candidate. Why not?

A Cycle of Preferences

In the last example A beats B and B beats C, but C beats A.

Exercise. Why is it impossible for a Condorcet method to pick a winner of this election without one of the losers being a spoiler candidate?

Important to Understand

• Every voting method sometimes has spoiler candidates or sometimes fails to elect a Condorcet candidate.

• That doesn’t mean that all voting methods are bad or always have problems.

• Most methods work pretty well most of the time.

The Story So Far

*There are actually several Condorcet methods, not all satisfy the Monotonicity criterion.

Approval Voting

It would be nice to have a method that doesn’t have spoiler candidates. Several voting methods have that property including one very simple method.

Approval Voting Example

The preference table below does not have enough information to figure out the approval voting winner. Why not?

Approval Voting Example

Suppose that the first 13 voters split into two groups. Ten of them are okay with chocolate and vanilla, but 3 only like chocolate. Then that column could split into:

Approval Voting Example

Suppose that the first 13 voters split into two groups. Ten of them are okay with chocolate and vanilla, but 3 only like chocolate. Then that column could split into:

Approval Voting Example

The next group of voters might split evenly into two groups, one that approves their top two flavors, and one that only wants their top choice.

Approval Voting Example

The next group of voters might split evenly into two groups, one that approves their top two flavors, and one that only wants their top choice.

Approval Voting Example

The next 5 voters all like every flavor about equally; and the one guy at the end only approves of vanilla.

Approval Voting Example

The next 5 voters all like every flavor about equally; and the one guy at the end only approves of vanilla.

The Story with Approval Voting