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. 

Don't let the meanest people win

With a general election coming in the UK I've been thinking about voting systems. There are lots of interesting proofs showing what's possible (or rather, what's impossible) to achieve in voting systems. One example is that every reasonable voting system (with three or more candidates) is susceptible to tactical voting (the Gibbard--Satterthwaite theorem), where in some circumstances voting for the wrong candidate can improve your chances of a preferred outcome. Similarly, no rank-based voting system for choosing between three or more candidates meets four basic criteria (Arrow's impossibility theorem): roughly that

• we don't just let one person decide the outcome
• every set of votes translates into a single winner, in a predictable (not random) manner
• candidate C entering the race can't change the result from a win for A to a win for B
• if every voter prefers A to B then B can't win the election. 

(In the first version of this post condition 2 said "every election result is in principle possible" which is not the correct condition: it's in fact implied by the fourth condition.)

The last two conditions are the most nuanced; let's explore the last condition in the context of a chilli cook-off. The takeaway point will be careless scoring systems punish nice people. [My sister tells me that 'Come dine with me' is a good example of this: the winner is often not the best chef but the stingiest voter.] Here's the setup.   

You're hosting a chilli cook competition with some friends. Each participant scores the other chillis out of 10, but of course can't vote on their own.

As a first exercise, suppose everyone submits literally the same chilli. Who will win? Think about it a bit (I've already hinted at the answer!).