Consider the preference schedule below, in which a company’s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.

Initial votes

If this was a plurality election, note that B would be the winner with [latex]9[/latex] first-choice votes, compared to [latex]6[/latex] for D, [latex]4[/latex] for C, and [latex]1[/latex] for E.

There are total of [latex]3+4+4+6+2+1 = 20[/latex] votes. A majority would be [latex]11[/latex] votes. No one yet has a majority, so we proceed to elimination rounds.

Round 1: We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice

We then shift everyone’s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.

Round 2: We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone’s options to fill the gaps.

Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.

Now B has [latex]9[/latex] first-choice votes, C has [latex]4[/latex] votes, and D has [latex]7[/latex] votes. Still no majority, so we eliminate again.

Round 3: We make our third elimination. C has the fewest votes.

Condensing this down:

D has now gained a majority, and is declared the winner under IRV.

Let’s return to our City Council Election.

In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The [latex]214[/latex] people who voted for Don have their votes transferred to their second choice, Key.

So Key is the winner under the IRV method.

We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don’s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.

Consider the voting system below.

In this election, Carter would be eliminated in the first round, and Adams would be the winner with [latex]66[/latex] votes to [latex]34[/latex] for Brown.

Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to “jump on the bandwagon,” [latex]10[/latex] of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.

In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with [latex]51[/latex] votes to Adams’ [latex]49[/latex] votes! Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn’t seem right, and introduces our second fairness criterion.