An impractical guide to tactical voting

Last time I mentioned Arrow's theorem, which roughly says it's in general impossible to coherently combine people's individual rankings of candidates into a single order of preference, and the Gibbard--Satterthwaite theorem, which says any reasonable voting system is susceptible to tactical voting. In this post I will explain various voting systems and for each
-- explain which condition of Arrow's theorem fails
-- explain under which circumstances you should tactically vote.

Why "should" you do anything?

When the future is uncertain, the measure mathematicians and, particularly, economists, use to decide what you should do is expected utility: how likely your decision is to lead to a preferred outcome (weighted by how much you prefer an outcome). Deciding whether or not to take part in a lottery, you might feel your regret in the (extremely) unlikely event of choosing the correct numbers but not entering would be so large that you'd rather take the near certainty of the loss of your entry fee than the small chance of missing the jackpot. Similarly, a whole range of factors can influence how good you feel about your decision given outcome of an election: for example you may choose to vote for the green party even in a constituency where they are unlikely to win because you think a higher national percentage of the vote will make the main parties more likely to adopt greener policies. I will not discuss the philosophy of tactical voting further; instead I will focus on scenarios where changing your vote leads to a better candidate (from your perspective) winning the constituency/election.

Here are the conditions of Arrow's theorem again:

1. We don't just let one person decide the outcome
2. Every set of votes translates into a single winner, in a predictable (not random) manner
3. If every voter prefers A to B then B can't win the election. 
4. Candidate C entering the race can't change the result from a win for A to a win for B

And some quick remarks:

• The eagle eyed among you may have noticed I've rewritted condition 2 since last post; the old version, "every election result is in principle possible" was misrepresentative: it's implied by condition 3.
• In the following I ignore the possibility of exact ties between two candidates (which fail condition 2 above, but only because I've paraphrased the real conditions of Arrow's theorem).
• As you read the examples you'll notice that tactical voting is closely tied to condition 4. This makes sense: if an extra candidate (who you like best) standing can change the outcome from your second to your least preferred option, then it's in your interests to vote as if they had never entered the race. 

Here are some election systems.

Dictatorship: what Kweku says goes

• This fails condition 1 but meets the other criteria.
• There is no need to tactically vote: either you're me, and will get exactly what you want, or you're not, and your vote doesn't matter anyway.

 "Neverendum": if no candidate gets an absolute majority, run the election again, and again, and again....

• This fails condition 2 but meets the other criteria.
• There is no need to tactically vote (except that if you don't the election may never end!)

Artificial standoff: only two candidates are allowed to win the election, all other candidates are taken out

• This fails condition 3 but meets the other criteria: the best candidate may not be one of the two permitted.
• There is no need to tactically vote: you should vote for your preferred of the two allowed candidates.

First past the post: whichever candidate has the most votes wins, even if they have less than 50% support

• This fails condition 4 but meets the other conditions: imagine I would collect 60% of the vote to my opponent's 40%, but then my identical twin enters the election with the almost exact same promises. Now we each get around 30%, hence we both fail to win, even though either of us would win head to head against the opponent.
• When should you vote tactically? In the above situation, if my supporters swap to my twin they get a better outcome than sticking (even if they prefer my very slightly different promises). A tactical voter may thus choose their preferred of the two most likely candidates to win (though obviously, without being able to know for sure who they are); there are no other scenarios in which to tactically vote.

Runoff voting: each round the candidate with the fewest votes is eliminated, and the election repeats with the remaining candidates untill someone wins 50% support ("alternative vote" or "instant runoff voting" runs the rounds automatically by collecting everyone's full preference rankings)

For this I use the example of the Conservative leadership election of 2019. This was not exactly conducted according to runoff voting, because the electorate changed: in early rounds only Conservative MPs could vote, but in the final round members could vote, but the principles still apply.
Boris Johnson expected that he would defeat Jeremy Hunt in a final round but (allegedly) feared a loss to Michael Gove. His supporters thus may have voted for Hunt instead of Johnson to ensure that Hunt reached the final round.
This demonstrates that this voting system fails condition 4: Hunt entering the race may have turned it from a win for Gove to a win for Johnson. It also demonstrates one of the two circumstances in which you should vote tactically, as supporters of Johnson may have done. The other situtation in which you might vote tactically is as a supporter of Hunt who would prefer Gove to Johnson: by lending your support to Gove and eliminating your own preferred candidate, you can help your second favourite defeat your least favourite.

Proportional representation: parliamentary seats nationwide are allocated according to nationwide share of the vote. Single transferable vote (the voting system ideally wanted by the Lib Dems under Clegg) attempts to achieve proportional representation while mainting some local accountability of representatives.

This does not strictly fit the setting of Arrow's theorem, since it does not produce a single winner, and so the results of the election are able to near-perfectly reflect the choices of voters. However, you can view it as a system for ultimately choosing a prime minister. Then it fails condition 2: you can't always predict, based on which parties get in, who will be chosen as prime minister. It also potentially fails condition 4, if the preferred prime minister of a party does not match that of some of the party's supporters (for example consider the many Lib Dem voters who were unhappy -- despite his promise in advance to seek a deal with whichever party had more of the seats and more of the vote -- with Nick Clegg seeking a coalition with the Conservatives rather than Labour).

Condorcet methods such as Copeland's method or the Minimax Condorcet method: these seek to ensure that anyone who would win, head to head, against any other candidate also wins the election.

The issue for such methods is that there isn't always a candidate who would win head to head against any other candidate: voters may prefer rock to scissors, scissors to paper and paper to rock. This is achieved if 1/3 of voters have rock as first choice, scissors as second choice and paper as third choice (represeneted by RSP), 1/3 vote SPR, 1/3 vote PRS, since then 2/3 vote rock ahead of scissors, 2/3 vote scissors ahead of paper and 2/3 vote paper ahead of rock. In such a scenario, different Condorcet methods decide in different ways and fail to meet the conditions of Arrow's theorem in different ways. Copeland's method (elect whoever wins the most head-to-head match ups) fails condition 2 (it very often leads to ties). The minimax condorcet method (elect whoever's worst defeat is narrowest) fails condition 4 (an example demonstrating this has candidates rock, paper, scissors and other. If other narrowly loses in a head to head against each of rock, paper, or scissors -- by less than 1/3 of the vote -- then other will win in an election where all stand, since rock's worst loss (to paper) is by more than other's worst loss. So paper and scissors entering the election turns it from a win for rock to a win for other.)

Which electoral system should we use?

This is far from an exhaustive list of voting systems, and far from an exhaustive list of criteria to judge them on. Of the above, I would argue that only proportional representation or a constituency based condorcet method should be considered: each has its downsides, but they are not directly comparable, while I would argue that condorcet methods can be directly compared to first past the post or alternative vote, and are better. Which of the many possible Condorcet methods, or whether to choose another system entirely is something I have not thought about.