Your vote makes a difference only when it is pivotal. Now don’t worry, I am not bringing this up to sort through the tired old arguments about whether you should go to the polls today. You should! That settled, let’s talk about what it implies for how you should vote once you get there.
Because if your vote only makes a difference when it breaks a tie (or makes a tie), then when it comes time to decide how to vote, you might as well assume your vote will be pivotal. And ask yourself how would you vote if your vote was going to make or break a tie.
Be careful. This is not the same as the question “How would you vote if you were the dictator?” Indeed quite often your vote should not be the vote you would cast if yours was the only vote. That’s because when your vote is pivotal you learn something that a dictator doesn’t. You learn that all of the other voters were (almost) perfectly split and and that implies something very specific about the other voters and what they must know about the candidates (or propositions) on the ballot.
Quite often that information is crucial for determining how you want to vote. Let me give you a simple example. Judges are almost always re-elected. Pretty much the only time a judge is voted off the bench is if that judge is completely incompetent. Now you haven’t bothered to read anything about the judges on your ballot. You know nothing about them individually but you know that most judges are doing just fine and should be re-elected.
If you were the dictator (an uninformed dictator!) you would vote yes for every judge. But things turn completely upside-down in an election when you factor in the information you learn from your vote being pivotal. Since all competent judges are easily re-elected, the only way it could have happened that all the other voters are split is that this judge is not competent! Knowing that, and knowing that your vote will decide whether an incompetent judge is re-elected, you should vote no. Against every judge.
Now, the smart readers of this blog have already thought one step ahead and noticed that this logic is self-defeating. Because if everyone figured this out, then everyone is voting against every judge and then every judge is voted down, not just the incompetent ones. Here’s where the theory takes one of two paths, use your judgement.
First you might not believe that the electorate in general is as sophisticated as you are. The vast majority of voters don’t understand the logic of pivotalness and they are naively voting the way they would if they were dictators. In that case, the argument I have laid out works as written and you should vote against every judge.
On the other hand you might believe that a signifcant fraction of voters do understand the strategic subtleties of voting. Then we have an equilibrium to find. For starters we take as given that the judge himself and all of his friends will vote for him. So he has a head start. Now there’s a small group of do-gooders who have read up on this judge and know whether he is competent. They vote as if they are dictators, with good reason now because they are informed. They vote yes if he is competent and no if he is not.
The rest of us know nothing. Until, that is, we take into account what we can infer from being pivotal. And if it were just the informed and the judge’s friends who were voting then what we can infer is that enough of the informed are voting no to counteract the judge’s head start. That is, the judge is incompetent.
In equilibrium none of us uninformed voters vote yes. Because if any of us are voting yes, then effectively the judge has an even bigger head start and that makes it even worse news that the no votes caught up with the head start. But not all of us vote no. Some of us do, but most of us abstain. Enough of us that it remains a valid inference that a pivotal vote means that enough of the informed voted no to make it optimal for us to vote no.
This is the logic of The Swing Voter’s Curse.