Strategic Voting with Ranked Ballots

I used to think that ranked ballots were a simple way to eliminate the need for strategic voting. But it turns out that, even with ranked ballots, strategic voting can still arise! This fact is well-known by social choice theorists (see e.g. the Gibbard-Satterthwaite theorem). But it was new to me. Here is a simple example.

Recall that ranked ballots (also called ranked choice voting, equivalent to instant-runoff voting or to the single-member case of single tranferable vote) allow each voter to rank as many candidates as they wish, in order from 1 (best) downwards. To determine the winner, in each round the candidate with the lowest rank-1 votes is eliminated and their voters' subsequent choices are counted instead, with such rounds repeated until one candidate obtains a majority of the total votes.

This voting system is often thought to eliminate the need for strategic voting (i.e., voting for a "lesser evil" candidate instead of your favourite candidate to prevent a worse candidate from winning), since voters can rank their favourite choice first and their "lesser evil" choices subsequently in case their favourite choice is eliminated. But actually, even with ranked ballots, strategic voting can still arise!

For a specific example, suppose a riding consists of: 39 people who rank NDP #1, 30 people who rank Liberals #1 and NDP #2, and 29 people who rank Conservatives #1 and Liberals #2. Suppose two additional voters want the NDP to win, and are trying to decide how to vote. (I need to use two voters, not just one, in order to avoid ties which are a challenge for every voting system.)

If these two additional voters rank their favourite party (the NDP) as their #1 choice, then: The round-1 counts will be NDP 41 and Lib 30 and Con 29, so the Conservaties will be eliminated. And then, the round-2 counts will be NDP 41 and Lib 59, so the Liberals will win.

By contrast, if these two additional voters instead rank the Conservatives as their #1 choice, then: The round-1 counts will be NDP 39 and Con 31 and Lib 30, so the Liberals will be eliminated. Then, the round-2 counts will be NDP 69 and Con 31, so the NDP will win.

So, even though the two additional voters prefer the NDP, they are better off voting "strategically" for the Conservatives, in order to make the NDP win. Hence, strategic voting is still an issue, even with ranked ballots (though perhaps somewhat less so than with our current first-past-the-post system). So, it is not correct to write (as some do) that "With a ranked ballots and an Instant Runoff vote, you can always vote with your heart". This fact is well-known, but was a surprise to me.

-- Jeffrey Rosenthal, University of Toronto, November 2019